ME 459 Dynamics of Machinery
Homework Set #14 – Mode Shapes and Natural Frequencies
1. Calculate the natural frequencies and mode shapes of
the two degree-of-freedom spring-mass-damper
system of Homework #13, problem 4 using the
following values for the physical parameters.
m1 = 1 (kg)
k1 = 50 (N/m)
m2 = 0.25 (kg)
k2 = 10.4 (N/m)
Describe the motion of each mode.
2. For the system from Homework #12, problem 4 and
Homework #13, problem 2, a) linearize the equation
of motion about θ eq = 90 (deg) , and b) linearize the
equation about θ eq = 36.06 (deg) .
Determine the
stability of small motions about both of the
equilibrium positions. As given previously, use the
following values for the physical parameters
m = 80 (kg)
A = 3 (m)
k = 2000 (N/m)
3. For the system shown of Homework #13, problem 5,
a) show that θ = x = 0 is an equilibrium position,
b) linearize the equations of motion about the
equilibrium position, and c) calculate the natural
frequencies and mode shapes of the system using the
following physical data.
m1 = 0.3 (slugs) m2 = 0.15 (slugs)
k = 300 (lb/ft) A = 2 (ft)
Describe the motion of each mode.