DOM Problem based learning (PBL)
Free Vibration of Single-Degree of Freedom Systems
P1. A machine of mass m= 500 kg is mounted on a simply supported steel beam of length l= 2m having a
rectangular cross section (depth= 0.1 m, width = 1.2 m) and Young’s modulus E=2.06 x 10 11 N/m2 .To
reduce the vertical deflection of the beam, a spring of stiffness k is attached at mid-span, as shown in Fig.
1. Determine the value of k needed to reduce the deflection of the beam by,
a) 25 percent of its original value.
b) 50 percent of its original value.
c) 75 percent of its original value.
Assume that the mass of the beam is negligible.
Figure:1
Free Vibration of an Undamped Translational System
P2. A loaded mine cart, weighing 2300 kg, is being lifted by a frictionless pulley and a wire rope, as
shown in Fig. 2. Find the natural frequency of vibration of the cart in the given position.
Figure:2
P3. A sledgehammer strikes an anvil with a velocity of 16 m/sec (Fig. 3). The hammer and the anvil
weigh 5.5 kg and 45 kg, respectively. The anvil is supported on four springs, each of stiffness k= 10000
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
N/mm. Find the resulting motion of the anvil (a) if the hammer remains in contact with the anvil and (b) if
the hammer does not remain in contact with the anvil after the initial impact.
Figure:3
P4. Derive the expression for the natural frequency of the system shown in Fig. 4. Note that the load W is
applied at the tip of beam 1 and midpoint of beam 2.
Figure:4
P5. A scissors jack is used to lift a load W. The links of the jack are rigid and the collars can slide freely
on the shaft against the springs of stiffness’s k 1 and k 2 (see Fig. 5). Find the natural frequency of vibration
of the weight in the vertical direction.
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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Figure:5
P6. A bungee jumper weighing 72 kg ties one end of an elastic rope of length 200 ft and stiffness 115
kg/mm to a bridge and the other end to himself and jumps from the bridge (Fig. 6). Assuming the bridge
to be rigid, determine the vibratory motion of the jumper about his static equilibrium position.
Figure:6
P7. The crate, of mass 250 kg, hanging from a helicopter (shown in Fig. 7(a)) can be modeled as shown
in Fig. 7 (b). The rotor blades of the helicopter rotate at 300 rpm. Find the diameter of the steel cables so
that the natural frequency of vibration of the crate is at least twice the frequency of the rotor blades.
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
Figure:7
P8. A TV antenna tower is braced by four cables, as shown in Fig. 8. Each cable is under tension and is
made of steel with a cross-sectional area of 0.5 inch2 .The antenna tower can be modeled as a steel beam
of square section of side 1 in. for estimating its mass and stiffness. Find the tower s natural frequency of
bending vibration about the y-axis.
Figure:8
P9. An oil drum of diameter 1 m and a mass of 500 kg floats in a bath of salt water of density ρw = 1050
kg/m3. Considering small displacements of the drum in the vertical direction (x), determine the natural
frequency of vibration of the system.
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
P10. A steel hollow cylindrical post is welded to a steel rectangular traffic sign as shown in Fig.10 with
the following data:
Dimensions: l = 2 m, r0 = 0.050 m, ri = 0.045 m, b = 0.75 m, d = 0.40 m, t = 0.005 m;
Material properties: ρ (specific weight) = 76.50 kN/m3, E = 207 GPa, G = 79.3 GPa
Find the natural frequencies of the system in transverse vibration in the yz- and xz-planes by considering
the masses of both the post and the sign.
Hint: Consider the post as a cantilever beam in transverse vibration in the appropriate plane.
Figure:10
Free Vibration of an Undamped Torsional System
P11.A heavy ring of mass moment of inertia 1.0 kg-m2 is attached at the end of a two-layered hollow
shaft of length 2 m (Fig. 11). If the two layers of the shaft are made of steel and brass, determine the
natural time period of torsional vibration of the heavy ring.
Figure:11
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
Free Vibration with Viscous Damping
P12. A railroad car of mass 2,000 kg traveling at a velocity v = 10 m/s is stopped at the end of the tracks
by a spring-damper system, as shown in Fig. 12. If the stiffness of the spring is k = 80 N/mm and the
damping constant is c = 20 N-s/mm, determine (a) the maximum displacement of the car after engaging
the springs and damper and (b) the time taken to reach the maximum displacement.
Figure:12
P13.A boy riding a bicycle can be modeled as a spring-mass-damper system with an equivalent weight,
stiffness, and damping constant of 800 N, 50,000 N/m, and 1,000 N-s/m, respectively. The differential
setting of the concrete blocks on the road caused the level surface to decrease suddenly, as indicated in
Fig. 13. If the speed of the bicycle is 5 m/s (18 km/hr), determine the displacement of the boy in the
vertical direction. Assume that the bicycle is free of vertical vibration before encountering the step change
in the vertical displacement.
Figure:13
P14. Figure 14, shows a uniform rigid bar of mass m and length l, pivoted at one end (point O) and
carrying a circular disk of mass M and mass moment of inertia J (about its rotational axis) at the other end
(point P). The circular disk is connected to a spring of stiffness k and a viscous damper of damping
constant c as indicated.
a. Derive the equation of motion of the system for small angular displacements of the rigid bar about the
pivot point O and express it in the form:
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
b. Derive conditions corresponding to the stable, unstable, and marginally stable behavior of the system.
Figure:14
P15. The mass of a spring-mass system vibrates on a dry surface inclined at 300 to the horizontal as
shown in Fig. 15.
a. Derive the equation of motion.
b. Find the response of the system for the following data:
m = 20 kg, k = 1,000 N/m, μ = 0.1, x 0 = 0.1 m, ̇
5 m/s.
Figure:15
Harmonically Excited Vibration
P16. A three-bladed wind turbine (Fig. 16) has a small unbalanced mass m located at a radius r in the
plane of the blades. The blades are located from the central vertical (y) axis at a distance R and rotate at an
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
angular velocity of ω. If the supporting truss can be modeled as a hollow steel shaft of outer diameter 0.1
m and inner diameter 0.08 m, determine the maximum stresses developed at the base of the support (point
A). The mass moment of inertia of the turbine system about the vertical (y) axis is J0 .Assume R = 0.5 m,
m = 0.1 kg, r = 0.1 m, J 0 = 100 kg-m2 , h = 8 m, and ω = 31.416 rad/s.
Figure:16
P17. In the cam-follower system shown in Fig. 17, the rotation of the cam imparts a vertical motion to the
follower. The pushrod, which acts as a spring, has been compressed by an amount x0 before assembly.
Determine the following: (a) equation of motion of the follower, including the gravitational force; (b)
force exerted on the follower by the cam; and (c) conditions under which the follower loses contact with
the cam.
Figure:17
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DOM Problem based learning (PBL)
P18. A video camera, of mass 2.0 kg, is mounted on the top of a bank building for surveillance. The video
camera is fixed at one end of a tubular aluminum rod whose other end is fixed to the building as shown in
Fig. 18. The wind-induced force acting on the video camera, f(t) , is found to be harmonic with f(t) = 25
cos(75.3984t) N. Determine the cross-sectional dimensions of the aluminum tube if the maximum
amplitude of vibration of the video camera is to be limited to 0.005 m.
Figure:18
P19. The landing gear of an airplane can be idealized as the spring-mass-damper system shown in Fig. 19.
If the runway surface is described y(t) = y0 cos ωt, Determine the values of k and c that limit the
amplitude of vibration of the airplane (x) to 0.1 m. Assume m = 2000 kg, y0 = 0.2 m and ω = 157.08
rad/s.
Figure:19
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
P20. A single-cylinder air compressor of mass 100 kg is mounted on rubber mounts, as shown in Fig. 20.
The stiffness and damping constants of the rubber mounts are given by 10 6 N/m and 2000 N-s/m,
respectively. If the unbalance of the compressor is equivalent to a mass 0.1 kg located at the end of the
crank (point A), determine the response of the compressor at a crank speed of 3000 rpm. Assume r = 10
cm and l = 40 cm.
Figure:20
Two-Degree-of-Freedom Systems
P21. A machine tool, having a mass of m = 1000 kg and a mass moment of inertia J0 = 300 kg-m2, is
supported on elastic supports, as shown in Fig. 21. If the stiffness’s of the supports are given by k1 = 2000
N/mm and k2= 3000 N/mm, and the supports are located at l1 = 0.8 m and l2= 0.5 m, find the natural
frequencies and mode shapes of the machine tool.
Figure:21
P22. An overhead traveling crane can be modeled as indicated in Fig. 22. Assuming that the girder has a
span of 40 m, an area moment of inertia (I) of 0.02 m4,and a modulus of elasticity (E) of 2.06 x 1011 N/m2
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
the trolley has a mass (m1) of 1000 kg, the load being lifted has a mass (m2) of 5000 kg, and the cable
through which the mass (m2) is lifted has a stiffness (k) of 3.0 x 105 N/m, determine the natural
frequencies and mode shapes of the system.
Figure:22
P23. The drilling machine shown in Fig. 23 (a) can be modeled as a two-degree-of-freedom system as
indicated in Fig. 23(b). Since a transverse force applied to mass m1 or mass m2 causes both the masses to
deflect, the system exhibits elastic coupling. The bending stiffnesses of the column are given by (see
Section 6.4 for the definition of stiffness influence coefficients)
Determine the natural frequencies of the drilling machine.
Figure:23
P24. One of the wheels and leaf springs of an automobile, traveling over a rough road as shown in Fig.
24.The automobile has a mass of m1 = 1000 kg and the leaf springs have a total stiffness of k 1 = 400
kN/m. The wheels and axles have a mass of m2 = 300 kg and the tires have a stiffness of k 2 = 500 kN/m.
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
If the road surface varies sinusoidally with an amplitude of Y = 0.1 m and a period of l = 6 m, find the
critical velocities of the automobile.
Figure:24
P25. A hoisting drum, having a weight W1 is mounted at the end of a steel cantilever beam of thickness t,
width a, and length b, as shown in Fig.25. The wire rope is made of steel and has a diameter of d and a
suspended length of l. If the load hanging at the end of the rope is W2, derive expressions for the natural
frequencies of the system.
Figure:25
Batch
Problems to solve
Batch
Problems to solve
Batch
Problems to solve
T2
P16 or P21.
T7
P18 or P22.
T15
P20 or P24.
T3
P13 or P17.
T8
P19 or P23.
T16
P14 or P25.
References:
1. Mechanical Vibrations, Singiresu S. Rao, 5th Edition, Prentice Hall.
2. Solutions manual, Mechanical Vibrations, Singiresu S. Rao, 5th Edition, Prentice Hall.
Prepared by: Prof. Ganesh D. Shrigandhi, School of Mechanical Engineering, MIT WPU, Pune.
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DOM Problem based learning (PBL)
MATLAB Programs:
T2 Batch:
Example E1: Write a MATLAB script for plotting
(a) the non-dimensional response magnitude for a system with harmonically moving base shown in
Fig. E1.
(b) the response phase angle for system with harmonically moving base.
Fig. E1. Single degree of freedom system w ith m oving base
T3 Batch:
Example E2: Write MATLAB script for plotting the magnitude of the frequency response of a system
with rotating unbalanced masses as shown in Fig. E2.
Fig. E2. Single degree of freedom system w ith rotating eccentric m ass
T7 Batch:
Example E3: A simplified model of an automobile suspension system is shown in Fig. E3 as a two
degree of freedom system. Write a MATLAB script to determine the natural frequencies of this model.
Fig. E3. Sim plified m odel of an autom obile
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DOM Problem based learning (PBL)
T8 Batch:
Example E4: Two gears A and B in mesh are mounted on two uniform circular shafts of equal stiffness
GJ/L. If the gear A is subjected to a torque M0cosωt, derive an expression for angular motion of B.
Assume the radius ratio as: RA/RB = n. Here L is length of each shaft. Write a MATLAB script to plot the
response.
T15 Batch:
Example E5: A two story building is undergoing a horizontal motion y(t) = Y0.sinωt. Derive expression
for displacement of second floor. Write MATLAB script to plot the response. Assume appropriate values
of stiffness and mass of the system.
Equations of motion for building can be written as:
T16 Batch:
Example E6: An analytical expression for the response of an damped single degree of freedom system
(Fig. E6) to given initial displacement and velocity is given by,
Where C and φ represent the amplitude and phase angle of the response, respectively having the values.
Fig. E6.
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DOM Problem based learning (PBL)
Plot the response of the system using MATLAB for ωn = 5rad/s, ζ = 0.05, 0.1, 0.2 subjected to the initial
conditions x(0) = 0, x(0) = v0 = 60 cm/s.
Reference:
1. MATLAB An Introduction with Applications, Rao V. Dukkipati, New Age International (P) Ltd.,
Publishers, pp No.549-645.
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