1. Given that r = rî = xâ+ yŷ + z2 and that f(r) is a scalar function, prove the following relations:

ASSIGNMENT: MATLAB Using Matlab software, do the analysis on the system in Figure 1. 1. If P-Controller, G₁ = K, Obtain the Value of K for damping ratio, & = 0.707? Calculate the maximum overshoot. What is the natural frequency of the system? Plot the time response of this system if it is excited with a step input signal. 2. If I-Controller, Gc= is used, what is the value of K; for damping ratio, { = 0.707? Calculate the maximum overshoot. What is the natural frequency of the system? Plot the time response of this system if it is excited with a step input signal. 3. If PI-Controller, G₁ = K(s+K) is used, what is the value of K and K, for damping ratio, { = 0.707? Calculate the maximum overshoot. What is the natural frequency of the system? Plot the time response of this system if it is excited with a step input signal. 4. Superimpose all the three output responses obtained above and make comparison. Discuss the effect of K; on the root locus and the transient response. U(s) + Gc Figure 1 1 (s + 1)(s + 2)(S + 4) Y(s)

Question 1 20 pts For cruise control, the longitudinal motion of a vehicle on a flat road can be modeled by the first-order nonlinear differential equation mvu - Kv - Kav², where m is the vehicle's mass, v is its speed, u is the tractive force generated by the engine, is the viscous friction force, and is the aerodynamic drag. Suppose m = 4500 lbs, Kf = 2.5N/(m/s), and K = 0.8N/(m/s)². (3+7 + 10 = 20 pts) 1. Define equilibrium values for all variables of interest for the desired equilibrium point where the equilibrium speed is ū=65 mph. 2. Linearize the system around this equilibrium point, defining all variables in the equation clearly, and specify the linearized transfer function (numerical values). 3. Suppose that the custom is at/n3. Suppose that the system is at equilibrium (at 65mph), and the road grade suddenly increases to 3% (see Elevation Grade Calculator (omnicalculator.com) if you need help with this term). The equations will now change to mi = u - K₁v-K₁v² - mgsin(0), and the speed will drop. Can you use linearization to design a control law of the form u=ū+ Au, Au=-kAv, Av=v- i , where k is the gain of the controller that you will choose by "experimentation" (too small a gain and it won't have much effect, too large a gain might cause the throttle to saturate in a real-world scenario) to bring the car speed back (close) to the equilibrium value of 65mph ? If so, show me a simulation for 10 min of the system where the grade abruptly changes from 0 to 3% at 5mins, and then drops back to zero at 7mins. Include 2 subplots, one where there is no feedback control. I.e.. Ava - SO/nequilibrium value of 65mph ? If so, show me a simulation for 10 min of the system where the grade abruptly changes from 0 to 3% at 5mins, and then drops back to zero at 7mins. Include 2 subplots, one where there is no feedback control, i.e., Au = 0, so that u =ū, and a second subplot where you have designed the above controller for an appropriately chosen gain k. Are you able to make the error converge to zero? Why do you think the above controller is unable to do so ? Later on in the course, we will see how to use integral control to make the "steady-state error" (i.e, the error as t → ∞o) zero. Upload Choose a File

Problem 1. (25 pts) Use the AE5031_HMW1_1.m code to solve the ODE df/dt = -f, f(0) = 1. a) [5 pts] Find the exact solution. b) [10 pts] Use the code for various timesteps 4t and calculate the error= | Exact- Numerical/Exact as a function of the Dt at time time t = 1. c) [10 pts] Plot the error versus the At on a logarithmic scale to confirm that the error decreases proportionally to the 4t (1st order accuracy)

3. Go to the [File] menu and move down to [New] and select [M-file]. An M- file is just a text file that contains MATLAB commands. Type in the commands given below, which create a function called GetLine, for returning the equation of a line that connects two input points. function [y,A,B) GetLine (x1, y1, x2, y2); function [y, A, B] = GetLine (x1, y1, x2, y2); If only one set of coordinates is entered, assume the second point is the origin at (0,0). if (nargin=-2) x2 = 0; y2 = 0; end m = (y2-yl)/(x2-x1);&m=slope of line by-axis intercept This function takes two sets of points: (x1, y1) and (x2, y2) and returns the equation of the line connecting them in the variable y. A and B are optional parameters that represent the smallest and largest x coordinates of the points entered. by2m*x2; syms x; y = m*x + b; A B return W- min (x1, x2); max (x1, x2); After you have typed in the commands, save this file as GetLine.m in the directory you created called mymatlab. When calling this function, notice that it returns three variables. If only one variable is given to hold the return value, only the formula of the line is returned. For example: » w = GetLine (2, 5, 1,7) A- -2*x+9 But if the lower and upper limits are needed, then call the functions using: » [w, A, B] = GetLine (2, 5, 1,7) -2*x+9 B = Define x as symbolic y the equation of line 1 A smallest x coordinate B = largest x coordinate 2

2. Start running MATLAB at your computer. The first window that MATLAB loads up is the command window, which is blank except for the prompt »>. Change the current working directory to the mymatlab directory, i.e. type: >> cd c:\temp\mymatlab

The simple all-revolute 4-bar linkage shown below has an input angle of 60 degrees. Write a MATLAB.m file that performs the Newton-Raphson's method to solve for 3 and 4. Submit your code and your answer for full credit. 90'0 2 0.15 m 0.18 m 3 4 0₂ 3 0.08 m Ө.

Throughout the next several weeks you will be developing a computer program to perform a complete analysis of a four-bar linkage. This current assignment only requires the position analysis. Velocity, acceleration, and force analyses will be added in the future. You may use any programming language. I suggest that you will want a procedural language (eg, Matlab, C++, Fortran). Input the rigid dimensions of a four-bar linkage. Your program should be flexible enough to readily accept any dimensions, ie, it should not be hard-wired for only one specific set of dimensions. Perform a position analysis of the linkage for the complete range of motion of the input link. Allow for choice of form of assembly. Inputs: R₁, R₂, R3, R4. Bp. p.Open or crossed, increment of 2 Outputs: 3, 4, and the absolute position of P, all tabulated for the entire range of 2 (b) (c) Rp (d) R₂ Homework 4 ME 3313 Fall 2023 Assigned: 9/5/2023 Bp Y answers. Program listing: P 0₂ 03 R₁ Each programming submission should include the following, submitted as a scanned image or pdf file: (a) R4 04 X Cover memo: This memo should briefly summarize what follows. Describe any known bugs or incorrect This is a text document of the program code, not the MATLAB executable. The program must be documented. Make it easy to understand what is going on. Sample run: Show a sample run with the output of the program using the following data: R₁ = 0.6 m; R₂= 0.2 m; R3 = 0.3 m; R4= 0.4 m; Bp = 0.2 m; p= 35° Verification of answers: It is important to verify your answers. In addition to the penalty for incorrect answers, you can lose another 10% for not knowing that your answers are incorrect. There is no excuse for not checking the validity of your numbers. Don't skip or skimp this part of the assignment. Include documentation that shows your verification. You will want to spot check the results for at least one position, including open and crossed configuration. You will also want to confirm that your range of motion is correct.

Problem #1: Finite element beam problem L=1m Radius of the beam r=2 cm, and elastic modulus E = 200 GPa. Assume the beam is undergoing a small linearly elastic deformation. 8 a) Using MATLAB, analytically compute the deflection at the free end of the beam using the following equation: P = 100 N Px² 6EI (3L - x) b) Using the finite element method (FEM) with five beam elements of equal length: Generate the individual stiffness matrices K₂ for each element of the beam and assemble them into the global stiffness matrix Kg. Print each of the individual matrices and the global stiffness matrix. Calculate and print the displacements and rotations (in degree) for each node of the beam. Calculate and print the reaction forces and moments at each node of the beam. c) On the same figure, plot the vertical deflection throughout the beam for both analytical and FEM solution. Include title, axis labels, legends, and grid. Compare and discuss the two solutions. d) Discuss how you would increase the bending stiffness of the beam when the cross-sectional area and material must be kept the same. e) Perform Model (Frequency) Analysis of the beam in Solidworks. Take screen shot of the first three modal shapes along with their frequencies (make sure the values are readable). f) Validate the 1st natural frequency from part (e) using analytical modal analysis for cantilever beam

Problem 4. (10 pts) Determine the approximate forward difference representation for which is of the 0 (Ax), given evenly spaced grid points as shown in Figure below by means of: a) Taylor series expansion b) Forward difference operators X₂ f₁+2 X1+2 AX-XX X1+5 X

Problem 3. (15 pts) Consider the advection equation du + c(x)=0 with c(x) = x - 1/2 a) [10 pts] Find and sketch the characteristic for the computational domain 0 < x < 1 and 0 < t < 1 b) [5pts] Find the correct number of initial and/ or boundary conditions to specify at x = 0,1 and t = 0,1