vibration measurement lab procedures - university of...

ME 4600:483 – Lab Manual Revised August 2015

Vibration Measurement of 15

Vibration Measurement

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 .................................................................................................................................. 7

A. Transient Decay of a Vibrating Cantilever Beam .............................................................. 7

B. Steady State Forced Vibration of Cantilever Beam ........................................................... 9

V. Required Data Analysis .......................................................................................................... 12

VI. References .............................................................................................................................. 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 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

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 to the beam in an axial direction,

4. Four pound set of weights with pan and 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,

WARNING: PART OF THIS LAB USES STROBE LIGHTS!



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 nth order:

F(t) =y a + dt

dya ..... +

dt

y da +

dt

y da 01-1n

-1n

-1nn

n

n

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:

KF(t) =y ; F(t) =y a0

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:

e KA) - y( + KA = y(t) ; F(t) =y a + dt

dy a

t -

001

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.



Second-Order Systems

A second-order system is modeled by the equation:

F(t) =y a + dt

dy a +

dt

y d a 012

2

2

which can be rearranged as:

KF(t) =y + dt

dy

2 +

dt

y d

1

n2

2

2n

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:

e C + e C = y(t) t 2

t 1

21

2) If =1, then

e t C + e C = y(t) t 2

t 1

21

3) If <1, we get

)t ( e C = y(t) dt - n sin

where

2

nd

2

nn ,21 - 1 = ; 1 - + - =

The first case gives exponential time delays similar to the first-order system and is called

overdamped. The second case is critically damped, and gives the smallest pure time-delayed

response possible for the system. The last case is underdamped. Here the solution is oscillatory

at the damped natural frequency d. A second-order system can thus oscillate. Mechanical

oscillation is called vibration.

Vibrations are an important aspect of mechanical engineering. Vibration is present at some level

in all engineering machines and structures. Vibration can be either helpful, as in the case of

vibrating sifters and mixers, or harmful, such as in the case of the Tacoma Narrows Bridge, which

collapsed due to unstable torsional vibrations. One of the first steps required in the analysis of any

vibration problem is determining the natural frequencies of vibration of a part, or a system

containing the part, in question.



Part A) Free Vibration of a Single Degree of Freedom System

Free vibration is a condition that exists when a system, starting from some initial condition, is

allowed to vibrate without the effect of an external forcing function. In the first part of this lab,

we will study the free vibration of a cantilever beam with a mass on the end, shown in Figure 1.

We will model the freely vibrating mass as a single degree-of-freedom system, as shown in

Figure 2. This is an approximation because the cantilever beam, although stiff and light

compared to the mass, may bend and introduce additional degrees of freedom. The degree of

freedom is equal to the number of independent coordinates that must be used to fully describe the

motion of the system as it vibrates. The spring-mass-damper model in Figure 2 is the simplest

model which can be used to represent all single degree-of-freedom (or point-mass) systems and is

the basis of the modeling equations given earlier.

Consider the spring-mass-damper in an initial state of equilibrium. When the system is perturbed

by external forces, it absorbs a certain amount of energy. The energy storage may be in the form

of potential energy of the spring, or kinetic energy of the mass. As the perturbing force is

removed, the system tends to return to its equilibrium position and dissipate energy in the

process. Depending on the damping in the system, this process may be characterized by a simple

asymptotic return to its original position (overdamped or critically damped) or by exponentially

decaying vibrations at a single characteristic natural frequency about the equilibrium position

(underdamped).

The underdamped system is characterized by the amplitude of vibration, the damping ratio and

the natural frequency. The amplitude depends on the initial condition and will decrease to zero as

equilibrium is approached, but the frequency and the damping ratio should remain constant. They

are not affected by the nature of the initial disturbance (this is true only for purely linear systems,

non-linear systems may change frequency or damping with amplitude but our experiment behaves

quite linear).

The deflection of the vibrating beam, or the spring-mass-damper system, in our experiment will

be characterized by a damped sinusoidal motion, shown in Figure 3, of the form:

) + t ( eA = (t) x d

t - 1

-d

2

sin

Where :

M

k = ; - 1 =frequency natural damped =

eq

eq

n

2

nd

: damping ratio

: offset phase angle

keq: equivalent spring rate ( or stiffness) = Ebh3/4l3)

Meq: equivalent mass = mass of weight + 33/140 * mass of the overhung portion of the

beam ( this means that 33/140=23.5% of the beam mass participate in the motion)

Note that as approaches zero (i.e. the system is undamped) d approaches n, the undamped

natural frequency.

The natural log of the amplitude decay of each successive period in Figure 3 is called the

logarithmic decrement, , and it is related to the damping ratio, , by the equation



2dd - 1

2 =]

)T n + (t x

(t) x[

n

1 =]

)T + (t x

(t) x[ = lnln

Where: x(t): displacement at time t

Td: period of the damped sinusoidal wave

n: any arbitrary integer

Thus, the damping ratio, , can be calculated by measuring the decay in amplitude after any

number of oscillations, n , and plugging it into the above equation. Notice that as the damping

ratio approaches unity, goes to infinity. This corresponds to the critical damping ratio.

Part B) Forced Vibration of a Distributed-Mass Cantilever Beam System

Forced vibrations occur when a system is subjected to some type of continuing external excitation

that adds energy to the system. In this lab, we will examine the effect of a steady-state sinusoidal

excitation of a distributed-mass cantilever beam system. With the hanging mass removed, a

cantilever beam will exhibit higher modes of vibration just like the guitar string in our first lab.

The higher modes of vibration had a negligible effect on the initial overhung mass system, and

thus were neglected in the single degree-of-freedom analysis. In the second part of the lab,

however, you will see that these higher modes have a significant effect on the dynamics of the

distributed-mass cantilever beam. Continuous systems like this possess an infinite number of

degrees-of-freedom. In this lab we will examine the characteristics of the first three natural

frequencies. Infinitely many higher modes exist, but their amplitude becomes too small to

measure.

An electrically driven mini-shaker will provide excitation to the system by vibrating the end of

the cantilever beam. The vibration is produced by applying a sinusoidal alternating current to an

inductive coil surrounding a central plunger. By adjusting the frequency of oscillation of this

current with a sine-wave signal generator, different forcing frequencies can be applied to the

beam.

The vibration of the beam used in this lab will be characterized by the first three mode shapes

shown in Table 1, plus an additional spurious mode ???, which may occur (or may not, depending

on boundary condition on your particular system) somewhere between modes 1 and 2. This extra

mode is not predicted by the first-order analysis of the cantilever beam. It is caused by the fact

that the shaker mount does not exhibit perfect translation, but allows for a certain amount of

rotation. The location of the mystery mode will be dependent on the stiffness of the bearing

system - a boundary condition. You may not observe this mode on your particular setup. You

should always think about the effects of system boundaries when designing an experiment as they

can modify what you think you're measuring. Note that the mystery node is not really a

“mystery”, it is simply not predicted by the theory, because the theory assumes a perfectly

clamped boundary condition.

The primary natural frequencies of the mode shapes (except spurious mode) are determined by

(use consistent units, that means force, mass, time and length unit can not be independently

chosen and are related via Newton’s Law)

l A

I E )l ( =

4

2

nn



Where: n: nth - harmonic natural frequency [ rad / sec ]

: density of the beam

A: cross-sectional area of the beam

l: length of the beam

E: Modulus of Elasticity

I: Area Moment of Inertia

n: mode number ( n = 1 for the 1st fundamental mode )

n: mode constant for mode n given in terms of the product of (n l ) in Table 1

In general, the vibration will depend upon

1) the natural frequencies of the system,

2) the damping inherent in the system, and

3) the frequency and amplitude of excitation.

One phenomenon, which you will notice is that the magnitude of the oscillation becomes very

large when the frequency of the excitation matches one of the natural frequencies of vibration of

the beam calculated above. In addition, the cantilever beam will visibly assume each of the above

mode shapes at the respective natural frequencies. This phenomenon can be observed with the

use of the strobe light. Note that in the second and third mode shapes, there are points where

there is no displacement; these points are called nodes (denoted N). The points corresponding to

maximum displacement are likewise called antinodes (denoted A).

n

n l

Mode Shapes (N: node, A: anti-node)

1

1.8751

?????

??????

2

4.6941

3

7.8548

Table 1. A schematic of the mode shapes for a cantilever beam subjected to forced

vibration with the corresponding nodal and anti-nodal positions.



IV. Procedure

In Part A, we will use the resistance strain gage

transducer to monitor the time-dependent deflection

of a cantilever beam with a mass hung from the free

end. The vibration is sensed with a strain gage. The

change in resistance of the strain gage is measured

with a Wheatstone bridge circuit similar to the strain

experiment. The strain output is recorded with the

data acquisition system so that the decay rate of the

vibration can be plotted and evaluated. After setting

up the data acquisition, the beam will be given an

initial disturbance and allowed to come to rest while

the strain versus time is recorded.

A. Transient Decay of a Vibrating Cantilever Beam

1. Connect axial half bridge strain gage wires to the data acquisition system (refer back to

strain lab)

2. Modify the looping version of the VI to read strain inputs

a. Open the DAQ Assist properties window.

b. Under “Channel Settings”, click the red “X” to remove the voltage input channel.

c. Click the blue “+” to add a strain channel (9219, ai0).

d. Set gage factor to 2.09, gage resistance to 120.5, and configuration to Half Bridge II.

e. Add about 3 lbs of weight to the beam.

f. Calibrate the system the same way as in the strain lab.

g. Resave the VI as FreeVibration.vi

3. Assemble transient vibration experiment apparatus

a. Verify that the beam is clamped in the base perpendicular to the edge with the

weight-pan eyebolt located 6 1/2 inches from the edge of the base

b. Measure the physical dimensions of the beam. You'll need these values to find the

natural vibration frequency for the beam.

c. Identify and record the material composition of the beam

4. Record vibration decay data for beam

a. Set the data acquisition to take 300 samples at 100 samples/sec.

Beam Tip Oscillation

Added Mass



b. Run the VI.

c. Quickly push the beam down with a hard metal object (screwdriver) and release

sharply, however, click the Stop button right before pushing on the beam. This will

freeze the graphs on the damped vibration trace. Clicking stop afterwards will freeze

the graphs one data period too late.

Don't push it too hard or the beam will undergo plastic deformation and you'll

have to start all over again with a new apparatus!

d. Repeat this process until you acquire a good trace of the decaying oscillation of the

system. Adjust the DAQ parameters or rezero the bridge as necessary.

e. Export the data from the two graphs to Excel.

5. Evaluate potential errors in the measurements a. Look at the equations on page 4. Identify and evaluate possible sources of error.

Remember the model is for a cantilever beam, but your experiment has more

complex boundary conditions. Can you change the frequency of oscillation by

adjusting any uncontrolled boundary condition in the experiment? Can you affect the

rate-of-decay of the oscillations?

6. Cleanup experiment A. Carefully dismantle and store all equipment. Close the VI.



B. Steady State Forced Vibration of Cantilever Beam

In Part B, we will use a strobe light

and a magnetic-reluctance velocity

transducer to monitor the forced

vibration of a long, distributed-

mass, cantilever beam. The

magnetic transducer is effectively a

small magnet with a surrounding

coil of wire. When a ferrous beam

breaks the magnetic forcelines of

the sensor, it induces a voltage in

the coil, just like an electrical

generator. The voltage generated

is proportional to the rate at which

forcelines are broken, so the

transducer actually responds to the

velocity of the beam in the direction normal to the magnet surface. It cannot distinguish between

velocity away from or toward the sensor (sign of velocity). Because the beam is metal and moves

in the magnetic field of the transducer, the transducer disturbs the system. This is a common

concern with any measurement system; the sensor may actually interfere to some degree with

what you are trying to measure. Always try to keep this effect to a minimum.

The transducer signal will be recorded with the data acquisition system, and the fundamental

vibration frequency and any harmonics will be evaluated with the FFT function of LABVIEW as

in the first lab. Since the beam is in simple harmonic motion, dividing the magnitude of the

beam's maximum velocity at each point by the fundamental frequency will yield the maximum

displacement of the beam at that point. The conversion from measured voltage to velocity

depends on the gap between beam and transducer and is not linear, so the calibration curve given

in Figure 4 must be used. The transducer will be moved along the length of the beam in one inch

increments so that data can be taken along the length of the beam at each of the three lowest

natural frequencies of vibration of the beam. This data can then be used to plot the mode shapes

of vibration at these three natural frequencies. In addition, the strobe light will be used to view

the beam's vibration at these three resonant conditions in a darkened laboratory. The frequency

of the beam vibration can be found by "stopping" the motion with the stroboscopic illumination.

Locate the equipment for experiment B, place it on top of the counter.

1. Measure beam Geometry

a. Measure the beam length, width and thickness as accurately as possible. The first

two theoretical resonant frequencies will be calculated in the data analysis from the

given equation and Table 1. During lab, assume the two frequencies are 20Hz and

150Hz.

Shaker, drives beam

motion

Magnetic Reluctance Velocity Transducer, measures

velocity along the beam axis

WARNING: THIS PART OF THE LAB USES STROBE LIGHTS!



2. Assemble equipment for force vibration equipment.

a. Refer to figure 5 for a schematic of the experiment. Connect the function generator to

the Bogen Audio Amplifier (AUX1). Connect its output to the mini-shaker. The

magnetic pickup connects back to the data acquisition board.

b. Remove the cardboard shim on the shaker.

c. Position the magnetic transducer at the end of the beam and use the gap gauge to set

the transducer-to-beam spacing. Measure and record the gap gauge thickness for use

in Data Analysis Question 3.

d. On the amplifier, turn bass to cut and turn treble to zero. Maximize the signal from

signal generator so you don’t have to increase the amplification too much.

3. Modify the looping version of the VI to display the velocity amplitude

a. Find the maximum and minimum of the voltage signal.

Array → Max and min

Wire into data out of DAQ Assist before the Vrms block

b. Add a formula to calculate velocity amplitude.

Mathematics → Script and Formula → Formula

Change the X1 label to max, and the X2 label to min

Subtract max from min and divide by 2 to get the amplitude of the signal

c. Wire the maximum and minimum values of the array into the formula.

d. On the front panel, add a numeric indicator and rename it “Velocity Amplitude”

e. Wire the result of the formula to the indicator. The result is in V.

f. Rename the VI ForcedVibration.vi

New



4. Observe vibration of beam

a. Turn on the equipment (sine wave on function generator at 10 Hz) and begin

continuous data acquisition. Start with a frame size of 2000 at 2000 Hz but you may

want to change this later.

b. Set the signal generator to 10 Hz. Adjust the Aux 1 setting on the amplifier until you

see a corresponding frequency spike on the FFT display.

c. Slowly increase the frequency of the Signal Generator to approach the Mode 1

resonant frequency assumed above. Stop when you see a significant increase/decrease

in the amplitude as you sweep the frequency. You can judge the amplitude visually

but to fine-tune the amplitude, use the data acquisition display and look for the

maximum amplitude on the signal (see next step). Go beyond the calculated resonant

frequency if necessary.

d. Determine the actual resonant frequency by fine-tuning the frequency of the signal

generator to obtain the maximum amplitude on the display. Adjust AUX 1 if

necessary to prevent the beam from hitting the transducer as the oscillation increases.

e. Record the frequency and amplitude of the fundamental vibration indicated by the data

acquisition display. Also record the indicated frequency on the signal generator.

f. Consider the boundary conditions of the experiment. Does the indicated resonance

frequency (the frequency of maximum vibration amplitude) change, if the transducer

is moved nearer or farther from the beam or along the beam length? (find the

resonance for the following modifications: change the gap between transducer and

beam and in a second step move the transducer to about the middle of the beam).

Record any effects and the estimated range of uncertainty for use in the error analysis.

g. Use the strobe light to observe the beam motion at the resonance. First, move the

transducer away from the beam in case it alters the resonant frequency. You can adjust

the strobe frequency to stop the image at even integer multiples of the actual

frequency, with a doubled image of the beam visible, or at even integer fractions of

the actual frequency, with a single image visible. To find the true frequency, you must

start high, with a doubled image, and then decrease the flash rate until the first single

image is observed. Turn off the lights in the lab to adequately observe the

phenomenon. Think about how strobing the beam is really the same as “visually

sampling” it. By flashing it twice as fast you see two images, top and bottom. By

flashing it with the vibration frequency you make it stop in time, that is, you

“visually” alias the vibration into zero frequency (remember the folding diagram).

h. Sketch the observed beam shape in your notebook.

5. Recording mode shapes with magnetic transducer

a. Place the magnetic transducer centerline 1/2 inch from the free end of the beam.

b. Verify that the gap is set with the plastic gap gauge. Measure and record the gap

thickness.

c. Set the AUX 1 dial on the amplifier to a low value

d. Set the FREQUENCY of the signal generator to obtain the mode 1 resonance found in

step 3e and fine-tune the frequency to obtain the maximum amplitude as before.

e. Increase AUX 1 to obtain good signal amplitude, but be careful to maintain the setting

low enough to prevent the beam from striking the magnetic pick-up. Adjust the gain

of the DAQ system if necessary to get a good signal. If you have to move the

transducer closer or farther from the beam, make sure you measure the gap.

f. Record the velocity amplitude (in V).

g. Turn the amplifier OFF, but do not change the FREQUENCY or AUX 1 settings.

Move the magnetic transducer towards the shaker at one-inch interval and repeat the



velocity amplitude measurement for a total of seven to eight positions along the beam.

At each position verify that the gap is maintained with the test gauge.

6. Repeat steps 3 and 4 for Mode 2

7. Clean up

a. Replace the cardboard shim on the shaker.

b. Carefully dismantle the equipment and return it to its original state for use by the next

laboratory group

V. Required Data Analysis

1 a. Plot the time-series strain gage output data stored in Step A4. Measure the amplitude

of the first two successive peaks of the decaying vibration of the approximate point-

mass system. From the amplitude of the peaks find the logarithmic decrement and

thus the damping ratio . Then calculate the undamped natural frequency using the

damped natural frequency from the FFT measurement and the damping ratio just

found.

b. Compare the experimental natural frequency to the theoretical natural frequency of the

beam.

c. Repeat the above calculation for two other successive peaks near the end of oscillation.

d. Evaluate the uncertainty in the damped natural frequency d, resulting from

uncertainties in the beam dimensions and measurements. Don't forget boundary

conditions that affect d and .

2 a. Calculate the first two theoretical mode resonance frequencies of the continuous mass

system using Table 1. Assume that = 0.283 lbm/in3 and modulus is 29x106 psi. Use

consistent units in your calculation. Compare these values to the measured resonant

frequencies.

b. The actual resonant frequencies can be evaluated from the readout of the signal

generator, from the FFT analysis of the magnetic reluctance transducer measurement,

and by stopping the beam motion with the strobe light. Discuss the effectiveness of

these measurements by listing the potential errors in each. Consider how each device

is calibrated, the effect each device has on the experiment itself, and any boundary

conditions. Evaluate these errors from your data if possible. Evaluate which of the

readings is most trusted.

3. a. Convert your readings of beam velocity amplitude in millivolts to absolute velocity in

cm/s using Figure 4. Divide each beam velocity by the resonant frequency (in

radians) to get beam deflection at each point the transducer was placed. Plot the

displacement as a function of length along the beam for each of the three modes on the

same plot. Note that you will have to determine the sign (+/-) of each point by

comparison to Table 1 because the sign of the displacement is not given by the

measurement. Compare your results to the mode shapes you sketched in step 3h.



VI. References

1. Theory and Design for Mechanical Measurements, R.S. Figliola and D.E. Beasley,

Wiley, (1991)

2. Theory of Vibration with Applications, W. T. Thomson, Prentice Hall, 2nd edition,

1981, Appendix D, pp. 464 - 474.

3. Vibration of Mechanical and Structural Systems, M. L. James, Harper & Row, New

York, (1989).

4. Vibration Problems in Engineering, S. Timoshenko, Wiley, New York, (1974).

VII. Figures

Figure 1. Schematic of a cantilever beam with overhanging weight.

Figure 2. Schematic of an equivalent SDOF spring-mass damper system.



Figure 3. Free vibrating oscillation of a damped single degree of freedom (SDOF) system.

Figure 4. Conversion chart from voltage to velocity as a function of spacing for the

magnetic velocity transducer.



Figure 5. Schematic of the overall experimental apparatus.

Part 1

Part 2

NI 9219 Data Acquisition Module

Half-Bridge Configuration

LabVIEW VI

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