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Q1:Problem 5: System of Nonlinear Equations (MATLAB) Given a system of nonlinear equations: u(x, y, z) = x(x + 2y + 5z - 4) = 0, v(x, y, z) = y(5x + y + 2z - 4) = 0, w(x, y, z) = 2(2x + 5y +z - 4) = 0, (5) (6) (7) 1. Derive all real solutions to this system of equations by hand. Clearly show your work. 2. Write a MATLAB script to obtain the root of this system of non-linear equations using the Newton-Raphson method. Use an initial guess of (x, y, z) = (4.4,-0.8,0.4). How many iterations are needed for the approximate error ca to go below € = 0.1%? 3. Use your solution to Problem 5.2 to generate all the analytical solutions from Problem 5.1 by modifying your initial guesses for (x, y, z).See Answer
Q2:Problem 3: Newton-Raphson method (MATLAB) Use the Newton-Raphson method to find the root of f(x) = c 0.5x (4-x) - 2. Employ initial guesses of ₁,1 = 2, ₁,2 = 6, and ₁,3 = 8. Explain your results. (3)See Answer
Q3:3. Develop a function that implements the Newton-Raphsom method, called NRM_[your last name].m, and upload it to canvas.See Answer
Q4:2. Develop a function that implements the Secant method, called Secant_[your last name].m. The initial guesses should be input as a vector such that if the user inputs [xl, xu] it uses the regular secant method, if the user inputs a single value, [x0], the program uses the modified secant method with. Upload the .m file to canvas.See Answer
Q5:4. Develop a function that implements the Inverse Quadratic Interpolation method, called IQI_[your last name].m, and upload it to canvas.See Answer
Q6: Given the following block diagram, with G_{f}=10, G_{c}=20, G_{p 1}=\frac{2.0}{1.0 s^{2}+10.0 s+20.0}, G_{p 2}=\frac{5.0}{1.0 s^{2}+5.0 s+25.0}, H=1.0 s+6.0 а.(50%) Convert this system into state-space form, accounting for the multiple inputs and out-puts. There are several ways to transfer the block diagram into state-space, but let's try to remodelthe state-space from the beginning. \text { - The plant transfer functions give us the plant dynamics } \dot{x}=A x+B u \text { and } u=[u d]^{\wedge} T^{\prime \prime} The sensor H(s) relates the state to the output: y = Cx- - The feedback defines the control law: U(s) = G.(Gf * R(s) - Y(s)), which can be substituted intothe plant dynamics to find the closed-loop form. С.(50%) Simulate the response of the system using basic numerical integration. To a- Reference: r(t) = 5 - Random Noise, normally distributed, zero mean with o = 2: d(t) = 2 * rand() Note you already have the model of the system from the state-space, and this is a linear system.You can calculate the derivative and propagate the system directly. See Answer
Q7: Given a 2-DOF system with the following nonlinear equations of motion 3 x_{1}-2 x_{2}+12 \dot{x}_{1} \dot{x}_{2}-11 \dot{x}_{1}+20 \ddot{x}_{1}=5 f_{1}+3 f_{2} 2 x_{1}+0.9 x_{2}^{2}+12 \dot{x}_{2}+11 \ddot{x}_{2}=f_{1}+5 f_{2} a. (25%) Put the nonlinear equations in a vector form i = f(x, t, u). b. (25%) Simulate the natural response of the nonlinear system with the following initial condi- \text { tions } x_{0}=\left[\begin{array}{llll} 0.0 & 0.05 & 0.05 & 0.0 \end{array}\right]^{T} \text { for } t=0: 15 s c. (25%) Linearize the system (ignore the square and coupling terms) and put the system into the state space form x = Ax + Bu d. (25%) Design a full-state feedback controller using pole placement method, to stabilize the system. Test your controller on the linearized system first using Isim, then simulate the closed-loop system response using the linear and nonlinear model in a numerical integration setup.Assume you can control both inputs (What is the dimension of the gain matrix?) Your controller is a regulator, so observe the closed-loop response with r = 0 and initial conditions xo = | 15.0 0.7 0.510.0 Plot the response of all the 4 states, for both the linear and non-linear system in one subplot.Can you control the non-linear (actual) system with the same controller? Explain why or why not.See Answer
Q8: On February 18th, 2021, NASA's Mars 2020 Preserverance Rover is planned for landing on Mars.A landing animation video can be seen here http://bit.ly/PreserveranceThe Sky Crane, which is responsible for gracefuly landing the rover on the designated site, can be modeled as a rigid body with thruster forces being controlled by a gimbal to produce thrusta t angles 0 as shown. a. (20%) Derive the equations of motion for the system (3 directions)The thruster angles and thrust forces are all independent input variables now.We wish to design a state-space controller to help the SkyCrane navigate to a desired position in space. b. (30%) Linearize the system and put it in state-space form, then design a controller via pole placement to achieve the following transient response characteristics - Ts = 2 s -\omega_{d}=20 \mathrm{rad} / \mathrm{s} c. (25%) Apply your controller on the nonlinear system (numerical integration) d. (25%) In reality, there are limits to the inputs. The thrust can not be negative, and there is a minimum thrust once the engine is ignited. The thruster gimbal can only operate within a specific angular range. Repeat part c with the following saturation limits: - Thrusts: 100N < F < 2000N Gimbal Angle Range: -45° < 0 < 45°-See Answer
Q9: Rise:0~25 mm in 90° Dwell:at 25 mm in 45° Fall:25-0 mm in 90° Dwell:for the remainder Set up appropriate boundary conditions and determine the coefficients C, for the 7th-degree polynomial s=\sum_{i=0}^{7} C_{i}\left(\frac{\theta}{b}\right)^{n} \text { where } \theta \text { is the cam angle, } b \text { is the total angle of any segment, rise, fall, or dwell } (b) Develop a MATLAB program to compute and plot svaj diagrams, assume the cycle takes 2 sec.Parameters: The mass of the follower is 1.0 kg. The spring has a rate of 10 N/m, a damping ratio of 0.10,and a preload of 1.0 N. Find the follower force over one revolution. If there is a follower jump, re-specify the spring rate and preload to eliminate it. 1: Design a double-dwell cam to move a follower:See Answer
Q10: Problem 2: The figure shows a 4-bar linkage. Write a Matlab calculate and plot the angular displacements, angular velocities and angular accelerations of links 3 and 4, and x and y acceleration components of the coupler point P over the maximum range of motion possible. Given: w2 = 20rpm and az =0.program to See Answer
Q11: Problem 5:Write an M-file to evaluate the definite integral below using the composite trapezoidal rule with 1, 2 5,and 10 intervals. Compute the true relative percent error (s.) for each result. Provide a printout of your m-file and the command window showing the results. I=\int_{a}^{n}(8+4 \cos x) d xSee Answer
Q12: (20 pts) Find view factor between each surface for the given enclosures. A bar in the shape of an Isosceles triangle 30 mm on each side, centered inside a D = 40 mm tube: A rectangular enclosure with dimensions as shown: A regular pentagonal enclosure as shown: A hemispherical shell over a 25 mm circular disk: See Answer
Q13: Controller Design via Pole Placement (20pts) \text { Given the state space model } \dot{\mathbf{x}}=\left[\begin{array}{ccc} 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0 \\ -23.0 & -5.0 & -1.0 \end{array}\right] \mathbf{x}+\left[\begin{array}{c} 0.0 \\ 1.0 \\ 0.0 \end{array}\right] u y=\left[\begin{array}{lll} 1 & 0 & 0 \end{array}\right] \mathbf{x} Design a controller via pole placement, to achieve the following performance specifications - Ts = 2s - Š = 0.866 - Zero steady-state error Hint: Design the full-state feedback controller first for the transient response, then add the integral controller.See Answer
Q14: Problem 3: The offset crank-slider linkage has the dimensions given in the figure. Develop a MATLAB program to calculate and plot the accelerations AAand Ag in the global coordinate system for w 25 rad/sec CW and a2= 0rad/sec?. See Answer
Q15:Project Part-1: 1. Implement and test (show execution of) the continuous-time component representing the dynamic model of a car given in the Textbook. Use the following values in the model: m= 1450 kg, -63. Simulate the response for the case F-0, with initial conditions (0)-0, v(0)-15 m/sec; and the case F-550 N with initial conditions x(0)-0), and v(0)=0. Use Trapezoidal discrete approximation of derivative with simulation step At 0.10 sec. Plot the component responses generated from your simulation. 2. Now add the effect of graded road to the above car model and regenerate the car responses to road grade of 0-5deg, and 0-10deg and the case F 5SON with initial conditions x(0)-0, and v(0)-0 only. Plot the component responses generated by your simulation.See Answer
Q16:Q1 a) Use the colon operator to create the following row vector [ 1 2 3 4 5]. Save this as variable a but suppress the output in the command window. b) Use the transpose operator on a to create a column vector b, suppress the output in the command window. c) Calculate a x b using matrix multiplication and display the answer clearly in the command window. See Answer
Q17:Q2 a) Create a 3 x 3 matrix of random integers between 5 and 20. Save this as variable A and display it clearly in the command window. b) Create a copy of A called B (do not display it in the command window yet). Modify B so that the middle element is equal to 50, and display B clearly in the command window. c) Use elementwise operations to calculate 4²/B and display the answer clearly in the command window.See Answer
Q18:Figure explanation (see also comments in the script) The figure above shows three curves plotted in the same figure window. The blue curve y1 is the plot of the rational function The above formula estimates exp(-x) for small values of x. The red curve y2 is the plot of the decaying exponential function exp(-x). Finally, the black dash-dotted line y3 shows a horizontal line with y-coordinate 0.See Answer
Q19:Q2 a) Create a 3 x 3 matrix of random integers between 5 and 20. Save this as variable A and display it clearly in the command window. b) Create a copy of A called B (do not display it in the command window yet). Modify B so that the middle element is equal to 50, and display B clearly in the command window. c) Use elementwise operations to calculate 4²/B and display the answer clearly in the command window.See Answer
Q20:Q1 a) Use the colon operator to create the following row vector [ 1 2 3 4 5]. Save this as variable a but suppress the output in the command window. b) Use the transpose operator on a to create a column vector b, suppress the output in the command window. c) Calculate a x b using matrix multiplication and display the answer clearly in the command window. See Answer

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