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A mass m = 4 kg is attached to both a spring with spring constant k = 485 N/m and a dash-pot with damping constant c = 4 N. s/m. The mass is started in motion with initial position zo = 4 m and initial velocity vo = 6 m/s. Determine the position function (t) in meters. r(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form a(t) = Cie cos(w₁ta₁). Determine C₁, w₁,₁and p. C₁ = = الا 01 p= Graph the function (t) together with the "amplitude envelope" curves x = -Cie Pt and x = C₁e pt. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t) = Cocos(wotao). Determine Co, wo and co. Co= WI= (assume 00₁ < 2TT) α0 = (assume 0 < a < 2π) Finally, graph both function z(t) and u(t) in the same window to illustrate the effect of damping.

A mass m = 4 kg is attached to both a spring with spring constant k = 485 N/m and a dash-pot with damping constant c = 4 N. s/m. The mass is started in motion with initial position zo = 4 m and initial velocity vo = 6 m/s. Determine the position function (t) in meters. r(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form a(t) = Cie cos(w₁ta₁). Determine C₁, w₁,₁and p. C₁ = = الا 01 p= Graph the function (t) together with the "amplitude envelope" curves x = -Cie Pt and x = C₁e pt. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t) = Cocos(wotao). Determine Co, wo and co. Co= WI= (assume 00₁ < 2TT) α0 = (assume 0 < a < 2π) Finally, graph both function z(t) and u(t) in the same window to illustrate the effect of damping.

Advanced Engineering Mathematics
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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A mass m = 4 kg is attached to both a spring with spring constant k = 485 N/m and a dash-pot with damping constant c = The mass is started in motion with initial position zo = 4 m and initial velocity vo = 6 m/s Determine the position function (t) in meters. x(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form z(t) = C₁e Pt cos(wita₁). Determine C₁, w₁,0₁and p. C₁ = W₁ = 01 = P = Co = Graph the function z(t) together with the "amplitude envelope" curves x = -C₁e Pt and x = Cie pt. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t) = Cocos(wotao). Determine Co, wo and co- wo = (assume 0 ≤0₁2) α0 = 4 N-s/m. (assume 0 < a < 2π) Finally, graph both function (t) and u(t) in the same window to illustrate the effect of damping.
Transcribed Image Text:A mass m = 4 kg is attached to both a spring with spring constant k = 485 N/m and a dash-pot with damping constant c = The mass is started in motion with initial position zo = 4 m and initial velocity vo = 6 m/s Determine the position function (t) in meters. x(t) = Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form z(t) = C₁e Pt cos(wita₁). Determine C₁, w₁,0₁and p. C₁ = W₁ = 01 = P = Co = Graph the function z(t) together with the "amplitude envelope" curves x = -C₁e Pt and x = Cie pt. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t) = Cocos(wotao). Determine Co, wo and co- wo = (assume 0 ≤0₁2) α0 = 4 N-s/m. (assume 0 < a < 2π) Finally, graph both function (t) and u(t) in the same window to illustrate the effect of damping.
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