Altair Students Guide - CAE and Multi Body Dynamics

CAE and Multi Body Dynamics

Introduction

Contents Introduction ......................................................................................................2 About This Series ...........................................................................................2 About This Book .............................................................................................2 Supporting Material ........................................................................................3 Mechanics, Mechanisms and Machines ................................................................4 What is Multi-Body Dynamics? ........................................................................5 Learning MBD - Different Approaches ..............................................................6 Putting It All Together ....................................................................................6 Typical Design Issues.........................................................................................8 Product Liability..............................................................................................8 Some Application Areas ..................................................................................9 The MBD Modeling Philosophy ...................................................................... 13 Summary ..................................................................................................... 16 Theory: Basic, Essential and Advanced.............................................................. 17 Theory … ..................................................................................................... 18 … and Practice ............................................................................................. 30 The Virtuous Circle ....................................................................................... 31 Working with MBD Models................................................................................ 33 Different Strokes for Different Folks............................................................... 33 Basic Building Blocks .................................................................................... 34 Solution Control ........................................................................................... 37 Results - Verification and Validation............................................................... 38 Optimization ................................................................................................ 39 MBD Simulation with HyperWorks ..................................................................... 41 The Simulation Process................................................................................. 41 The Anatomy of a Model............................................................................... 43 Solution and Results ..................................................................................... 45 Integration with HyperWorks ........................................................................ 47 Advanced Topics ............................................................................................. 48 Flexibility ..................................................................................................... 48 Contact........................................................................................................ 50 Control Systems ........................................................................................... 52 Cams, Gears and other Higher Pairs .............................................................. 54 Glossary And References.................................................................................. 58 References................................................................................................... 58 Other Resources........................................................................................... 58 Types of Analyses ........................................................................................ 58 Formulae for the Moments of Inertia ............................................................. 59 Common Coefficients of Friction .................................................................... 61 1

Introduction

CAE and Multi Body Dynamics

Introduction About This Series To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments. Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book This volume introduces techniques to model and analyze mechanisms, which lie at the heart of machines. If product design is your area of interest, you will find the companion volumes, CAE And Design Optimization – Basics and CAE And Design Optimization – Advanced useful. The techniques outlined in this book are usually applied at the very early stage in product design, to be followed up at a later stage in the design cycle with detailed analyses and optimization, both to improve peak performance and to introduce robustness. While it’s not essential, a good grasp of the basic principles of vector mathematics will help you tremendously. Several essential aspects are covered in this book, although in a qualitative fashion. You may want to treat the chapter titled Theory: Basic, Essential and Advanced as a reference. If you choose to adopt this approach, at least a cursory reading of this chapter is strongly recommended.

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Introduction

The various references cited in the book will probably be most useful after you have worked through your project and are interpreting the results.

Supporting Material Your instructor will have the Instructor’s Manual that accompanies these volumes – it should certainly be made use of. Further reading and references are indicated both in this book and in the Instructor’s Manual. If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

My robots were machines designed by engineers, not pseudo-men created by blasphemers. Isaac Asimov

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Mechanics, Mechanisms and Machines The study of mechanisms can be a joy, if done properly. The devil, unfortunately, lies in that “if”. Much of the mathematics of the subject is tedious when done by hand, and a beginner can be excused for feeling lost in the headlong rush of vector notations and vector manipulations. Most engineers end up treating mechanisms like poisonous snakes: worthy of a great deal of respect, and safe only when viewed from a distance. Even more unfortunately for a core discipline in mechanical engineering, the study of mechanisms at the undergraduate level has probably benefited the least from the widespread developments in CAE1 software and related technologies. In fact, a mechanical engineer who uses an Internet-search engine to look for material on “mechanisms” is likely to give up the exercise as counterproductive. Most search engines return references from economics and game theory, making a challenging subject even more confusing! This is a shame. The use of the word “mechanisms” in Game Theory is a very good illustration of the power of the various theories and approaches used in the design of machines. The methods used to construct building blocks that allow the modeling of complicated machines are appealing in their simplicity, and often stunning in their power. Hence their use in fields as far flung as Economics, where a mechanism is simply “the agency or means by 1

The South Pointing Chariot is widely regarded as the most complex geared mechanism of the ancient Chinese civilization. Invented sometime around 2600BC in China by the Yellow Emperor Huang Di, the first historical version was created by Ma Jun (c. 200-265 AD). The chariot is a two-wheeled vehicle, upon which is a pointing figure connected to the wheels by means of differential gearing. Through careful selection of wheel size, track and gear ratios, the figure atop the chariot will always point in the same direction, hence acting as a non-magnetic compass vehicle. After being mocked that he could not reproduce a non-historical and nonsensical pursuit, Ma Jun retorted "Empty arguments with words cannot (in any way) compare with a test which will show practical results". After inventing the device and proving those who were doubtful wrong, he was praised by many. The differential in the gear system integrates the difference in wheel rotation between the two wheels and thus detects the rotation of the base of the chariot. The mechanism compensates this rotation by rotating the pointer in the opposite direction.

Adapted from The Wikipedia See Wikipedia, South Pointing Chariot

Short for Computer Aided Engineering – a term that usually covers design, analysis, 3D modeling, and testing in the course of product-design.

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which an effect is produced or a purpose is achieved”.

What is Multi-Body Dynamics? At first glance, there are few design issues common between a fighter plane in supersonic flight, a car rolling over as it crashes, a ship pitching in the stormy seas and the micro-precision movement of the read / write head of a hard-disk drive. If we apply the wider definition of “mechanisms” that we have just seen, the effects are different, the purposes are different and the agencies used are different. A little consideration, however, shows that all of these involve the investigation of the movement of, and impact of, multiple bodies. The jet plane shoots missiles at other targets, and may even be a target itself. The car bounces off the road or crashes into other vehicles or obstacles. The ship contains several bodies – machinery, passengers, cargo, etc. And, as anyone who has mistimed a dive into a swimming pool knows, at high velocity water can be “hard” enough that it can be treated as a single body, rather than a collection of droplets. Every computer owner knows that sooner or later the disk-drive will “crash” – one form of this is a literal crash, when the head makes contact with the platters themselves, destroying the disk and any data the unfortunate user has placed there. While the scale of movements, the sizes of the bodies and the forces involved vary widely between these applications, in all these cases designers need to understand how the forces affect the movement of the body, and vice versa. And, of course, there are multiple bodies involved. This aspect, together with advances in software technology over the recent past has, in fact, led to the widespread adoption of the title Multi-body Dynamics2 in the place of phrases like “Rigid Body Mechanics” and “Mechanism Design”. MBD finds applications in almost any field where there are moving mechanical components: machine tools, packaging equipment, conveyor belts, engines, road vehicles, elevators, railways, stereos, washing machines, aircraft, spacecraft, pumps, robotics – the list can go on almost indefinitely. One application that’s sometimes dismissed as trivial but throws up several exquisite applications of this remarkable science is the design of

2

Often abbreviated to “MBD” 5

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toys, as can be attested to by anyone who has puzzled over the internal workings of Rubik’s Cube. The problems that designers grapple with are introduced in the next chapter, but common to all of them is the need to deal with one or more of the forces, displacements, velocities and accelerations of different parts of the system. Some designers analyze mechanisms: that is, they find out the values of parameters of interest under different operating conditions. Still others synthesize mechanisms: they come up with designs that will provide required movement. The subject is often multi-disciplinary. For instance, the source of motion – the actuator – could be hydraulic. Study of the mechanical links or components requires a strong hold on mechanics. The control system could be electronic, while the sensors could be piezoelectric.

Learning MBD - Different Approaches There are two ways, then, to gain a command over the capabilities of MBD tools. One approach is to focus on the theory, drawing comfort from the fact that a robust theory can be applied widely, provided the fine-print is followed meticulously. Another approach is to pick a specific application and pay attention to the assumptions and data specific to this application. The use of general-purpose MBD software for CAE mirrors these approaches. At the “theoretical” level, all bodies can be modeled using a few basic building blocks. At the applied level, each of these building blocks is adapted to the requirements of the specific field. For instance pneumatics and hydraulics both use similar building blocks – valves, pistons, etc. – but the specific behaviors of the fluids varies. In several industries this “specific behavior” is treated as intellectual property. It is fiercely guarded, since it is arrived at over the course of much trial and error, and can make the critical difference between performance that’s “just good enough” and performance that makes the product a pleasure to use!

Putting It All Together MBD, then, is about more than just machines and mechanisms, but definitely involves mechanics. Unlike “simple” mechanism design, though, it often involves a lot more. If the study of the approaches that MBD tools take 6

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seems overwhelming, it is useful to remember the old saying take care of

the pennies and the pounds take care of themselves. Analysis problems are solved using the divide-and-rule approach: break down complex objects into simpler blocks, and these into even simpler blocks, and so on. The synthesis problems are solved by starting with known blocks, and looking for ways to put them together to achieve complex behaviors. Our goal, then, is easy to state and vast in it’s coverage: we expect to be able to design any mechanical system with moving parts!

The secret of getting ahead is getting started. The secret of getting started is breaking your complex overwhelming tasks into small manageable tasks, and then starting on the first one. Mark Twain

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Typical Design Issues Practitioners in the engineering industry often complain that software tools are used just for the sake of using the tools. All too often, these complaints are justified. What is the point of spending time and effort, in addition to money, on usage of tools if the results do not help the designer? To make matters worse, the usage of the tools may even draw resources away from the actual design goals! Of course, this potential criticism of CAE tools applies to all the tools covered in this series of books. What’s special about MBD? Why should we pay special attention to this aspect when studying MBD? As we have already seen, the spread of applications that MBD addresses is extremely wide. And we will see later that the MBD modeling approach takes a relatively abstract view of the behavior. It’s this abstraction that makes it that much more important for you to keep track of what the design issues are. Your entire model-building approach and results-interpretation should be tailored to suit these. Accordingly, before reviewing the underlying theory, it is useful to review some areas of application and the related design issues as relevant to MBD modeling and analysis.

Product Liability The lot of a product designer is often stressful, and not just because of pressures on time, cost and quality. Laws in several countries are extremely demanding, and the trend is towards stronger legal safeguards against faulty products. In a review of the impact of legislative reforms on productliability risks in the Asia-Pacific region3,

100% of insurers/brokers thought that there had been an increase in the number of product liability claims in the Asia-Pacific region since the Reforms. One hundred percent reported that there had been an increase in settlements.

3

Kellam, J and Nottage, L: "Report on Clayton Utz Asia-Pacific Product Liability Survey" (2006) 17 (9) APLR 121, published in the Australian Product Liability Reporter. 8

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Failures that can cause loss of life or grievous harm are often identified and publicized voluntarily by the manufacturers themselves. Such product recalls can be expensive both in terms of actual expenditure to fix the flaws and in terms of the damage to the reputation of the company involved. Legal protection for consumers mean that designers need to be alert even to failures that are potentially less exacting, as illustrated in the extracts from US Consumer Product Safety Commission’s recall notice reproduced below4:

In cooperation with the U.S. Consumer Product Safety Commission (CPSC), JB Research Inc., of Bellflower, Calif., had voluntarily recalled about 15,000 back massagers sold under the Relaxor, Deep Knead™ Shiatsu brand name. The motor for the massager's Deep Knead™ mechanism can jam and overheat. This will cause scorching to the foam and fabric on back of the unit, presenting a potential fire hazard to consumers…JB Research Inc. has received 46 reports of units overheating. No fires or injuries have been reported. Consumers should stop using the recalled massagers immediately. Since JB Research Inc. is no longer in business, recalled massagers should be discarded or destroyed to prevent fires and injuries.”

Some Application Areas Machine Tools Machine tools are often thought of as “old” technology, which in a sense they are: the growth in use of various types of machine tools dates back to the 100 years between 1860 and 1960. But that does not mean the technology is trivial or that design is easy. High-precision jig-boring machines, even in the 1940s, were designed so as to account for the effects of heat generated by human operators and even the slightest tremors of the earth. Common to all machine tools is the principal goal: a specified degree of precision.

4

The complete recall notice can be found at http://www.cpsc.gov 9

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To achieve this, the designer should produce variable movements, and provide control over these movements. Conditions of operation can usually be maintained within specified ranges, particularly if the designer can demonstrate a link between the operating conditions and the precision of the machine. Attendant design issues are the life of the machine, and its cost. From a designer’s point of view, models that can calculate and predict forces, loci of various points, and times of motion are particularly critical.

Packaging Machinery The term itself usually covers machines that can do one or more of wrapping, palletizing, taping, capping, filling, labeling and printing. If you consider that almost any goods – from toothpaste to automobiles – need to be packed, the size of the industry is extremely large. Environmental concerns are prompting changes in the materials used, prompting designers to exercise their ingenuity. From one point of view, packaging machinery is similar to machine tools. Conditions of operation can usually be controlled, and the movement needs to be controlled automatically. The differences stem mainly from the scale of usage. Packaging machinery is usually critical for mass-produced items, where the volume of production is extremely high. This means the design approach can often afford to sacrifice versatility of motion for economy and precision – in a sense, this is similar to the design approach that underlies Special Purpose Machines. And since the scale of production of the goods being packaged is very large, a lot of design focus is on the time of motion. A design that can reduce the filling time by 1 second can be much more attractive if the filling time per package is of the order of seconds! An example of this approach is highlighted by a product manufacturer’s brochure, which says “machine kinematics have been designed to allow a

very high speed, while at the same time leaving more time for the most delicate phases of the filling process”. In the pharmaceutical industry, speeds of 200,000 capsules per hour are not uncommon.

Engines Most mechanical engineers are familiar with, if not extremely comfortable with the working of, IC engines. What we sometimes fail to remember is 10

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that engines themselves can vary tremendously: from the enormous diesel engines to the rotary-piston Wankel engine. The picture shows engineers installing the thin-shell bearings for the Wartsila-Sulzer RTA96-C turbocharged two-stroke diesel engine – an engine with a stroke of over 8 feet. The technicians shown in the picture show its size! Engine design is a multi-disciplinary area, covering heat transfer, vibration, combustion, etc. Conditions of operation are less predictable than for machine tools, so designers often have to investigate and allow for harsh operations. Recommended ranges of operation are usually provided, such as the “red-line” speed limit for IC engines. From an MBD perspective, the red-line speed is an interesting parameter. If a 4-stroke engine is run at a higher-than-recommended speed, the force exerted on the return-springs may be high enough that the valves “float”. That is, the valve-lifters lose contact with the lobes of the cam. This, in turn, leads to lost horse-power. Unlike machine tools, the interest is not so much in providing variable movement as in calculating component forces at various operating conditions. These forces are then used to perform stress and fatigue verifications. The baseline science scenario [for the design Advanced engines also require quite of a Mars airplane mission architecture] sophisticated forms of motion-control. requires completion of a controlled aerial High performance engines, for example, survey, spanning a flight range of 500 km at alter the valve timings and lift as the an altitude below 2 km. These requirements drive selection of a powered airplane as well engine speed changes.

Vehicles Lumping cars, trucks, buses, motorcycles, bicycles, ships, aircraft and spacecraft, etc. into one group is obviously a simplification, but one that is quite effective from our current perspective. The degrees of complexity vary – from a few dozen parts in a bicycle to several thousand parts in larger vehicles – but the vehicles themselves have fairly similar requirements: stability, safety, comfort and (in most, but not all, cases) economy of operation.

as the airplane propulsion and navigation systems and aerodynamic configuration. The entire sequence of events (including pullout) is approximately 5 minutes in duration. Airplane extraction is initiated 7 seconds after heatshield release.

Six-degree of freedom multi-body simulations have confirmed the 7-second delay is sufficient to mitigate the potential for recontact between the airplane and the heatshield. From

The Mars Airplane IEEE, 2004

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The first requirement, stability, is of particular interest. Since the operating conditions tend to vary widely, a lot of design effort centers around issues of providing proper control over the behavior of the motion, rather than the motion itself. Remember that the motion is reasonably predictable. What is unpredictable is the acceleration that the driver applies, the conditions that the surface provides, and so on. Car designers, for example, pay attention to the “toe curve”, since it is a critical measure of drive quality of the vehicle. Modern designs have seen a steady increase in the amount of “on board” electronics used to help steer the vehicle safely. Many road vehicles, for instance, come with anti-lock brakes, where a control system senses the motion and automatically adjusts the brake pressure to prevent a skid. Vehicle stability, in fact, is currently receiving the attention of lawmakers in some countries. Electronics stability controls may soon be mandatory in many vehicles5.

Robotics In his book “Inside the Robot Kingdom”, F.L.Schodt paints an impressive picture of the Fanuc factory in Japan, where, under Mount Fiji, robots work unattended at night – making other robots! The point, of course, is that robots are not just inhabitants of Science-Fiction worlds. They are very much here to stay. Robotics, as a discipline, poses problems that are hard to categorize, since robots themselves are so hard to categorize. Some work in controlled environments, such as factories, while others are designed for harsh and unpredictable environments such as the depths of the oceans. Some require precision, such as those designed to help weak or ill humans, while others are designed to work with more forgiving payloads. The mechanical side of the robot is the classical mechanisms problem of synthesis: how to assemble mechanical elements that can describe various motions. In several cases, inspiration is drawn from biology, mimicking human or animal joints.

5

See http://www.consumeraffairs.com/news04/2006/09/nhtsa_stability.html, for example. 12

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A multi-body designer, then, needs to generate paths of motion, predict velocity of different parts of the assembly, and to predict forces that will be experienced and that can be generated by the robot. A large degree of integration with electronic control systems is also essential, given the current state of technology in robotics design. Issues of stability, which have to be handled by the control systems, have recently been addressed well enough for a two-legged robot to climb stairs or catch a ball thrown at it.

The MBD Modeling Philosophy There are many other areas, of course, where designers seek to understand the forces experienced by and caused by multiple bodies. There are also many modeling and analysis techniques other than MBD, some of which are covered in other books in this series. As the figure below6 emphasizes, any model is a part of a larger system, and can in turn be broken into smaller sub-systems – all the way down to quantum mechanics

MBD methods are reduced order models that are best applied at the “Product” and “Assembly” stages7.

6

Courtesy of Prof. Bert Bras, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA. 7 The Finite Element Method, covered in A Designr’s Guide to FEA, is most relevant at the “Component” and “Assembly” stages. 13

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From a product design perspective, just as product design follows a sequence – starting with a concept, going on to a rough design, and finally the detailed design – it is useful to group the design issues into system level issues and component level issues. Simulation of component behavior is often done using the Finite Element method. Here, the analyst requires the forces on the component as data for the model. Simulation of system level behavior is best done using the MBD approach. Obviously, one benefit is that the forces calculated from an MBD analysis can be used to provide data for a Finite Element analysis. However, there are other reasons that make this a natural way to address several complex design issues. For one, MBD models take a ”lumped” approach. That is, the behavior of an arbitrarily complex component or assembly is abstracted as a single element. The abstraction may represent a single rigid link, the suspension assembly of an automobile, or the undercarriage of an aircraft. In all these cases, some accuracy is traded for speed of analysis. Where a Finite Element analysis frequently requires minutes, if not hours or days of CPU time, an MBD analysis is often complete in seconds. Next, simple MBD models are used to build more complex models. In an approach that follows the engineering practice of using simple tools to build more complex tools, this provides the capability to quickly build complex models that yield useful results without taking an inordinate amount of time. Take, for example, a bearing that supports a rotating shaft. Both theory and practice tell us that there are losses within the shaft, but it is well near impossible to get an accurate model that can predict the losses in a production-quality bearing within reasonable times and at reasonable expense. The MBD approach is quite practical, yielding a usable model while taking into account the absence of detailed mechanics8: The bearing module is used to determine the bearing moments due to friction for all of the bearings in the transmission. The input signals to the bearing module consist of the torque at each shaft location. The module then calculates the torque loss due to friction with the relation:

8

From Design And Analysis Of A Modified Power Split Continuously Variable Transmission, A.J.Fox, West Virginia University 14

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Tloss,n = Fndµ where Tloss,n is the torque loss in bearing n, Fn is the force on bearing n, and µ is the bearing coefficient of friction. Fn is calculated from the force analysis on the shaft due to tangential gear forces and component weight. The bearing module solves this relation for each bearing location, and the output signals consist of the torque loss at each bearing location.

Note how simple the equation is. The simplicity is justified because there is no complete theory of the specific mechanics that also lends itself to quick calculations. Actual usage of this bearing module would only require that the coefficient of friction be fed in, since the forces are calculated using the equations of equilibrium. The model is not only simple, it is effective, since several of these bearing modules can be employed in the model of the overall transmission. Another example of this approach is the construction of computer models for animated movies – such as the dinosaurs in Jurassic Park and its sequels. Designers concentrate on capturing an adequate behavior of selected joints, not on the body as a whole. Once they have the individual joints behaving the way they want them to, they can assemble these to get the complete body – and can be sure that the assembly will move in a “realistic” fashion. A publication from the Aalborg University9 puts it well:

Depending on the type of joint, kinematic pairs are either referred to as force-closed or form-closed, where the fastening of the pair in a form-closed pair is maintained by the shape of the bones themselves, and the fastening of a force-closed pair is maintained by a superficially applied forces, such as a tendon. […] To limit the amount of joints, [… some] joints have been combined. Bone structure of [the figure] refers to the bone controlling the belly of the orc, which makes it possible to animate a “jumping” belly whenever the orc moves, walks, runs or jumps. This approach – complex models constructed from simpler models – lies at the heart of multi-body dynamics. Once we look at the building blocks that are commonly available, our study will be almost complete!

9

An Orc’s Tale – Animation of a Virtual World, Aalborg University, Copenhagen 15

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Summary Engineers working on CAE have several sources of worry, ranging from potential legal complications to essential product performance. To make things worse, the expectations change almost continuously during any product-design project. Almost invariably, the quality of results expected from the designers tends to get raised as the project progresses. One benefit, however, is that early in the design cycle the analyses need not be very precise. Later, when the detailed-design phases is undertaken, results need to be more accurate – but at the early stage, quick results are often of more value than accurate results. The MBD approach is tailor-made for this. And if it can be coupled with detailed-design tools such as Finite Element Analysis and Design Optimization, the design engineer really can’t ask for very much more.

The fact that the man who gave the world electric light, motion pictures, talking machines, and the Edison storage battery was responsible for this utterly useless device should encourage inventors whose first attempts have failed. George Lee Dowd Jr in Popular Science Monthly,1930, on Edison’s unsuccessful Helicopter

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Theory: Basic, Essential and Advanced

Theory: Basic, Essential and Advanced The availability of easy-to-use and reliable simulation software, coupled with high quality graphics, makes it easier to grasp the principles embodied in much of engineering and engineering mathematics. A picture can be worth a thousand lectures. This does not mean, of course, that the software can be used without a good grasp of the underlying theory. The previous chapter outlined the importance of focusing on the requirements of design and on the importance of proper abstraction of behavior. Using software without an understanding of the fundamentals is an invitation to disaster, not to mention being a waste of time, effort and money! This chapter is intended to serve as a quick reference, not as a complete discussion. Our emphasis is on providing definitions with a minimum of equations or other mathematical notations. The references listed at the end of this book are an excellent (and strongly recommended) source of complete theory, and should be referred to if any of the intentionally brief definitions presented below are ambiguous or incomplete. A quick review of the adoption of 3D CAD tools by the industry is illuminating. In the early years, users had to understand internal details like the equations of splines or the algorithms used to calculate surface intersections. As usage and software matured, a lot of this could be taken for granted: just as the way you can today drive a car without understanding how an IC engine works. Of course, if the engine breaks down, you either need to call for an expert, or develop the expertise yourself! In a similar fashion, if the software fails to achieve a particular task, a grasp of the theory used always helps. While MBD tools have not been adopted as widely by the industry as CAD tools, for a variety of reasons, the fact is that MBD tools today are both capable and robust. The benefits are clear, the applications are clear, and the tools are available. To use MBD tools effectively, of course, you should make sure that you pay attention to detail. When you work through the assignments that accompany this book, you may want to turn back either to this chapter or to the

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references listed at the end of this book to make sure that you are correctly correlating the software’s features with the theoretical underpinnings. Finally, classroom studies are often limited to problems that can be solved using trigonometry and vector algebra. Usually, this means the coverage is restricted to Cams, Gears and 4-bar linkages. Non-linear equations, complex numbers and mechanisms with more links are often omitted simply because they are not tractable enough for hand-calculations. MBD software makes such problems tractable, thereby making it easy for meaningful problems to be modeled and analyzed even at the learning level.

Theory … Basic Definitions Statics, Kinetics, Kinematics and Dynamics Mechanics (or, more correctly, Solid Mechanics10) has three branches: Statics, Kinetics and Kinematics. Statics covers the effects of forces on bodies in the absence of motion. Kinetics is the study of the action of forces on bodies in motion. Kinematics is the study of the relative motion between bodies. Kinetics and kinematics together are often referred to as dynamics. Often designers use kinematics to determine the initial design to achieve the required motion. Kinetics is then applied to investigate and improve weight, stability, cost, control, etc. Kinematics is sometimes called the geometry of pure motion – because there is no reference to mass or forces. Most CAD packages use this approach to animate assemblies. Kinematics, for example, can be used to calculate the motion required for a robot to pick up an object, while dynamics can tell you the forces required for this. A kinematic diagram is used to represent a mechanism – the figure on the right is the kinematic diagram of the folding steel chair on the left. The procedure to derive a kinematic diagram from a mechanism can be confusing, as can the process of visualizing a mechanism represented by a given kinematic diagram. MBD software is very useful here, since it allows you to add graphics to the kinematic diagram.

10

The kinematics of fluids is normally not called kinematics, and is considerably more complicated than that of solids. 18

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Mechanisms Engineering Mechanics differentiates between a structure and a mechanism by calculating the mobility of the assembly. If the assembly is such that no movement is possible, it is a structure. A good example is a simple plane truss, where the dimensions of the links define the only relative position that can be achieved. A mechanism is also sometimes defined as a device that transfers force / motion from a source to a destination. From our point of view, a mechanism consists of links and joints. Machines There is no clear-cut difference between a mechanism and a machine. Some define a machine as a mechanism that does useful work, but that distinction is not relevant to our study. One of the dictionary meanings for “machine” is, not surprisingly, “mechanism”. Conservation of Linear Momentum In the absence of external forces, the momentum of a body or set of bodies remains constant. When applied to linear motion, this results in the familiar equation F = m • a . Another way of stating the same, of course, is to say that force involved in a collision is equal to the rate of change of momentum. Collisions can be either elastic or inelastic. Elastic collisions conserve kinetic energy but inelastic collisions don’t. Both, of course, conserve momentum. The coefficient of restitution is a measure of the elasticity of the collision. It has no units, since it is the ratio of the differences in velocities before and after collision. It is 1 for a perfectly elastic collision and 0 for a perfectly inelastic collision – that is, one in which the bodies stick together after the collision. Conservation of Angular Momentum If the law of conservation of momentum is applied to angular motion, it leads the equation T = I • α , where T is the torque, I is the mass moment of inertia about the axis of rotation, and α is the angular acceleration. Calculation of the mass moments of inertia is often confusing, particularly to beginners who often forget the reliance on the axes (or coordinate systems) used, particularly if the values are calculated automatically by a CAD 19

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package. See the Glossary and References section for more details on Moments of Inertia. Links A link is a body that is a part of a mechanism. Some definitions of links require that they be treated as rigid bodies (i.e. those that cannot deform under the action of forces) but MBD removes this necessity. Links are often classified as binary, ternary and quaternary, depending on how many other links they’re attached to. A binary link is attached to two other links, a ternary link to 3 other links, and a quaternary link to 4. Nodes A node is the point at which one link is attached to another in a kinematic drawing. (Do not confuse this with a node in a Finite Element model!). A binary link has 2 nodes, a ternary link has 3, and a quaternary link has 4. Degrees of Freedom The DOFs (which is how the phrase degrees of freedom is usually abbreviated) of a link is the number of independent inputs required to determine its position with respect to the ground. The DOFs of a mechanism are the number of independent inputs required to determine the positions of all links (with respect to the ground) that make up the mechanism. Note that a joint “eliminates” one or more degrees of freedom, as described below. Calculating the DOFs of a mechanism is not a trivial task, as we will see when we discuss Gruebler’s Equation. A rigid body in 3D space has 6 DOFs – translations along the 3 axes, and rotations about the 3 axes. In 2D space (textbooks often refer to planar mechanisms, which are mechanisms restricted to 2D space) a rigid body has 3 degrees of freedom: translations along the 2 axis and rotation about the third axis (which is the cross product of the two axes of translation). Constraints A constraint is a condition that removes one or more DOFs. In MBD, a constraint is usually imposed by defining a joint. For instance, if a system consists of 2 links that are not connected to each other, the system has 12 dofs (6 for each link). If they are connected by a 20

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joint, however, the dofs will be less than 12. Which dofs are eliminated by the joint is often a source of confusion to a beginner. If the number of constraints is more than the dofs of the system, the system is described as over-constrained. An over-constrained system cannot be analyzed using MBD. If presented with an over-constrained system, many programs arbitrarily discard as many constraints as necessary. A designer should beware of such situations! It is far better to correct the joint definitions yourself than to leave it to the software. Joints From a mathematical perspective, a joint is just a constraint – it relates the motion between one or more DOFs of one or more bodies. In the context of MBD modeling, a joint is usually defined using a physical equivalent. Most joints eliminate one or more DOFs. However if the joint is redundant, it does not affect the dofs of the system. Redundant constraints are also called passive constraints: their presence or absence does not make any difference to the behavior of the mechanism11. Pairs, Higher and Lower Two links connected by a joint are called a pair. Pairs are classified as lower pairs and higher pairs. A lower pair is one where interchanging the links does not alter relative motion. For a higher pair, exchanging the links alters relative motion. There are only 6 lower pairs, while there are infinite types of higher pairs. The 6 lower pairs are revolute (or pin-joint), prismatic (or slider joint), helical (as in a nut-and-screw), cylindrical (as a shaft in a collar), spherical (or ball joint) and planar12. Cams, gears, belt-drives, etc. are higher pairs. Higher pairs normally need additional equations, such as the gear-ratio, to fully-define them. Higher pairs are also more susceptible to drawbacks such as backlash, slip, creep and friction losses. Of course, manufacturing tolerances can introduce error into lower pairs too. 11

Passive constraints can cause trouble when manufacturing tolerances are taken into account. 12 Planar, Revolute and Prismatic pairs can be treated as special cases of helical pairs. A zero lead makes it a revolute joint, an infinite lead makes it a prismatic joint. Moving the center of the helical pair to infinity gives a planar joint. 21

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Closure Some joints ensure contacts between the links by means of the elements themselves – a revolute joint is a good example. Such joints have form closure. Other joints, such as cams, require external forces to maintain contact, and have force closure. The external force can be supplied via a spring, or by gravity, etc. Chains A chain is a series of pairs connected together, without a grounded link. A chain is called a mechanism only if at least one link is grounded. This is because force-transmission makes sense only if the “ground” provides the support for the reactions that Newton’s Third Law guarantees. A chain can be either closed or open. Two binary links connected by a joint are called a dyad. Inversion The behavior of some mechanisms can change dramatically depending on which links in the chain are fixed and which are left free to move. An excellent example of this is the epicyclic gear train. Gruebler’s Equation and the Kutzbach Criterion Calculating the DOFs of an assembly is not easy. If movement is restricted to a plane (that is, if we have a planar mechanism), we can use Gruebler’s Equation:

F = 3(n − 1) − 2l − h where F is the total degrees of freedom of the mechanism, n is the number of links (remember to include the ground or frame), l is the number of lower pairs and h is the number of higher pairs. Be careful when using the formula – it is not foolproof in the sense that it cannot be applied blindly, but needs some judgment. The mechanism shown below has 1 DOF although Gruebler’s equation would say it has none!13

13

n = 5 since there are 5 links including the ground, l = 6 since there are 6 lower pairs. The formula fails due to redundancy: removal of the middle link has no affect on the mechanism. The correct values of n and l should be 4 and 4, respectively, which gives 1 dof. 22

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The DOFs of a mechanism are also called its mobility. This term is used when we want to count the number of input parameters that must be controlled independently to achieve a particular motion or position. The Kutzbach Criterion, which is used to calculate the mobility allows for the elimination of partial DOFs by a joint. Also remember that there’s a difference between mechanisms an 3D space and mechanisms in 2D space. The equation used in 3D has a slightly different form14. If the 2D equation is applied to a 3D mechanism, the answer can be misleading. Take, for instance, the slider-crank mechanism. If restricted to 2D, there are 4 links in all, with 3 dof each, for a total of 12 dof for the system. If link is grounded, that leaves 9 dofs. The three revolute joints remove 2 dof each, since they only permit rotation about the axis. This leaves 9 – 6 = 3 dof. The slider joint too removes 2 dofs, since it only permits translation along one axis. The system, then, has 1 dof.

If the same calculation is conducted in 3D space, we start with 18 dofs (6 dofs for each of the free links). The 3 revolute joints and the 1 slider joint remove 5 dof each. As a result, the mechanism is over-constrained!

14

For details on the 3D form, see page 551 of Advanced Mechanism Design, Volume 2, Erdman and Sandor. 23

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It is easy to see that a slider-crank mechanism in 3D should, of course, use spherical joints to avoid this situation15.

Essential Theory Analysis vs. Synthesis Analysis involves calculating items of interest for a given mechanism or system. Synthesis, on the other hand, involves finding a mechanism that provides a required behavior. Synthesis can be extremely challenging, since it means choosing both the types and dimensions of links and joints – called type synthesis and dimensional synthesis. Type synthesis is sometimes referred to as number synthesis since it determines the number of links in the mechanism. There are infinite possible mechanisms that can satisfy any given set of behaviors, so the chosen mechanism usually depends on the experience of the designer or the available links and joints. Complexity of synthesis rises dramatically as the number of links increases. Even for 4-bar mechanisms, Hrones and Nelson’s Atlas of Curves, which presented several thousand coupler loci, is often the starting point even today. The Atlas itself was compiled in the 1950s! Depending on the specifications for synthesis, the goal of the designer is one or more of function, path and motion. Function generation involves synchronization of the motion of input and output links, as in a Pantograph, for example. In path generation, a point is required to trace a path with respect to a reference frame. In some cases, timings can also be a part of the specification. A cricket-bowling machine is a good example of 15

Computational complexity theory is the study of the complexity of problems - that is, the difficulty of solving them. Some problems are difficult to solve, while others are easy. Take the traveling salesman problem, for example. If the network of cities grows by 1, the time needed to solve the problem - that is, construct the shortest route that visits every city exactly once - is multiplied by a factor of c, hence the time needed to find the route grows exponentially.

Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. These problems might require large amounts of time or an inordinate amount of space.

There exist a certain class of problems that require so much time or space that it is not practical to attempt to solve them although they are solvable in principle. These problems are called Intractable.

From The Wikipedia http://en.wikipedia.org/wiki/Complexity

See, for example, page 612 of Advanced Mechanism Design, Volume 2, Erdman and Sandor.

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this. Motion generation involves guiding the entire body (or link) through a prescribed sequence of motion. Consider the movement of the bucket of a tipper machine. Not only must the bucket follow a particular path, the rotation of the bucket must also be controlled. Types of Analysis Depending on the scenario being investigated, the analysis is classified as one of the following: •

Static – usually used to find the equilibrium position of a mechanism

•

Quasi-Static – used when the inertial forces are not important

•

Kinematic – used if there are zero dofs in the system. That is, all possible movements are specified either by joints or by input motion.

•

Dynamic (or Transient Dynamic)

•

Linear – most MBD models are non-linear. Linear analyses are used mainly to calculate eigenvalues16 or for design of control systems.

The data required to construct the model, the methods used to solve the problem, and the type of results that can be computed vary according to the type of the analysis. The last chapter of this book, Glossary and References, contains a table that summarizes these. Forward and Inverse Kinematics In forward kinematics, given the forces and positions of some links, we want to estimate the location / velocity / acceleration of the bodies or points of interest. In inverse kinematics, we want to know what forces to apply to achieve a required position / velocity / acceleration. The latter is often required in robotics.

16

Eigenvalues are not discussed in this book. See the companion volume A Designer’s Guide To Finite Element Analysis for more details on how and why we

calculate eigenvalues. 25

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Quaternions and Euler Angles Any spatial movement can be expressed as a combination of rotations and translations along 3 axes. Rotations, unfortunately, are not commutative – that is, the final configuration depends on the order of the rotations about different axes. Rotations and translations are usually represented by 3x3 transformation matrices, and Euler Angles are one convention used to specify angles of rotation. A quaternion17 is an alternative representation of the transformation. Instantaneous Center of Rotation This is useful to determine the relative velocities between two bodies. The instantaneous center of rotation for two bodies in plane motion is a point, common to both bodies, which has the same instantaneous velocity in each body. The point can be a “virtual” point, located off the two bodies. Damping Coefficient Vibrating bodies experience damping, a force that retards movement. While this is sometimes an adverse affect, in other cases it can be useful – as in the case of shock-absorbers on a car. Damping coefficients are hard to characterize. Testing is frequently used to establish reliable values. A damped system is non-conservative. This means that energy is “lost” (from the mechanical system) – it is converted to other forms such as sound or heat. Numerical calculations sometimes introduce numerical damping. This is a loss of energy due to the finite precision of computer-arithmetic or because of other truncation errors. Numerical Integration The differential equation of motion is

ma + cv + ku = p where m is the mass, c is the damping coefficient, and k is the stiffness. a, v and u are the acceleration, velocity and displacement, respectively, while p is the external force.

17

Discovered, invented or defined by W.R.Hamilton in 1843. The utility of the quaternion has been a subject of lively debate since then, but most engineers encounter H, the set of all quaternions, named for the illustrious mathematician.

26

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A common form of the same equation is

m&x& + cx& + kx = f (t ) where x(t) is the displacement vector. Given the initial state of the body or bodies, we need to calculate the configuration of the bodies as we move forwards in time. The initial state is called the initial condition. This is normally done by numerically integrating the equation of motion. That is, we use the finite difference equations to replace the derivatives with differences. For instance, we can write

v=

du ∆u u j − ui = = dt ∆t t j − ti

and a similar equation between a and u. To get started, i is at initial time. That is, at t = 0. At the initial time, if we know the velocity vt=o, we can solve the equation for uj. Then, substituting for in the equilibrium equation we solve for acceleration. Now that the values at time tj are known, we follow the same method to step forward. There are many numerical integration schemes available to solve such problems. The numerical integration methods are classified as explicit and implicit, single-step and multi-step, and corrector-predictor methods. For a complete discussion, see the references listed at the end of this book.

Related Topics Most mechanical components are relatively easy to deal with. You can touch, see and feel the components, and what you see is at least an indicator of what you will get. Electricity is less tangible18.

Approximately ten separate things have the name "electricity." There is no single stuff called "electricity." ELECTRICITY DOES NOT EXIST. Franklin, Edison, Thompson, and millions of science teachers should've had a long talk with Mrs.McCave before they decided to give a variety of independent science concepts just one single name. 18

From an article by William J. Beaty. See http://amasci.com/miscon/whatis.html 27

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Mrs.McCave was invented by Dr.Seuss. She had twenty three sons. She named them all "Dave." Whenever we ask "WHAT IS ELECTRICITY," that's just like asking Mrs.McCave "WHO IS DAVE?" How can she describe her son? There can be no answer since the question itself is wrong. It's wrong to ask "who is Dave?" because we are assuming that there is only one Dave, when actually there are many different people. They all just happen to be named Dave. Who is Dave? Mrs.McCave cannot answer us until she first corrects our misunderstanding. Understanding electrical systems is not essential for MBD, but can help when working with the control of mechanisms. Electricity is a common means of transmission of power, so in this section we’ll briefly review a few salient aspects related to electrical actuation and control. Transfer Function A transfer function is simply an equation linking input and output of a system. It is probably more common in electrical engineering than in mechanical, but the principle is applicable to any system modeling. For MBD systems, transfer functions are commonly used to represent the effect of controllers – sensors and actuators, for example – which are often electrical in nature. AC, DC, Stepper and Servo Motors In many mechanisms, the source of motion or force is an electrical motor. Mechanical sources (hydraulic, pneumatic, etc.) are also used, but are often less precise than electrical controllers. The inner workings of motors are beyond the scope of our discussion, but it is useful to be familiar with the characteristics of the most common types of electrical motors. For example, you may need to calculate the time taken for a motor to reach operating speed, to define the force-time variation in your MBD model. AC motors (the familiar three-phase induction motors, also called squirrelcage motors) typically run at 1500 rpm. They are simple, reliable and relatively inexpensive. One drawback is that they need a transmissionsystem to achieve different speeds. Speed and torque control are easier with the more expensive DC motors, since the speed and torque are directly 28

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linked to the voltage and current. The disadvantage is that the brushes wear out with usage. Brushless DC Motors (BLDCs) address this, but are even more expensive. Both these motors are less than ideal if position control is important. For accurate position control, Servo or Stepper Motors are better choices. The position of the armature of a servo motor is controlled by the electrical input. They are widely used in robotics and radio-controlled toys. Stepper motors are often used in open-loop systems. Sensors Applications in which precision is important usually use closed-loop control systems. That is, the output values of parameters of interest are used to decide the input signal. A sensor is a device that monitors the parameters of interest. Sensors often, but not always, use electrical signals. Gyrocompasses, for instance, are mechanical devises that provide visual feedback. Sensors can also be used to detect events. An excellent example of an event sensor is a limit switch: you can use it to shut off input when a particular position has been achieved. Sampling Frequency Sensors measure the parameters of interest. If the parameters vary with time, the Nyquist criterion dictates the minimum sampling rate that should be used. That is, if the interval between measurements is not small enough, the sensed values will be unreliable. PID Control After acquiring the signal from a sensor, how should the input signal be corrected? If we define the error as the difference between the sensed value of the parameter and the desired value, positive feedback means input signal is increased in proportion to the error while negative feedback means the input signal is decreased in proportion to the error. This is called Proportional control. In some cases, the cumulative value of the error and the rate of change of the error are also important. In this case, simple proportional control is not enough. We also use Integral and Derivative control – that is, the input signal is modified based on the integral of the error (thereby taking into 29

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account the cumulative error) and the derivative of the error (which is nothing but the rate of change of the error). Closed loop systems often come with PID Controllers. These incorporate all three elements – proportional, integral and derivative – and need to be tuned. Depending on the actual situation, the gain for each measure of error is chosen or adjusted.

… and Practice Remember that MBD models are models of physical systems. Their value lies in their ability to represent the behavior of systems in the real world. The models, however, are built using theory, and as we know theory is always built on some assumptions19. As the bumble-bee paradox illustrates, our knowledge of theory, while powerful and useful, is far from complete. It’s important, therefore, to keep in mind the reasons that a model of the physical world can differ from actual behavior.

Precision Points Consider, for instance, function generation. That is, you have to correlate the motion of input and output links. A 4-bar mechanism is often chosen because it can be synthesized easily (relative to linkages with more links, that is!), and is often easy to construct. Unfortunately, a 4-bar mechanism is not capable of error-free generation of arbitrary curves.

“Conventional aerodynamics seemed to suggest that the insect should not generate enough lift to fly. The bees stayed resolutely airborne and the sums caused consternation. The underlying problem turned out to be treating a wing as if it was fixed, like in an aeroplane and, thanks to studies over the past few years, including the construction of robotic bees, this "bumble-bee paradox" has been solved: extra lift comes when flexible insect wings slice through the air at a high angle of attack, creating a large swirling vortex at their leading edge. In this way, insect wings produce the vortices – spinning masses of air – which generate lift and help them move. Today, Prof Ismet Gursul of the University of Bath will describe another step on the way for engineers to make air vehicles smaller than a human hand that can be used for detecting chemicals leaks and reconnaissance.”

As an acceptable solution, we settle for correlation at a selected set of points. These points are called the precision points. The location and spacing of these points can be 19

See Godel, Escher, Bach: an Eternal Golden Braid by D.R.Hofstadter for an entertaining, challenging and comprehensive discussion of the Incompleteness Theorem. 30

Roger Highfield, Science Editor The Telegraph

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calculated by a variety of methods, none of which can be addressed in this book20. Remember to keep this in mind when working with either the synthesis of a mechanism or the verification of a proposed mechanism. Since a linkage only has finite significant dimensions, there can only be a finite number of precision points.

Engineering Data and Robust Design Beginners have a tendency to take published data as the Gospel Truth. This is, quite simply, wrong. At the other end of the spectrum is a stubborn and unreasonable refusal to build a model unless all data is available at the required precision. Designers, like the rest of the human race, have to live in a world that is less than perfect. Data is not always available at the right time. It may be insufficient. It may be unreliable. And so on. To deal with this, one approach is to look for robust designs. In this approach, we look for a design that will produce the required output even if the specified inputs vary. Obviously, this is not always possible. This is particularly true if we are looking for an optimum design – one that provides the best possible performance at the least possible cost. Various techniques are available to verify the robustness of designs, as well as to build robust and optimal designs. More details on these can be found in the companion volumes in this series. One approach in particular is very useful for MBD modeling: the method called parameter identification21. This refers to the extraction of information about a system using measured input-and-output data. It is particularly useful when the transfer-function approach is used, or if a high degree of abstraction is involved.

The Virtuous Circle MBD tools are easier to use if the fundamentals are clear, and fundamentals are easier to grasp if MBD tools can be used to demonstrate the 20

See Advanced Mechanism Design: Analysis and Synthesis by G.N.Sandor and A.G.Erdman for an excellent discussion. 21 Covered in the companion volume CAE and Design Optimization - Advanced 31

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applications. It’s this aspect that makes it so enjoyable and so challenging to study and use MBD tools for CAE!

In theory there is no difference between theory and practice. In practice there is. Yogi Berra

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Working with MBD Models

Working with MBD Models Based on the previous chapters, we now have an idea as to the design issues involved in several different types of MBD scenarios, and are familiar with the terminology and underlying theory. In this chapter, we’ll look at how software helps us eliminate a lot of the tedium from the process of applying the theory to design problems.

Different Strokes for Different Folks A Machine-design Approach MCAE, short for Mechanical Computer Aided Engineering, covers the various tasks involved with design of mechanical components. Solid modeling, CNC tool-path generation, creation of manufacturing drawings, Finite Element Analysis, etc. For all of these, the 3D solid model usually serves as the basis. That is, the 3D model is the central source of data on which all other applications draw. MBD too falls under the umbrella of MCAE, particularly when the approach or goal is to design a mechanism. It’s only natural, then, for a designer to expect to follow the same approach: to expect to use the 3D model as the starting point for MBD modeling and analysis. As we have seen in the previous chapter, solid models are indeed useful when it comes to tasks like the calculation of the mass moments of inertia of geometrically complex objects. But as we have also seen, a lot of mechanisms-theory uses kinematic representations to perform various calculations. In this approach, the detailed shape of the body is immaterial. You only need to specify the locations of the nodes and the moments of inertia of each of the links22. This means the detailed 3D model is often a dispensable overhead. There are situations where the 3D model is essential, as will see subsequently when we study problems that involve contact. But MBD models

22

Those familiar with FEM will recognize the parallel with beam elements. 33

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for many scenarios can be built without using any 3D graphics (of the “shaded” variety) whatsoever. The use of 3D graphics, though, often helps interpretation dramatically. Provided the tools support it, it is useful to include 3D graphics even if the modeling does not require it, simply because it makes it so much easier to visualize the performance and to catch modeling errors.

A Control-System Approach There are some elements, however, particularly those representing electrical systems such as motors, where 3D graphics hurts more than it helps. It makes little sense to take the effort of building even a representative model of a motor when the only data to be visualized is the rotation of a shaft! Also, unlike mechanical power transmission which involves visible, tangible conduits like links or pipes, electrical power transmission cables need no 3D modeling. Electrical designers, in fact, often prefer to use symbolic tools for design simulation.

And Ever The Twain Shall Meet The thing to remember, then, is that there are two distinct parts to any MBD model, and effective usage brings both together. One is the equivalent of the kinematic model: locations of nodes, properties of links or transfer functions, and constraints between them. This is sufficient for all electrical components, and is often used by experienced mechanical designers to quickly build MBD models of mechanical parts too. The other is the 3D graphics, consisting of visually appealing images of the bodies that are represented by the links. This is rarely, if ever, required for an electrical component. The graphics are usually derived from a CAD model, but the task of integrating these into the MBD model requires an understanding of the abstractions involved in and represented by the various elements of the model.

Basic Building Blocks Just as CAD modelers give you the power to use simple construction primitives to build intricate and accurate models of almost any geometry, one of the best things about MBD tools for CAE is that we can start with

34

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simple systems and put these together to achieve remarkably useful simulations of very complicated assemblies. Once you understand the data required for the “primitives”, it is easy to work forwards. Pay attention, therefore, not only to the building blocks required, but also to the data that is required for these.

Bodies A body is the same as a link. Graphics can be associated with a body if required, but it’s not essential. The mass properties of the body are essential. These properties consist of the mass and the 6 mass moments of inertia and the coordinates of the center of gravity of the body. Further, for a dynamic analysis, the initial velocity of the body must be specified. The initial position is defined by the joints, while the accelerations are computed as a part of the solution. In some cases, the body may have no appreciable moment of inertia. This occurs when the mass is so closely concentrated at the center of gravity compared to the overall dimensions involved in the model. In conventional analyses, bodies are considered to be rigid, but current technology includes the capability to define bodies as flexible23.

Constraints or Joints A joint represents a constraint on the bodies that are connected to it. A revolute joint, for example, only leaves 1 dof free – the bodies can only rotate with respect to each other about the axis of rotation of the joint. The 6 lower pairs are essential for modeling. Higher pairs are not essential, since they do not form a finite set. In the absence of available elements, they can sometimes be constructed using combinations of other building blocks.

Forces A concentrated force, at its most basic, can de defined as a vector: the magnitude, orientation and point of application are enough to completely specify the force. Other forces, require more general definitions, since not all forces can be modeled as point forces. 23

This aspect is covered in more detail subsequently. 35

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For modeling of physical systems, for instance, one force that is widely required is gravity. In mechanics, gravity is called a body force since it acts at all points in the body, and the force experienced by the body depends on the distribution of mass within the body. From a numerical-calculation point of view, the “smoother” the force is, the easier it is to calculate the solution. The smoothness of a function is usually measured by its continuity: a function with n derivates that exist is smoother than one that has n-1 derivatives that exist. This smoothness definition, of course, applies both to spatial derivatives and time derivatives. Since MBD models take a lumped-approach by abstracting the bodies as pairs (that is, links connected at nodes and constrained by joints), for MBD the smoothness normally relates to the time-derivatives of the prescribed forces. A force that has a sudden jump in time like a stepfunction is less tractable than a ramp function, for example. The derivatives of the step function, shown in red, are not defined24 at the time where the force steps up.

Motions A prescribed motion, strictly speaking, is a constraint: it removes the necessity to calculate one or more dofs. Since the motion dictates how the dofs move, the dofs are no longer “free”. Usually specified for a joint, prescribed motion can be arbitrarily complex both in space and in time.

Sensors In several situations the forces acting on the body depend on how it responds to the force applied initially. It is not always reasonable to expect the statement of the problem to specify the values of forces against time: what the statement should include is what forces will be applied if certain events taken place. Sensors, therefore, are required to detect events – such as the closing of a gap that can trigger a change in one or more forces – or variables such as the velocity, so that a control system can change state based on the value.

24

Strictly speaking, the derivative at these points is the Dirac Delta function, which defines the integral of the derivative – not the derivative itself. 36

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Controllers Control systems are usually defined by transfer functions. The transfer function can be quite complex, but broadly fall into two categories: SISO and MIMO. The former is short for Single-Input-Single-Output, while the latter stands for Multiple-Input-Multiple-Output.

Reference Entities Coordinate systems are critical in MBD analyses. Displacements of bodies are almost always large, so bodies may change orientation during the period of interest. We know that properties like the moments of inertia are strongly dependent on the coordinate system used. In such a situation specifying data with respect to an immutable “global” coordinate system does not make sense. Local coordinate systems associated with a joint or a body are usually used instead.

User routines While not essential, user routines are often useful, particularly if the behavior of a design-entity needs to be protected from external scrutiny. If a user-routine is used, it is often saved as a Dynamic Link Library (or DLL) that is called by the MBD solver at each time step.

Other Entities Design-chains nowadays consist of an Original Equipment Manufacturer (OEM) and several levels of suppliers, often called Tier-1 suppliers, Tier-2 suppliers, and so on in decreasing order of design complexity, with component suppliers at the end of the chain. The MBD approach encourages a build-and-reuse strategy. Specific industry segments or applications tend to use standard definitions, and share these definitions across several levels of the design-chain. Such data can be “published” and shared as special purpose entities.

Solution Control The first part of the MBD-Simulation cycle involves building the model. The second part, which we have largely taken for granted in our discussion, relates to the methods used to solve the equations of motion. A complete discussion of the various methodologies employed is beyond the scope of this book, but one aspect is worth discussing. 37

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The three measures of performance of a solution algorithm are stability, accuracy and efficiency. Efficiency is easy to understand and measure. It is the amount of CPU time and disk space the solver requires to carry out the calculations. Stability and accuracy are a little harder both to understand and to measure. A stable method need not be accurate, while a method can be unstable and still yield accurate results at particular combinations of circumstances. To understand this, reflect for a moment on which you would rather have – a watch that has stopped or a watch that loses 5 seconds a day? The former is 100% accurate twice a day, but you have no idea when. The latter gets more and more inaccurate as time passes, but you can always tell how inaccurate it is, and therefore correct accordingly. Stability25 of a method allows us to refine the parameters safe in the knowledge that we can predict the effect on accuracy. Accuracy without stability is meaningless. In MBD simulation, stability relates to performance of the algorithm used for numerical integration. Apart from the choice of the algorithm itself, the time step (∆t) used for the integration also matters. Since MBD models involve a fairly high level of abstraction, numerical solution algorithms tend to be “tuned” for specific applications.

Results - Verification and Validation MBD analysis is a fairly simple in the types of outputs generated: animations of movement and plots of forces, accelerations, velocities, displacements against time are the main data generated. What’s challenging, though, is to check whether the data is realistic. It can be very tempting to accept the results as “right”, when it is usually more correct to use them to gain an insight. Accepting the results of an MBD simulation without some correlation with other sources of information like physical tests is rarely a good idea. As in all other forms of engineering analysis, verification and validation are distinct, though they go hand in hand. 25

38

Monotonic convergence of series is a related topic.

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Validation consists of asking whether the right equations have been solved. Verification involves checking whether the equations have been solved correctly. Data like the coefficient of restitution or the coefficients of friction should not be taken for granted. It is always sensible to check the design for performance over a range of values, rather than for single values of data. A question raised by a user of simulation software serves to drive the point home:

My simulation results aren't matching up with test results. My model contains springs which naturally have some damping. How would I go about determining the damping coefficient? The answer, of course, lies in a judicious mix of theory and practice. A test can suggest values, but tests are rarely repeatable. A wise designer would first check the theory underlying the model used in the solver, then “tune” the model by selecting a value that does a good job of reproducing the test results without violating the assumptions implicit in the theory. System Identification and Design Of Experiment are techniques well suited to this task26.

Optimization The Monaco Grand Prix, an annual event that is one of the more celebrated races on the Formula 1 circuit, puts drivers and their racers on the streets of Monaco. Normal traffic is off the roads, of course, while the drivers attain speeds that are breathtaking even to the casual observer. For engineers on the design teams, they represent a formidable challenge. MBD tools help calculate critical parameters such as the lateral forces on the tires, of course, but it’s not enough to just predict the values. What the Formula 1 driver, who’s pitting his or her life against equally skilled and competitive drivers, needs is an optimum design. Stories of losing drivers 26

Both are discussed in the companion volume CAE and Design Optimization -

Advanced 39

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showing their ire on the vehicle and its designers are a part of the folklore of Formula 1 racing! The task that the design engineers face, then, is not only to analyze the forces, but to calculate how much which parameters need to be changed to give the driver that edge that can make the difference between winning and losing. And, in some cases, between life and death. Spacecraft are similarly challenging to design. Even if unmanned craft do not raise the scepter of life-and-death, the loss of a craft can mean a huge loss in terms of money and time. Fortunately for the health of design engineers, not all MBD designs are so demanding. But in a competitive marketplace, the struggle to design better products cheaper and faster is never ending. Techniques to achieve such designs are discussed in the other books27 in this series.

Explanations should never multiply assumptions without necessity. When two explanations are offered for a phenomenon, the simplest full explanation is preferable. Occam’s Razor, a principle popularized by William of Ockham

27

See CAE and Design Optimization – Basics and CAE and Design Optimization –

Advanced 40

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MBD Simulation with HyperWorks

MBD Simulation with HyperWorks We now have a fair understanding of both the design issues and the theoretical basis for MBD simulation. We also know that MBD problems originate from 3D CAD in several, but not all, cases. We have seen the 3 distinct phases involved in the process. We start by creating the model, go on to solve it in the second step, and then move on to the third stage, results-interpretation. Finally, we have seen that it would be nice to have the ability to optimize a design and to check how robust it is in terms of tolerance to deviations in operating conditions or data. With this background, let’s look at the problem from a HyperWorks point of view.

The Simulation Process The various modules of HyperWorks can be used in a variety of ways for MBD Simulation, depending on the problem specification and complexity. We’ll review the different approaches, and correlate these with our understanding of how MBD is used in the Product Design cycle.

Model Preparation – MotionView and HyperMesh The first stage, model preparation, can be done using either HyperMesh or MotionView. HyperMesh is perhaps better used if the components are mechanical in nature and are flexible. That is, if stresses in the components are to be calculated in addition to the forces and velocities. MotionView is more of a “traditional” MBD preprocessor. It is less 3D-graphics-intensive than HyperMesh, since it allows the definition of models using MDL – the Model Definition Language – in which graphical representations of the surfaces and volumes of the bodies are an option, not a necessity. This capability, which is similar to using kinematic diagrams, is not available in HyperMesh. Since MotionView also allows us to simulate the behavior of flexible bodies and add graphics, our focus in this book and in the accompanying exercises is on MotionView, not on HyperMesh28.

28

See A Designer’s Guide to Finite Element Analysis – Student Project Summaries for an example of how HyperMesh can be used for the analysis of flexible MBD systems. 41

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Solving the Equations – MotionSolve and OptiStruct Once the model has been built, the center of attention shifts to the solver. This is where the equations of motion are drawn up, parameters chosen for the numerical algorithms, and the numbers are crunched. This is done by MotionSolve. It is usually invoked from within MotionView but can also be used as a standalone application that reads an input file and generates output data. OptiStruct, as detailed in other books in this series, is intended for optimization of linear problems and for linear finite element analysis. However if a problem statement requires the kinds of analyses that are a part of MBD, OptiStruct can call MotionSolve. The process by which OptiStruct invokes MotionSolve and interacts with it is transparent to the user: except for records in the log file, there is nothing you need to do to manage this process. Since our goal is to see how MBD theory and MBD practice come together, we will restrict our attention to MotionSolve. If you’re comfortable with Finite Element Analysis, the companion volumes in this series discuss the use of OptiStruct for MBD problems.

Results – HyperView and HyperGraph The data generated by the solver depends not just on the statement of the problem, but also on what you have asked to review. For a dynamics problem for example, you may choose to generate printed results at specific time steps. Both HyperView and HyperGraph are useful in this respect. HyperView provides a variety of facilities for 3D viewing: animation, vector plots, and so on. HyperGraph comes in handy to generate plots. For cam design for instance, you need to plot the velocity-vs. time.

Optimization – OptiStruct and HyperStudy As we have seen above, OptiStruct can invoke MotionSolve transparently, as needed. This is useful not only to when the MBD model contains flexible links, but also to run optimization. Look up the online documentation for a discussion on techniques like the Equivalent Static Load Method for the optimization of problems involving dynamic stresses.

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Optimization of MBD models, of course, is not always related to stress or mass. In the case of synthesis of mechanisms, position control may be a more important objective. HyperStudy, which supports DOE and other techniques for non-linear optimization and robust design, can be used with MotionSolve to address these requirements. A discussion of this approach is contained in CAE and Design Optimization – Advanced.

The Anatomy of a Model MotionView provides all the building blocks we listed as essential and desirable for MBD modeling. Remember that one of the recommended approaches in MBD is to build validated libraries of systems, and to use these as sub-systems in the construction of more complex assemblies. To promote this approach, MotionView stores model-definitions in files.

What the Files Contain The principle storage structure for MBD models follows the MDL (Model Definition Language) format. An MDL file, which usually carries the suffix mdl is an ASCII file that can be opened using any text editor29. These files can be created without using MotionView at all. This approach requires that you be familiar with the syntax of the MDL statements. For instance, a revolute joint is defined using the statement *RevJoint(…) where the items in brackets should be replaced with the relevant values, as shown in the annotated file displayed alongside. 29

Statements 1 to8 are definition-statements. Every entity is defined by at least a name and a label. Entities such as joints require additional data. 1. *BeginMDL (pendulum, "Pendulum Model") 2. *point (p_pendu_pivot, "Pivot Point") 3. *point ( p_pendu_cm, "Pendulum CM") 4. *Body (b_link, "Ball", p_pendu_cm) The joint connects the ground (default name is B_Ground) and the body defined on line 4. The axis of rotation is the X axis, centered at the point defined on line 2 5. *RevJoint (j_joint, "New Joint", B_Ground, b_link, p_pendu_pivot, V_Global_X) Graphics are purely for visual appeal. The sphere’s radius is set to 1, and the cylinder’s radius to 0.5. 6. *Graphic (gr_sphere, "pendulum sphere graphic", SPHERE, b_link, p_pendu_cm, 1 ) 7. *Graphic (gr_link, "pendulum link graphic", CYLINDER, b_link, p_pendu_pivot, p_pendu_cm, 0.5, CAPBOTH ) This is where we specify the output: we want the displacement history of the link defined on line 4 8. *Output (o_pendu, "Disp Output", DISP, b_link) Here we assign coordinates to the points, and mass and moments of inertia to the link 9. *setsystem (MODEL) 10. *setpoint (p_pendu_pivot, 0, 5, 5) 11. *setpoint (p_pendu_cm, 0, 10, 10) 12. *setbody ( b_link, 1, 1000, 1000, 1000, 0, 0, 0) With this, we have finished defining our model 13. *EndMDL ()

MDL files can be encrypted, to protect their contents. 43

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In the MDL syntax, the name is used by other MDL statements, while the label is used in the interactive-editor. For example, in the annotated MDL file, the name of the point defined on line 3 (p_pendu_cm) is used in the link definition on line 4. In MotionView, you would see it referred to by its label (that is, as Pendulum CM). You will see that there are two types of statements for each entity. The first names it, the second assigns data to it. The definition statement must always precede the assignation statement, of course. It is customary, but not essential, to group all definition statements followed by all assignment statements. It is also customary, but not essential, that names follow a pattern. This makes it easier to read an MDL file, as you will have to from time to time. In the annotated example, the first letter of the variable name indicates its type – p for points, b for bodies, and so on. Note that the “ball” of the pendulum is not modeled as a link at all from a kinematic point of view. To make the graphic display realistic, however, graphic primitives are assigned to the link. In general, graphics can be assigned either from predefined primitives (such as the cylinder and sphere used in the example) or by importing graphics from files. The latter is common for complex geometry, and ways to do this are covered in the accompanying projects. Since MotionView is an interactive graphics editor, and since model construction may well take more than one session, it is often useful to save the definition of the “desktop” – the windows, their contents, the last view of the model, and so on. These items are relevant only to the interactive graphics environment. They are of no use to the construction of hierarchical systems (systems that are built using other systems). So MotionView uses a different structure, the Session File format, to save this data. Session files usually have the suffix mvw, and contain the complete MDL definition of the model in addition to the state of the desktop. The MDL statements can be saved either in the MVW file or as a separate MDL file that is referenced by the MVW file.

Summary of Modeling Entities MotionView and MotionSolve together provide several building blocks, some of which are more than the “basic” blocks we discussed in the earlier chapter. Some of these are essential for the modeling of higher pairs. For 44

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example, you need point-to-surface or point-to-curve constraints30 to model a cam and its follower. A complete list of the entities and their properties is contained in the online documentation. The table below summarizes some of the more commonly used entities. Entity

Data Required#

Notes

Point Body

Coordinates Mass, Moments of Inertia

Spring

Stiffness, Damping, Preload, Free length / angle Names of the two bodies connected by the joint, and the axis of revolution Names of the two bodies connected by the joint, and the axis of sliding Names of the two bodies connected by the joint, and the center of rotation Body it is connected to, origin and orientation

Usually used to define nodes Local Coordinate Systems can be defined if the moment of inertia’s origin is not the same as the centerof-mass Springs can be either helical or torsional Leaves only 1 free dof – rotation about the axis

Revolute Joint

Translational Joint

Ball Joint

Marker

System

Points of attachment, orientation and initial conditions.

Leaves only 1 dof free – translation along the axis Leaves 3 dofs free – rotations about the 3 axes A marker is a Local Coordinate System, but is treated as a distinct entity. You can attach a marker to a point on a link, and request output for that marker! These are roughly similar to subroutines in a programming language. A system can be saved in a separate MDL file

# - All entities require a name and label

Several more entities – joints, bodies, forces, etc. – are supported. See the online documentation for details.

Solution and Results MotionSolve can solve several different classes of problems. The most general class consists of problems in dynamic analysis, where the system can have more than one uncontrolled dof.

30

These are called PTSF and PTCV constraints, respectively. 45

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Static analysis is most often used to compute the equilibrium configuration of a mechanism. Kinematic analysis, used for systems that have no uncontrolled dofs, is typically used early in the design cycle, at the concept stage. Quasi-static analysis is applicable when the forces change with time, but do so slowly. This means inertial forces can be ignored, and the staticequilibrium equations can be solved at each instant of time. Stability analysis is a good example of its usage. MotionSolve does not read MDL or MVW files. Instead, MotionView creates an XML file that is used as input by MotionSolve. Several different output files can be generated. The important ones are: •

Log files (.log) contain the history of the solution. It’s a good practice to review the log files after every analysis, checking for errors or warnings.

•

Altair binary files (.abf) are used to generated animations in MotionView

•

HyperView 3D Player files (.h3d) can be viewed without HyperWorks, using the free player

•

Plot files (.h3d) are used to generate graphs with HyperGraph

Remember that MotionSolve has to solve non-linear equations, and has to numerically integrate the differential equations of motion. The numerical solution of the non-linear equations is iterative. That is, the solver first guesses at a set of values and checks whether these form the solution at that instant of time or not. If there’s an error, the software corrects the guess and repeats the cycle until the error is within an acceptable tolerance. Once this happens, the software concludes that the iteration has converged, and moves onto the next time step. If the error does not fall within the specified tolerance within a specified set of iterations, the software concludes that the solution has diverged (i.e. not converged) and gives up the hunt for the answer. MotionSolve’s default settings for the numerical algorithms are usually adequate, but in advanced situations, you will need to choose between the Maximum Kinetic Energy Attrition method and the Force Imbalance method, between the Adams-Bashforth-Adams-Moulton and VSTIFF / MSTIFF integrators, the integration-time-step size, the iteration tolerance etc. 46

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Familiarity with the mathematics is, of course, essential for proper choice of these settings. The online documentation and the references listed at the end of this book are a good place to cover these topics. One warning, however, is that the default settings work well for a wide range of physically realistic problems. That is, for problems where the properties of various entities in the system are realistic. Entering meaningless values, or neglecting to check for consistency in units are the first things to check for if MotionSolve fails to converge.

Integration with HyperWorks MotionView and MotionSolve are quite closely integrated with the other HyperWorks applications. For instance, it is possible to use MotionView, together with HyperForm, to construct extremely realistic and useful simulations of stamping transfer presses. Such examples of integration are presented at technical conferences the world over. Several samples can be found on the website listed at the end of this book.

Freedom is not worth having if it does not include the freedom to make mistakes. Mahatma Gandhi

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Advanced Topics Modeling physical systems can be extremely satisfying, but it is quite a challenging task. As Tom Clancy put it31, “The difference between fiction and reality? Fiction has to make sense.” All simulation models are, of course, fiction. Their proximity to reality is limited by several things: the assumptions inherent in the model that generates the equations to be solved, the algorithms used to solve the equations, the precision of the computer if numerical methods are used, and so on. Since MBD models involve a high level of abstraction, several complications can be swept under the carpet. That is, the errors introduced by the abstraction can be compensated for by tuning the model, as outlined earlier. However there are some situations in which more detail has to be included, for the simulation to be realistic and useful. Some of these are discussed briefly in the pages that follow.

Flexibility When asked “Is light a particle or a wave?”, Einstein is supposed to have answered, wholly seriously, “Yes”. Wondering whether a body is rigid or flexible is a similar question, and deserves the same answer. The sheer complexity of including the effects of the elasticity of links has led to the widely used assumption of rigid links but that is not always accurate enough.

Compliant Joints The pin-joint in a link can be a major source of error, as any designer who has analyzed tolerance stackup can attest. As we have seen, designers usually try to reduce the number of links in a chain. In some applications such as the “scissor linkage” or in several open loop mechanisms, however, the number of links is deliberately large. In cases like these, or in case where precision is extremely important, including the compliance of a joint in the model can make a significant difference. 31

In a similar vein, Mark Twain observed that “It's no wonder that truth is stranger than fiction. Fiction has to make sense.”

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Advanced Topics

High Speed Mechanisms At high speeds, the effects of inertial forces are large enough that the deformation of the links may be significant. Neglecting these may well lead to failure of the mechanism: it may jam, vibrate too much, generate too much noise, and so on.

Compliant Links An interest in high precision makes it preferable that the deflection of the links because of elastic deformation be included in the model. In mechanisms that involve bodies of different materials, some materials may be much less stiff than others – which means the stiffer ones can be considered rigid, while the more flexible ones should, preferably, be compliant.

MBD and FEA Finite Element Analysis is widely used to calculate stresses and deformations due to elastic effects, and it is only natural that an interface between FEA and MBD is the preferred way to include the effect of link-compliance in MD simulation. One challenge, of course, is that the very approaches of FEM and MBD are different. One uses a distributed model while the other uses a lumped model. The first results in partial differential equations while the latter yields ordinary differential equations. But there is one approach, used even in “pure” FEA to reduce the size of the problem, that allows us to elegantly mix the two methods. Called Component Mode Synthesis (CMS), it involves representing a set of elements as a black-box. That is, the set of elements is reduced to a matrix, the size of which is defined by the number of modes that are employed in the abstraction. A complete discussion of the theory of the method is beyond the scope of this book. An excellent description that is both complete and very comprehensible can be found in Structural Dynamics, by R.R.Craig32.

32

The Craig-Bampton and Craig-Chang methods, the most widely used CMS methods, both bear his name. 49

Advanced Topics

CAE and Multi Body Dynamics

Contact The very nature of MBD means that in many cases bodies move through large distances during the periods of interest. The movement may cause contact to occur between different bodies, or between different surfaces of the same body. In turn, the contacts give rise to forces. Examples of contact abound, of course. Electric switches, for instance, are designed to make and break contact. The duration of contact is a critical parameter, particularly for high-voltage equipment. The problem, then, is for the simulation tool to figure out whether contact has been made or contact has been broken. This necessarily complicates the MBD modeling approach, since such a calculation is based on a knowledge of where one body ends and another begins. In other words, the definitions of the surfaces that make up the outer volumes of the bodies are essential. This is quite a departure from the approach we have seen so far, where the surface definitions of the body are dispensable for the calculations. In the absence of contact, the inclusion of the surfaces is mainly to aid visualization. If contact has to be included in the analysis, however, the boundary surfaces are no longer optional. They are an essential part of the problem definition. One common mistake is to use contact where a constraint would suffice. If you are sure that two links are always going to be in contact, then it is more efficient to use a constraint such as a point-to-curve or a point-to-surface constraint. It is when you are unsure of whether the bodies will be in contact with each other or not that a full modeling of contact becomes essential. The coefficient of restitution (COR) is an important property in any collision. And since the establishment of contact is a collision, the COR must be specified whenever contact is used. There’s another bit of data that makes a larger difference to the solution than the COR. And unfortunately, is even harder to characterize. This is the coefficient of friction.

50

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Advanced Topics

Slip and No-Slip When one object rolls against another, it is important to establish whether slip is involved or the motion is pure rolling. For involute gears, for instance, slip is involved at all points of contact except at the pitch point. If slip is involved, the coefficient of friction is different than if the motion is pure rolling.

Friction: Static, Dynamic and Stiction The “Laws” of solid friction are probably better referred to as “Theories” of solid friction. The study of the mechanics of friction dates back at least to Leonardo da Vinci’s times, but the accepted “Law” of friction is not as useful as we would like it to be. David Kessler put it quite clearly when he wrote33 “It is one of the dirty little secrets of physics that while we physicists can tell you a lot about quarks, quasars and other exotica, there is still no universally accepted explanation of the basic laws of friction." Coulomb’s Law of friction is simple, and widely used. In this, friction is of two types: static and dynamic. Static Friction occurs when there is no relative motion between the surfaces in contact with each other, while Dynamic Friction applies if there is relative motion. What of the transition zone between static and dynamic friction? This is sometimes referred to as Stiction, probably derived from “sticky friction”, which is seen when a body is just beginning to move: it is also sometimes called the Limiting Friction.

In any case, we calculate the frictional force using the formula

F = µ • RN where µ is the coefficient of friction, and RN is the normal reaction at the point or surface of contact. 33

In the journal Nature (413, 285-288, 20 September 2001) 51

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The approach used by MotionSolve to model friction is shown in the figure, where µs is the static coefficient of friction, µd is the dynamic coefficient of friction, vs is the stiction transition velocity, and vd is the friction transition velocity.

Control Systems In 196os, several spacecraft, the Ranger series, were dispatched to explore the surface of the moon. The craft were supposed to rough-land on the moon, so needed some way to stabilize and control their descent from second-stage ejection till the lunar landing. Signals from Earth were used to control the system, but one of the problems34 in particular is relevant to our discussion. The craft was designed with a gyro as the control system. Given the gyro’s time constant and inertia, the designer’s problem is to estimate the gain so that the response of the Ranger to a step input (sent from earth) would overshoot by less than 5%. For more down to earth (literally!) applications, consider the “automated manual transmission” systems used in high-performance cars. This is of a manual transmission, but without a clutch pedal. When the driver shifts gears, a control system manages the clutch – the actuation force is usually provided using either electronic or hydraulic actuators. This approach reduces the time it takes to change gears. Problems such as these make it essential for us to include control systems in the MBD model. Including the control system and supplying input to the system is more realistic than omitting it from the model and applying the motion or forces directly to the mechanical component. For an effective use of control systems, a quick revision of two essential concepts in Control Theory is in order.

34

52

Described here in simplified terms

CAE and Multi Body Dynamics

Advanced Topics

State Space Models If the behavior of a system is represented by an nth order differential equation, the State Space approach involves reducing this to a set of n coupled first order differential equations. The forms are entirely equivalent, but the latter is better suited for computer simulation. For instance, the equilibrium equation for a damped spring-mass system is

m&x& + cx& + kx = f (t ) which is a second order differential equation. The state-space model for this is the equations

x& = v v& = f (t ) − k

( m)x − (b m)v

Here, x and v are the state variables, and the set of equations involves only first derivatives of the state variables. (The second equation is obtained from the equilibrium equation by simple substitution for x& and &x& , followed by rearrangement of terms to leave only v& on the left hand side). Typical calculations performed using the state variables include evaluating the response to specified inputs, and calculation of the transfer-function. It is particularly convenient when the system has multiple inputs and multiple outputs – a MIMO control system. The state-space model, when written in a standard (or canonical) matrix form, uses 4 matrices named A, B, C and D. This nomenclature is used by MotionSolve for the definition of MIMO systems.

Laplace Transforms The Laplace transform of a function is defined by the equation ∞

Y ( s ) = L( y (t )) = ∫ y (τ )e − sτ dτ 0

53

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Transfer functions are often represented using Laplace transforms, which have several advantages including the fact that they are distributive, which means

L[r (t ) + s (t )] = L[r (t )] + L( s (t )} Laplace transforms are particularly useful for control systems since differentiation of a signal is equivalent to multiplication of its Laplace transform by s, while its integration is equivalent to multiplying its transform by s.

Block Diagrams The Laplace transform of the transfer functions of the various elements of a control-loop are usually represented by a block diagram, such as that shown below:

Block diagrams are commonly used for modeling process-plants and electrical-systems. They are less common in modeling of mechanical systems. With MotionView and MotionSolve, you can include control systems in your model, though not as a block diagram. Look up the online documentation for MotionSolve for details on how to build MBD models that include SingleInput-Single-Output (SISO) and Multiple Input Multiple Output (MIMO) systems using the Laplace transform and state-space representations.

Cams, Gears and other Higher Pairs There are only 6 lower pairs, but any number of higher pairs can be constructed. Several higher pairs are fairly esoteric, which means their applications are restricted to specific domains. Modeling elements for tires, for instance, are called for almost exclusively by vehicle-dynamics designers. Some higher pairs can be constructed using simpler modeling elements, if the modeling tool supports programmatic control. For instance, a one-way 54

CAE and Multi Body Dynamics

Advanced Topics

clutch can be modeled using a bush together with an “if” statement to change properties based on the direction of rotation35. Two higher-pairs that are extremely common are cams and gears.

Cams A cam rotates about an axis and pushes a follower. The cam usually rotates at a uniform speed, and the profile of the cam is chosen so as to deliver the required motion to the follower. There are various classifications of both cams and followers, most of which reflect the topology or shape of the respective elements36. The follower is usually spring loaded to ensure that it stays in contact with the cam all through the rotation cycle. Design interest centers principally around two things: 1. the profile the cam should have to achieve a required motion – the rise, dwell and return 2. the velocities and accelerations of the follower, and the resulting forces on the various components in the assembly

Positive Return Cam, from the KMODDL

The first is usually the more interesting problem, but the second is no less challenging. Sometimes the cam profile is determined to match a specified follower-motion, but such cams can be expensive to manufacture. Often a predetermined cam profile is chosen and the follower of the motion is to be determined so that the design of the rest of the assembly can be tailored accordingly. In 4-stroke IC engines, for instance, designers need to determine the forces on the tappet. The joint between the cam and its follower is maintained by contact. General contact can be used, but this approach is subject to the difficulties discussed above, in the section on Contact. It is usually more computationally efficient to use point-to-curve (PTCV) or point-to-surface (PTSF) constraints. This approach does sacrifice some of the generality offered by a full-fledged contact model. For instance, the PTCV constraint does not allow for contact to be broken. But at the concept design stage, the analysis is usually a kinematic analysis, since the goal is to derive the required profile of the cam. Once this is done constraints like the PTCV can be used to verify that there 35

MotionView provides support both for bushes and for programmatic control. See the companion volume Managing the CAE Process – Basics. 36 Details can be found in any undergraduate-text on Machine Design. 55

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has been no loss of contact. If there is indeed loss of contact, full fledged contact modeling is essential. Contact between the cam and follower can break if the spring-load is not enough to compensate for the inertial forces (that is, forces due to the accelerations the bodies experience). In engine-design this commonly called valve float, because cams are mainly used in the engine to control the valvetiming of four-stroke engines. The term lift-off is also used in several applications.

Gears There are two distinct problems posed by gears, which serve to transmit torque between different axes of rotation. The transmission of torque is by positive engagement of the teeth. Accordingly, the tooth itself needs to be designed for strength. The design of gear teeth is a subject that is normally not covered by MBD simulation. MBD analysis can help calculate the tooth-loads, and these loads can then be used as input for a stress analysis program – usually using Finite Element Analysis. The other main class of problems deals with the design of the gear train itself. Gear trains range from the aptly named simple gear trains to the amazingly complex epicyclic gear trains. In these cases, analyzing the motion of the output shaft and calculating the ratio of input and output torques are the main areas of interest. An excellent range of models and animations at the KMODDL shows how complex the motion of gear trains can be. The images of a 4-bar mechanism with two gears, taken from an animation at the KMODDL, illustrate how complex the motion can be. Designers of planetary gear trains need to calculate the loads on each gear. Several gearboxes allow for multiple inversions of the gear train – that is, different gears are held “fixed” to generate different motion. MBD models go a long way towards eliminating the tedium and error in this demanding task. MBD models also make it easier to estimate the efficiency of the gear train. A detailed discussion of this aspect is beyond the scope of this book37.

37

See, for instance, Gear Handbook: The Design, Manufacture, and Application of Gears by Dudley, D

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Epicyclic gears are over 300 years old, and are widely used today in a variety of applications, ranging from almost all propeller and turbine driven aircraft to lawn-mowers. While they are more challenging to design, the present a host of advantages, principally a lower weight and volume. Calculating the efficiency of the gear train is an important but tedious task even for gears whose axes of rotation are fixed, like the worm-driven helicalrack-and-pinion shown alongside38. Gear models in MBD are relatively easy to build. Revolute joints define the axes of rotation of the shafts, while the gear joint represents the constraint between the two revolute joints.

If everything seems under control, you're just not going fast enough Mario Andretti

38

This image too, is from a model at the KMODDL. 57

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Glossary And References References Applied Kinematics, Kurt Hain Modern Control Engineering, Katsuhiko Ogata Mechanism Design: Analysis and Synthesis, Volume 1, A.G.Sandor, G.N.Erdman Advanced Mechanism Design: Analysis and Synthesis, Volume 2, A.G.Sandor, G.N.Erdman Design of Machinery, An Introduction to the Synthesis and Analysis of Machines and Mechanisms , Robert L. Norton Handbook of Numerical Applications, Jaroslav Pachner

Other Resources www.altair-india.com/edu, which is periodically updated, contains case studies of actual usage. It also carries tips on software usage. The Kinematic Models for Design - Digital Library (http://kmoddl.library.cornell.edu) is an excellent resource both for a historical coverage of mechanisms, animations of models and for several ebooks, including da Vinci’s Codex Madrid I and Hartenburg’s Kinematic Synthesis of Linkages.

Types of Analyses The table below39, is a convenient way to summarize the types of analyses, the data required for each, the principles involved in finding the solutions, and the types of results that can be calculated. Statics

39

58

Method Kinetostatics

From Advanced Mechanism Design, Erdman and Sandor

Dynamics

CAE and Multi Body Dynamics Masses / Inertias

Glossary and References Weight of links may be required but the inertia is not

Required

Required

Loading

Specified

Specified at each position

Specified in terms of position, velocity, and / or time

Motion

Positions specified

Position, velocity and acceleration specified

Unknown

Output Information

Force required to balance load, mechanical advantage at each position, reactions in joints

Force required to sustain assumed motion, reactions in joints

Position, velocity and acceleration of each member as a function of time – that is, the actual motion

Required analytical tools

Statics, Linear Algebra

D’Alembert’s principle, statics, linear algebra

Differential equations of motion

Input Information and assumptions

Formulae for the Moments of Inertia In these days of 3D CAD, we often pay little attention to the geometric and mass properties of the bodies we’re working with. Most CAD packages can quickly and accurately give you these properties even for complicated shapes. However this reliance on CAD calculations often leads to mistakes which can critically affect the analysis. The most common mistake is to forget that the Moments of Inertia are strongly orientation dependent. A moment’s reflection will remind you that this is only to be expected, since Mass Moments of Inertia are related to angular acceleration by40

T = Iα

40

Similar to F = ma for linear acceleration. Look up Euler’s Equations of Motion for a more complete treatment of the variables involved. 59

Glossary and References

CAE and Multi Body Dynamics

where T is the torque, I is the moment of inertia and α is the angular acceleration. Which Moment of Inertia should be chosen depends on the axis of rotation. The equations for the mass Moments of Inertia are

I xx = ∫∫∫ ( y 2 + z 2 )dm

I yy = ∫∫∫ ( x 2 + z 2 )dm I zz = ∫∫∫ ( y 2 + x 2 )dm I xy = ∫∫∫ z 2 dm I xz = ∫∫∫ y 2 dm I yz = ∫∫∫ x 2 dm Ixx = Izx + Ixy, Iyy = Iyz + Ixy, Izz = Iyz + Izx The radius of gyration is given by

rx =

Ix mass

When you build a model, it’s useful to run a first analysis with approximate bodies – cylinders, boxes, etc. – both to reduce computation time and to verify that the range that the properties lie in is acceptable to the Solver’s default settings. The Moments of Inertia of some “primitives” are listed below. All the values are about the center of gravity. Refer to any text on Statics for details – see, for example, Theoretical Mechanics by P.F.Smith and W.R.Longley. Note that the units are mass*length2. In SI units, therefore, the mass moment of inertia would be in kg-m2. Mass moments of inertia should not be confused with the area moments of inertia, used for example in the formulae for beam bending. The area moment of inertia uses a different formula, and has the units m4.

60

CAE and Multi Body Dynamics

Glossary and References

Cylinder with open ends The z axis is along the axis of the cylinder. The x and y axes are any diameters.

1 2 2 m(r1 + r2 ) 2 1 2 2 I x = I y = m(3r1 + 3r2 + h 2 ) 12

Iz =

where m is the mass, r1 is the inner diameter, r2 is the outer diameter, and h is the height. Solid Sphere

2mr 2 I= 5 where m is the mass and r is the radius. Cuboid

1 m( w 2 + d 2 ) 12 1 I d = m( w 2 + h 2 ) 12 1 I w = m( h 2 + d 2 ) 12 Ih =

where m is the mass, and h, d and w are the dimensions along the 3 principal directions. The origin of the 3 axes is at the center of mass of the cuboid.

Common Coefficients of Friction Friction coefficients are extremely sensitive to the presence / absence of lubrication, as well as to other factors like the pressure between the surfaces, surface finish, etc. The values in this table should be treated with corresponding care. Several websites provide similar information (see, for instance, http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm) which are useful for preliminary design. For further analyses, nothing beats lab tests. 61

Glossary and References

CAE and Multi Body Dynamics

Coefficient Of Friction Material 1

Material 2

Dry Static

62

Greasy

Sliding St at ic Sliding

Aluminum

Aluminum

1.05-

1.4

0.3

Aluminum

Mild Steel

0.61

0.47

Brake Material

Cast Iron

0.4

Brake Material

Cast Iron (Wet)

0.2

Bronze

Cast Iron

Bronze

Steel

Cadmium

Cadmium

Cadmium

Mild Steel

Cast Iron

Cast Iron

1.1

Chromium

Chromium

0.41

Copper

Cast Iron

1.05

Copper

Copper

1.0

Copper

Mild Steel

0.53

Copper

Steel

Copper

Steel (304 stainless)

0.23

Copper-Lead Alloy

Steel

0.22

Glass

Glass

0.9 - 1.0

Glass

Metal

0.5 - 0.7

Glass

Nickel

0.78

Graphite

Steel

0.1

0.1

Plexiglas

Plexiglas

0.8

0.8

Plexiglas

Steel

0.4 - 0.5

0.4 -

Polystyrene

Polystyrene

0.5

0.5

Steel

Brass

0.35

0.19

Steel

Cast Iron

0.4

0.21

Steel

Phos Bros

0.35

Steel(Hard)

Polystyrene

0.3-0.35

Steel (Mild)

Steel (Mild)

0.74

0.57

Steel(Hard)

Steel (Hard)

0.78

0.42

Teflon

Steel

Teflon

Teflon

0.22 0.16 0.5

0.05 0.46 0.15

0.07 0.34

0.29 0.08 0.36

0.18

0.8 0.21 0.4

0.1 -

0.09-

0.2 0.56

0.30.090.05 -

0.029-

0.04

0.04

0.04

0.04

0.04

0.04