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Question 35641

posted 1 years ago

1. (2.5 points) Solve the following ODE using the Laplace Transform approach.
\ddot{y}(t)+7 \dot{y}(t)+10 y(t)=4
y(0)=\dot{y}(t)=0
\text { Note: } \mathcal{L}(1)=\frac{1}{s} \text { and } \mathcal{L}\left(e^{-a t}\right)=\frac{1}{s+a} \text {. }

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Question 35644

posted 1 years ago

Problem 4. The closed-loop system is shown below. WVe want to draw its root-locus and design the positive constant K to achieve closed-loop stability.
Find the departure angles at pi and p2, and the arrival angle at 21.
\text { Departure angle: } \phi_{d e p}=\sum_{j=1}^{m} \psi_{j}-\sum_{i \neq d e p}^{n} \phi_{i}-180(2 k+1)
\text { Arrival angle: } \psi_{a r r}=\sum_{i=1} \dot{\phi}_{i}-\sum_{j \neq a r r}^{m} \psi_{j}+180(2 k+1)
(2) (2.5 points) Find the range of K for closed-loop stability using the Routh stability criterion.

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Question 35643

posted 1 years ago

whcre K1 and K2 arc the positive constants.
(2) (2.5 points) Determine K, and K2 such that wn = 4 rad/sec, and t, = 1 sec. Note:uhere u and t are the natural freguency and damning ratio respectivel:
Consider the closed-loop control system shown below:
Derive the closed-loop sensitivity function: S(s) = E(s)/R(e).
t_{s}=\frac{4}{\omega_{n}} \text {, where } \omega_{n} \text { and } \zeta \text { are the natural frequency and damping ratio, respectively. }

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Question 35645

posted 1 years ago

Problem 5. (5 points) Consider the dynamic system that has negative real poles only.Determine the transfer function from the asymptote of the Bode magnitude plot shown below:

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Question 35642

posted 1 years ago

Problem 2. (1) (2.5 points) DErive the Equations of motion of a quarter-car model shown below. (2) (2.5 points) Obtain the state-space model. (3) (2.5 points) Obtain the transfer function: G(s) = Y(s)/R(s).

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Question 36903

posted 1 years ago

IFigure P10.1 shows a general feedback control system with forward-path transfer functions Ge(s) (controller) and Gp(s) (plant) and feedback transfer functions H(s). Given the following transfer functions,determine the closed-loop transfer function T(s) = Y(s)/R(s).

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Question 36905

posted 1 years ago

10.4 Figure Pl0.4 shows a closed-loop control system.
a. Compute the controller gain Kp so that the undamped natural frequency of the closed-loop system is w, = 4 rad/s.
b. Compute the controller gain Kp so that the damping ratio of the closed-loop system is = 0.7.
Compute the steady-state output for a step reference input r(t)=4u(t) and controller gain kr=2

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Question 36904

posted 1 years ago

Figure P10.3 shows a general closed-loop control system. The plant transfer function is
a. Determine whether the closed-loop system is stable for control gain K, 2.
b. Compute the controller gain Kp so that step response shows 25% overshoot.
c. Estimate the settling time for a step reference input if the control gain is K, 0.5.
G_{p}(s)=\frac{1}{s^{2}+6 s+8}

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Question 33982

posted 1 years ago

Consider again the simple RL circuit shown in Fig. 9.5 (Example 9.1). The transfer function of the RLeircuit is
G(s)=\frac{I(s)}{E_{\text {in }}(s)}=\frac{1}{L s+R}
where the output is current /() and the input is source voltage e (). If the system parameters are L 0.02Hand R = 1.5 2, determine the bandwidth (in hertz, Hz) of the RL circuit.

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Question 33983

posted 1 years ago

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.

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