Free Vibration Experiment
Interesting fact: Free vibrations of skyscrapers were observed in Japan and in
Taiwan during the earthquake in Tohoku, Japan on 11th March, 2011, with some
buildings swaying for a duration of 10 to 30 minutes due to their low inherent
damping. Vibration spectra recorded by seismic transducers in TAIPEI 101, the
second tallest building in the world clearly showed the fundamental mode of
vibration at 0.15 Hz, as well as higher mode vibrations. Magnification factors of
ground accelerations for the fundamental mode vibrations in the building to those at
a nearby borehole station were found to be as high as 110 and 146, respectively, on
the 74th and 90th floors above ground. The frequency content of accelerograms
recorded below the ground was found to be in a frequency band of 0.015–0.1 Hz,
clearly proving it was free vibration and not forced excitation due to ground motion
which caused high vibration in the TAIPEI 101 skyscraper.
Aim:- To determine the first natural frequency and damping ratio of a cantilever
beam by free vibration test.
Equipment:- Aluminium beam, non contact electromagnetic velocity transducer with
ferromagnetic strip, storage oscilloscope/ recorder, rigid clamp.
Theory:- Free vibration takes place when an elastic system not aided upon by any
steady excitation force is disturbed from its mean equilibrium position. When a
system is allowed to vibrate freely, vibrations die out gradually over a few cycles of
motion because the energy imparted to the system initially is dissipated during the
motion. The damping in most mechanical systems is so small that it practically has no
influence on the natural frequency of the system. The response of a single degree of
freedom damped system is as shown in Figure 1.
Figure 1 Single degree of freedom damped system response
The displacement from the mean equilibrium position at any point of time is given by
the equation
x(t) = Xe
−ζω n t
sin( 1− ζ 2 ω n t + φ )
(1)
where 𝑥(𝑡) = amplitude of vibration at any point of time.
X = Maximum amplitude of vibration
!
ζ = Damping ratio =!
!
ω n = Natural frequency without damping
2
ω d = Natural frequency with damping= ω n 1− ζ
t = instant of time considered
φ = phase angle
𝐶 = Damping coefficient
𝐶" = Critical damping coefficient = 2mwn
m = mass of the system
The time period of vibration (with damping) is given by
Td =
2π
ωd
(2)
From this equation, the natural frequency can be calculated. For most mechanical
systems ζ lies in the range 0.001 to 0.05 and therefore ω d » ω n and hence while
calculating the natural frequency we can neglect damping. Considering two
successive amplitudes X1 and X2 at times t1 and t1+Td we have
X 1 = Xe
−ζω n t1
X 2 = Xe
sin( 1− ζ 2 ω n t1 + φ )
−ζω n (t1+Td )
sin( 1− ζ 2 ω n (t1 + Td ) + φ )
(3)
(4)
Since they are exponential functions, it is convenient to define a logarithmic
decrement in two successive amplitudes as
!X $
2πζ
δ = ln ## 1 && = ζω nTd =
" X2 %
1− ζ 2
δ ≈ 2πζ
(5)
The logarithmic decrement can also be defined as d =
1 é X1 ù
ln ê
ú
n ë X n +1 û
(6)
Test Setup and Procedure:One end of the test beam is firmly clamped between two massive steel blocks and the
other end is free to vibrate as shown in Fig. 2. The electromagnetic pickup is fixed
close to the free end of the beam. The rap test consists of simply impacting the beam
by hand or by an impulse hammer and storing the response from the vibration pickup
on the oscilloscope. From the decaying vibration response, the period of oscillation is
noted and the natural frequency obtained. The damping ratio may be obtained from
eqns. (5) and (6) by noting amplitudes X1 and Xn+1 which are n cycles apart.
Oscilloscope
Electromagnetic
pickup
Figure 2 Test Setup
Transducers used
The principle of operation of an electromagnetic transducer is explained below. This
particular transducer has been suggested for this experiment since it is a very
inexpensive transducer of the non-contact type and does not need any additional
signal conditioning amplifier. Its output voltage may be straightaway fed to a storage
oscilloscope or recorder. Alternately a non-contact capacitance transducer may be
used in conjunction with a displacement measuring unit. One could also use a
miniature piezoelectric transducer with minimal mass loading along with a charge
amplifier. If one is interested only in the natural frequency and damping ratio and not
in the amplitude of vibration a simple strain gauge pasted at the free end will serve the
purpose. This will have to be used in conjunction with a carrier frequency amplifier.
Electromagnetic transducer: This is a very inexpensive and useful vibration pick-up
of the non-contact type. It is an active transducer. In the electromagnetic transducer, a
coil is wound directly on the core of a permanent magnet as shown in Fig. 3(a).
Figure 3(b) shows a photograph of the same. When a ferromagnetic body placed
opposite to one of the poles of the magnet is moved with respect to the magnet, a
change in the flux f linking the coil is obtained. This causes a voltage proportional to
df / dt to be generated in the coil. The voltage may be expressed as
df
dt
(7a)
=k
df dy
dy dt
(7b)
=k
df d ( y0 + x )
dy
dt
(7c)
=k
df dx
dy dt
(7d)
e0 = k
where y is the instantaneous gap, y0 the average gap between the face of the magnet
and the ferromagnetic body and k is a proportionality constant. As can be seen from
Eqs. (7a) to (7d), the induced voltage is directly proportional not only to the velocity
of vibration, but is also dependent on the gap y . When the pick-up is located near a
vibrating ferromagnetic body, a voltage proportional to the velocity of vibration will
be induced and as the average distance increases, sensitivity reduces. Thus df / dy
forms the sensitivity factor indicating that the sensitivity is inversely proportional to
the gap. In principle, this method can be employed to measure absolute or relative
velocities. The air gap should be at least five times the vibration amplitude for
distortion to be less than 2%.
Ferrous Plate
x
y
y0
Magnet
Coil C
Fig. 3 Electromagnetic pick-up: (a) schematic, (b) photograph (Courtesy of Brüel &
Kjaer)
Analytical Validation:The experimental results obtained may be verified from the closed form expression
for the natural frequencies of beams and is given by
ω n = ( βn l) 2
EI
ρl 4
(8)
Here r is the mass per unit length of the beam, EI is the flexural rigidity, l is the
length, n is the mode number and βn depends on the boundary conditions of the beam.
For a beam in cantilever configuration, ( β1l) 2 = 3.52 and ( β1l) 2 = 22.0.
Results may be tabulated as shown in Table 1.
Table 1: Results of free vibration experiment
Questions
1. Will the natural frequencies and damping ratios change for other boundary
conditions?
2. How would you get damping ratios in the case of structural damping?
3. What would be the discrepancy in damping ratio got from the exact and
approximate formulae in Eqn. (5)?
4. Why is the transducer kept near the free end?