Solution Found!
Forced Vibrations. As discussed in Section 4.1, a
Chapter 4, Problem 42E(choose chapter or problem)
Forced Vibrations. As discussed in Section 4.1, a vibrating spring with damping that is under external force can be modeled by
(15) \(m y^{\prime \prime}+b y^{\prime}+k y=g(t)\),
where \(m>0\) is the mass of the spring system, \(b>0\) is the damping constant, \(k>0\) is the spring constant, \(g(t)\) is the force on the system at time \(t\), and \(y(t)\)is the displacement from the equilibrium of the spring system at timet. Assume \(b^{2}<4 m k\).
(a) Determine the form of the equation of motion for the spring system when \(g(t)=\sin \beta t\) by finding a general solution to equation (15).
(b) Discuss the long-term behavior of this system. [Hint: Consider what happens to the general solution obtained in part (a) as \(t \rightarrow+\infty\).]
Equation Transcription:
Text Transcription:
my''+by'+ky=g(t)
m>0
b>0
k>0
g(t)
t
y(t)
b^2<4mk
g(t)=sin beta t
t right arrow + infinity
Questions & Answers
QUESTION:
Forced Vibrations. As discussed in Section 4.1, a vibrating spring with damping that is under external force can be modeled by
(15) \(m y^{\prime \prime}+b y^{\prime}+k y=g(t)\),
where \(m>0\) is the mass of the spring system, \(b>0\) is the damping constant, \(k>0\) is the spring constant, \(g(t)\) is the force on the system at time \(t\), and \(y(t)\)is the displacement from the equilibrium of the spring system at timet. Assume \(b^{2}<4 m k\).
(a) Determine the form of the equation of motion for the spring system when \(g(t)=\sin \beta t\) by finding a general solution to equation (15).
(b) Discuss the long-term behavior of this system. [Hint: Consider what happens to the general solution obtained in part (a) as \(t \rightarrow+\infty\).]
Equation Transcription:
Text Transcription:
my''+by'+ky=g(t)
m>0
b>0
k>0
g(t)
t
y(t)
b^2<4mk
g(t)=sin beta t
t right arrow + infinity
ANSWER:
Solution:
Step 1:
In this problem, we have to determine the form of the equation of motion for the spring system.