Emm3522 Lab Manual

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February February 201 201₅₅

EMM 35

EMM 3522

22

MECHANIC

MECHANICAL VIBRATIONS

AL VIBRATIONS

LAB MANUAL

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1. FREE VIBRATION OF AN UNDAMPED

SIMPLY SUPPORTED BEAM

Objectives:

1. To determine the frequency and period of vibration of an undamped simply-supported beam subjected to free vibration.

2. To determine the stiffness of an undamped simply-supported beam subjected to free vibration.

Introduction:

Determining the natural frequency of any system helps to find how the system will behave when just disturbed and left (free vibration), and to find what kind of excitation frequency to be avoided in the system. Ignoring the effect of self weight, for a simply supported beam loaded at the centre, the beam stiffness, k , is given by,

k = 48 EI / L (1) where;

E is the modulus of elasticity o f the beam material

I is the second moment of area of the beam

Theoretically, the fundamental natural frequency of oscillation, ω (rad/s), of a simply-supported beam is

given by;

ω

= (

k /m) (2)

where;

k is the beam stiffness

m is the loaded mass

Procedure:

1. Set up the apparatus with consideration of the following: a. Length of beam

b. Masses to be fixed to beam

2. Affix the low-voltage displacement transducer (LVDT) appropriately for measurements to be taken.

3. Run the data recorder (quickDAQ software) and follow instructions as given by the demonstrator for recording of experimental values.

4. Attach a predetermined mass to the loading rod centered between the ends of the beam. 5. Initiate the data recorder. Displace the beam slightly and release to allow the beam to oscillate

naturally. Stop the recorder once oscillation dies.

6. Save all data into a MS Excel spreadsheet for later reference. 7. Repeat the experiment for different loadings.

Results/Observations:

1. Experimental Frequency and Period of Oscillation for Each Mass

a. Plot graphs of the displacement of the beam vs. time for each experiment. (Refer to Appendix A of this manual to convert LVDT voltage output to displacement).

b. From the graphs, determine the frequency of vibration of the beam under the different loadings.

Repeat the above procedures for all loadings. Obtain the average angular frequencies and average periods of oscillations of the beam for all the loading conditions. Tabulate the results

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Faculty of Engineering, Universiti Putra Malaysia EMM 3504 – MECHANICS OF MACHINES

For all the loadings selected, compare the experimental angular frequencies obtained to those calculated using Eq. (2). Tabulate appropriately.

2. Theoretical and Experimental Stiffness of the Oscillating Beam Taking the log of Eq. (2) gives,

log (ω) = ½ log (k ) - ½ log (m) (3)

Equation (3) is a linear graph in the for m of,

y =ax +c

where,

y = log (ω)

ax = - ½ log (m)

c = ½ log (k ) is the intercept on the y-axis. Using the experimental values obtained, complete the following table:

Table 1: Experimental results Angular frequency, ω

(rad/s)

Loaded mass,m

(kg) log (ω) log (m)

From the data in Table 1, plot the graph of log (ω) vs. log ( I ). From the plot, locate the intercept on

the y-axis to determine the beam stiffness, k (N/m). Compare the experimental value obtained fro m the plot with that obtained from Eq. (1).

Figure 1: Graph of log ωvs. log m

Discussion

1. Discuss the differences (if any) between the theoretical and experimental results. 2. Describe the shortcomings of the experiment performed.

3. Although there is no dashpot attached to the system to provide damping, why did the system come to rest after a period of time?

θ

½ log

k l o g ω

log

m Graph oflogω vs. log m

• k is the stiffness of the beam •

The slope θ represents the

variation of m with respect to

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2 _.

_{UNDAMPED SPRING-MASS SYSTEMS}

Objective:

To compare the theoretical natural frequency of an undamped spring-mass system

calculated for various masses to the values obtained by measurement.

Theory:

Refer to Fig. 1:

Figure 1: Free-body diagram of

forces acting on the mass

Figure 2: Description of apparatus

Setting-up the equation of motion involves establishing equilibrium of forces at the

mass:

∑

F

=

m

x

_

=

−

F

_c

+

mg

The spring load F

c

is calculated from deflection x and spring constant c:

cx

F

_c =

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Faculty of Engineering, Universiti Putra Malaysia EMM 3522_{– MECHANIC} AL VIBRATIONS

g x m

c x



+ =

Solving the equation gives harmonic oscillations with natural angular frequency

ω_o

or natural frequency f :

(

t

)

x

( )

t

x

= _o

cos

ω _o m c o = 2 ω

,

m

c

f

π 2 1

=

The period is:

c

m

T

=

2π

As can be seen, the period/natural frequency can easily be adjusted by altering the

mass.

Procedure:

1. Start the recorder.

2. With no mass attached to the system, deflect carriage downwards by hand

and allow it to oscillate freely until it comes to rest.

3. Stop the recorder.

4. Repeat experiment with other additional masses.

Results:

Fill in the following table:

Experiment

No.

Additional

masses

(kg)

Total mass

(kg)

Experimental

natural frequency

(Hz)

Theoretical natural

frequency

(Hz)

1

0

1.250

2

3

4

6

5

8

6

10

*Note: spring constant = 1710 N/m

Do not forget to include SAMPLES of your calculations in the report. Refer to

Appendix A of this manual to convert LVDT output voltage to displacement readings.

Discussion:

Compare the calculated and theoretical natural frequencies obtained and discuss the

variation between them, if any.

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3 _.

_{DAMPED SPRING-MASS SYSTEMS}

Objective:

To investigate the influence of damping on free oscillation of a spring-mass system.

Theory:

Refer to Fig. 1:

Figure 1: Free-body diagram of forces acting on the mass

Figure 2: Description of apparatus

The equilibrium of forces at the mass is the basis used for setting up the equation of motion. This time a speed-proportional damper force F_d_{is additionally introduced. As the constant force due to the}

weightmg has no influence on oscillation behaviour, it is ignored here

∑

F mx = =−F − _d

c F



The damper forceF d results from the velocity x



and thedamper constant d :

x d F

d =



This produces the following homogeneous differential equation for the equation of motion:

0 = + + x m c x m d x__ _

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or, with D asdegree of damping andω_o as natural angular frequency:

0 2

+

2

=

+

D x x x ω _o o



ω



o m d D ω 2 = , m c o = 2 ω

Solving the equation gives decaying harmonic oscillations with frequency:

2

1 D

o d =ω − ω

It can be seen that at D ≥1 _{oscillation is no longer possible. The angular frequency} ω_d approaches

zero or becomes imaginary.

Procedure:

1. Fit 5 additional weights (mg = 11.25 kg) and secure with knurled nut.

2. Use adjuster to align carriage with center of plot. 3. Start recorder.

4. Deflect carriage downwards by hand and allow it to oscillate freely. 5. Stop recorder.

6. Repeat experiment with different damper settings.

Results:

Provide the oscillation curves for the following experiments:

Experiment No. Needle valve setting 1 Open 8 turns 2 Open 4 turns 3 Open 2 turns 4 Open 1 turns

5 Closed

*Note: spring constant, c = 1710 N/m

Determine the damped natural frequency, ω_d for each of the experiments. Do not forget to include

SAMPLES of your calculations in your report. Refer to Appendix A of this manual to convert LVDT output voltage to displacement readings.

Discussion:

Discuss each damped natural frequencies obtained with their respective oscillation curves. Show all calculations in your report.

Note:

1. Assume a linear relationship between the no. of turns of the damper knob and the range of damping constants (15 – 300 Ns/m), i.e. a fully-closed knob will apply a 300 Ns/m damping, a fully-opened knob will have 15 Ns/m damping.

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4 _{. FREE TORSIONAL VIBRATION}

Objectives:

1. To determine the frequency and period of vibration of a shaft and mass undergoing torsional vibration.

2. To experimentally determine the stiffness of the shaft.

Introduction:

Consider a system with a circular disk at the end of cylindrical shaft (Fig. 1). The hanging disk has a mass moment of inertia I about the axis of rotation. The cylindrical shaft has a torsional stiffness k . If the mass is rotated through an angle θ ₀_{and released, torsional vibration results. Typically the inertia of}

the shaft can be ignored.

Figure 1: Shaft undergoing free torsional vibration

Procedure:

1. Set up the apparatus with consideration of the following: a. Length of rod

b. Mass(es) to be fixed on end of rod

2. Affix the low-voltage displacement transducer (LVDT) appropriately for measurements to be taken.

3. Run the quickDAQ software and follow instructions as given by the demonstrator for recording of experimental values.

4. Save all data into a MS Excel spreadsheet for later reference.

Results:

1. Frequency and Period of Oscillation for Each Setup

a. Plot graphs of displacement vs. time for each experiment. Refer to Appendix A of this manual to convert LVDT output voltage to displacement readings.

b. From the graphs, determine the frequency of vibration of the rod under the different rod lengths and mass setups.

c. Tabulate your findings accordingly.

Theoretically, the angular frequency, ω _{(rad/s), of the oscillating rod is given by:}

(

q I

)

=

ω ₍₁₎

where q is the torsional stiffness (Nm) of the rod and I is the mass moment of inertia of the oscillating body in kg·m2.Compare the experimental and theoretical angular frequencies obtained for all the experimental setups. Tabulate accordingly.

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2. Comparison between Theoretical Stiffness and Experimental Stiffness Taking the log of equation (1) gives;

( )

q

log

( )

I

2

1 log

2

1 log

ω = − (2)

Equation (2) is a linear graph in the for m of

c

ax

y

= + where, y = log (ω₎ ax = _log

_{( )}

_I 2 1 − c = _log

_{( )}

_q 2

1 _{is the intercept on the}_y_-axis.

Using the experimental data obtained, complete the table below: Table 1: Table of data

Angular frequency of oscillating rod,ω

(rad/sec)

Mass moment of inertia of oscillating body, I

(kgm2)

log (ω₎ _{log (}_I₎

From the data in the previous table plot the graph of log (ω_{) vs. log (}_I_{) (Refer Fig. 2). From the}

intercept on the y-axis calculate the torsional stiffness, q (Nm). The experimental value obtained shall now be compared to theoretical values which are a function of the rod itself (material type, length and diameter).

Figure 2: Graph of log ω_{vs. log}_I

θ

½ log

q l o g ω

log

I Graph of

log

ω

vs. log

I

• q is the torsional stiffness of

the rod

• The slope θ represents the

variation of I with respect to

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Discussion/Conclusion

1. Make comparisons between the theoretical and experimental frequencies of oscillations and stiffness.

2. Describe the probable causes of error in your experiment.

Appendix: Additional information

Experimental Setup

Calculation of torsional stiffness,q

From the torsion equation:

L G J T θ = or; L GJ T = θ where; L GJ q =

The polar moment of inertia of the shaft, J ₌

( )

π d 4 32

All symbols have their usual meaning.

Mass moment of inertia of the fl ywheel of radiusr , I _{f =}mr 2 2

Mass moment of inertia of additional mass about the rotation axis, I

2 h m I _{f +} _a = where;

ma = the additional mass

h = the distance between the centre of the additional mass and the axis of rotation. h = 300 mm

Fixed End Specimen Additional Mass Arm to Increase Mass Moment Of Inertia Flywheel

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5 _.

_{FORCED, DAMPED AND UNDAMPED VIBRATION}

OF A SIMPLY SUPPORTED BEAM

Objective:

To study the effect of damping on a simply-supported beam subjected to forced vibration and determine the damping factor for a given damping medium.

Introduction:

When a dynamic system is subjected to a steady-state harmonic excitation, it is forced to vibrate at the same frequency as that of the excitation. Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machinery, forces produced by the reciprocating machines, or the motion of machine itself. Although pure harmonic excitation is less likely to occur than the periodic or other types of excitation, understanding the behaviour of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system. The harmonic excitation can be given in many ways like with constant frequency and variable frequency or a swept-sine frequency, in which the frequency changes from the initial to final values of frequencies with a given time-rate ( i.e., ramp).

Procedure:

Setting up the apparatus

1. Check the position of the pinned supports to make sure that they are at the same level.

2. Take the beam and mount it on the two supports. Make sure that the beam is held in position at one support, by slightly turning the screw such that it touches the beam. Ensure that the other end of the beam is free to slide over the s upport.

3. Locate the centre of the beam and fix the motor with the out-of-balance mass at this location. 4. Decide on the mass to be used for loading the beam.

5. Anchor the weight stopper tightly to the loading rod.

6. Insert the loading rod through the hole at the centre of the mass.

7. Insert the displacement measuring plate at the top of the mass. Anchor it tightly to the rod. 8. Screw the top end of the loading rod to the hole at the bottom of the motor mounting.

9. Fix the LVDT to the LVDT stand and its probe is on the displacement measuring plate. Ensure that the probe is clear of all objects.

10. Connect the LVDT to one of the channels at the data acquisition terminal. 11. Press the ‘RUN’ button to set the LVDT reading to 0.

12. Adjust the position of the LVDT’s probe until the signal from the LVDT displayed on the screen is at ‘0’ position. Anchor the LVDT tightly.

Running the experiment

The experiment will be executed first in the undamped mode followed by the d amped mode. (a) Undamped experiment

1. Decide on 3 speeds to be used for this experiment. The chosen speeds should produce significant amplitude of vibration.

2. Switch on the motor and set the speeds as decided.

3. Press the ‘RUN’ button to record the resulting oscillations. 4. After a few seconds press the stop button.

5. Save the data in .csv format.

6. Increase the speed of the motor to the next chosen speed. 7. Repeat steps 3 to 6 until all 3 speeds are executed.

(b) Damped experiment

1. Attach the damping pot close to the motor.

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3. Press the ‘RUN’ button to check the position of the LVDT probe. If it is not a ‘0’ adjust the LVDT position until the signal from the LVDT is ‘0’ or approximately ‘0’.

4. Switch on the motor and set the speed as decided in the undamped case. 5. Press the ‘RUN’ button to record the resulting oscillations.

6. After a few seconds press the stop button. 7. Save the data in .csv format.

8. Increase the speed of the motor to the next chosen speed. 9. Repeat steps 5 to 8 until all 4 speeds are executed.

(c) To determine the damping factor

1. With the damping pot attached, choose a speed close to the resonance frequency so as to obtain a reasonable displacement.

2. Press the ‘RUN’ button to check the position of the LVDT probe. If it is not at ‘0’ position adjust the LVDT position until the signal from the ‘LVDT is ‘0’ or approximately ‘0’.

3. Switch on the motor and set the speed as decided.

4. Press the ‘RUN’ button to record the resulting oscillations.

5. Switch off the motor but let the data acquisition ‘RUN’ until the oscillations die down. 6. Press the stop button.

7. Scroll the screen to the position where the motor stops, i.e. the oscillation starts to decrease. 8. Save the data from this point onwards. This is the free vibration state.

Results/Observations:

1. For each speed and condition (undamped or damped) determine the maximum amplitude of vibration from the recorded data and record it in the table below. Refer to Appendix A of this manual to convert LVDT output voltage to displacement readings.

2. For each speed, using the captured data plot the oscillations for the damped and undamped cases on the same graph to show the effect of damping on the amplitude of oscillation.

Table 1: Speed = _____________ rev/s

Undamped Damped

Cycle no. Amplitude (mm) Cycle no. Amplitude (mm)

1 1 2 2 3 3 4 4 5 5 6 6 7 7 Average Average Table 2: Average amplitude for each selected speed

Motor speed (rev/s) Undamped Damped

3. Using the data from the free vibration mode, determine the damping factor for the system using the logarithmic decrement method. The logarithmic decrement, δ is given by:

n x x n 1 ln 1 =

δ

(1) where; δ = logarithmic decrement n ₌ _{cycle no.}

x₁ ₌ _{amplitude for cycle no. 1} x_n ₌ _{amplitude for cycle}n

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(

)

(

2

)

1

2

πζ

ζ

δ

= − (2)

where;

ζ is the damping factor

Table 3: Determination of the damping factor

Cycle no. Amplitude, x

(mm)

Amplitude ratio,

x₁_/ x_n

ln x1 / x n logarithmic

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