MATH 238_Project 3_11

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Jan 9, 2024

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Olivia Campbell 12213131 MATH 238 – Project 3 11/28/23 Part 1 : A mass-spring motion is governed by the ordinary differential equation where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x (0) = 0. Assume the following numerical ′ values for this part of the project: m = 1, k = 1/5, b = 1/5, and F(t) = cos γt. (a) Read section 4.10. Explain what the resonance frequency is, and then compute the resonance frequency for this mass-spring system. The resonance frequency of a mass-spring system is the natural frequency where a medium vibrates at the highest amplitude. The resonance frequency of this mass spring system is f=0.65 rad/s at ? =4.2 (b) The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the provided MATLAB code to plot the solutions and estimate the amplitude A of the steady response for γ = 0.2, 0.42, 0.6, and 0.8. ? = 0. 2 ? =6.1 ? = 0. 4 ? =11.2 ? = 0. 6 ? = 5 ? = 0. 8 ? = 2.1 (c) Plot the graph of A versus γ. For what frequency γ is the amplitude the greatest? Is it equal to that you obtained in (a)?
Amplitude is greatest at ? =4.3, where A=11.5. This is very close to what I solved for in part a. Part 2 : (Variable Coefficient Equations) A linear second-order differential equation that can be expressed in the form where a, b, and c are constants, is called a Cauchy-Euler equation. We consider the homogeneous case (f(t) = 0) in this part. Note that the solutions should have the form because then (a) Show that substituting y = t^r into the Cauchy-Euler equation yields the characteristic equation ar2 + (b − a)r + c = 0. y=t^r y’=rt^(r-1)= (ry)/t y’’=(r^2)(r-1)t^(r-2)=(r(r-1)y)/t^2 Plug into equation: at^(2)*(r(r-1)t^(r-2))+bt((r)*t^(r-1))+c(t^r)=0 ar(r-1)(t^r)+brt^r+ct^r=0 (ar(r-1)+br+c)t^r=0 t^r=0 a(r^2)-1ar+br+c=0 a(r^2)+(b-a)r+c=0 (b) Find the roots of the characteristic equations of t^2y + 7ty − 7y = 0, then fnd the ′′ ′ general solution to this equation. a=1, b=7, c=-7
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