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Obtain the state space representation of the mechanical system shown. The moment of inertia with respect to the center of the disk is J, and both the displacement z and the angle of rotation theta are measured from their respective equilibrium positions. The input force is u=F, and the outputs are y1=force in the spring and y2=force in the damper.
For the given transfer function G(s)=Y/U, (a) Obtain a state space representation (including both state equation and output equation) using the following state variables: x1=y, x2=ydot, x3=ydoubledot. (b) Obtain a state space representation using the partial fraction expansion approach. (c) Obtain a state space representation using tf2ss command in MATLAB.
Consider the mechanical system shown in the figure. The system is initially at rest. The displacements x1 and x2 are measured from their respective equilibrium positions before the input u is applied. Assume that b1=Ns/m, b2=10Ns/m, k1=4N/m, and k2=20N/m. Obtain the displacement x2(t) when u is a step force input of 2 N. Derive the transfer function and then plot the response for 5 s using step command. The step input force u has a magnitude of 2 N.
In the mechanical system shown, assume that the rod is massless, perfectly rigid, and pivoted at point P. The displacement x is measured from the equilibrium position. Assuming that x is small, that the weight mg at the end of the rod is 5 N, and that the spring constant k is 400 N/m, find the natural frequency of the system. Use the energy method to derive the equation of motion and then find the natural frequency.
The mechanical system shown consists of a uniform disk (mass M and radius R) and a block (mass m) connected to a spring and a damper. Derive the equation of motion of the system, and then find its natural frequency. The moment of inertia J is about the center of the disk, and the mass of the small pulley is negligible.
Derive the inverse Laplace transform of F(s)=1/(s^2(s^2+w^2))
Use symbolic MATLAB to verify your answers in problems 1, 2, and 3.
Consider the mechanical system shown in the figure in the textbook. Find the amplitude of the steady state displacement response using the following system parameters: m=5 kg, b2= 20 N/sm, k1=k2=k=360 n/m, and input parameters: p=180 N, and f=4 Kz for the excitation frequency.
A free vibration of the mechanical system shown in the figure indicates that the amplitude of vibration decreases to 25% of the value at t=t0 after four consecutive cycles of motion, as the figure shows. Determine the viscous-friction coefficient b of the system if m=1 kg and k=500 N/m.
A tank of capacitance 2.5 m^2 is initially at steady state with a steady head of 4 m. The outflow through the valve is turbulent and it is related to the head by q=0.2h^1/2. If the inflow is increased by 0.005 m^3/s, determine the change in head using the linearized equations of motion.
Obtain the transfer functino Eo(s)/Ei(s) of the system shown in the figure.
Compute z for the following cases and express the answers in both rectangular form and polar (or Euler) form. Note: Show your detailed step-by-step work; writing the final answers directly from the calculator is not acceptable. (a) z=-6+15i-10e^i-130 degrees (b) z=((-3^1/2+3i)^4(e^i(-5pi/6)))/(2+2i)^2 .
In the given simplified dc servomotor the inductance in the armature circuit can be neglected. Also motor damping (c1) and motor spring constant (k1) are both small and negligible. Obtain the transfer function G(s) between the output rotation theta2 and input voltage ea using the shown set of parameters.
At steady state, the flow rate throughout the liquid-level system shown in the figure is Qbar, and the heads of tanks 1 and 2 are H1bar and H2bar, respectively. At t=0, the inflow rate is changed from Qbar to Qbar+q, where q is small. The resulting changes in the heads (h1 and h2) and flow rates (q1 and q2) are assumed to be small as well. The capacitances of tanks 1 and 2 are C1 and C2, respectively. The resistance of the outflow valve of tank 1 is R1 and that of tank 2 is R2. Obtain the transfer function for the system when q is the input and q2 the output.
Consider the mechanical vibratory system shown in the figure. Assume that the displacement x is measured from the equilibrium position in the absence of the sinusoidal excitation force. The initial conditions are x(0)=0 and xdot(0)=0, and the input force p(t)=Psinwt is applied at t=0. Assume that m=2 kg, b=24 Ns/m, k=200N/m, P=5N, and w=6 rad/s. Obtain only the steady state response Xss(t) of the system.
Find the Laplace transform of the given function: f(t)=5-2t^3-e^(-4t)+t^2e^-6t-4sin5t+tcost-9e^-7tsin8t .
Find the inverse Laplace transform for the given function F(s)=(s^2+7s+2)/((s+1)(s+3)(s+5))
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