System Dynamics Homework Help

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:

If the equations are linearized around the normal operating point, they can be put into matrix form as \dot{\hat{q}}=A \hat{q}+B \hat{u} where q=\left[q_{1} q_{2} q_{3}\right]^{T} What is the value of the A matrix?

1. Consider the closed-loop system shown below: where K, > 0, K, > 0 and T > 0. Determine stability of the closed-loop system using Routh's stability criterion.

Problem 5. In the Problem 4, calculate the steady-state errors with the unit step (r(t) = 1)(5 points) and the unit ramp (r(t)3Dt) (5 points) references.

Problem 1. (a) A first order instrument is described by the following mathematical model: 2 \frac{d y}{d t}+30 y=120 The time constant of the system is 15 seconds. (b) The following spring-mass-damper system is: * Zero order * 1st Order * 2nd Order * 3rd Order * YES * NO (c) The natural frequency of the above system is: \omega_{\mathrm{n}}=\sqrt{\frac{M}{K}} * YES * NO (d) The damping ratio of the above system is: \zeta=\frac{f_{v}}{2 \sqrt{K M}} * YES *NO (e) A thermometer is an analog device (f) A pressure gauge is a digital device * YES * NO (g) The following signal has a frequency of 10-Hz. * YES * NO (h)A design stage uncertainty analysis is performed when there is a finite set of data available. * TRUE * FALSE

4. Give the differential equation to model the height h1 of the fluid system below. (Include cases where h1 > D and h1 < D.)

2. For the fluid system below а.Give the dynamic systems model for h1, h2, and h3 in state variable matrix form. b. Determine the height of the water in each tank in steady state, assuming that A1 = 2, A2 = 3,A3 = 1, R1 = R2 = R3 = 1, qmi = 5 kg / s, g = 10 m² / s, p = 0.1 kg / m³.

The thin disk below turns on a massless arm B by way of a smooth bearing that keeps the axes of D and B aligned. The arm is hinged to the shaft driven at a constant angular speed N by a motor. The system is set up so that the arm is horizontal when the disk contacts the ground as shown. Assuming the disk of mass m rolls without slipping, find the force reactions and the moment reactions exerted by the arm on the bearings at point C in terms of the given dimensions, the mass,the normal force acting on the disk, the friction forces acting on the disk and the angular speed N.

Recall, the Eulerian angles that we defined in class as shown below. The axes (i, j, k) are fixed in body frame B and the axes (Î, Ĵ, K)of B with respect to mathF is represented through the angles (ø, 0, v) using a sequence about intermediate z, intermediate y and intermediate z-axis again to obtain body-fixed frame B in the final configuration, from the inertial reference frame F. This is often referred to as 3-2-3 sequence are fixed in the inertial reference frame F. The orientation or z-y-z sequence. You are hired as an control and navigation engineer at a satellite manufacturing firm. The satellite is equipped with thrusters which can rotate it about all the possible axes in the intermediate body-fixed frame at any given instant instead of just the y and z-axes. Now, there is a requirement to use a 2-1-3 sequence or the y-x-z sequence of successive rotations about the intermediate axes, \boldsymbol{F} \stackrel{\hat{J} \text { or } \hat{n}_{12}, \phi}{\longrightarrow} \mathbf{F}_{1} \stackrel{\hat{n}_{11} \text { or } \hat{n}_{21}, \theta}{\longrightarrow} \mathbf{F}_{2} \stackrel{\hat{n}_{23} \text { or } \hat{j}, \psi}{\longrightarrow} \mathbf{B} 1. Write down the individual rotation matrices for each of the Eulerian angles, i.e. Tó, Tạ, Ty. 2. Write the simplest angular velocity vector wB/F in terms of Euler angles according to the2-1-3 rotation sequence. Remember, the simplest expression is always written using basis vectors of intermediate reference frames. 3. Express the angular velocity vector wB/F in the second intermediate reference frame, F2, i.e.in terms of (în21, Ñ22, îÑ23)

Problem 3. In the Problem 2, find the range of K such that the closed-loop system is stable using Routh's stability criterion if it exists.

Knowing that the tension in cable BC is 805 N, determine the resultant of the three forces exerted at Point B of beam AB. The magnitude of the resultant of the three forces is C The direction of the resultant of the three forces is y

5. Consider the system shown in Fig. 5, and derive an equation of motion for the block of mass mi. Assume that: m1 < m2 and that both pulley shave negligible friction and negligible inertia.

Knowing that a = 21°, determine the resultant of the three forces shown. The resultant of the three forces is----NYO°----. (Please provide the angle with reference to the negative x-axis.)

3. A car moving with a constant speed on a straight road is approaching an intersection.The driver hits the brake. By looking at the skid marks, it is evident it took 30 ft to the car to stop. Considering that the speed limit on that road was 45 mph, was the car speeding?Consider the friction coefficient of asphalt to be 0.65. Show your calculations.

7. Find the natural frequency of the system shown below. Show your calculations.

1. Find the Inverse Laplace Transform of the following functions: \text { a. } F(s)=\frac{5 s+2}{(s+2)(s+1)^{2}} \text { b. } F(s)=\frac{2 s^{2}+4 s+2}{s(s+2)} \text { c. } F(s)=\frac{s+w}{s^{2}\left(s^{2}+w^{2}\right)} Show your calculations.

4. Consider the pulley system shown in Fig. 4, and derive an equation of motion for the mass m1. Assume that: m1 > m2 and that the pulley 1 has negligible friction and negligible inertia – that is, the tensions on the cable are the same on both sides of this pulley. The pulley with moment of inertia I2 has radius R2.

The setup for controlling a radio telescope dish is shown in Figure Q2. The desired angular position is given by 0,(s), while the actual angular position is given by 0(s). The error between actual and desired position is amplified and used to actuate a motor. The dish is acted upon by the torque provided by the motor Um (s), and also by torque from external wind disturbances Ua(s). The actual position 0 (s) is measured and fed back by a sensor which has gain K2. Derive the closed-loop transfer function of the system when there are no wind disturbances, theta (S)/thetar (s) Show that if there is a unit step change in 0,r, then the condition for zero steady-state error in position is K1 = K,K2 + 20 In addition, it is desired that a steady-state error of exactly 0.1 rad/s is observed-when the desired position of the dish is adjusted by a ramp input of 4 rad/s. By-substituting the condition derived in part (b) above into the closed-loop transfer-function, calculate suitable values for the gains K, and K2. Calculate the steady-state position 0ss resulting from a step change in wind disturbance torque Ua of 5Nm when the reference position 0, is zero.(Find an expression in terms of the gains K1 and K2, and then substitute the values from part (c))