Consider a horizontal Spring mass system m attached with a dashpot. The stiffness/ spring constant damping c. Let x(t) denote the position function at time t. is denoted by k, the constant is denoted by 1. A mass on a spring without damping is acted on by the external force F(t) = FO cos^(3) wt. Show that there are two values of w for which resonance occurs, and find both. One can use the formula cos 3t = -3 cost + 4 cos^(3)t, and then rewrite cos^(3)t as a function of cost and cos 3t. 2. If there is no external force and the system is critically damped, show that x(t) = (x0+v0t+ c/(2m)*x0t)e^((- c/2m) t) Here x0 = x(0) and v0 = x'(0) denotes the initial displacement and velocity.

Consider a horizontal Spring mass system m attached with a dashpot. The stiffness/ spring constant damping c. Let x(t) denote the position function at time t. is denoted by k, the constant is denoted by 1. A mass on a spring without damping is acted on by the external force F(t) = FO cos^(3) wt. Show that there are two values of w for which resonance occurs, and find both. One can use the formula cos 3t = -3 cost + 4 cos^(3)t, and then rewrite cos^(3)t as a function of cost and cos 3t. 2. If there is no external force and the system is critically damped, show that x(t) = (x0+v0t+ c/(2m)*x0t)e^((- c/2m) t) Here x0 = x(0) and v0 = x'(0) denotes the initial displacement and velocity.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

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Consider a horizontal Spring mass
system m attached with a
dashpot. The stiffness/ spring
constant is denoted by k, the
damping constant is denoted by
c. Let x(t) denote the position
function at time t.
1. A mass on a spring without
damping is acted on by the
external force F(t) = FO cos^(3) wt.
Show that there are two values of
w for which resonance occurs,
and find both.
=
One can use the formula cos 3t
-3 cost + 4 cos^(3)t, and then
rewrite cos^(3)t as a function of
cost and cos 3t.
2. If there is no external force and
the system is critically damped,
show that x(t) = (x0+v0t+
c/(2m)*x0t)e^((- c/2m) t)
Here x0 = x(0) and v0 = x'(0)
denotes the initial displacement
and velocity.

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