The solution to the differential equation , subject to initial position and initial velocity, is the equation of motion of a mass, m, hanging on a spring exhibiting free damped motion, where is the damping constant and k is the spring constant. Dividing by m and letting , we can analyze the following three cases:

Case 1 If > 0, then the system is said to be overdamped.

A large damping coefficient, , in comparison with the spring constant k.

Case II If = 0, then the system is said to be critically damped.

Resulting motion is oscillatory with a slight decrease in the damping coefficient.

Case III If < 0, then the system is said to be underdamped.

A small damping coefficient in comparison with the spring constant, k.

 

• Example 1

A 32-lb weight is attached to a spring whose spring constant is 4 lb/ft. The medium through which the weight moves offers a resistance numerically equal to 5 times the instantaneous velocity. If the weight is released 1 foot below the equilibrium position with an upward velocity of 1 ft/sec, find the equation of motion and graph it.

The IVP for the system is: x'' + 5x' + 4x = 0, x( 0 ) = 1, x'( 0 ) = -1.

• Solution using DSolve

Gives {{ x[ t ] -> E^-t }} with a plot of:

This is an overdamped system where = 5 and= 4. Notice that the damping is so great that even with an upwards initial velocity, the velocity is not large enough to force the spring through the equilibrium point. Recall that the system is set up so that up is negative, down is positive, and the t-axis ( or x = 0 ) is the equilibrium position.

1. "Varying the Damping Constant"

Redo Example 1 for = 8, 4, 2 and 1. Draw conclusions about the effect of damping on the system.

2. "Forced Damped Motion"

a) Redo Example 1 with the addition that an external force, f( t ) = cos( 2t ), is acting on the spring. This changes the linear DE from homogeneous to nonhomogeneous.

b) Now use f( t ) = t, instead.

3. "Changing Initial Conditions"

Redo Example 1 with the weight released 1/2 ft above the equilibrium position with a downward velocity of 5 ft/sec.

4. "Analyzing the Graph of the Equation of Motion"

Solve and graph the system having the IVP: x'' + 2x' + 10x = 0, x( 0 ) = 0, x'( 0 ) = 1

a) Approximate the maximum displacement of the mass from the equilibrium position.

b) Approximate the second time the mass passes through the equilibrium position in an upward direction.