Question

The forced oscillations of a body of mass m on a spring of modulus k (as shown in Figure 2) are governed by the ODE: my'' + cy' + ky = r(t) where y = y(t) is the displacement from rest, c is the damping constant, k is the spring constant (spring modulus), and r(t) is the external force depending on time, t. (a) Let m = 1 gm, c = 0.05 gms^-1 and k = 25 gms^-2. Write the ODE of the system. (b) Given the external force r(t) (measured in gm cm s^-1): r(t) = { t + pi/2, -pi < t < 0; -t + pi/2, 0 < t < pi } r(t + 2pi) = r(t). Plot the external force r(t). Then, determine the Fourier Series of r(t). (c) Based on (a) and (b), find the steady-state solution yn(t) = An cos nt + Bn sin nt by showing that An = 4(25 - n^2) / (n^2 pi Dn), Bn = 0.2 / (n pi Dn); Dn = (25 - n^2)^2 + (0.05n)^2 (d) Plot the steady-state solution yn(t).

          The forced oscillations of a body of mass m on a spring of modulus k (as shown in Figure 2) are governed by the ODE:
my'' + cy' + ky = r(t)
where y = y(t) is the displacement from rest, c is the damping constant, k is the spring constant (spring modulus), and r(t) is the external force depending on time, t.
(a) Let m = 1 gm, c = 0.05 gms^-1 and k = 25 gms^-2. Write the ODE of the system.
(b) Given the external force r(t) (measured in gm cm s^-1):
r(t) = { t + pi/2, -pi < t < 0; -t + pi/2, 0 < t < pi }
r(t + 2pi) = r(t).
Plot the external force r(t). Then, determine the Fourier Series of r(t).
(c) Based on (a) and (b), find the steady-state solution yn(t) = An cos nt + Bn sin nt by showing that
An = 4(25 - n^2) / (n^2 pi Dn), Bn = 0.2 / (n pi Dn); Dn = (25 - n^2)^2 + (0.05n)^2
(d) Plot the steady-state solution yn(t).

The forced oscillations of a body of mass m on a spring of modulus k (as shown in Figure 2) are governed by the ODE:
my” + cy' + ky = r(t)
where y = y(t) is the displacement from rest, c is the damping constant, k is the spring constant (spring modulus), and r(t) is the external force depending on time, t.
(a) Let m = 1 gm, c = 0.05 gms^-1 and k = 25 gms^-2. Write the ODE of the system.
(b) Given the external force r(t) (measured in gm cm s^-1):
r(t) = t + pi/2, -pi < t < 0; -t + pi/2, 0 < t < pi
r(t + 2pi) = r(t).
Plot the external force r(t). Then, determine the Fourier Series of r(t).
(c) Based on (a) and (b), find the steady-state solution yn(t) = An cos nt + Bn sin nt by showing that
An = 4(25 - n^2) / (n^2 pi Dn), Bn = 0.2 / (n pi Dn); Dn = (25 - n^2)^2 + (0.05n)^2
(d) Plot the steady-state solution yn(t).

Added by Ricky D.

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