Forced Vibration of Multi-dof Cantilever Beam

EXPERIMENT 5

FORCED VIBRATIONS OF CANTILEVER BEAM

(CONTINUOUS SYSTEM)

5.1 Objective of the Experiment

The aim of the experiment is to analyze the forced vibrations of the continuous cantilever beam, the

phenomena of resonances, the phase of the vibration signal and to obtain the fundamental natural

frequency and damping ratio of the system, and compare the results with theoretically calculated values.

The basic aim of these experiments is to provide a feel of actual experiments

along with learning of basic components while performing the virtual experiments.

5.2 Basic Definitions

Forced vibration: 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. 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).

If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of

resonance is encountered and dangerously large oscillations may result, which results in failure of major

structures, i.e., bridges, buildings, or airplane wings etc. Hence, the natural frequency of the system is the

frequency at which the resonance occurs. At the point of resonance the displacement of the system is a

maximum.

Thus calculation of natural frequencies is of major importance in the study of vibrations. Because of

friction & other resistances vibrating systems are subjected to damping to some degree due to dissipation

of energy. 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 eigen function in the continuous system. For every natural frequency there would be a

corresponding eigen function. 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 for higher modes the number of

nodes increases. The mode shape changes for different boundary conditions of the beam.

5.3 Mathematical analysis

Fixed support

Free end

Beam

Figure 5.1 (a) A cantilever beam

Exiciter

L1

Figure 5.1 (b) The beam under forced vibrations

Fig 5.1(a) is showing a cantilever beam which is fixed at one end and other end is fixed, having

rectangular cross-section.

Fig 5.1(b) is showing the cantilever beam which is subjected to forced vibration. An exciter is

used to give excitation to the system. The exciter is capable to generate different type of forcing

signal e.g. sine, swept sine, rectangular, triangular etc.

Continuous Beam Model: In actual case, the beam is a continuous system, i.e. the mass along

with the stiffness is distributed throughout the beam. The equation of motion in this case will be

(Meirovitch, 1967)

2 2 2

12 2 2

( , ) ( , )( ) ( ) ( ) ( )

y x t y x tEI x m x f t x L

x x t

(5.1)

where, E is the modulus of rigidity of beam material, I is the moment of inertia of the beam cross-section,

y(x, t) is displacement in y direction at distance x from fixed end, m is the mass per unit length,

( )m A x , is the material density, A(x) is the area of cross-section of the beam, ( )f t is the forced

applied to the system at x = L1 .

Free Vibration Solution: We have following boundary conditions for a cantilever beam (Fig. 5.1)

at ( )

0, 0, 0dY x

x Y xdx

(5.3)

and

at

2 3

2 3

( ) ( ), 0, 0d Y x d Y x

x ldx dx

(5.4)

For a uniform beam under free vibration from equation (5.1), we get

44

4

( )( ) 0

d Y xY x

dx (5.5)

with

24 m

EI

A closed form of the circular natural frequency nf , from above equation of motion for first mode can

be written as (Meirovitch, 1967)

2

41.875nf

EI

AL

rad/sec (5.6)

Second natural frequency

2

2 44.694nf

EI

AL

Third natural frequency

2

3 47.855nf

EI

AL

For different boundary conditions these expressions would be different. The natural frequency is related

with the circular natural frequency as

2

nf

nff

Hz (5.7)

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

4

64I d

(5.8)

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

3

12

bdI

(5.9)

where b and d are the breadth and width of the beam cross-section as shown in the figure.

b

d

Fig 5.3 The first three undamped natural frequencies and mode shape of cantilever beam.

5.4 Experimental setup

Data acquisition

system

Accelerometer

Cantilever beam

Fixed end

Free end

Exciter

Controller

Figure 5.4 An experimental setup for the forced vibration of a cantilever beam

The experimental setup consists of a cantilever beam (Picture), an exciter (Picture),

controller/amplifier(Picture), two transducers (e.g., accelerometer (Picture)and laser vibrometer), a data-

acquisition system (Picture)and a computer with signal display and processing software The schematic of

the experimental setup is shown in Fig. 5.4. Different types of beam materials and its properties are listed

in Table 5.1. Different combinations of beam geometries for each of the beam material are summarized in

Table 5.1

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

encodes it in 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, etc.) it display on the computer screen by using

analysis software.

When we perform the experiment, the exciter keep the excitation force in such a way that the

displacement increases almost linearly Fig. 5.10(b) . Because of this reason we get continuously

increasing displacement-frequency graph Fig. 5.11(a) but the force- frequency graph Fig. 5.10(a)

decreases up to the resonance than it increases. The controller of the exciter minimizes the force

amplitude so that at resonance large oscillations does not take place.

Table 5.1 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 5.2 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

5.5 Photos of experimental setup

Figure 5.5 Experimental setup of a cantilever beam for forced vibration.

Figure 5.5 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

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

response of the cantilever beam, it is very important to ensure that clamp is tightened properly, otherwise

it may not give fixed end conditions and relative sliding may take place.

Fig 5.6 Exciter

Exciter is used to give desired excitation to the beam. The power is given to the exciter by

controller which is connected with a computer to select the excitation parameter. The different

type of excitation can be genrated by exciter e.g. sine, swept sine , rectengular, tringilar etc.

In case of forced vibration we use swept sine force signal, in which user have to select the initial

and final frequency and the sweep rate.

Figure 5.7 A close view of the fixed end of the cantilever beam

Figure 5.8 A close view of an accelerometer mounted on the beam at the free end

An accelerometer (Fig. 5.8) is a time-dependent (dynamic) vibration measuring device. It is a contacting

type transducer. It is a transducer, which converts the acceleration of vibration into equivalent voltage

signal, and sends it to the data acquisition system (Fig. 5.9).

Figure 5.9 Data acquisition system

Data acquisition system receives voltage signal from the accelerometer and calibrate the data into

equivalent accelerometer scale through its sensitivity (give value with units?) and send it to computer

where by using a software these data can be analyzed as time history (displacement-time) and frequency

domain (i.e., using FFT).

Figure 5.10 time-force and time-response graph

(a) Excitation force against Time

(b) Response against Time

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

mechanical vibration data and send it to computer where by using the vibration measurement software, it

can be plotted as shown in above Figure and can be used for further analysis.

6.6 Experimental procedure

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

(e.g., accelerometer or laser vibrometer).

2. Clamp one end of the beam as the cantilever beam support (see Fig. ?).

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

the forced vibration response (acceleration).

4. Place the exciter stinger on the beam slightly offset from the middle of the beam (Fig.?). Ensure

the connection of the exciter with the controller and the level of input power.

5. Connect another accelerometer at the exciter to measure the forced vibration response

(acceleration) away from exciter (Picture/Fig. ?) location.

6. Make a proper connection of accelerometer with data acquisition system and with computer to

capture the vibration data.

7. During setting of the swept-sine parameter make sure that in the vibration measurement software

the time duration should be greater than the total time of excitation .

8. Make the settings to generate swept-sine signal from exciter.

9. Start the experiment by giving force signal to the exciter and allow the beam to forced vibrate.

10. Record all the data obtained from the chosen transducer in the form of variation of the vibration

response with time.

11. Repeat the experiments for five to ten times to check the repeatability of the experimentation (i.e.,

vibration data).

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

13. Record the whole set of data in a data base for further processing and analysis.

5.7 Experimental Identification of Resonance Conditions

It is relatively easy to obtain the resonance condition by observing the behavior of the system at

varying frequency (e.g., swept-sine). During the experiment the frequency of excitation force is

swept between a chosen range of frequency with a fixed time-rate. The frequency range is

selected in such a way so that the natural frequency of system lies in within the frequency range

and the swept rate is selected by several trials of experiments (i.e., increment of 2 Hz to 3 Hz per

sec). As we know that in case of forced vibration the system vibrates with the same frequency as

that of exciter. At the point of resonance we get sudden increase in the response, the frequency of

excitation corresponding to the resonance is the natural frequency of the system. but in our case it

is linear because of property of exciter to generate force signal in such a manner to get linearly

increasing displacement.

(a)

(b)

(c)

Fig. 5.11 (a) Displacement against frequency

(b) magnification factor against frequency ratio.

(c) change phase value between 0 and 180 degree at resonance

At the resonance the phase difference between the force and response changes by 180 degree,

which also implies the indication of resonance and corresponding frequency is the natural frequency of

the system.

5.8 Calculation of Damping Ratio

The damping ratio can be calculated with the help of phase difference between the force and the response

at any excitation frequency especially which is close to the natural frequency of the system (Fig. 5.11(c)).

The damping ratio can be calculated as follows. We have

2

2tan

1

(5.10)

where, is phase difference between the force and the response, and

nf

The above equation can be rewritten as

2(1 )tan

2

(5.11)

Damping ratio can be calculated from above formula directly once we know the natural frequency, and

phase difference at a particular excitation frequency.

Example 5.2:- From fig 5.11(b) and also from fig 5.11(c) the natural frequency is 13.2 Hz, let =8

Hz and at that point phase difference between force and response is 10 degree.

The frequency ratio is

80.615

13

Hence, the damping ratio is

2(1 0.615 )tan10 0.0890

2 0.615

5.9 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. 5.7 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 5.8 Overview of measurement based virtual experiments

5.10 Discussions

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

correction for the mass of the sensor will improve the correlation better. 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. 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. User is suggested to use sensors and other measuring instruments with high sensitivity

and minimize the noise in measuring data, it minimize the error and improve the result. User is also

suggested to repeat the experiment with patience.

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

3. The mass of the accelerometer should be small as compared to beam mass.

4. The power of excitation should be proper, too less power may cause of insufficient vibration and

too high power may cause of damage of the system.

5. Select the sweep rate which should not either very less or very high. Usually sweep rate 2 or 3

Hz/sec is suggested.

6. Placement of sensors and exciter stinger, should not be at node.

5.12 Questions

1. What is the difference between the free and forced vibrations?

2. In a damped vibratory system, what will be the phase difference between the input force and

the output displacement when the excitation frequency is exactly same as the natural

frequency of the system?

3. What will be the phase difference between the input force and the output displacement when

the excitation frequency is less than the natural frequency of the system, and when its is

greater than the natural frequency of the system?

4. How would you determine the natural frequency of a vibratory system in case of forced

vibrations.

5. Give some examples based on the daily-life observations where the harmonic force

generates?

6. Give some practical examples where we can observe resonance.

7. How many distinct resonance conditions can exist for an ‘N’ degree of freedom system?

5.13 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.