A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e ¯P' cos (@,t-a,). Also, find the undamped position function u(t) = Co cos (@ot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 1 c= 3, k = 8, xo =7, vo = 0 m = 4'

A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e ¯P' cos (@,t-a,). Also, find the undamped position function u(t) = Co cos (@ot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 1 c= 3, k = 8, xo =7, vo = 0 m = 4'

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 Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

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**Educational Content on Damped Harmonic Motion**

A mass \( m \) is attached to both a spring (with given spring constant \( k \)) and a dashpot (with given damping constant \( c \)). The mass is set in motion with initial position \( x_0 \) and initial velocity \( v_0 \). Find the position function \( x(t) \) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form:

\[ x(t) = C_1 e^{-pt} \cos(\omega_1 t - \alpha_1) \]

Also, find the undamped position function \( u(t) = C_0 \cos(\omega_0 t - \alpha_0) \) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so \( c = 0 \)). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of \( x(t) \) and \( u(t) \).

Given:
\[
m = \frac{1}{4}, \, c = 3, \, k = 8, \, x_0 = 7, \, v_0 = 0
\]

Determine:

\[
x(t) = \boxed{\phantom{x}} , \text{ which means the system is } \boxed{\text{(Select one: overdamped, critically damped, or underdamped)}}
\]

Instructions:
- Use integers or decimals for any numbers in the expression.
- Round to four decimal places as needed.
- Type any angle measures in radians.
- Use angle measures greater than or equal to 0 and less than or equal to \( 2\pi \).

There are no graphs or diagrams provided. The instructions indicate constructing a figure comparing graphs of \( x(t) \) and \( u(t) \) to analyze damping effects.

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