Question

A mass weighing 8 pounds stretches a spring 1 foot. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. Find the equation of motion (solve the IVP) if the mass is driven by an external force equal to f(t) = 5 cos(2t).

Graph the solution. What part of the graph shows the transient behavior? What is the steady-state behavior?

Answer 1

weight is w=8 pound

gravity is 32 m/s²

mass is given by

slug

.

a force of 8 pounds, stretches a spring 1 feet

so x=1

from the Hooke's law, spring constant k is

.

.

damping force numerically equal to 2 times the instantaneous velocity

so damping constant is

.

force is

DE is given by

for the homogeneous system find roots

for complex roots general solution is

....................(1)

here we have

so assume that a particular solution is

..................(1)

take derivative

take derivative

put all values in DE

compare coefficient both sides

put both constant in equation 1.

.

general solution is

....................(2)

here mass is initially released from rest from a point 2 foot below the equilibrium position.

so y(0)=2

.................put it back in equation 2

.....................(3)

take derivative

here initial velocity is zero so y'(0)=0

.................put it back in equation 3

.

.

.

here transient terms are

.

steady-state terms are