Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement ofa mass-spring oscillator resting on a frictionless surface,governed by the ODEAy + By = f(t)(i) Suppose A=B= 1, and the force f(t) represents a push to the left of the mass over a time period fromt = 2π to t = 2π + afor some a > 27.Let f(t) bef(t)= - [u(t-2π)-u (t-(2π + a))]such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function ult) is the unitstep function.Use Laplace Transform to solve the initial value problem16(ii) Suppose A =32= f(t)reelley(0) = 1ÿ + 2y + 2y =B = 2 and the force f(t) represents an impulse force. Write down the governing equations for thedisplacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and thenstruck by 4 force units at t = 27 time units later. [Solve the governing equation!]Consider now the more realistic scenario of a frictional surface, so that the displacement of the mass-spring oscillator is damped.(d) Find an expression for the underdamped displacement y(t) in terms of a convolution integral for some real constants k. and K.given·ylo) = Ko}(1)g(0) = 0y (o)= k₁Other oscillating systems (e.g., circulating-fuel reactors) are governed by integro-differential equations like that below, with u(t)being the unit step function3 [[y(7) - 8 (T-A)] d7u(t-A)dy tay +B!(e) Find y(t), given y(0)=0, a=4, A=π and B=1.

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A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity with the Lagrangian L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency ω and note that the transformed Hamiltonian yields a constant of motion. Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).
Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.
If the mass is below the equilibrium position at the given time t, what can we say about the value of the corresponding displacement function, a (t)? O z (t) 0
An electron undergoes simple harmonic motion with the acceleration shown below: ax(t)=−amaxsin(2t/T) with amax=5839 ms2 and T=316 seconds. Assuming that the only motion is oscillatory (ignoring overall translation), what is the maximum speed of the electron? What is the amplitude of the electron's position?
The quantities A and φ (called the amplitude and the phase) are undetermined by the differential equation. They are determined by initial conditions -- specifically, the initial position and the initial velocity -- usually at t = 0, but sometimes at another time. In the oscillating part of the experiment, I measured only the time of 30 periods. I measured no position or velocity. Consequently, A and φ (and also y0) are irrelevant in the problem. We only compare the period T or the frequency ω with the theoretical prediction. You have (hopefully) derived (or maybe looked up) the relation between ω and k and m. This final question relates ω and T. If ω = 8.2*102 rad/s, calculate T in seconds. (Remember, that a radian equals one.) T might be a fraction of a second.
Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.
Calculate the energy, corrected to first order, of a harmonic oscillator with potential:
A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity. Show that the following Lagrangian gives the desired equation of motion: L = e2γt(1/2 * mq˙2 − V (q))