Contact Course AS2080 Vibrations: Problem Set 03
Topic:- Two-Degree-Of-Freedom Systems
References:
[1] Mechanical Vibrations (Fifth Edition) by Singiresu S. Rao
[2] Elements of Vibration Analysis (Second Edition) by Leonard Meirovitch
[3] Fundamentals of Vibrations (International Edition 2002) by Leonard Meirovitch
[Q-1] Derive the equations of motion of the Two DOF system shown in the figure below.
[Q-2] Two disks of mass polar moments of inertia I1 and I2 are mounted on a circular massless shaft
consisting of two segments of torsional stiffness GJ1 and GJ2, as shown in the figure below. Derive the
differential equations for the angular displacements of the disks.
[Q-3] Derive the equations of motion for the two-degrees-of-freedom system shown in the figure below,
and then (a) Obtain the natural frequencies and natural modes, (b) Plot the mode shapes and write the
general solution.
[Q-4] The system shown in the figure below consists of two lumped masses connected by an
inextensible string and undergoing the vertical displacements. Assume that the displacements are small,
so that the sine and the tangent of an angle can be approximated by the angle itself, and that the string
tension T is constant. Considering that the excitation forces are equal to zero, derive and solve the
eigenvalue problem for the case in which m1 = m, m2 = 2m, L1 = L2 = L, L3 = 0.5L. Also, plot and
explain the natural mode shapes.
[Q-5] A two-mass system consists of a piston of mass m1, connected by two elastic springs, that moves
inside a tube as shown in the figure below. A pendulum of length l and end mass m2 is connected to the
piston as shown in the figure.
(a) Derive the equations of motion of the system in terms of x1(t) and θ(t).
(b) Derive the equations of motion of the system in terms of the x1(t) and x2(t).
(c) Find the natural frequencies of vibration of the system.
[Q-6] 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 the figure below. If the stiffnesses of the supports are given
by k1 = 3000 N/mm and k2 = 2000 N/mm, and the supports are located at l1 = 0.5 m and l2 = 0.8 m, find
the natural frequencies and mode shapes of the machine tool.
[Q-7] An automobile is modeled with a capability of pitch and bounce motions, as shown in the figure
below. It travels on a rough road whose surface varies sinusoidally with an amplitude of 0.05 m and a
wavelength of 10 m. Derive the equations of motion of the automobile for the following data: Mass =
1000 kg, Radius of gyration = 0.9 m, l1 = 1.0 m, l2 = 1.5 m, kf = 18 kN/m, kr = 22 kN/m and velocity 50
km/hr.
[Q-8] A two-story building frame is modeled as shown in the figure below. The girders are assumed to
be rigid, and the columns have flexural rigidities EI1 and EI2, with negligible masses. For m1 = 2m, m2
= m, h1 = h2 = h, and EI1 = EI2 = EI, determine the natural frequencies and mode shapes of the frame.
[Q-9] Two identical pendulums, each with mass m and length l, are connected by a spring of stiffness k
at a distance d from the fixed end, see the figure below. a. Derive the equations of motion of the two
masses. b. Find the natural frequencies and mode shapes of the system.
[Q10] Find the response of the system shown in the figure below with the following initial conditions:
x1(0) = 0.05, x2(0) = 0.10, 𝑥𝑥̇ 1(0) = 0, 𝑥𝑥̇ 2(0) = 0.
