A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant e). 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" cos (a,t-a,) Also, find the undamped position function ut) Co cos (o -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 c0). 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 -6 CHD x(t) = which means the system is (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 2x.)
A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant e). 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" cos (a,t-a,) Also, find the undamped position function ut) Co cos (o -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 c0). 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 -6 CHD x(t) = which means the system is (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 2x.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A mass m is attached to both a spring (with given spring oconstant k) and a dashpot (with given damping constant o). 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 cos (u,t-a,) Also, find the undamped position function u(t) = C, cos (o "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).
m= 1, c=6, k=9, Xo " 6, Vo = -6
x(t) = which means the system is
%3D
(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 2x.)
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