Consider the damped harmonic oscillator shown below and modeled by the system: k - ()*+ (-) m (1) with mass m = 1 kg, spring stiffness k = 100 N/m, and initial conditions x(to) = 0.1 m and x(to) = 0, where x(t) is the displacement of the mass from the nominal (unstretched) spring position. Four dashpots are being considered for this system: D1, D2, D3 and D4 have damping coefficients b₁ = 3, b₂ = 10, b3 = 20, and b4 = 50 N/(m/s), respectively. Pneumatic Dashpots: D1 D2 end *+ D3 D4 x=0 Assume: Horizontal Plane (no gravity) x(t) bu m www k You are tasked with selecting the correct dashpot that renders the system critically damped and to determine the response of the system over a two second interval from the initial condition. To perform your analysis, include the following: • A single plot with all five curves showing the response of the system to the initial conditions for each dashpot over a two second interval, and for the case of no dashpot b = 0. Each curve should be plotted with a different color with time on the abscissa and the displace- ment x(t) on the ordinate axis. Label your axes and include a legend. • Determine the damping ratio (numerical value) for Case 0 (no dashpot) and for Cases 1-4 with dashpots D1, D2, D3, and D4. For each case specify whether it as critically damped, overdamped, undamped, or underdamped. • Make your final recommendation on which dashpot to choose MATLAB Hints (optional): • To avoid repeating the same code for each of the five cases you are encouraged to encapsu- late it in a function of the form below. function xhist = simulate MassSpring Damper (tvals,x0, xdo to, k, m, b) % simulate system (your code here) This function can then be called five times in your main script to simulate each case (with a different input value for b).
Consider the damped harmonic oscillator shown below and modeled by the system: k - ()*+ (-) m (1) with mass m = 1 kg, spring stiffness k = 100 N/m, and initial conditions x(to) = 0.1 m and x(to) = 0, where x(t) is the displacement of the mass from the nominal (unstretched) spring position. Four dashpots are being considered for this system: D1, D2, D3 and D4 have damping coefficients b₁ = 3, b₂ = 10, b3 = 20, and b4 = 50 N/(m/s), respectively. Pneumatic Dashpots: D1 D2 end *+ D3 D4 x=0 Assume: Horizontal Plane (no gravity) x(t) bu m www k You are tasked with selecting the correct dashpot that renders the system critically damped and to determine the response of the system over a two second interval from the initial condition. To perform your analysis, include the following: • A single plot with all five curves showing the response of the system to the initial conditions for each dashpot over a two second interval, and for the case of no dashpot b = 0. Each curve should be plotted with a different color with time on the abscissa and the displace- ment x(t) on the ordinate axis. Label your axes and include a legend. • Determine the damping ratio (numerical value) for Case 0 (no dashpot) and for Cases 1-4 with dashpots D1, D2, D3, and D4. For each case specify whether it as critically damped, overdamped, undamped, or underdamped. • Make your final recommendation on which dashpot to choose MATLAB Hints (optional): • To avoid repeating the same code for each of the five cases you are encouraged to encapsu- late it in a function of the form below. function xhist = simulate MassSpring Damper (tvals,x0, xdo to, k, m, b) % simulate system (your code here) This function can then be called five times in your main script to simulate each case (with a different input value for b).
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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Transcribed Image Text:5 Problem
Consider the damped harmonic oscillator shown below and modeled by the system:
b
m
(1)
with mass m = 1 kg, spring stiffness k = 100 N/m, and initial conditions x(to) = 0.1 m and
x(to) = 0, where x(t) is the displacement of the mass from the nominal (unstretched) spring
position. Four dashpots are being considered for this system: D1, D2, D3 and D4 have damping
coefficients b₁ = 3, b2 = 10, b3 = 20, and b4 = 50 N/(m/s), respectively.
Pneumatic Dashpots:
D1
D2
end
* +
D3
D4
x +
m
x=0
Assume: Horizontal Plane (no gravity)
x(t)
bu
m
www
k
You are tasked with selecting the correct dashpot that renders the system critically damped and
to determine the response of the system over a two second interval from the initial condition. To
perform your analysis, include the following:
• A single plot with all five curves showing the response of the system to the initial conditions
for each dashpot over a two second interval, and for the case of no dashpot b = 0. Each
curve should be plotted with a different color with time on the abscissa and the displace-
ment x(t) on the ordinate axis. Label your axes and include a legend.
• Determine the damping ratio (numerical value) for Case 0 (no dashpot) and for Cases 1-4
with dashpots D1, D2, D3, and D4. For each case specify whether it as critically damped,
overdamped, undamped, or underdamped.
• Make your final recommendation on which dashpot to choose
MATLAB Hints (optional):
• To avoid repeating the same code for each of the five cases you are encouraged to encapsu-
late it in a function of the form below.
function xhist = simulateMass Spring Damper (tvals, x0, xdot0, k, m, b)
% simulate system (your code here)
This function can then be called five times in your main script to simulate each case (with
a different input value for b).
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