## Transcribed Text

1. A system is vibrating with very small angle 0, which is shown in figure 1 with massless lever
a =0.8 m and b = 1 m and mass m1 = 1.5 kg and m2 = 2.0 kg. A stiffness k = 100 N/m and a
damper c = 0.5 N/m-s are vertically connected to mi and m2, respectively. The force f = 10 N
cos(2t)
/
/
m2
k
f
b
c
m,
a
45°
/
7
4
X
Figure 1. Problem 1
a) Derive the equation of motion in terms of coordinate y.
b) Find the natural frequency and damping ratio.
2. A machine is subjected to harmonic ground excitation. The driving frequency f = 2 Hz. The
machine with base isolator can be modeled a SDOF system. The mass is, m = 500kg and the
damping ratio is E = 0.05.
a) Calculate the damping coefficient.
b) Calculate the corresponding relative displacement
c) Determine the stiffness of the base isolator so that the acceleration of the machine can be
reduced to 1/2 of the level when the stiffness is considered to be infinity.
3. An under-damped 2-DOF system has mass and stiffness matrix M and K. We also know that
the system has damping ratio E1 = 0.5 and E2 = 0.4. The damping matrix is written as
a) Try to find the coefficients a., ß and Y-
b) Find if this system proportionally damped?
4. Write the equation of motion of the system shown in figure 2.
a) Calculate the natural frequency and modal shape by assuming c = 0, where the rotating
stiffness k1 = k2 = k; the moment of inertia J1 = 3 J2.
01
w
02
.
V
k1
J1
k2
J2
2/1 c
Figure 2 Problem 4
5. A system has mass matrix
M = 0 1 4 0 and eigenvector matrix U = "1 U2 0.9673
0.2535 0.9979
-0.0654
And the natural frequencies of the system are: Onl = 1.3582 and On2 = 6.3761.
a) Normalize U with respect to M. (Hint: Uc is an eigenvector of M-K. Let =1)
7. A system has
= and K = -100 400 -100 100
and its natural frequencies are 50 and
150 respectively.
Check which of the following can be eigenvectors of matrix M-1/2 KM-1/2
2
Suppose this system has initial conditions: x(0) 2 and x(0) = {-0.05} 0
=
Calculate the responses of displacements.
8. a) Find the dynamic magnification factor (amplitude of the transfer function) for a first order
vibration system
= = - -
(p8)
and E are respectively the natural frequency and damping ratio of this system.
b) if equation (p8) is the equation of a modal response of one part of a complex conjugate pair,
then can you find the transfer function of this particular mode? plot the transfer function by
letting E = 0.3 and On = 10.

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