FAST 2) An oscillator is described by # +2 +5x f(t), where x is the output...
Problem 1 (25 points): Consider a system described by the differential equation: +0)-at)y(t) = 3ú(1); where y) is the system output, u) is the system input, and a(t)is a function of time t. o) (10 points): Is the system linear? Why? P(15 points): Ifa(t) 2, find the state space equations?
04. (25 pts)(Fourier Analysis) A periodically driven oscillator and the forcing function is shown tbelow. F(t) The governing equation of the system shown above can be written as mx" + cx' +kx = F(t) where m, c and k are some constants. Considering a forcing function defined as a pulse below for 0 T/2 t 2 for π /2 < t <3m/2 , or 3π which is periodic with a period of 2π in the interval of OSK o Find...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t) Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. a. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer. 2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t)...
Given the System described by the differential equation below: D^2 y(t) + 3 Dy(t) + 2y(t) = x(t) where D=d/dt and D^2 is the second derivative 1) find its Transfer Function H(s) (assume all initial conditions are zero) 2) use the first procedural way of getting A, B, C, D from H(s) to find the corresponding state space representations . Then do the reverse step of finding H(s) from the A, B, C, D representation just found (i.e. check that...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
For a Mechanical Engineering System Dynamics class 2. i) Obtain the state model for the reduced-form model 28 +62 + 12x = 10y(t). Use x, and xz as the state variables. Put the equations in standard form and find [A] and [B] matrices. Given the state variable model x = x; – 5x, + f (1) * = -30x, +10f2(1) where f(t) and f (t) are the inputs, and the output equations y = x, - x2 + f,0 Y2...
Let a linear system with input x(t) and output y(t) be described by the differential equation . (a) Compute the simplest math function form of the impulse response h(t) for this system. HINT: Remember that with zero initial conditions, the following Laplace transform pairs hold: Let the time-domain function p(t) be given by p(t) = g(3 − 0.5 t). (a) Compute the simplest piecewise math form for p(t). (b) Plot p(t) over the range 0 ≤ t ≤ 10 ....
(5) For the system described by the following difference equation y(n)= 0.9051y(n 1) 0.598y(n 2) -0.29y(n 3) 0.1958y(n - 4) +0.207r(n)0.413r(n 2)+0.207a(n - 4) (a) Plot the magnitude and phase responses of the above system. What is the type of this filter? (b) (b) Find and plot the response of the system to the input signal given by /6)sin(w2n +T /4) u(n), where w 0.25m and ws 0.45m a(n) 4cos(win -T = (c) Determine the steady-state output and hence find...
10.Represent the translational mechanical system shown in the Figure in state- space, where xX3(t) is the output IN- 11.Find the state equations and output equation for the phase-variable representation of the transfer function G(s) 2s+1/(s2+7s+ 9) 12. Convert the state and output equations shown to a transfer function. -1.5 2 u(t) X = X 4 0 Y [1.5 0.625]x 13. For each system shown, write the state equations and the output equation for the phase- variable representation 8s10 sh25 t26...