Free Vibration of Single Degree of Freedom Systems

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write theoretical Background and Application in one page. The topic is Free Vibration of Single Degree of Freedom Systems and I am going to attach a lab file.

Faculty of Engineering and Applied Science
Mechanical Vibrations
MECE 3210U
Experiment # 2
Free Vibration of Single Degree of Freedom Systems
(Rotational and Torsional Systems)
Revised by
Dr. A. Mohany, N. Arafa
Winter 2014
Part (A): Bifilar Suspension System
Introduction
The bifilar suspension can determine the moment of inertia about an axis by suspending two
parallel cords of equal length through the mass centre of bodies, as shown in Figure 1. Angular
displacement of the body about the vertical axis through the mass centre G is by angle θ, which
is sensibly small.
Figure 1: Schematic of the bifilar suspension system
This equation of the angular motion is:

gb
which may be written as:
̈
gb

g
The motion is clearly simple harmonic and the period is:

g

gb

where I is moment of inertia about swing axis through the center of gravity ( g

).
Knowing the periodic time and the magnitudes of the various parameters, the radius of gyration
and therefore the value of the moment of inertia can be determined.
Apparatus
Figure 2 shows the apparatus and consists of a uniform rectangular bar B7 suspended by fine
wires from the small chucks. Drawing the two wires through the chucks and tightening alters the
lengths of the suspension. The bar is drilled at regular intervals along its length so that two 1.85
kg masses may be pegged at varying points along it.
Figure 2: The bifilar suspension system
Procedure
With the bar is suspended by the wires, adjust length L to a convenient size, and measure the
distance between the wires, b. Displace the bar through a small angle and measure the time taken
for 10 complete oscillations. From this, calculate the periodic time. Adjust the length of the
wires, L, and measure the time taken for a further 10 swings. Increase the inertia of the body by
placing two masses symmetrically on either side of the centreline distance apart, x, and repeating
the procedure for various values of L and the distance between the masses. Calculate the radius
of gyration of the system as previously outlined. Tabulate the results in Table 1.
Table 1: Results of the bifilar suspension system
Test
#
L
(m)
x
(m)
Time for 10
oscillations (s)

(s)

(m)
(kg.m2
)
1
2
3
4
Further Considerations
1. Some noteworthy points will have arisen as a result of performing this experiment.
Summaries your conclusions about this experiment and point out the sources of error that
maybe associated with it.
2. Compare the values of the moment of inertia for each case with the ones obtained
analytically.
Part (B): Torsional Oscillations of a Single Rotor
Introduction
This is an example of simple harmonic angular motion. The system is comprised of a rigid rotor
at one end of an elas ic shaf is ‘ orsional vibra ion’ ue o he twisting action along the shaft
axis. Analysis of this situation is analogous to the mass-spring system. The torsional deflection
angle ( ) replaces deflection x, and the stiffness, k, is now torsional stiffness of the shaft. The
polar moment of inertia of the rotor, I, replaces mass, M. The equation of motion is:
̈
which results in a simple harmonic motion. It can be shown that the time perio , , is:

where:
L = Effective length of the shaft;
G = Modulus of rigidity of the material of the shaft;
J = Polar moment of area of the shaft section.

Apparatus
For experiments on un-damped torsional vibrations, the inertia is provided by two heavy rotors,
cylindrical in shape, one 168 mm diameter the other 254 mm diameter. Figure 3 shows the
smaller diameter rotor, H2. The rotor mounts on a short axle, which fits in either of the vertical
members of the portal frame, and secures by a knurled knob. The rotor is fitted with a chuck
designed to accept shafts of different diameter. An identical chuck rigidly clamps the shaft,
which is an integral part of a bracket (I1). This is at the same height as the flywheel chuck and
adjustable, relative to the base of the portal frame. Three steel test shafts are supplied (3.18, 4.76
and 6.35 mm in diameter, each 965 mm long).
Figure 3: The single rotor setup showing the added weights (H2)
By bolting two pairs of steel arms to each side and attaching heavy masses at each end, we can
increase the inertia of the smaller rotor. Two pairs of masses are available of approximately 1800
g and 3200 g.
Procedure
Pass the shaft through the bracket centre hole (so that it enters the chuck on the flywheel and
then tighten). Move the bracket along the slotted base until the distance between the jaws of the
chuck corresponds to the required length L. Tighten the chuck on the bracket. Ensure that the
jaws securely grip the shaft. Displace the rotor (flywheel) angularly and record the time for 10
oscillations. Vary the distance between the chucks in suitable increments by sliding the bracket,
and record and tabulate the values of periodic time for the various shaft lengths.
Test # L
(m)
Time for 10
oscillations (s)
τ
(s)
τ
2
(s2
)
1
2
3
4
Further Consideration
1. Plo a graph of 2
against L. determine the moment of inertia I from the slope.
References:
Please note that the experimental is reproduced with express permission of TQ Education and
Training Limited.
“TM16, Universal Vibration, TQ Education and Training Limited.”


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