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)=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). 1 m= ₁ c = 5₁ k = 12₁ x₁ = 6, V₁ = 0
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)=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). 1 m= ₁ c = 5₁ k = 12₁ x₁ = 6, V₁ = 0
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text: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).
1
m=2, c= 5, k = 12, xo = 6, Vo = 0
x(t)=, which means the system is underdamped.
(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 2π.)
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