2142-391 Engineering Mechanical Laboratory
Vibration of Beam
Nopdanai Ajavakom
Chanat Ratanasumawong
1. Introduction
Vibration is the branch of engineering that deals with repetitive motion of mechanical
systems. A few examples of vibration problems are the wing fluttering, the response of an
engineering structure to earthquakes, the vibration of an unbalanced rotating machine, the
time response of the plucked string of a musical instrument, and the quality of ride of an
automobile or motorcycle. Most practical vibrating systems are very complex, and it is
impossible to consider all the details for a mathematical analysis. Only the most important
features are considered in the analysis to predict the behavior of the system under specified
input conditions. Often the overall behavior of the system can be determined by considering
even a simple model of the complex physical system. Thus the analysis of a vibrating system
usually involves mathematical modeling, derivation of the governing equations, solution of
the equations, and interpretation of the results.
1.1. Motivation
In general the vibrating system is simplified to a simple model including three basic
elements. Three basic elements are 1) the element restoring or releasing kinetic energy that is
a mass or a mass moment of inertia, 2) the element restoring or releasing potential energy that
is an elastic component or a spring, and 3) the element dissipating energy that is a damper. To
analyze the vibration problem, the quantities of these elements must be known. Although the
mass and the spring stiffness of the vibrating system are able to determine easily by a scale or
by static test, the damper are difficult to measure. To determine these quantities, the dynamic
measurement is required. For our study, we simplify the wing fluttering problem into the
vibration of rigid beam. To predict the behavior of beam vibration, we have to know the
quantities of three basic elements: the mass moment of inertia of the beam, the stiffness of
spring and the damping of the dashpot, put them in the governing equation and solve for
response. Some static and dynamic methods to determine these quantities are studied and the
measurements are also done.
1.2. Objective/Problem Statement
Broad description of the family of systems and experimental conditions
We shall focus on the single-DOF system, and will test both free vibration case and
forced vibration case.
Specific aspects of interest in these systems
The aspect of current interest is the quantities of the mass moment of inertia of beam, the
elastic stiffness of spring and the damping coefficients of dampers.
Problem Statements
Determine the time response of the displacement at the end of the beam at various
conditions. [Plot displacement VS time] Then find the quantities of the basic elements as well
as other important vibration-related parameters.
2. Approach
2.1.General
Measure and plot the time response of the displacement, then calculate the other
dynamic parameters using theoretical relations.
2.2.Brief Description and Diagram of Apparatus
The diagram of apparatus is shown in Fig.1. The apparatus consists of a beam that at
its one end is hinged on a trunnion attached to the vertical frame. The beam is hung to lie
horizontally with a spring that can be changed to vary the stiffness. A damper can be added
into the system by attaching a dashpot to the beam. The damping coefficient of the damper
can also be varied by adjusting the diameter of oil orifices inside the dashpot. For forced
vibration experiment, an unbalance motor will be attached to the beam. The rotational speed
of the motor can be adjusted by an inverter. The vibration of beam can be measured by the
displacement sensor attached close to the end of the beam, and then the measured signal is
sent to a computer to be processed further.
2.3.Brief Description of Measurands and Measurement Instruments
The data measured from the displacement sensor are sent to the real-time controller
unit to process the signal and subsequently sent to demonstrate and collect at PC.
3. Experimental Results
3.1.Graphical Results
Displacement VS time and Displacement VS frequency plots.
3.2.Questions Pertaining to Experimental Results
3.2.1. What is the frequency of the oscillation?
3.2.2. What are the inertia, the stiffness constant, and the damping ratio of the
system’s elements and the natural frequency of the system? How can you find
them?
3.2.2. If you know the dynamic parameters of the system, how can you predict the
response of the system with another initial conditions and/or forcing
functions?
References and Recommended Readings
1. Daniel J. Inman, Engineering Vibration, Prentice Hall.
2. William T. Thomson and Marie Dillon Dahleh, Theory of Vibration with
Applications, Prentice Hall.
3. Singiresu S. Rao, Mechanical Vibrations, Prentice Hall.
4. Kelly J., Fundamentals of Mechanical Vibrations, McGraw Hill.
5. ESSOM, Instruction manual MM 320 Universal vibration apparatus.
Frame
Unbalanced
motor
Spring
Displacement sensor
Dash pot
Beam
Control box
Figure 1 Universal Vibration Apparatus
2145-392 Engineering Mechanical Laboratory
Vibration of Beam
I. Theoretical background
1. Free Vibration
The single-degree of freedom beam system is generally modeled as shown in Fig.2. IO
is the mass moment of inertia of beam and motor, c is the damping coefficient, and k is
stiffness of the spring. The equation of motion of this system can be derived by using
Newton’s second law of motion, and written in Eq.(1).
I O  ca 2  kb2  0
(1)
Equation (1) can be written in the form of natural frequency and the damping ratio as shown
in Eq.(2).
  2   2   0
n
n
(2)
Where natural frequency n  kb2 IO , and damping ratio   ca 2 / 2 IO kb2
The solution of Eq.(2) that shows the response of the torsional system varies due to
the value of damping ratio  . The responses of vibration corresponding with Eq.(2) can be
separated into 3 cases that are underdamped motion (   1 ), overdamped motion (   1 ) and
critically damped motion (   1 ). These responses can be shown by Eqs.(3)-(5).
(t )  Ae   nt (sin(  d t  ))
Underdamped motion:
where
(3)
d  n 1   2
(t )  e   n t (a1e  n  1t  a2e  n
Overdamped motion:
Critically damped motion
(t )  (a1  a2t )e nt
2
c
o
 2 1t
)
(4)
(5)
k

a
am
b
L
Figure 2 The single-degree-of-freedom beam system
The constants A, , a1, and a2 can be determined, if the initial condition and/or boundary
condition are specified.
2. Logarithmic decrement
The damping coefficient or, alternatively, the damping ratio is the difficult quantity to
determine. One of the common ways to determine the amount of damping is to measure the
rate of decay of free oscillations. The larger the damping, the greater will be the rate of decay.
The logarithmic decrement is defined as the natural logarithm of the ratio of any two
successive amplitudes (see Fig.3). The expression for the logarithmic decrement then
becomes
  ln
(t )
Ae  n t sin( d t  )
 ln
(t  T )
Ae  n (t T ) sin( d (t  T )  )
(6)
Figure 3 Underdamped response used to measure damping
where T is the period of oscillation. Since the values of sines are equal when the time is
increased by the damped period T, the preceding relation reduces to
  ln
e  n t
e  n (t T )
 ln e nT  nT
(7)
By substituting the damped period, T  2 ( n 1   2 ) , the expression for the logarithmic
decrement becomes

2
(8)
1  2
Solving this expression for  yields


4  
2
2
(9)
Peak measurements can be used over any integer multiple of the period to increase the
accuracy over measurements taken at adjacent peaks. The expression of the logarithmic
decrement corresponding to n-period peak measurement is shown in Eq. (10)
1
(t )
  ln
n (t  nT )
(10)
3. Forced Harmonic Vibration
Consider the beam system shown in Fig.2, if this system is excited by an unbalance force
from the motor F0 cos t , the differential equation of motion becomes
I O  ca 2  kb2  am F0 cos t
(11)
The solution to this equation consists of two parts, the free vibration response as shown in
Eqs.(3)-(5), and the force vibration response of the same frequency  as that of the excitation.
The force vibration response is observed to be in the form
 p   cos(t  )
(12)
Where  is the amplitude of oscillation and  is the phase angle of the displacement with
respect to the exciting force.
The total solution of Eq.(11) is the sum of free vibration response and force vibration
response and is shown in Eq.(13) for the case of underdamped motion.
(t )  Ae  n t (sin( d t  ))   cos(t  )
(13)
Note that the term of free vibration response can be changed to be overdamped response
or critically damped response depended on the value of damping ratio of the system.
From Eq.(13), it is evident that for the large values of t, the term of free vibration
response approaches zero and the total solution approaches the force vibration response. Thus
the term of force vibration response is called the steady-state response and the term of free
vibration response is called the transient response.
Consider the steady-state response shown in Eq.(12), the Amplitude  and phase  can
be found by substituting Eq.(12) into the differential equation of motion Eq.(11). The
Amplitude  and phase  are shown in Eqs.(14) and (15).

and
am F0
(kb  I O 2 ) 2  (ca 2) 2
2
  tan 1
ca 2
kb2  I O 2
(14)
(15)
Equations (14) and (15) can be expressed in the nondimensional form as shown in Eqs.(16)
and (17)
kb2
am F0

Figure 4
Plot of the normalize amplitude and phase of the steady-state response
of a damped system versus the frequency ratio
kb2
1

2
am F0
(1  r ) 2  (2r ) 2
2 r
1 r2
where r is the frequency ratio r    n .
and
  tan 1
(16)
(17)
The plots of Eqs.(16) and (17) for several values of the damping ratio  are shown in
Fig.4. Note that as the driving frequency approaches the undamped natural frequency (r 
1) the magnitude approaches a maximum value for the cases that are light damping. Also note
that at this point, the phase shift crosses through 90º. This point is defined resonance point for
the damped case.
II. Details of apparatus
The apparatus may be adjusted into a variety of configurations to study the effect of mass
moment of inertia, spring stiffness and damping coefficient to the vibration of beam both in
the case of free vibration and force vibration experiment. For force vibration, the rotational
speed of unbalance motor can be adjusted to excite the beam to vibrate at wide frequency
range.
The system consists of three subsystems:
1. The main apparatus shown in Fig. 5. It is comprised of the beam, spring, adjustable
damper, unbalance motor, and the displacement sensor. The positions that attached
spring, damper and measuring vibration can be adjusted, and spring can be changed to
vary the stiffness.
2. The control box that is used to convert analog signal to digital signal and control the
rotational speed of the motor.
3. The system interface software, which runs on a PC in windows XP. The program is
the user’s interface to the system and supports data acquisition, plotting, system
execution commands, etc. Here the program “Freedamp” is used in both free vibration
experiment and forced vibration experiment.
Figure 5 Universal Vibration Apparatus
Some important specifications
1. The total sampling rate of all channels must not be higher than 48,000 samples/sec.
The number of sampling rate is divided equally for each channel, if multiple channels
are used.
2. The sensing range of the displacement sensor is between 30 to 300 mm. The Power
LED (left LED) must be green and the output LED (right LED) must be yellow in
normal operation.
3. Sensor sensitivity is approx. 70 mV/mm
To start up
Turn on the PC
Turn on the power to the Control Box
To shut down
Turn off the power to the Control Box
Turn off the PC
Safety Precautions
1. Be sure to firmly tighten (but not overtight) the bolts that fasten the beam, damper,
and the motor.
2. Stay clear of and do not touch any part of the mechanism while it is moving,
especially the running motor.
3. Do not run the motor with too high speed.
4. Do not drive the motor with excessive power.
III. Experiments
1. Free vibration
This section gives a procedure for identifying the system parameters. The approach will
be indirectly measure the spring stiffness constant, and damping constants by making
measurements of the movement of the beam. The mass moment of inertia of the system
can be determined by weight the beam and the motor, and measure beam dimensions, and
then the mass moment of inertia of the beam can be calculated.
Procedures
1. Set up the apparatus to be the same as shown in Fig.2. Be careful to adjust the beam to
lie horizontally.
2. Set the positions of the attached damper “a”, spring “b” and motor “am” according to
the instructor.
3. Power up the system and open the system interface program “Freedamp”.
4. Measure the length of the beam, the weight of the beam, and the weight of the motor.
Input the distances that attach spring and damper and the masses of beam and motor
in the “figure” tab.
5. In the “signal measurement” tab, input the data into the following windows
5.1 Time to wait before get signal (s): input delay time from press “Get signal” button
until measurement start. [0 s]
5.2 Sample rate: input the number of samples measured in one second. [200+]
5.3 Time of samples (s): input measured time [4 s]
5.4 Number of samples: Number of measured samples (= Sample rate  Time of
samples)
6.
7.
8.
9.
5.5 Time out (s): The time that the interface program still actives. If there are not any
responses until the setting time is passed, program will becomes inactive.
5.6 Elapsed time: The time before measurement start.
Press “Get signal” button.
Displace the end of the beam down approximately 30 mm and then release. Be careful
that the vibration of beam is completely collected and recorded.
Plot the beam time response (displacement versus time graph). Save the raw data text
or excel file format.
Determine the damped natural frequency of beam d by choosing several
consecutive cycles and dividing the number of cycles n by the time taken to complete
the cycles tcycles .
Td 
t cycles

tn  t0
n
(18a)
n
2
d 
(18b)
Td
10. Determine the logarithmic decrement by measuring the reduction from the initial
cycle amplitude  0 to the last cycle amplitude  n for the n cycles measured in step 10.
Using relationship:
1  
  ln  0 
n  n 
11. Determine the damping ratio from  
(19)

.
4 2   2
12. Calculate the natural frequency of the system using
d  n 1   2
(20)
13. Change the position that the damper is attached to the beam “ b ”.
14. Questions and Discussion: Do you think whether the values of d ,  , n change
after the position of the damper changed? If so, how do they change (increase or
decrease)?
15. Redo steps 7-13 to obtain Td , d   and n . Discuss the any differences found in the
new results compared to the previous ones.
16. Now you can calculate the mass moment of inertia of the beam and motor system I O
and stiffness of spring k from (solving two equations for two unknowns):
n 
kb2
and n 
IO
kb  2
IO
Then the damping coefficient is c 
2 I  
2 I 0n
and c  0 2 n . They should be very
2
a
a
close to each other.
17. Compare the experimental mass moment of inertia of the beam and motor system I O
1
to the theoretical one from I O  mb l 2  mm a m2 .
3
18. Questions and Discussion: Can you suggest an alternative approach to find the
stiffness of the spring? Hint: The approach uses static displacement of the beam. Use
the approach to find the stiffness of the spring and compare them to the one from the
previous experiment.
2. Forced Vibration
This section will study the response of the system due to harmonic force input from the
unbalance motor. The frequency of harmonic force is adjusted by varying the rotational speed
of the motor.
Procedures
1. Set up the apparatus to be the same as shown in Fig.2. Be careful to adjust the beam to
lie horizontally.
2. Set the positions of the attached spring, damper and motor identical to the first
experiment.
3. Open the system interface program: “Freedamp”.
4. In the “figure” tab, input the distances that attach spring and damper, the length of
beam, the masses of beam and motor, the stiffness of string k, and the damping
coefficient c. (k and c are known from the former experiment)
5. In the “signal meas” tab, input the data into the windows
6. Start motor, and adjust to operate at low rotational speed.
7. Wait for the system become steady state, and then press the “Get signal” button to
start to collect the data.
8. Save the collected data into a file.
9. Plot the displacement vs time, and collect steady state output amplitude.
10. Increase rotational speed of motor and repeat steps 7-9 to get a total of at least 10
points. Make sure that the range of speed of the motor covers the natural frequency of
the system.
11. Change the damping coefficient by adjusting the orifice area inside the dashpot, and
repeat steps 7-10.
12. Plot the steady state amplitude vs frequency to see the amplitude response of the
system. Then, determine roughly at what frequency the amplitude of the response is
highest. Is it near the natural frequency? Discuss.
13. Also plot the theoretical graph using the parameters obtained from free vibration
experiment.
14. Discussion: Compare three graphs and discuss on the differences.