System Dynamics

For each of the following ODEs determine if the eigenvalues are (a) real and distinct, (b) repeated, (c) complex conjugate pairs:

9.11 Figure P9.11 shows a 1-DOF mechanical system driven by the displacement of the left end, (), which could be supplied by a rotating cam and follower (see Problem 2.2). When displacements X() = 0 and r = 0 the spring k is neither compressed nor stretched. The system purameters are Im = 2 kg. k = 50K) N/m,and h = 20N-s/m. Determine the frequency response if the position input is x(1) = 0.04 sin 50r m.

2. A system with input R(s) and response Y(s) is described by the following equations: X_{1}(s)=R(s)-Y(s) X_{2}(s)=G_{A}(s) X_{1}(s) X_{3}(s)=X_{2}(s)+X_{4}(s)-Y(s) X_{4}(s)=X_{3}(s) G_{B}(s) Y(s)=X_{4}(s) G_{C}(s) Assemble a block diagram representation of the system, clearly showing and labelling all intermediate signals X1(s), X2(s), X3(s), X4(s), and all transfer functions GA(s), GB(s), Gc(s).

\dot{x}=\left[\begin{array}{cc} 0 & 1 \\ -20 & -4 \end{array}\right] x+\left[\begin{array}{c} 0 \\ 0.2 \end{array}\right] u \quad y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x a. Obtain the 1/O equation for this system where y is the output and u is the input. b. Obtain the transfer function for this system. a. Obtain the 1/O equation for this system where y is the output and u is the input.

6. x + x = sin(3t) with no initial conditions (x(0) = 0 and x(0) = 0)

For the system to a sinusoidal input with unit amplitude and zero initial conditions: y(0) = 0, y(0) = 0, and 4 (t) = sin(t) The response is y(t)=e^{-t}+t e^{-t}-\cos t D True O False

Consider the differential equation below. m \ddot{y}=k(z(t)-y)+b \frac{d}{d t}(z(t)-y) where y(t) = Yejwt and z(t) = Zejwt Ya) Solve for the frequency response, i.e., the ratio Y/Z ) Assuming that the damping ratio is 3 = 0.05 and that m = 1, plot the magnitude of the transfer function vs. frequency of the response of part (a), i.e., plot Y/Z vs.(W/WN). Use a logarithmic scale for the x-axis. ) Use the plot in part (b) to explain for which range of frequencies one can ignore z(t) in the original differential equation (i.e. set z(t) to zero). Assuming that the damping ratio is = 0.05 and that m = 1, plot the phase angle of the transfer function vs. frequency of the response of part (a) (i.e. the phase angle of Y/Z vs. (W/WN). Use a logarithmic scale for the x-axis. (e) Assuming that 3 = 0.1, and m = 1, use the plots in parts (b) and (d) to explain for which range of frequencies one can ignore all the derivative terms in the original differential equation. ) With the same assumptions as in part (e), use the solution for part (a) to find y(t)when z(t)10sin(3t).= Consider a car moving at a constant speed Vo on a wavy road where the "waves"have a peak-to-peak distance (wavelength) of L, and an amplitude of Z, inches.Show that this system will result in the same differential equation provided at the beginning of the problem.

5.1Derive the state-variable equations for the system that is modeled by the following ODES where a, w, and z are the dynamic variables and v is the input. 0.4 \dot{\alpha}-3 w+\alpha=0 0.25 z+4 z-0.5 z w=0 \ddot{w}+6 w+0.3 w^{3}-2 \alpha=8 v

2. Given the differential equation n ÿ + 3ỷ + y = 5ġ – 2g, where g(t) is the input function and y(t) is the response (output), find the transfer function Y(s)/G(s).

1. The equations of motion of this system are +3y + 4y -32-4Z = 0 2 +52 +62-5ý-6y = f(t)