Mechanics 2 (Dynamics) Lecture 2: Absolute Motion Analysis Dr Aziza Mahomed
[email protected] Office: S21
Mechanical Engineering Lecture notes adapted from “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
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Mechanics 2: Lecture Topics Semester 2: Mechanics of Dynamic Systems lectures Week
Lecture
Lecture Topic
Tutorial Topic
1
1
Translation and rotation
Translation and rotation
2
2
Absolute motion analysis
Absolute motion analysis
3
3
Relative motion analysis-velocity
Relative motion analysis-velocity
4
4
Instantaneous centre of zero velocity
Instantaneous centre of zero velocity
5
5
Relative motion analysis-acceleration
Relative motion analysis-acceleration
Reading week (no lecture)
Reading week (no lecture)
6
7
6
Relative motion analysis using rotating axes
Relative motion analysis using rotating axes
8
7
Moment of inertia
Moment of inertia
9
8
Planar kinetic equations of motion-translation
Planar kinetic equations of motion-translation
10
9
Equations of motion- rotation about a fixed axis
Equations of motion- rotation about a fixed axis
11
10
Equations of motion- General plane of motion
Equations of motion- General plane of motion
QUIZ 1. The fan blades suddenly experience an angular acceleration of 2 rad/s2. If the blades are rotating with an initial angular velocity of 4 rad/s, determine the speed of point P when the blades have turned 2 revolutions (when ω = 8.14 rad/s). A) 4.27 m/s
B) 5.31 m/s
C) 6.93 m/s
D) 8.0 m/s
2. Determine the magnitude of the acceleration at P when the blades have turned the 2 revolutions. A)
0 m/s2
C) 34.79 m/s2
B) 1.05 m/s2 D) 34.81 m/s2
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Plane Motion Translation Motion of the body specified by motion of any point in the body. No rotation of any line in body.
Mechanics 2: Revision Lecture 1 Plane Kinematics of Rigid Bodies
Rotation about a Fixed Axis All the particles of the body, except those on the axis of rotation, move along circular paths in planes perpendicular to the axis of rotation and rotate through the same angle.
General Planar Motion The body undergoes simultaneous translation and rotation The actual paths of all particles in the body are projected on to a single plane of motion.
Extracted from : J. L. Meriam and L. G. Kraige, Engineering Mechanics, Vol II Dynamics, 7th Ed, © John Wiley & Sons, 2012. All rights reserved.
ROTATION ABOUT A FIXED AXIS : PROCEDURE (SUMMARY)
Establish a positive sign convention along the axis of rotation and label it clearly alongside each kinematic equation If a relationship is known between any two of the variables (α,ω,θ or t), the other variables can be determined from the equations: 𝑑𝜃 𝜔= (rad/s) + (Eq 1-1) 𝑑𝑡
𝛼=
𝑑𝜔 𝑑𝑡
(rad /s2) 𝑑𝜔
+ (Eq 11-2) or 𝛼𝑑𝜃 = 𝜔𝑑𝜔
𝛼 = 𝜔 𝑑𝜃 (rad /s2)
+ (Eq 1-4)
If α is constant, use the equations for constant angular acceleration To determine the motion of a point, these scalar equations can be used: 𝑣 = 𝜔𝑟, at = αr and an = ω2r and 𝑎 = 𝑎𝑛2 + 𝑎𝑡2
Alternatively, the vector form of the equations can be used (with i, j, k components) 𝐯 = 𝛚 𝐱 𝒓𝑷 = 𝛚 𝐱 𝒓 (Eq 1-8) 𝒂 = 𝛂 𝐱 𝒓𝑃 + 𝝎 𝐱 (𝝎 𝐱 𝒓𝑷 ) 𝒂 = 𝒂𝑡 + 𝒂𝒏 𝒂 = 𝛂 𝐱 𝐫 − ω2 𝐫
SIGN CONVENTION
Angular position of a moving line is specified by a anti- clockwise angle, θ Angular velocity, ω, is defined as positive in the anti-clockwise direction Angular acceleration, α, is defined as positive in the anti-clockwise direction
Extracted from : J. L. Meriam and L. G. Kraige, Engineering Mechanics, Vol II Dynamics, 7th Ed, © John Wiley & Sons, 2012. All rights reserved.
Week 2- Planar Rigid Body Motion: Absolute Motion Analysis Hibbeler: Chapter 16.4
Lecture Objectives Students will be able to determine the velocity and acceleration of a rigid body undergoing general plane motion using an absolute motion analysis Learning Outcomes You should be able to mathematically analyse/calculate the kinematics of a rigid body undergoing general plane motion using an absolute motion analysis Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis
Absolute motion analysis relates the position of a point, P, on a rigid body to the angular position, θ, of a line contained in the body The velocity and acceleration of point P are obtained in terms of angular velocity, ω, and angular acceleration, α, of the rigid body If a body is represented by a thin slab, the slab translates in the plane and rotates about an axis perpendicular to the plane
P
Extracted from : J. L. Meriam and L. G. Kraige, Engineering Mechanics, Vol II Dynamics, 7th Ed, © John Wiley & Sons, 2012. All rights reserved.
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis
The motion can be completely specified by knowing both the angular rotation of a line fixed in the body and the motion of a point on the body For example, use : Rectilinear coordinate (s) to locate the point along its path Angular position coordinate (θ) to specify the orientation of the line Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
PROCEDURE FOR ANALYSIS The velocity and acceleration of a point P undergoing rectilinear motion can be related to the angular velocity and angular acceleration of a line contained within the body using the following procedure: Position Coordinate Equation: Locate point P using a position coordinate s, which is measured from a fixed origin and is directed along the straightline path of motion of point P Measure from a fixed reference line the angular position θ of a line lying in the body From the dimensions of the body, relate s to θ , s= f(θ), using geometry and trigonometry Time derivatives: Take the first derivative of s= f(θ) with respect to time to get a relationship between ν and ω i.e. f(θ, ω) Take the second derivative with respect to time to get a relationship between a and α i.e. f(θ, ω, α) Usually the chain rule of calculus must be used when taking the derivatives of the position coordinate equation
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis
Consider the skip on the truck. It rotates about a fixed axis and passes through A. The hydraulic cylinder (BC) extends to perform the operation. As part of the design process for the truck, you as an engineer have to relate the speed (v) at which the hydraulic cylinder extends to the angular velocity (ω) of the skip
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis
Noting that: - the angular position is defined as θ - The position of point C is defined using the rectilinear coordinate, s Thus, since a and b are fixed lengths, then using the cosine law: 𝑠 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠𝜃 Use the chain rule, to obtain:
𝑣=
1 2
𝑎 2 + 𝑏2 − 2𝑎𝑏 cos θ
−1/2
(2𝑎𝑏 sin 𝜃)𝜔
(This is a relation of the speed to which the hydraulic cylinder extends to the angular velocity of the skip)
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis
The stroke of the piston is defined as the total distance moved by the piston as the crank angle varies from 0 to 180°. How does the length of crank AB affect the stroke?
The position of the piston, x, can be defined as a function of the angular position of the crank, θ. By differentiating x with respect to time, the velocity of the piston can be related to the angular velocity, ω, of the crank
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 1
Given: Crank AB rotates at a constant velocity of w = 150 rad/s Find: Velocity of P when q = 30° Plan: Define x as a function of q and differentiate with respect to time. Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 1
Solution: xP = 0.2 cos q +
(0.75)2 – (0.2 sin q)2
vP = -0.2w sin q + (0.5)[(0.75)2 – (0.2sin q)2]-0.5(-2)(0.2sin q)(0.2cos q)w vP = -0.2w sin q – [0.5(0.2)2sin2q w] / (0.75)2 – (0.2 sin q)2 At q = 30°, w = 150 rad/s and vP = -18.5 m/s = 18.5 m/s Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Mechanics 2:Planar Rigid Body Motion-Absolute Motion Analysis The
rolling of a cylinder is an example of general plane motion. During this motion, the cylinder rotates clockwise while it translates to the right The position of the center, G, is related to the angular position, θ, by: 𝑆𝐺 = 𝑟𝜃 if the cylinder rolls without slipping. Can you relate the translational velocity of G and the angular velocity of the cylinder?
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
Cylinder
EXAMPLE 2
At
a given instant, the cylinder of radius r, has an angular velocity, ω, and angular acceleration, α. Determine the velocity and acceleration of its centre G, if the cylinder rolls without slipping Extracted from : J. L. Meriam and L. G. Kraige, Engineering Mechanics, Vol II Dynamics, 7th Ed, © John Wiley & Sons, 2012. All rights reserved.
Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 3 Given: Two slider blocks are connected by a rod of length 2 m. Also, vA = 8 m/s and aA = 0.
vA=10 m/s
Find: Angular velocity, w, and angular acceleration, a, of the rod when q = 60°.
Plan: Choose a fixed reference point and define the position of the slider A in terms of the parameter q. Notice from the position vector of A, positive angular position q is measured clockwise. Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 3 (continued) Solution: By geometry, sA = 2 cos q reference
q A
By differentiating with respect to time, vA = -2 w sin q
sA Using q = 60° and vA = 8 m/s and solving for w: w = 8/(-2 sin 60°) = - 4.62 rad/s (The negative sign means the rod rotates counterclockwise as point A goes to the right.) Differentiating vA and solving for a, aA = -2a sin q – 2w2 cos q = 0 a = - w2/tan q = -12.32 rad/s2 Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 4 Given: The w and a of the disk and the dimensions as shown. Find: The velocity and acceleration of cylinder B in terms of q. Plan: Relate s, the length of cable between A and C, to the angular position, q. The velocity of cylinder B is equal to the time rate of change of s. Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
EXAMPLE 4 (continued) Solution: Law of cosines: s = (3)2 + (5)2 – 2(3)(5) cos q vB = (0.5)[34 – 30 cosq]-0.5(30 sinq)w vB = [15 sin q w]/
34 – 30 cos q
(15w2 cosq + 15a sinq) (-0.5)(15w sinq)(30w sinq) + aB = 34 - 30cosq (34 - 30cosq)3/2 2cosq + asinq) 2sin2q 15(w 225w aB = (34 - 30 cosq)0.5 (34 - 30 cosq)3/2 Extracted from: “Mechanics for Engineers: Dynamics, 13th SI Edition (2013), Russell C. Hibbeler, Kai Beng Yap, Pearson Education Centre,” MasteringEngineering Instructor Resources. Images © Pearson Education South Asia Pte Ltd 2013. All rights reserved.
PROCEDURE FOR ANALYSIS The absolute motion analysis method (also called the parametric method) relates the position of a point, P, on a rigid body undergoing rectilinear motion to the angular position, q (parameter), of a line contained in the body. (Often this line is a link in a machine.) Once a relationship in the form of sP = f(q) is established, the velocity and acceleration of point P are obtained in terms of the angular velocity, w, and angular acceleration, a, of the rigid body by taking the first and second time derivatives of the position function. Usually the chain rule must be used when taking the derivatives of the position coordinate equation.
Mechanics 2 Further reading for Semester 2 Chapter 16 of Hibbeler (Dynamics) 16.1: Planar Rigid-Body Motion 16.2: Translation 16.3: Rotation about a Fixed Axis 16.4: Absolute Motion Analysis 16.5: Relative Motion Analysis: Velocity 16.6: Instantaneous Center of Zero Velocity 16.7: Relative Motion Analysis: Acceleration 16.8: Relative Motion Analysis using Rotating Axes Chapter 17 of Hibbeler (Dynamics) 17.1: Mass Moment of Inertia 17.2: Planar Kinetic Equations of Motion 17.3: Equations of Motion: Translation 17.4: Equations of Motion : Rotation about a Fixed Axis 17.5: Equations of Motion : Genera l Plane Motion
