Name _________________________________

(Who are you?)

1

ME 540: Advanced Mechanical Vibrations

Exam 1: Monday, October 7, 2013

QUESTION 1:

Consider the inverted pendulum connected to two equal springs, shown below. Assume the

springs are undeflected when in the vertical position and that the mass m of the ball at the end of

the pendulum rod is substantially larger than the mass of the rod itself, so that the rod is

considered to be massless. The rod length is l and the springs are attached at the point l/2.

1.) Draw a free body diagram (FBD) and obtain the equation of motion (Hint: use θ as

independent variable and complete the diagram below as your FBD)

Name _________________________________

(Who are you?)

2

2.) Linearize about the equilibrium position and find the linearized equations of motion.

Name _________________________________

(Who are you?)

3

QUESTION 2:

1.) Consider the system arrangement below:

Assume k1 = 10 N/m, k2 = 1 N/m, k3 = 3 N/m, k4 = 2 N/m, k5 = 4 N/m, and the mass

m = 10 kg. Calculate the natural frequency of the system.

Name _________________________________

(Who are you?)

4

QUESTION 2: (continued)

2.) Now assume we add a viscous damping as shown below of c = 1 kg/s.

Is the system overdamped, critically damped, or undedamped?

Name _________________________________

(Who are you?)

5

3.) Without doing any calculations, provide a qualitative (very rough) plot of the response of the

mass m to a non-zero, positive initial condition (no external forces except for gravity.)

Name _________________________________

(Who are you?)

6

QUESTION 3:

A clockmaker builds a clock based upon the frequency of oscillations of the mass-spring system

below:

1. He wants to attach a mass if 1 kg to a spring. What should the spring stiffness be for this

vibratory system to “beat” the second, i.e., the period of oscillation is T = 2 seconds,

assuming a mass less spring?

Name _________________________________

(Who are you?)

7

2. It turned out that the mass of the spring of 300 g was not negligible compared to the 1

kg mass. Should he add or remove mass to the system to keep the clock accurate? By how

much?

3. What is the period of oscillation of this system on the planet Mars where the

gravitational constant is 3.8 m/s2.

Name _________________________________

(Who are you?)

8

(extra work sheet)