Derive an expression for the acceleration for the base

Question

Instructions

Show all your work and do not take short cuts

If you need to use a computational package, state which package you are using. Do each problem on a set of sheets of paper. Start each of the problem on a new sheet of paper.

Write legibly so your paper can be graded properly.

Problem 1

For a spring-mass system with base excitation, derive the following expression for the relative motion z(t) = x(t) - y(t), for an undamped system:

z(t)= -1/ωn∫₀^ty¨ (τ)sinωn(t-τ )d τ               (1)

Now, assume that the base motion is specified by a velocity pulse of the base given by:

v (t)=(v0-v0(t /t0))=v0(U (t)-t/t 0)             (2)

where U(t) is the Unit Step Function

Consider that the velocity of the base has a jump from zero to v0 instantly at t=0.

Derive an expression for the acceleration for the base motion from the velocity equation above

Derive an expression for z(t) using equation and the acceleration term you have calculated for the base

If the peak amplitude of z occurs at a t < t1, derive an expression for the maximum amplitude z.

Problem 2

Given the forcing periodic function shown, construct the steady state Fourier Series solution. To accomplish this noble goal, you need o find the Fourier coefficients for the forcing function and then follow that with constructing the steady sate response. Construct the general solution with the initial conditions x(0)=0.1, and x'(0)=0. If it becomes rather difficult to apply the initial condition to the full solution, consider other options but state your assumptions.

The period is T=5 s. The mass-spring-damper properties are

m = 10; k = 1000; β = 50;

The periodic forcing function is given by,

F(t)=F₀ t²/(T /2)².......0

F(t)=-2F₀(t/T)+2 F0.........T /2

Problem 3

Consider the spring-mass system undergoing Coulomb friction and drag with a harmonic forcing function described by the following:

m x^- + μ m gsgn(x?)+α sgn(x?)(x?)²+kx=F₀sin(ω t)

Use the equivalence of energy dissipation to compute the equivalent damping factor.

Find the driving frequency at which the Coulomb and drag contribution become equal.

Use the equivalent damping ratio to solve for the maximum steady state displacement.

Problem 4

For a single degree of freedom spring mass damper system (k=1000 N/m, m=10 kg, β=100 kg/s) driven by random input force F(t), calculate and plot the power spectral density of the response if the PSD of the applied force is given by the expression below with S0 = 500 and S1=1000

Sf=S0+S₁ω²

Calculate the Expectation E[(x2)]