EXPERIMENT 2

FREE VIBRATIONS OF A CANTILEVER BEAM

WITH A LUMPED MASS AT FREE END

2.1 Objective of the experiment:

To experimentally obtain the fundamental natural frequency and the damping ratio of a cantilever beam

having lumped mass at free end and to analyze the free vibration response of a cantilever beam subjected

to an initial disturbance. This virtual experiment is based on a theme that the actual experimental

measured vibration data are used.

2.2 Basic Definitions

Free vibration takes place when a system oscillates under the action of forces inherent in the system itself

due to initial disturbance, and when the externally applied forces are absent. The system under free

vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamical

system, established by its mass and stiffness distribution.

In actual practice there is always some damping (e.g., the internal molecular friction, viscous damping,

aero-dynamical damping, etc.) present in the system which cause the gradual dissipation of vibration

energy and it result gradual decay of amplitude of the free vibration. Damping has very little effect on

natural frequency of the system, and hence the calculations for natural frequencies are generally made on

the basis of no damping. Damping is of great importance in limiting the amplitude of oscillation at

resonance.

The relative displacement configuration of the vibrating system for a particular natural frequency is

known as the mode shape (or eigen function in continuous system). The mode shape corresponding to

lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental (or the first)

mode. The displacements at some points may be zero. These points are known as nodes. Generally nth

mode has (n-1) nodes (excluding end points).The mode shape changes for different boundary conditions

of the beam.

2.3 Theoretical natural frequency for cantilever beam

Figure 2.1 (a) A cantilever beam

Figure 2.1 (b) The beam under free vibration (without mass at free end)

A cantilever beam with rectangular cross-section is shown in Figure 2.1(a), the bending vibration can be

generated by giving an initial displacement at the free end of the beam. Figure 2.1(b) shows a cantilever

beam under the free vibration.

When a system is subjected to free vibration and the system is considered as a discrete system in which

the beam is considered as mass-less and the whole mass is concentrated at the free end of the beam. The

governing equation of motion for such system will be

..

0m y t ky t (1.1)

where m is a concentrated mass at the free end of the beam and k is the stiffness of the system. The

transverse stiffness of a cantilever beam is given as (using strength of materials deflection formula,

Timoshenko and Young, 1961)

3

3EIk

l (1.2)

Where E is the Young’s modulus of the beam material (it can be obtained by the tensile test of the

standard specimen). The fundamental undamped circular natural frequency of the system is given as

nf

k

m (1.3)

Where, m is an equivalent mass placed at the free end of the cantilever beam (of the beam and sensor

masses), on substituting equation (1.2) into equation (1.3) we get

3

3nf

EI

ml (1.4)

The undamped natural frequency is related with the circular natural frequency as

2

nf

nff (1.5)

I the moment of inertia of the beam cross-section and for a circular cross-section it is given as

4

64I d (1.6)

Where, d is the diameter of cross section, and for a rectangular cross section

3

12

bdI (1.7)

Where b and d are the breadth and width of the beam cross-section as shown in Figure 2.2. Dimensions of

the beam material are given in Table 2.2.

b

d

Fig 2.2 A rectangular cross-cross of the beam

In case of the test specimen, the beam mass is distributed over the length. By taking a one-third of the

total mass of beam at the free end (Thompson. 1961), the system can be assumed as discrete system.

Hence,

33

140bm m (1.8)

Where bm is the mass of beam and is given as

bm V bdl

Where,

is the mass density of the beam material and V is the volume of the beam from the fixed end to

the free end.

The equivalent tip mass of a cantilever beam can be obtained as follows. Consider a cantilever beam as

shown in Fig.2.3 (a). Let 1m be the mass of the beam per unit length, l is the length of the beam,

1bm m l is total mass of the beam, and

maxv is the transverse velocity of the free end of beam and f is

the force applied, E is the young’s modulus of the beam and I is the moment of inertia of the beam.

Fig. 2.3(a) A cantilever beam with distributed mass Fig. 2.3(b) The cantilever beam with a tip mass

Consider a small element of length dx at a distance x from the free end (Fig.2.3 (a)). The beam

displacement at this point is given by (Timoshenko and Young, 1961)

3

2 3

3

1( ) 3

2 3

fly x lx x

l EI (1.9)

Here

3

3

fl

EI is the deflection at free end of the cantilever beam. Now the velocity of the small element at

distance x is given by

2 3

max3

3( )

2

lx xv x v

l

Hence, the kinetic energy of the element is given by

22 3

max3

1 3

2 2

lx xdT Adx v

l

and the total kinetic energy of the beam is

22 22 3 2 5 6 7

2 4 5 6max maxmax3 6 6

0 0 0

2 72 2maxmax max6

1 3 9 69 6

2 2 8 8 5 6 7

33 1 33 1 33

8 35 2 140 2 140

ll lAv Avlx x l x lx x

T A v dx l x lx x dxl l l

Av lAl v Al v

l

(1.10)

where 1 bm l m . If we place a mass of

33

140bm at the free end of the beam and the beam is assumed to be

of negligible mass, then

Total kinetic energy possessed by the beam = 2

max

1 33  

2 140bm v (1.11)

Hence two systems are dynamically same. Therefore the continuous system of cantilever beam can be

changed to single degree freedom system as shown in Fig.2.3(b) by adding the 33

140bm of mass to its free

end.

Values of the mass density for various beam materials are given in Table 2.1. If any contacting type of

transducer is used for the vibration measurement, it should be placed at end of the beam and then the mass

of transducer has to be added into the equivalent mass of the beam at the free end of the beam during the

natural frequency calculation. If tm is the mass of transducer, then the total mass at the free end of the

cantilever beam is given as

33

140b tm m m (1.12)

2.4 Experimental setup

Figure 2.4 An experimental setup for the free vibration of a cantilever beam

The experimental setup is consists of a cantilever beam, transducers (strain gauge, accelerometer, laser

vibrometer), a data-acquisition system and a computer with signal display and processing software (Fig.

2.4). Different types of beam materials and its properties are listed in Table 2.1. Different combinations of

beam geometries for each of the beam material are summarized in Table 2.2

Accelerometer is a sensing element (transducer) to measure the vibration response (i.e., acceleration,

velocity and displacement). Data acquisition system takes vibration signal from the accelerometer and

encode it digital form. Computer acts as a data storage and analysis system, it takes encoded data from

data acquisition system and after processing (e.g., FFT) it display on the computer screen by using

analysis software.

Table 2.1 Material properties of various beams

Material Density (kg/m3) Young’s modulus (N/m

2)

Steel 7850 2.1×1011

Copper 8933 1.2×1011

Aluminum 2700 0.69×1011

Table 2.2 Different geometries of the beam

Length, L, (m) Breadth, b, (m) Depth, h, (m)

0.45 m 0.02 m 0.003 m

0.65 m 0.04 m 0.003 m

Example 2.1 Obtain the undamped natural frequency of a steel beam with l = 0.45 m, d = 0.003

m, and b = 0.02 m. The mass of transducer at the free end =18.2 gm

33 33 33 (0.02 0.003 0.45) 7850( ) 0.0182 0.00681 kg

140 140 140b t tm m m bdl m

3 3 11 3

3 3 3 3

3 /123 3 3 2.1 10 0.02 0.00367.59 rad/sec

12 12 0.0681 0.45nf

E bdEI Ebd

ml ml ml

or

10.75 Hznf

2.5. Photos of experimental setup

Figure 2.5 Experimental setup of a cantilever beam

Figure 2.5 shows an experimental setup of the cantilever beam. It includes a beam specimen of a

particular geometry with a fixed end and at the free end an accelerometer is mounted to measure the free

vibration response. The fixed end of beam is gripped with the help of clamp. For getting precise free

vibration cantilever beam data, it is very important to ensure that clamp is tightened properly, otherwise it

may not give fixed end conditions in the free vibration data.

Figure 2.6 A close view of the fixed end of the cantilever beam

Accelerometer: It is the most common contacting type sensor for the vibration (i.e., acceleration,

velocity or displacement) measurement. It is available with connecting cable as-well-as wireless

type. It is pasted onto the surface by either using magnetic base or by using adhesive glue or by

threaded screw (Fig. 2.7).

Figure 2.7 A close view of an accelerometer mounted on the free end of the beam

The basic principle of the measurement by the accelerometer is that it measures the force exerted by a

body as a result of a change in the velocity of the body (i.e. which leads to acceleration). A moving body

possesses an inertia which tends to resist change in velocity. The force caused by vibration or a change in

motion causes the mass to "squeeze" the piezoelectric material which produces an electrical charge that is

proportional to the force exerted upon it. Since the charge is proportional to the force, and the mass is a

constant, hence the change is proportional to the acceleration.

A Laser Doppler Vibrometer (LDV) is an instrument (Fig. 2.8) that is used to make non-contact vibration

measurements of a surface. The laser beam from the LDV is directed at the surface of interest, and the

vibration amplitude and frequency are extracted from the Doppler shift of the laser beam frequency due to

the motion of the surface. The output of an LDV is generally a continuous analog voltage that is directly

proportional to the target velocity component along the direction of the laser beam (give a line diagram of

basic principle of the LDV)

Fig 2.8 A laser vibrometer system

Fig 2.9 A laser projector of a laser vibrometer fitted on a stand.

Fig 2.10 The controller for the laser vibrometer

Fig 2.11 The laser generator of a laser vibrometer

Figure (2.8) is a rotational laser vibrometer. The complete setup contains the projector (Fig. 2.9), the

controller (Fig 2.10), and laser generator (Fig 2.11). All the settings related to the measurement are done

in the controller then the laser beam is generated by the laser generator. The laser goes to the reflector by

an optical fiber cable. The beam is projected to the measurement surface and measurement signal is taken

into the computer through a data-acquisition system.

Rotational Laser Vibrometer (RLV): The optical measurement principle for the rotational vibrometer is

based on laser interferometry. Use of the RLV is not limited to cylindrical parts. By using a special

differential measurement process with two laser beams, independently of the shape of the object under

investigation, only the rotational movement component is acquired and translational vibrations are

predominantly suppressed. A schematic layout of the signal paths is shown in Fig. 2.12.

` Fig 2.12 Principle of measurement of rotational laser vibrometer

Dynamic acquisition of rotational vibrations is possible in a frequency range from 0 Hz to 10 kHz. It also

cover challenging measurement tasks e.g. in the order analysis in rotors. The interferometric process

works continuously, i.e. in principle there is no limit to the angular resolution as for example, this

limitation exist when using optical encoders with a finite number of divisions.

Data acquisition system: Data acquisition typically involves the conversion of analog signals and

waveforms into digital values, and processing the values to obtain desired information. Data acquisition

systems, as the name implies, are products and/or processes used to collect information to document or

analyze some phenomenon. The components of measurement and data acquisition systems include (see

Fig. 2.13) (i) Sensors that convert physical parameters to electrical signals, (ii) Signal conditioning

circuitry to coerce sensor signals into a form that can be converted to digital values, and (iii) Analog-to-

digital converters, which convert conditioned sensor signals to digital values.

Figure 2.13(a) An overall measurement system

Figure 2.13(b) Data acquisition system

Data acquisition system receives voltage signal from sensors (e.g., accelerometer) and calibrate the data

into equivalent physical quantity (e.g., acceleration) and send it to computer where by using a vibration

measurement software these data can be analyzed as time history (e.g., acceleration-time, velocity-time or

displacement-time) and in frequency domain (i.e., using FFT) Fig. 2.14.

Figure 2.14 A typical response with time and the corresponding FFT plot

When the voltage signal from the accelerometer is sent to the data-acquisition system, it converts the

signal to a mechanical vibration data (acceleration) and stores it to the computer. A typical screen-shot of

a captured vibration signal by using vibration measurement software is plotted as shown in Figure 2.14

and can be used in further analysis.

2.6 Experimental procedure

1. Choose a beam of a particular material (steel, aluminum or copper), dimensions (L, w, d) and

transducer (i.e., measuring device, e.g. strain gauge, accelerometer, laser vibrometer).

2. Clamp one end of the beam as the cantilever beam support.

3. Place an accelerometer (with magnetic base) at the free end of the cantilever beam, to measure

the free vibration response (acceleration).

4. Give an initial deflection at the free end of cantilever beam and allow it to oscillate its own. This

could be done by bending the beam from its static equilibrium position by applying a small static

force at the free end of the beam and suddenly releasing it, so that the beam oscillates its own

without any external force during the oscillation.

5. The free oscillation could also be started by giving a small tap at the free end of the beam.

6. Record the data obtained from the chosen transducer in the form of graph (variation of the

vibration response with time).

7. Repeat the procedure for 5 to 10 times to check the repeatability of the experimentation.

8. Repeat the whole experiment for different material, dimensions, and measuring devices.

9. Record the whole set of data in a data base.

2.7 Virtual experimentation

Virtual experimentation provides the interface which provides facility to perform experiments virtually. It

provides different options for material selection, instruments, and specimen dimensions. After making

desired selection and running the program it gives the result from a storage database for a particular

configuration selected by the user. Fig. 2.15 shows an overall flowchart for a virtual laboratory in which

several experiments remote users can perform through the internet with the help of already stored

measured data.

Figure 2.15 Overview of measurement based virtual experiments

2.8 Steps in virtual experimentation and its programming

The program of free vibration is divided in many sections. The step by step description of program is

given as follows

1. TITLE PAGE- This is the first page of the virtual experiment of the free vibration of a cantilever

beam. It includes the title of experiment, and a photo of the experimental setup (see Fig. 2.16).

Fig 2.16 Title Page

2. INTRODUCTION- This section contains aim of experiment, some important definitions related to

free vibration, damping ratio etc.

Fig 2.17 Introduction page (some basic definitions)

3. INPUT SECTION- This section contains various input options about the experiment configuration

the user can choose, i.e. the beam material, beam dimensions, the transducer for the vibration

measurement etc. User has to select proper input to proceed for the virtual experiment.

When user enters the input configuration, each parameter generates a specific number and form a

set of numbers. Based on the input configuration the virtual program takes the particular stored

measured vibration data, which are related to that particular configuration from the database.

For each experiment the database contains 10 set of files, which are chosen by the program by

randomly.

Fig 2.18 User information and selecting materials with different specifications

4. EXPERIMENT- The data from the files are read and are plotted in a particular sequence. First the

response-time graph is plotted then its FFT is plotted. The data from the file is taken as an array

and it is plotted in the loop one-by–one by using a script (computer code) inside the loop.

Fig 2.19 Response- Time graph

5. THEORETICAL CALCULATIONS- Theoretical calculations are done based on the input

configuration chosen by the user to compare with experiments results. Program decodes the input

and generates the related parameter like the Young’s modulus, the density and dimensions of the

beam. By using these values the theoretical calculation are done with the help of formula

provided in this report. Refer Section 2.3 for the theoretical formulations and calculations.

Fig. 2.20 Theoretical Calculation of natural frequency for the chosen configuration in input

section

6. EXPERIMENTAL CALCULATIONS- The experimental calculations are done by using the data

taken from already stored measurement data files. A waveform peak detector is used to get peak

values and its time locations from the freely decaying response. By using these peak locations the

damped natural frequency is calculated, and by using the sets of peak values the damping ratio is

obtained by using the logarithmic decrement. The natural frequency and the damping ratio are

calculated at different peaks and then we take the average damped natural frequency. The natural

frequency can also be obtained by using the FFT plot. Again a waveform peak detector is used to

get the peak location of FFT plot. The peak location is itself the damped natural frequency of the

system. The undamped natural frequency can be calculated by using a formula given in equation

2.17. Refer section 2.9 and 2.10 for formulae and the procedure for experimental calculation of

the undamped natural frequency and the damping ratio.

Fig 2.21 Experimental calculation of natural frequency

7. RESULTS – The result compares of theoretical and experimental results.

Fig 2.22 comparison of theoretical and experimental results

8. TESTS OF USERS FOR ASSESSMENT OF THEIR LEARNING- After the successful

completion of the experiment, the computer program offers a test. User has to go through it and it

is basically a student performance evaluation technique about his/her learning and understanding

of the subject.

Fig 2.23 Evaluation test page

9. EXERCISES- An exercise page with sets of further questions is given, which are based on the

basic theoretical concepts of the experiment for further evaluation of the user.

Fig 2.24 Exercise page (small test to the users)

10. FEEDBACK- At the end of the program, there is a feedback section which asks for the quick

feedback about the performance and usefulness of the overall experiments, learning, navigational

aspect, feel of performing aspect, testing, etc.

Fig 2.25 Feed back page and end of the experiment

The total procedure is described as a flow chart in fig 2.26

Fig. 2.26 A flow chart for virtual the vibration lab and its program execution

Fig. 2.26 describes virtual vibration laboratory and the flow of execution of the program. Virtual vibration

lab has number of experiment and user has to select one of them at a time. Once a particular experiment is

selected, first of all he/she will go through the lab manual and then only starts the virtual program for

performing the virtual experiment. At the beginning of the experiment, after the brief introduction, the

virtual program offers a simple pre-experiment test to the user. If user qualifies the test he can precede

further otherwise program will be terminated and user needs to go through manual again. After qualifying

the pre-experiment test, program gives input options related to the experiment, the user selects a

combination of input configuration and proceed to the experiment for that configuration of the

experiment. The virtual experiment leads the display of experimental results and its analysis. The

experimental results are then compared with the theoretical analysis results. The user can repeat the

experiment with different set of input configuration. It leads to the comparison with respect to different

configurations chosen by the user so as to analyze effects of chosen parameters on the results. At the end

the virtual program, it offers the student an evaluation test. Finally, it takes the feedback regarding the

program execution, control, navigation, ease of use, learning, feel of performing experiment, and all other

related components. This user feedback is very important for the further improvement of the existing

virtual experiments and overall laboratory.

Fig. 2.27 A flow chart for the virtual experiment and its virtualization

Figure 2.27 explains the actual experiment, its virtualization and application by using the internet. The

experiment is performed with the help of input configuration and sensing instruments. For this actual

experiments need to be performed with all possible input configuration and all data are stored in the

database by using the measurement technique and measurement software. A virtual programming is done

in a sequence which follows the same sequence as in case of actual experiment. This developed graphical

computer program is published into the internet by using the internet publishing tools of the virtualization

software. This allows the virtual experiments to be accessible by the remote users through internet.

2.9 Calculation of damping ratio

The logarithmic decrement is defined as

1

ln nnf d

n

XT

X (1.13)

With

2

2 2

(1 )d d

nf nf

T (1.14)

so that

2

1

2ln

(1 )

n

n

X

X

or

22

2

1

(1 )2 / ln n

n

X

X

Hence, finally we get

2

1

1

2 / ln n

n

X

X

(1.15)

Example 2.2 Obtain the damping ratio from the two consecutive peaks of a freely decaying vibration

signal as given in table below.

Displacement (nX ) Displacement (

1nX )

23.8 unit 21.4 unit

We have,

2 2

1

1 10.012

23.82 / ln2 / ln

21.4n

n

X

X

2.10 Calculation of experimental natural frequency

To calculate the natural frequency of the cantilever beam experimentally, conduct the experiment with the

specified cantilever beam specimen. Record the data of time history (time versus displacement), and plot

the graph as shown in Fig 2.28. Obtain the free vibration response peak values and corresponding time

instants.

Figure 2.28 Variation of an under-damped free response with time

Let nX is the peak value of

thn peak and 1nX is peak value of next consecutive peak and

nT and 1nT are

the corresponding time respectively. The experimental damped natural frequency is given as

1

1

( )

d

nf

n n

fT T

(1.16)

and undamped natural frequency is given as

21

d

nf

nf

ff (1.17)

Example 2.3 Obtain the damped natural frequency from time instances of two consecutive peaks of a

freely decaying signal as given in table below.

Time (1T ) of the first peak Time (

2T ) of the second peak

302 ms 208 ms

Hence, the damped natural frequency is given as

1

10.638(0.302 0.208)

d

nff Hz

and undamped natural frequency is given as

2

10.63810.6388

1 0.012nff Hz

2.11 Discussions

Good agreement of the theoretically calculated natural frequency with the experimental one is found. The

present theoretical calculation is based on the cantilever beam end conditions (i.e., one end is fixed end),

in actual practice it may not be always the case because of flexibility in support that may affect the natural

frequency and the variation of free vibration response of cantilever beam with respect to time to an initial

disturbance may be observed

2.12 Precautions during Experimentations and Analyses

1. Fixed end condition of the cantilever beam could be ensured by properly gripping one end of the

beam.

2. The beam should be given initial disturbance such that the first mode (Fig. 2.1(b)) is excited, i.e.,

a small deflection of the free end of the beam.

3. Care should be taken that the cables of accelerometer should not affect the beam motion.

4. Initial displacement of the beam should be small so that linearity assumption holds true.

5. The end of the beam should be fixed rigidly so that cantilever end condition prevails in

experiment.

6. By considering all the precaution and by using the procedure step by step with proper

coordination with the subsystem , measuring instruments , data acquisition system and vibration

measuring software, the result can be improved.

7. User is suggested to use sensors and other measuring instruments with high sensitivity and

minimize the noise in measuring data, by ensuring these it minimizes the error and improve the

result.

8. It is also suggested to user to repeat the experiment several times to confirm the consistency of

results.

2.13 Questions

1. The theoretical natural frequency obtained is damped or undamped natural frequency?

2. The experimental natural frequency obtained is damped or undamped natural frequency?

3. What would be the effect on the natural frequency when a notch is cut on the cantilever beam?

4. Where the location of the notch would have effect on the natural frequency of the cantilever

beam?

5. Do you think the sensor weight would affect the natural frequency of the beam?

6. What is the affect of damping on the natural frequency? Could you tell whether the obtained

natural frequency is damped natural frequency or undamped one?

7. What is the relationship between the damped and undamped natural frequencies?

8. Do you think the natural frequency and the damping ratio would change if we consider

subsequent consecutive free vibration amplitudes instead of first two amplitudes?

9. Do you think the beam tightening at the support (i.e., the flexibility of support) would affect the

natural frequency of the system?

10. What type of free vibration signal you would expect if the beam is given initial disturbance other

than the free end (i.e., if the initial disturbance excites higher modes of vibration also)?

11. How many distinct natural frequencies can exist for an ‘N’ degree of freedom vibratory

system?

12. Is the frequency of a damped free vibration smaller or greater than the natural frequency of a

system? Why?

13. What is the basic difference between the free responses exhibited by an under damped

system and an over damped system?

14. For the system given in example 2.1, if we add a mass of 500k gm at free end then find out

the natural frequency of that system?

15. How many number of degree of freedom has the given system if it is considered as discrete

system with N number of masses in Fig 2.1?

16. A system has n mode shapes. Then how many numbers of natural frequencies the system has?

17. Do you think other boundary conditions (e.g., the simply supported beam, fixed-fixed, or free-

free beam) have same natural frequency as fixed-free case (i.e. cantilever case)? Obtain the

theoretical natural frequencies for such case of the present beam and compare with the present

one.

18. Damped free oscillation amplitude is observed to be reduced to 20% of its initial amplitude in 100

complete cycles. Estimate the damping ratio ς.

2.14 References

1. Meirovitch, L, 1967, Analytical methods in vibration, Ccollier-MacMillan Ltd., London.

2. Thomson, W.T., 2007, Theory of vibration with application, Kindersley Publishing, Inc., London.

3. Rao, J. S, and Gupta, K., Introductory Course on Theory and Practice of Mechanical Vibrations,

New Age International, New Delhi.

4. S.Timoshenko, D.H. Young, 1961, Strength of Material, Stanford University, California.

5. Wahab, MA, 2008, Dynamics and Vibration, John Wiley & Sons Ltd., Ghent University, UK.