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# plugin-math245lab06 - Math245 Computer Lab set#6 Spring...

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Math245 Computer Lab set #6, Spring 2009 Simulation of the forced 2nd-order ODE Consider the forced, spring-mass-damper model described by the linear 2nd order ODE d 2 y dt 2 + 2 λω dy dt + ω 2 y = f 0 sin ω f t. (1) λ , ω , f 0 and ω f are all constant parameters (natural frequency: ω = p k/m , damping parameter: λ = c/ (2 mk ), forcing amplitude: f 0 = F 0 /m , forcing frequency: ω f ). Example 1 Choose ω = 1, λ = 1 / 8, f 0 = 1, ω f = 1 and y (0) = y 0 (0) = 0. Simulate the response of this system. By using the simulated results, compute the peak amplitude of the steady response and the phase shift between the steady response and the forcing function. Step #1: Let y = y 1 and y 0 = y 2 , and express our 2nd-order ODE by a set of 1st-order ODEs. y 0 1 = y 2 , (2) y 0 2 = - 2 λωy 2 - ω 2 y 1 + f 0 sin ω f t, (3) RHS of these equations are coded in Matlab function file fode.m as follows: function dy = fode(t,y,lmd,omg,omgf,f0) dy = [ y(2) -2*lmd*omg*y(2)-omg^2*y(1)+f0*sin(omgf*t) ]; where we set λ , ω , f 0 and ω f as parameter variables lmd , omg , f0 and omgf , respectively. Step #2: Log on to Blackboard. Download Matlab function file force.m from /Course docu- ment/Discussion Sections/ M-files/ to your current working directory. We will use force to compute the peak amplitude and phase shift from the simulated results. To see how to use force , type help force after Matlab command line. Following is the summary: USE: [ypeak, pshift] = force(t, y, omega_f) INPUT: t : independent variable vector obtained by ode45 t(1) should be zero (0).

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plugin-math245lab06 - Math245 Computer Lab set#6 Spring...

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