ME 240 Fall 1998

Problem Statement: A 3-kg block in a damped spring-mass system is subjected to a frictionless horizontal motion. In this assignment, Matlab is used to study the effects of (i) varying the damping ratios along with a variety of input conditions for free motion and (ii) the interactions of the excitation frequency and the natural frequency for forced vibrations.

System Parameters: The values for the mass and stiffness are:
m = 3 kg
k = 27 N/m
c = you set the value for the corresponding damping ratio N.s/m

Vary "c" to let the damping ratio (zeta) take on values 0, 0.1, and 1.1 for the following problem 1.

1. Determine the free response for these values of the damping ratio along with the three initial conditions.
   a) Initial Conditions: xo = 0.2 m, Vo = 0 m/s.
   b) Initial Conditions: xo = -0.2 m, Vo = - 2.5 m/s.
   c) Initial Conditions: xo = 0.2 m, Vo = - 2.5 m/s.

   Plot the responses for displacement and velocity and turn in the plots for each case, which will be nine plots in total.

   • How does qualitative nature of the response change with the three damping ratios?
   • How does qualitative nature of the response change with the initial conditions?
   • Does the damped natural frequency change noticeably?

2. Now consider a forced damped system with the harmonic excitation force of F(t)=30 sin 6t N, using the same mass and stiffness mentioned above. The motion is started from rest with zero displacement and zero velocity, but with the harmonic excitation force. Plot the responses for two damping ratios, zeta=0.28 and 0.028. The plots for each response must include the displacements for the homogeneous, particular, and total response.
   • What is the effect of damping on the amount of time taken to reach the steady state response?

In addition to the above, the assignment should include the following:
• Problem statement
• Mathematical model along with assumptions made.
• Analysis method (graphical or algebraic)
• Summary and conclusions.