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System Dynamics

Problem 2.

(a) A tachometer has an analog display dial graduated in 5 rpm increments.The user manual states an accuracy of 1% of reading. Estimate the uncertainty in the reading at 10 rpm, 500 rpm and 5000 rpm.

(b) A certain obstruction type flow meter (orifice, venturi, nozzle), shown in the following figure is used to measure the mass flow rate of air at low velocities. The relationship describing the flow rate is:

\dot{m}=C A \sqrt{\left[\frac{2 g_{c} p_{1}}{R T_{1}}\left(p_{1}-p_{2}\right)\right]}

where, * C = empirical-discharge coefficient. * A = flow area * T1 = upstream temperature * R = gas constant for air * Pi and p2 = upstream and downstream pressures, respectively.

Calculate the percent uncertainty in the mass flow rate for the following conditions:

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System Dynamics

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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System Dynamics

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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System Dynamics

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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System Dynamics

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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System Dynamics

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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System Dynamics

(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:

whcre K1 and K2 arc the positive constants.

Derive the closed-loop sensitivity function: S(s) = E(s)/R(e).

Consider the closed-loop control system shown below:

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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System Dynamics

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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System Dynamics

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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System Dynamics

a. Obtain the 1/O equation for this system where y is the output and u is the input.

a. Obtain the 1/O equation for this system where y is the output and u is the input.

\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

b. Obtain the transfer function for this system.

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System Dynamics

Given the following system equation

4 z+20 z+84 z=0.12 u

b. Derive the system transfer function G(s) = Z(s)/U(s)

a. Obtain a complete SSR with input u and output y 2.

Derive the transfer function Y(s)/U(s) where the output is y = ż.C.

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**Use of solution provided by us for unfair practice like cheating will result in action from our end which may include permanent termination of the defaulter’s account.Disclaimer:The website contains certain images which are not owned by the company/ website. Such images are used for indicative purposes only and is a third-party content. All credits go to its rightful owner including its copyright owner. It is also clarified that the use of any photograph on the website including the use of any photograph of any educational institute/ university is not intended to suggest any association, relationship, or sponsorship whatsoever between the company and the said educational institute/ university. Any such use is for representative purposes only and all intellectual property rights belong to the respective owners.

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