Project 4 Appendix

One End Insulated, One End Fixed Temperature In this model, one end of the wire is maintained at a fixed temperature (say 0°C), while the other end is insulated. du/dt = k d2u/dx2 Where k > 0 is a constant (the thermal conductivity of the material). Boundary Conditions and Initial Condition: At both ends of the wire, the temperature is fixed at zero: u (0,t) = 0 and u(L,t) = 0 The initial temperature distribution along the wire can be some function f(x), where: u(x,0)=f(x), for 0≤x≤L, u(x, 0) = f (x), foro≤x≤L 0 temperature u L x insulation

'p' (liquid density 'A' (tank cross sectional area) 'Qin' (Input Flow Rate) "Que P 'R' (restriction coefficient) '' (head of liquid) dh Qin = A + dt Figure 1 The single tank system (Figure 1) has been modelled by the first order differential equation given as equation. The equation describes the relationship between the input flow rate entering the tank and the head of liquid in the tank. ph R 5 m equation The following constants are provided: R = 40 Kpa s m², 4 = 10 m², p= 1001 kg m², and g = 9.81 m s A pump is suddenly switched on and provides a step input flow rate of 0.5 m³s¹. (1) Using Laplace Transforms, solve equation and provide an expression to show how the level in the tank will vary in time after the step input has been applied.

1. Verify Eqs. 1 through 5. Figure 1: mass spring damper In class, we have studied mechanical systems of this type. Here, the main results of our in-class analysis are reviewed. The dynamic behavior of this system is deter- mined from the linear second-order ordinary differential equation: where (1) where r(t) is the displacement of the mass, m is the mass, b is the damping coefficient, and k is the spring stiffness. Equations like Eq. 1 are often written in the "standard form" ď²x dt2 r(t) = = tan-1 d²r dt2 m. M +25wn +wn²x = 0 (2) The variable wn is the natural frequency of the system and is the damping ratio. If the system is underdamped, i.e. < < 1, and it has initial conditions (0) = zot-o = 0, then the solution to Eq. 2 is given by: IO √1 x(1) T₁ = +b+kr = 0 dt 2π dr. dt ل لها -(wat sin (wat +) and is the damped natural frequency. In Figure 2, the normalized plot of the response of this system reveals some useful information. Note that the amount of time Ta between peaks is…

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): 1. What is the order of this system?

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.

Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…

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b₂ k₂ www Radial direction Disk Disk rotation Parameters for the Optical Disk Drive System Laser k₁ www Pick-up head (PUH) Cart Spindle motor Optical Disk Drive System Parameter b₁ PUH mass, m Chassis PUH suspension stiffness, k PUH suspension friction, b, Chassis/cart mass, m Chassis suspension stiffness, k, Chassis suspension friction, by Numerical Value 6(10) kg 60 N/m 0.03 N-s/m 0.124 kg 1960 N/m 6.23 N-s/m Frame Rubber vibration Cart servo motor isolation mounts

3. A perfectly mixed stirred-tank heater, with a single feed stream and product stream, as shown below. F is the volumetric flowrate and T is the temperature. Given that, the density, p and heat capacity, C of the liquid is constant. There is also no heat loss from the process. Develop a dynamic model, dT/dt of the process that involved with mass and energy conservation principle. F F

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Manufacturing process with two variables x1,x2 described by the empirical model:   y=bo +b1 x1 + b2 x2 + b12 x1 x2 + b3 (x1)^2 +b4  (x2)^2    please refere to the image attached

2) A device used in a ground radar system has age to failure that is described approximately by a Weibull distribution with mean life 83 h, shape parameter 1.5, and location parameter zero. When it fails it takes on average 3.5 h to repair: a) Calculate the reliability over a 25 h period, and the 'steady state' availability of the device. b) Calculate the reliability over 25 h, and the ‘steady state' availability of a subsystem that consists of two of these devices in active parallel redundancy.

The relative displacement u(t) of a single-storey shear building subjected to an earthquake ground motion is represented by the following second-order linear ordinary differential equation: d?u dt2 du + c + ku = a, (t) m dt where m, c, and k are the mass (kg), damping constant (Ns/m), and stiffness of the structure (N/m), respectively. Meanwhile, a,(t) is the function of earthquake ground acceleration. Suppose the building with a mass of 2000 kg and supported by columns of combined stiffness of 32 x 103 N/m is subjected to earthquake with ground acceleration given by the following function: ag(t) = 36000 cos 2t Find the equation of the displacement, u(t), given the damping is not installed to the building. Then, find how much the building is displaced from t =30 seconds to t =60 seconds of earthquake.

An unmanned submarine is equipped with temperature and depth measuring instruments and has radio equipment that can transmit the output readings of these instruments back to the surface. The submarine is initially floating on the surface of the sea with the instrument output readings in steady state. The depth measuring instrument is approximately zero order and the temperature transducer first order with a time constant of 50 seconds. The water temperature on the sea surface, , is 20 °C and the temperature at a depth of x meters is given by the relation:  If the submarine starts diving at time zero, and thereafter goes down at a velocity of 0.5 meters / second, draw a table showing the temperature and depth measurements reported at intervals of 100 seconds over the first 500 seconds of travel. Show also in the table the error in each temperature reading.What temperature does the submarine report at a depth of 1000 meters?

Simulink Assignment- Filling a Tank In fluid mechanics, the equation that models the fluid level in a tank is: dh 1 -(min Ap - mout) dt h is the water level of the tank A is the surface area of the bottom of the tank pis the density of the fluid Mim and mout are the mass flow rates of water in and out of the tank, the flow out is a function of the water level h: 1 mout Vpgh Ris the resistance at the exit. Assume that the exit is at the bottom of the tank. g is the gravitational acceleration What to Do: 1. Build a model in Simulink to calculate the water level, h, as a function of time. Your model should stop running as soon as the tank is full. 2. Assuming the height of the tank is 2 meters, after how many seconds does the tank fill up? Use the following constants to test your model: • A: area of the bottom of the tank = 1 m? • p: the density of fluid = 1000 kg/ m? • R: the resistance at the exit = 3 (kg.m)-1/2 • ho: Initial height of the fluid in the tank = 0 m • g: the gravitational…

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Case study: Roots of equation (use any method under finding roots of equation) Problem: Mechanical engineers, as well as most other engineers, use thermodynamics extensively in their work. The following polynomial can be used to relate the zero-pressure specific heat of dry air, cp kJ/(kg K), to temperature (K): C, = 0.99403 + 1.671 × 10 *T + 9.7215 × 10 *7² -9.5838 × 10 "T³ + 1.9520 × 10¯Upt Determine the temperature that corresponds to a specific heat of 1.2 kJ/[kg K).

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Problem 2. A linearized state space model of a three blade speed wind turbine with a 15 m radius working at 12 m/s wind-speed and generating 220 V is -5 5 1 -10.5229 -1066.67 -3.38028 23.5107 993.804 3.125 -23.5107 Wgm 10 -10 Wgm y = [ 0 0 0 1.223 × 10° 0] Wgm where 3 is the pitch angle of the wind turbine blades, { is the relative angle of the sec- ondary shaft, i is the angular velocity of the secondary shaft, wg is the generator speed, Wgm is the generator measurement speed, u is the pitch angle reference, and y is the active power generated. (a) What is the maximum number of poles that the transfer function could have? How many poles does it have?

The Gilles & Retzbach model of a distillation column, the system model includes the dynamics of a boiler, is driven by the inputs of steam flow and the flow rate of the vapour side stream, and the measurements are the temperature changes at two different locations along the column. The state space model is given by: x = 0 00 -30.3 0.00012 -6.02 0 0 0 -3.77 00 0 -2.80 0 0 Is the system?: a. unstable b. C. not unstable x+ 6.15 0 0 0 0 3.04 0 0.052 not asymptotically stable d. asymptotically stable -1 u y = 0 0 0 0 -7.3 0 0 -25.0 X

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Determine the coefficients (a and b) for the baseflow (Qb) in using the data below. Find values for a and b that satisfy the given equation shown below.  Present the figures or plots utilized during the estimation and determination of a and b. Compare the obtained b value with theoretical values such as b = 1 and b = 1.5 for large time intervals (t), and b = 3 for small time intervals.

In a linear spring finite element model, as per the finite element model’s sign convention, which one of the following statements is true?   Select one: a. If Fe1 is positive then the spring is under tension b. If Fe2 is negative then the spring is under tension c. If Fe2 is positive then the spring is under compression d. If Fe1 is positive then the spring is under compression Clear my choice

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Consider a cruise control system shown in Figure 1. Where f is the engine force, v is the velocity, fb is the friction force, and fb  = bv. Requirements: Assuming M = 2000 kg, b = 80Nsec/m, and f = 600 N: Find the transfer function of the system.

You are the mechatronics engineer of a manufacturing plant. You decide to perform an analysis on a robot arm of the assembly line with the objective of optimizing its performance. After taking several readings of the speed of the arm’s end effector, you approximate its velocity to the function given below. v(t) = -t4 + 5t3 - 7t2 + 3t + 0.22                                       0 =< t =< 3 where the velocity is in ms-1 Use your knowledge of differentiation to sketch the velocity-time graph, clearly marking the critical points. Using the graph sketched in (a) above, estimate the velocity when t = 1.5 s Calculate the velocity of the function when t = 1.5 s by substituting to the velocity function. Compare this value to the value you estimated in b above.

You are the mechatronics engineer of a manufacturing plant. You decide to perform an analysis on a robot arm of the assembly line with the objective of optimizing its performance. After taking several readings of the speed of the arm’s end effector, you approximate its velocity to the function given below. v(t) = -t4 + 5t3 - 7t2 + 3t + 0.22                                       0 =< t =< 3 where the velocity is in ms-1   d) Knowing that the distance travelled by an object is the area under its velocity-time graph, determine the distance travelled by the end effector on the interval 0 =< t =< 1  by using the mid-ordinate rule. Simpson’s rule correct to 3 decimal places using four intervals. e) Calculate the same distance as in (d) above by using the appropriate definite integral. f) Compare the distances you calculated in (d) and (e) above and comment on the accuracy of the two methods you used in (d)

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The Johnson & Royce model of a distillation column is driven by the inputs of steam flow and its flow rate; the output measurement is the temperature change. The state-space model is given by: -2 x = [¯²¯¯¯7 | x + | | | u; y = [0_1]x - the system defining the model of the distillation column is a. Asymptotically stable O b. Unstable Oc. Marginally stable Od. Indeterminate in relation to its stability

Derive the seven general property equation of the following thermodynamics processes: a. ISOMETRICPROCESS

16:57 A O * l LTE2 i ethuto.cut.ac.za/webapps/asse 27 ¥ Question Completion Status: QUESTION 53 The following processes occurs in a reversible thermodynamic cycle: 1-2: Reversible polytropic compression at pressure 0.6 bar at volume 0.08 m3 to a pressure 10.2 bar and specific volume 0.4 m3/kg. The index of compression may be taken as n. 2-3: Reversibly expansion with expansion index of 2 to pressure 3 bar. 3-1: Reversible cooling at constant volume to the initial state. Calculate the value of n in the process to 3 decimal places. Click Save and Submit to save and submit. Click Save All Answers to save all answers. Save All Answers Close Window Save and Submit