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 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) =C₁ e -pt cos (w₁t-α₁). Also, find the undamped position function u(t) = Cocos (@ot -%o) 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). m = 1, c=6, k = 9, Xo = 6, Vo = - 14 x(t) = =, which means the system is critically damped.
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 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) =C₁ e -pt cos (w₁t-α₁). Also, find the undamped position function u(t) = Cocos (@ot -%o) 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). m = 1, c=6, k = 9, Xo = 6, Vo = - 14 x(t) = =, which means the system is critically damped.
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter12: Coupled Oscillations
Section: Chapter Questions
Problem 12.22P
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