(a) Show that the general solution of

Is

Where A = √c²₁ + c²₂ and the phase angles ϕ and θ are, respectively, defined by sin ϕ = c₁/A, cos ϕ = c₂/A and

b) The solution in part (a) has the form x(t) = x_c(t) + x_p(t). Inspection shows that x_c(t) is transient, and hence for large values of time, the solution is approximated by x_p(t) = γ(g) sin(γt + θ), where

Although the amplitude g(γ) of x_p(t) is bounded as b → ∞, show that the maximum oscillations will occur at the value γ₁ = √(ω² – 2)². What is the maximum value of g? The number √(ω² – 2)²/2π is said to be the resonance frequency of the system.

(c) When F₀ = 2, m = 1, and k = 4, g becomes

Construct a table of the values of γ₁ and g(γ₁) corresponding to the damping coefficients β = 2, β = 1, β = ¾, β = 1/2, and β = 1/4. Use a graphing utility to obtain the graphs of g corresponding to these damping coefficients. Use the same coordinate axes. This family of graphs is called the resonance curve or frequency response curve of the system. What is γ₁ approaching as β → 0? What is happening to the resonance curve as β → 0?

Transcribed Image Text:

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