Vibration Lab

ME 4600:483 –  4600:483 – Lab Lab Manual

Revised August 2015

Vibration Measurement Measurement WARNING: PART OF THIS LAB USES STROBE LIGHTS! Table of Contents

Vibration Measurement .................................. ................. .................................. ............................... .............. Error! Bookmark not defined. I. Objective ................................. ................ ................................... ................................... ................................... ................................... ................................... ............................ .......... 1 II. Apparatus ................................... .................. .................................. ................................... .................................... ................................... ................................... ......................... ....... 1 III. Principles and Background .................................. ................. ................................... ................................... ................................... .................................. ................ 2 System Dynamics ........................................................................................................................ 2 Zero-Order Systems................................................................................................................. 2 First-Order Systems ................................................................................................................. 2 Second-Order Systems ............................................................................................................ 3 Free Vibration of a Single Degree of Freedom System ................................... .................. ................................... ............................ .......... 4 Forced Vibration of a Distributed-Mass Cantilever Beam System ............................................. 5 IV. Procedure Pr ocedure ................................... .................. .................................. ................................... .................................... ................................... ................................... ......................... ....... 7 A. Transient Decay of a Vibrating Vibrati ng Cantilever Beam.................................. ................. ................................... ............................ .......... 7 B. Steady State Forced Vibration of Cantilever Beam .................................. ................ .................................... ......................... ....... 9 V. Required Data Analysis ................................. ................ ................................... ................................... ................................... .................................... .................... .. 12 VI. References Refer ences ................................. ................ .................................. ................................... .................................... ................................... ................................... ....................... ..... 13 VII. Figures ................................................................................................................................... 13 I. Objective

The purpose of this lab is to study and observe free  and forced  vibrations of simple cantilever  beams. 1. In the first part of the lab, the frequency and amplitude of oscillation will be recorded of a free vibrating system. Using the acquired data and known information about about the cantilever beam, the characteristics of the vibrating system will be computed and verified. 2. In the second part of the lab experiment, the frequency and amplitude of vibration of a be continuous mass system  will be measured. The first three natural frequencies will be theoretically calculated, experimentally determined using several methods, and observed. The three mode shapes will be subsequently graphed and comparisons between the theoretical and the experimentally observed shapes will be discussed. II.

Apparatus Part 1: 1. Cantilever beam mounted in a clamp support, 2. C-clamp to hold the base to the table, 3. Two strain gages mounted mounted to the beam in an axial direction, 4. Four pound set of weights with pan pan and hanger, hanger, 5. Ruler 6. 24-bit A/D converter and data acquisition software, Part 2: 8. Cantilever beam with a mounting plate and an adjustable transducer mount, 9. B&K 4818 Mini-Shaker, 10. Bogen 60 watt Audio Amplifier,

Vibration Measurement

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ME 4600:483 – Lab Manual

Revised August 2015

11. BK Precision Signal Generator, 12. Magnetic Reluctance Transducer MM0002, 13. Strobe light III. Principles and Background System Dynamics All physical systems have a characteristic response to any forcing function. If you put a thermometer in the sun, it heats up. If you hit a piece of metal, it makes a noise. These characteristics can be modeled by differential equations. We can use these system models to  predict how a system will behave even if we have not specifically tested the system response. This is true of systems we perform measurements on as well as systems we use to take the measurements.

A general model for an output variable y can be defined as a system of n th order: n

an

d   y n

dt 

+ an-1

n-1 d   y n-1

dt 

+ ..... a1

dy dt 

+ a0  y = F(t)

Where  F(t)  is a forcing function and n  is the highest derivative. We will consider cases where n=0, 1, and 2. Zero-Order Systems

A zero-order system has the following equation and solution:

a0  y = F(t) ;  y = KF(t) where, 1/a0=K is the system gain. From the solution it is clear that the system response is simply the input multiplied by K. All zero-order systems mimic their input with no time delays, oscillations or other dynamics. The BAM is an example of a zero-order system (unless the input frequency becomes high). A lever arm is another one (as long as the motion is slow enough to not excite the lever arm bending mode). First-Order Systems

A first-order system has a derivative in the model. The model equation and the solution for the response to a step change in F(t) of amplitude A is:

a1

dy dt 

+ a0  y = F(t) ;

t 

 y(t) = KA + (  y0 -  KA) e-  

where a1 /a0=, the system characteristic time constant. A first-order system will respond to an input with a multiplication by  K   and an exponential time delay. A thermometer is a first-order system. Exposed to a temperature, its display exponentially approaches that temperature. One may think of a first-order system as having the capability to store one form of energy.

Vibration Measurement

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ME 4600:483 – Lab Manual

Revised August 2015

Second-Order Systems

A second-order system is modeled by the equation: 2

a2

d   y 2

dt 

+ a1

dy dt 

+ a0  y = F(t)

which can be rearranged as:

1 d 2  y 2

2

 n dt 

+

2   dy  n dt 

+ y = KF(t)

where  n is the natural frequency of the system and    is the system damping ratio. The solution to this equation has three parts because complex numbers occur in the roots. 1) If  >1, then the homogeneous solution (solution for F(t)=0) is:

 y(t) = C 1 e 1 t + C 2 e 2 t  2) If  =1, then

 y(t) = C 1 e 1 t + C 2 t e 2 t  3) If 