Answer:
t= 1/8, pi/8, 2pi/8,3pi/8
Step-by-step explanation:
Given
m=(8/32) lb s^2/ft
K=8/(6/12)=16 lb/ft
Use the following equation and plug in values
mu''+ku=f(t)
1/4u''+16u=8sin8t
u''+64u=32sin8t
This equation corresponds to the following homogeneous equation
u''+64u=0
r=+/-8i
uc(t)=c1cos8t+c2sin8t
Now find the particular solution
u(t)=Atcos8t+Btsin8t
u'(t)=-8Atsin8t+Acos8t+B8tcos8t+Bsin8t
u''(t)=-8tAsin8t-64Atcos8t-8Asin8t+B8cos8t-64Btsin8t+8Bcos8t
Substitute these values into the original equation and solve for Aand B
A=-2 B=0
the particular solution is u(t)=-2tcos8t
the general solution is u=u1(t)+u(t)
u=c1cos8t+c2sin8t-2tcos8t
Use the initial conditions to solve for c1 andc2
c1+0=(1/4) 8c2-2=0
c1=(-1/4) c2=(1/4)
u=(1/4)[cos8t+sin8t-8tcos8t]
To solve the next step differentiate u
u'=-2sin8t+2cos8t-2cos8t+16tsin8t
= -2sin8t+16sin8t
= 2sin8t(8t-1)
Velocity=2sin8t(8t-1)
Set this equation equal to zero to solve for zero velocity
8t-1=0 t=1/8
t= 1/8, pi/8, 2pi/8,3pi/8
