1

UNIVERSITI TUNKU ABDUL RAHMAN

Faculty : Engineering & Science Unit Code : UEME1143

Course : Bachelor of Engineering

(Hons) Mechanical

Engineering

Unit Title : Dynamics

Year/

Semester

:

Year 1/ Semester 2 Lecturer

:

Session :

Experiment 2: Free Vibration of a Cantilever

Objective

The purpose of this experiment is to determine the natural frequency of a cantilever beam and

study both undamped and damped free vibration motion of a cantilever beam.

Principles

Free Vibration

If a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is

known as free vibration. No external force acts on the system. The oscillation of a simple

cantilever beam is an example of free vibration as shown in Figure 1.

The simple cantilever beam shown in Figure 1 can be modeled as a mass-spring system where

the governing equation of motion is given by

0or2

=+−= xxkxxm nω&&&& … (1.1)

where m is the mass of the system and k is the stiffness of the system

Figure 1

2

ωn is known as the natural circular frequency of the system and is given by

ωn = m

k

Equation (1) is a homogeneous second-order equation linear differential equation, has the

following general solution:

txtx

x nn

n

ωωω

cos)0(sin)0(

+=&

… (1.2)

The natural period of the oscillation is established from ωnτ = 2π or

k

mπτ 2= … (1.3)

The natural frequency of the system is

m

kfn

πτ 2

11== … (1.4)

Viscously Damped Vibration

Every mechanical system possesses some inherent degree of friction, which dissipates

mechanical energy. Precise mathematical models of the dissipative friction forces are usually

complex. Viscous damping force can be expressed by

xcFd&= … (1.5)

The equation of motion of a free-damped vibration system is given as 0=++ kxxcxm &&& .

The general solution is given as

tt nn eAeAxωζζωζζ )1(

2

)1(

1

22 −−−−+− += …(1.6)

The radicand (ζ2

– 1) may be positive, negative or zero, giving rise to three categories of damped

motion: ζ > 1 (over-damped, Figure 2), ζ = 1 (critically damped, Figure 2) and ζ < 1 (under-

damped, Figure 3).

Figure 2

3

Figure 3

The frequency of damped vibration ωd is given by

nd ωζω 21−= … (1.9)

Natural Frequency of A Cantilever Beam

Figure 4

The maximum deflection of the cantilever beam under a concentrated end force P is given by

k

P

EI

PLy ==

3

3

max … (1.10)

Therefore the stiffness of the beam is given by 3

3

L

EIk = … (1.11)

Where

L = length of the beam

I = moment of inertia, for rectangular area, 12

3bh

I =

b = width of the beam

h = height of the beam

E = modulus of elasticity, for aluminum, E = 70GPa

tnCeζω−−

tnCeζω−

4

The static deflection of a cantilever beam y(x) is given as

)3(2

)3(6

)( 32

3

max

2

xLxL

yxL

EI

Pxxy −=−=

Expressed as velocity variation, gives

)3(2

)( 32

3

max xLxL

yxy −=

&&

The maximum kinetic energy of the beam itself is given by

( )}{ 2

max

2

0

max140

33

2

1

2

1ymdxxy

l

mT

L

&&

== ∫

Compare it with the kinetic energy equation Tmax = ½ meqv2 and therefore the equivalent mass of

the beam is meq = (33 / 140) m … (1.12)

If a damper is added to the free end of the cantilever beam, the total equivalent mass is given by

meq = (33 / 140) m + mdamper … (1.13)

Apparatus and Materials

1. Cantilever beam apparatus

Modulus of elasticity of aluminum (E) : = 70 GPa

Dimension of the cantilever beam : = 926 x 19 x 6 mm

Mass of the cantilever beam : = 295 g

Mass of the damper : = 122 g

2. Strain gauge

3. Strain recorder

4. Viscous damper

Experiment Procedures

1. Switch on the computer and the strain recorder.

2. Start the strain recorder application software by double click on the “DC104REng”

shortcut icon on the computer desktop.

Strain

Recorder Strain gauge

Cantilever beam

PC USB

Figure 5

5

3. Figure 5 shows the experiment setup. Please refer to the operational manual for the

operation of the strain recorder and the recorder application software.

4. Remove the viscous damper if it is attached to the beam.

5. Displace and hold the beam, ymax (refer to Figure 4) by -20 mm, -15 mm, -10 mm, -5 mm,

0 mm, 5 mm, 10 mm, 15 mm and 20 mm and record the strain recorder reading for each

displacement value manually from the “Numerical Monitor” screen of the application

software.

6. Obtain the relationship of the displacement (of the free end of the beam) and the strain

recorder reading by plotting an appropriate graph using a spreadsheet.

7. Displace the beam by 30 mm and leave the beam to vibrate on its own. Record the strain

recorder reading by clicking on the “Play” and “Stop” button.

8. Retrieve the recorded file by clicking on the “Read USB” button.

9. Plot the graph of the beam displacement versus the time, t.

10. Repeat the experiment using beam displacement of 50 mm.

11. Connect the viscous damper. Repeat steps 7 and 10 using beam displacement of 30 mm

and 50 mm, respectively.

Discussion

1. Calculate the theoretical natural frequency of the cantilever beam. Comment on the

difference in values between experimental natural frequency and theoretical natural

frequency.

2. For the free-damped vibration in Step 5, calculate the damped period, damped natural

frequency and the damping ratio of the system.

3. Comment on the accuracy of the experiment for amplitude = 30 mm and amplitude

= 50mm in both the free-undamped and free-damped cases.

4. Comment on the accuracy of experimental results if the strain gauge is mounted on the

other end of the cantilever beam (refer to Figure 5).

5. Discuss any other findings from the experiment.