## MAE 384: Advanced Mathematical Methods for Engineers Spring 2015 Homework #8 Due: Wednesday, April 8, in or before class. Note: Problems 2 (extra credit) and 3 have to be solved by hand. Problems 1 and 5 require MATLAB. The item 1(a) must be shown by hand. Problem 4 can be done either in Matlab or by hand. 1. Consider the following ODE: d y d x = ?8 y with y(0) = 3 on 0 < x < 5, (a) Calculate the largest step size required to maintain stability of the numerical solution to this equation using explicit Euler method. (b) Choose a step size two times smaller than this value. Solve the ODE with explicit Euler method using this step size. (c) Choose a step size two times larger than this value. Solve the ODE with explicit Euler method using this step size. (d) Now repeat parts (b) and (c) with implicit Euler method. (e) Plot all the solutions, including the analytical solution to this problem, on the same plot. Discuss your results. 2. Extra credit. Investigate the stability of the following numerical schemes on the example of an ODE d y d x = ? y with > 0. Show whether the scheme is conditionally or unconditionally stable. Derive the stability threshold if the scheme is conditionally stable. (a) The semi-implicit trapezoidal method: yi+1 = yi + 1 2 (f(xi; yi) + f(xi+1; yi+1)) h (b) The explicit midpoint method: yi+1 = yi + f xi+1=2; yi + f(xi; yi) h 2 h 3. Solve Problem 25.1 from the textbook with third-order Runge-Kutta (page 734) and fourth-order Runge Kutta (page 735) methods with h = 0:5. Plot your results on the same plot. Also, include results from (a),(b),(c) from the two previous homeworks, on the same plot. 4. Solve Problem 25.2 from the textbook with third-order Runge-Kutta (page 734) and fourth-order Runge Kutta (page 735) methods with h = 0:25. Plot your results on the same plot. Also, include results from (a),(b),(c) from the two previous homeworks, on the same plot. There is a typo in this problem. The interval should be from t=0 to 1, not x=0 to 1. 5. For the following rst-order ODE d y d t = t2 ? 2 y t with y(1) = 2, the purpose will be to write MATLAB functions that solve this equation from t = 1 to t = 4 with 1 of 2 MAE 384: Advanced Mathematical Methods for Engineers Spring 2015 (a) Third-order Runge-Kutta (page 734) (b) Fourth-order Runge-Kutta (page 735) For each method, (a) Write the MATLAB function that solves the ODE by using the number of intervals N as an input argument. (b) Solve the ODE using your MATLAB function for N equal to 8, 16, 32, 64. Calculate the step size h inside the function. (c) Calculate the EL2 errors between the true solution and the numerical solution for each N (consult HW6 for the true solution). The following plots should be presented: 1. Plot your solutions for the methods (a), (b), along with the analytical solution, explicit Euler solution from HW6, and solutions to problem 5 (a) – (c) from HW7, on the same plot for N = 8. Do not print out the values at your grid points. 2. Plot your solutions for the methods (a), (b), along with the analytical solution, explicit Euler solution from HW6, and solutions to problem 5 (a) – (c) from HW7, on the same plot for N = 32. Do not print out the values at your grid points. 3. Plot the values of EL2 errors for the methods (a), (b), as well as for the explicit Euler method from HW6, and solutions to problem 5 (a) – (c) from HW7, as a function of h, on the same plot. What do you observe? 4. Plot the values of EL2 errors for all the methods (a)-(c), as well as for the explicit Euler method from HW6, and solutions to problem 5 (a) – (c) from HW7, as a function of h, on the same plot, but in log-log scale. Discuss how you can estimate the order of convergence for each method from this plot. Estimate the order of convergence for each method. 5. Plot the values of EL2 errors for all the methods (a)-(c), as well as for the explicit Euler method from HW6, and solutions to problem 5 (a) – (c) from HW7, as a function of N, on the same plot, but in log-log scale. Discuss how you can estimate the order of convergence for each method from this plot. Estimate the order of convergence for each method. 6. Discuss whether your convergence results for each method correspond to the known order of accuracy for each method. Explain why or why not. 2 of 2

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## ENGR 2010 (Section 02) – Assignment 7 Due: Wednesday November 25th, 11:59 pm Points: 20 Prof. Lei Reading: Sections 6.2-6.3 of Nilsson and Riedel, Electric Circuits, 9th Edition Submit electronic solutions (i.e. using Microsoft Word or a scanned copy of your written work) to the following problems on Blackboard. To receive credit, you must show work indicating how you arrived at each final answer. Problem 1 Consider the RC circuit on the right. and suppose that Vs(t) is a time-varying voltage input shown at the bottom. a) Suppose VC(0) = 0V. Plot VR(t) and VC(t) from 0ms to 300ms. Show your work in obtaining VR(t) and VC(t). b) Suppose the capacitance value is changed to 2μF, and VC(0) = 0V. Plot VR(t) and VC(t) from 0ms to 300ms. Show your work in obtaining VR(t) and VC(t). c) Explain how VC(t) qualitatively compares with Vs(t), and how VR(t) qualitatively compares with Vs(t). d) Explain how the capacitance value affects VC(t). t Vs(t) 1V -1V 50ms 100ms 150ms 200ms 250ms + – Vs(t) 100000 Ohms 1 uF + – VC(t) + – VR(t) 0ms 300ms Note: Capacitors are often used to protect against sudden changes in a voltage value, which could damage electronic components. Here, Vs(t) undergoes many sudden changes, but VC(t) undergoes less change. Problem 2 Using PSpice, perform two transient analysis simulations – one for the circuit in part (a), and one for the circuit in part(b) of problem 1 – to verify that your plots in problem 1 are correct. For each simulation, plot the traces for VR(t) and VC(t). Hint: You may need to perform arithmetic operations between simulation traces. Take a screenshot of your constructed circuits and the simulation traces for VR(t) and VC(t), which you will submit onto Blackboard. t Vs(t) 1V -1V 50ms 100ms 150ms 200ms 250ms + – Vs(t) 100000 Ohms 1 uF + – VC(t) + – VR(t) 0ms 300ms 1 uF or 2 uF Problem 3 Consider the Resistor-Diode circuit on the right, and suppose that Vs(t) is a time-varying voltage input shown at the bottom. Suppose that for the diode to turn on, it needs 0.7V between the positive and negative terminals. a) Plot VR(t) and VD(t) from 0ms to 300ms b) Explain how VD(t) qualitatively compares with Vs(t), and how VR(t) qualitatively compares with Vs(t). t Vs(t) 1V -1V 50ms 100ms 0ms 150ms 200ms 250ms 300ms + – Vs(t) 100000 Ohms + – VD(t) + – VR(t) Problem 4 Using PSpice, perform a transient analysis simulation for the circuit in problem 3 – to verify that your plots in problem 3 are correct. For the simulation, plot the traces for VR(t) and VD(t). To create the diode in PSpice, use the Dbreak component. After placing the component on the page, highlight the component, and edit the Pspice model (Edit -> PSpice Model) and set Rs to 0. Hint: You may need to perform arithmetic operations between simulation traces. Take a screenshot of your constructed circuit and the simulation traces for VR(t) and VD(t), which you will submit onto Blackboard. Note that your simulation trace plots may not be exactly the same as those from Problem 3, since the PSpice diode model has a turn-on voltage that’s not exactly 0.7V. t Vs(t) 1V -1V 50ms 100ms 0ms 150ms 200ms 250ms 300ms + – Vs(t) 100000 Ohms + – VD(t) + – VR(t) Problem 5 (Bonus: 5 points) In the circuit from problem 1 (shown on the right), write several sentences to explain why VC(t) is often referred to as the “low-pass filtered” output, and VR(t) is often referred to as the “high-pass filtered” output. You will need to look up the definitions for “low-pass” and “high-pass” filters. Examining your plots for VC(t) and VR(t) will help. t Vs(t) 1V -1V 50ms 100ms 150ms 200ms 250ms + – Vs(t) 100000 Ohms 1 uF + – VC(t) + – VR(t) 0ms 300ms

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## 1 Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3.1 Laboratory Objective The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid measurements. 3.2 Educational Objectives After performing this experiment, students should be able to: 1. Understand the properties of sinusoids. 2. Understand sinusoidal manipulation 3. Use a function generator 4. Obtain measurements using an oscilloscope 3.3 Background Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any oscillatory motion can be described using sinusoids. Many types of electrical signals such as square, triangle, and sawtooth waves are modeled using sinusoids. Their manipulation incurs the understanding of certain quantities that describe sinusoidal behavior. These quantities are described below. 3.3.1 Sinusoid Characteristics Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a sinusoid. It carries the physical units of what the sinusoid is describing (volts, amps, meters, etc.). Frequency There are two types of frequencies that can describe a sinusoid. The normal frequency f is how many times the sinusoid repeats per unit time. It has units of cycles per second (s-1) or Hertz (Hz). The angular frequency ω is how many radians pass per second. Consequently, ω has units of radians per second. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 2 Period The period T is how long a sinusoid takes to repeat one complete cycle. The period is measured in seconds. Phase The phase φ of a sinusoid causes a horizontal shift along the t-axis. The phase has units of radians. TimeShift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time measurement of the phase. The time shift has units of seconds. NOTE: A sine wave and a cosine wave only differ by a phase shift of 90° or ?2 radians. In reality, they are the same waveform but with a different φ value. 3.3.2 Sinusoidal Relationships Figure 3.1: Sinusoid The general equation of a sinusoid is given below and refers to Figure 3.1. ?(?) = ????(?? +?) (3.1) The angular frequency is related to the normal frequency by Equation 3.2. ?= 2?? (3.2) The angular frequency is also related to the period by Equation 3.3. ?=2?? (3.3) By inspection, the normal frequency is related to the period by Equation 3.4. ? =1? (3.4) ?? Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3 The time shift is related to the phase (radians) and the frequency by Equation 3.5. ??= ∅2?? (3.5) 3.3.3 Equipment 3.3.3.1 Inductors Inductors are electrical components that resist a change in the flow of current passing through them. They are essentially coils of wire. Inductors are electromagnets too. They are represented in schematics using the following symbol and physically using the following equipment (with or without exposed wire): Figure 3.2: Symbol and Physical Example for Inductors 3.3.3.2 Capacitors Capacitors are electrical components that store energy. This enables engineers to store electrical energy from an input source such as a battery. Some capacitors are polarized and therefore have a negative and positive plate. One plate is straight, representing the positive terminal on the device, and the other is curved, representing the negative one. Polarized capacitors are represented in schematics using the following symbol and physically using the following equipment: Figure 3.3: Symbol and Physical Example for Capacitors 3.3.3.3 Function Generator A function generator is used to create different types of electrical waveforms over a wide range of frequencies. It generates standard sine, square, and triangle waveforms and uses the analog output channel. 3.3.3.5 Oscilloscope An oscilloscope is a type of electronic test instrument that allows observation of constantly varying voltages, usually as a two-dimensional plot of one or more signals as a function of time. It displays voltage data over time for the analysis of one or two voltage measurements taken from the analog input channels of the Oscilloscope. The observed waveform can be analyzed for amplitude, frequency, time interval and more. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 4 3.4 Procedure Follow the steps outlined below after the instructor has explained how to use the laboratory equipment 3.4.1 Sinusoidal Measurements 1. Connect the output channel of the Function Generator to the channel one of the Oscilloscope. 2. Complete Table 3.1 using the given values for voltage and frequency. Table 3.1: Sinusoid Measurements Function Generator Oscilloscope (Measured) Calculated Voltage Amplitude, A (V ) Frequency (Hz) 2*A (Vp−p ) f (Hz) T (sec) ω (rad/sec) T (sec) 2.5 1000 3 5000 3.4.2 Circuit Measurements 1. Connect the circuit in figure 3.4 below with the given resistor and capacitor NOTE: Vs from the circuit comes from the Function Generator using a BNC connector. Figure 3.4: RC Circuit Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 5 2. Using the alligator to BNC cables, connect channel one of the Oscilloscope across the capacitor and complete Table 3.2 Table 3.2: Capacitor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) Vc (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 3. Disconnect channel one and connect channel two of the oscilloscope across the resistor and complete table 3.3. Table 3.3: Resistor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) VR (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 4. Leaving channel two connected across the resistor, clip the positive lead to the positive side of the capacitor and complete table 3.4 Table 3.4: Phase Difference Function Generator Oscilloscope (Measured) Calculated Vs (volts) Frequency (Hz) Divisions Time/Div (sec) ts (sec) ɸ (rad) ɸ (degrees) 2.5 100 5. Using the data from Tables 3.2, 3.3, and 3.4, plot the capacitor sinusoidal equation and the resistor sinusoidal equation on the same graph using MATLAB. HINT: Plot over one period. 6. Kirchoff’s Voltage Law states that ??(?)=??(?)+??(?). Calculate Vs by hand using the following equation and Tables 3.2 and 3.3 ??(?)=√??2+??2???(??−???−1(????)) Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 6 3.5 New MATLAB Commands hold on This command allows multiple graphs to be placed on the same XY axis and is placed after the first plot statement. legend (’string 1’, ’string2’, ‘string3’) This command adds a legend to the plot. Strings must be placed in the order as the plots were generated. plot (x, y, ‘line specifiers’) This command plots the data and uses line specifiers to differentiate between different plots on the same XY axis. In this lab, only use different line styles from the table below. Table 3.5: Line specifiers for the plot() command sqrt(X) This command produces the square root of the elements of X. NOTE: The “help” command in MATLAB can be used to find a description and example for functions such as input. For example, type “help input” in the command window to learn more about the input function. NOTE: Refer to section the “MATLAB Commands” sections from prior labs for previously discussed material that you may also need in order to complete this assignment. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 7 3.6 Lab Report Requirements 1. Complete Tables 3.1, 3.2, 3.3, 3.4 (5 points each) 2. Show hand calculations for all four tables. Insert after this page (5 points each) 3. Draw the two sinusoids by hand from table 3.1. Label amplitude, period, and phase. Insert after this page. (5 points) 4. Insert MATLAB plot of Vc and VR as obtained from data in Tables 3.2 and 3.3 after this page. (5 points each) 5. Show hand calculations for Vs(t). Insert after this page. (5 points) 6. Using the data from the Tables, write: (10 points) a) Vc(t) = b) VR(t) = 7. Also, ???(?)=2.5???(628?). Write your Vs below and give reasons why they are different. (10 points) a) Vs(t) = b) Reasons: 8. Write an executive summary for this lab describing what you have done, and learned. (20 points)

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## MAE 318: System Dynamics and Control Dr. Panagiotis K. Artemiadis MAE 318: System Dynamics and Control Homework 4 Problem 1: (Points: 25) The circuit shown in Fig. 1 is excited by an impulse of 0.015V. Assuming the capacitor is initially discharged, obtain an analytic expression of vO (t), and make a Matlab program that plots the system response to the impulse. Figure 1 Problem 2: Extra Credit (Points: 25) A winding oscillator consists of two steel spheres on each end of a long slender rod, as shown in Fig. 2. The rod is hung on a thin wire that can be twisted many revolutions without breaking. The device will be wound up 4000 degrees. Make a Matlab script that computes the system response and determine how long will it take until the motion decays to a swing of only 10 degrees? Assume that the thin wire has a rotational spring constant of 2 10?4Nm/rad and that the viscous friction coecient for the sphere in air is 2 10?4Nms/rad. Each sphere has a mass of 1Kg. Figure 2: Winding oscillator. Problem 3: (Points: 25) Find the equivalent transfer function T (s) = C(s) R(s) for the system shown in Fig. 3. Arizona State University. Fall 2015. Class # 73024. MAE 318. Homework 4: Page 1 of 4 MAE 318: System Dynamics and Control Dr. Panagiotis K. Artemiadis Figure 3 Problem 4: (Points: 25) Reduce the block diagram shown in Fig. 4 to a single transfer function T (s) = C(s) R(s) . Figure 4 Problem 5: (Points: 25) Consider the rotational mechanical system shown in Fig. 5. Represent the system as a block diagram. Arizona State University. Fall 2015. Class # 73024. MAE 318. Homework 4: Page 2 of 4 MAE 318: System Dynamics and Control Dr. Panagiotis K. Artemiadis Figure 5 Problem 6: (Points: 25) During ascent the space shuttle is steered by commands generated by the computer’s guidance calcu- lations. These commands are in the form of vehicle attitude, attitude rates, and attitude accelerations obtained through measurements made by the vehicle’s inertial measuring unit, rate gyro assembly, and accelerometer assembly, respectively. The ascent digital autopilot uses the errors between the actual and commanded attitude, rates, and accelerations to gimbal the space shuttle main engines (called thrust vectoring) and the solid rocket boosters to aect the desired vehicle attitude. The space shut- tle’s attitude control system employs the same method in the pitch, roll, and yaw control systems. A simplied model of the pitch control system is shown in Fig. 6. a) Find the closed-loop transfer function relating the actual pitch to commanded pitch. Assume all other inputs are zero. b) Find the closed-loop transfer function relating the actual pitch rate to commanded pitch rate. Assume all other inputs are zero. c) Find the closed-loop transfer function relating the actual pitch acceleration to commanded pitch acceleration. Assume all other inputs are zero. Figure 6: Space shuttle pitch control system (simplied). Arizona State University. Fall 2015. Class # 73024. MAE 318. Homework 4: Page 3 of 4 MAE 318: System Dynamics and Control Dr. Panagiotis K. Artemiadis Problem 7: (Extra Credit Points: 25) Extenders are robot manipulators that extend (i.e. increase) the strength of the human arm in load- maneuvering tasks (see Fig. 7). The system is represented by the transfer function Y (s) U(s) = G(s) = 30 s2+4s+3 where U (s) is the force of the human hand applied to the robot manipulator, and Y (s) is the force of the robot manipulator applied to the load. Assuming that the force of the human hand that is applied is given by u (t) = 5 sin (!t), create a MATLAB code that will compute and plot the dierence in magnitude and phase between the applied human force and the force of the robot manipulator applied to the load, as a function of the frequency !. Use 100 values for ! in the range ! 2 [0:01; 100] rad s for your two plots. See Fig. 8 on how to dene dierence in magnitude and phase between two signals. You need to include your code and the two resulted plots in your solution. Figure 7: Human extender. A B dt T: signal period magnitude difference phase difference B A Figure 8: Magnitude and phase dierence (deg) between two sinusoidal signals.

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## Computer Project Overview You are to perform a 2D heat transfer simulation of the thermal system shown in the schematic below. All items, including justification, plots, and code must be put WITHIN A SINGLE WORD DOCUMENT, USING THE FILENAME “lastname.firstname”. Your document must be submitted into Safeassign in bblearn (in the computer project folder) by the date/time listed above. You will have the option of doing either of 2 different geometries for your simulation. The first geometry is easier, but will have a maximum possible score of 80%. The other geometry is more complex and is the only way to get a full 100% on the project. Your code must be able to handle any mesh size that is a multiple of 10 (e.g. 10×10, 20×20, etc). Code that does not allow a variable mesh size will result in a zero grade for the project. This project may or may not be painful, but I assign it because this is the direction that engineering (not just heat transfer) is going. Additionally, many students have expressed appreciation for the project, although for most of them that was after the fact. I include a couple quotes below from previous students who emailed me after graduating: “My mentor, who is the Lead Engineer for the testing group at Orbital, was SUPER impressed when I told him about the MATLAB project I did for heat transfer. My tears and extra gray hairs were worth it. Tell that to the new seniors when they whine about the project next year. “ “side note, i learned A LOT. i knew very little about matlab prior to this. thank you” “I gotta say though, I think the main thing that pushed me into the realm of being qualified for this job is my capability in Matlab. Your HT project you had all of us do was paramount in kicking me into gear.” Deliverables Your submitted document must contain all of the following: A one paragraph justification of how you know that your simulation is valid. Your justification must address EACH of the following tests: o Heat flows on each individual node should add up to 0 on every node (note: if done correctly, your sums should be something like 10-10) o Net heat flow summed over the boundaries of your simulation should match the heat generated in the system (again, if correct these sums should match within 10-10) o Your temperature profile should match what is physically expected given EACH of the prescribed boundary conditions (describe SPECIFICALLY what EACH boundary condition should do to the temperature profile, and verify that this is reflected in your 3D temperature map) o Doubling your mesh density should not affect your solution (i.e. your calculated heat flows) by more than 0.5% o If you do not get a reasonable answer, show how your solution is NOT valid and discuss HOW you would use these to check whether your code is valid A table listing the quantity of heat transfer per unit depth along each of the non-insulated edges, using a positive value for heat transfer into the solid and negative for heat transfer out of the solid. This should be the TOTAL (summed) heat transfer along each edge. A 3D projection of the temperature distribution (the ‘surf’ command may prove useful), arranged in a way that clearly shows the full temperature distribution, in the correct orientation. Plots of the temperature profiles along each of the edges. Put these together into a single figure with a 3×2 subplot (not a single plot) using the subplot command. Begin with the bottom edge, then work your way clockwise around the geometry. The code you wrote to complete the project Deadlines While there are no deliverables associated with the deadlines below other than the final submission, I will not provide assistance to you on the listed subtasks after their associated deadlines, unless it is during my normal office hours AND there is no one else waiting to ask questions. Also make use of the Finite Difference Overview, MATLAB primer, and tips document which are posted in bblearn. Deadlines o Deadline 1: 11/11/2015. By this time, you should have done the following: Make sure you understand how FD simulations work, by a) reviewing your lecture notes; b) reading Chapter 4 of your textbook; and c) reviewing the Finite Difference Overview posted in bblearn Chosen a numbering scheme for your nodes (where is node ‘1’, how do the nodes increment, etc.) Identified each of the ‘types’ of nodes present and determined the appropriate energy balance relation for each of them Determined the LOGIC to identify the node ‘type’ for an arbitrary node number ‘i’ (based on your numbering scheme and the size of the mesh ‘n’) For a given arbitrary node ‘i’, have a way of identifying the node numbers of all neighboring nodes using only ‘i’ and ‘n’. Written pseudocode that shows the logic of your program. If you do not have pseudocode, I cannot help you with your coding. o Deadline 2: 11/18/2015. By this time, you should have done the following: Have code that can identify the node numbers for all neighboring nodes for an arbitrary node ‘i’ (this will probably involve a series of ‘if’ statements) Have code that can identify which energy balance equation should be applied for any arbitrary node ‘i’ (again, this will probably involve a series of ‘if’ statements) o Deadline 3: 11/20/2015. By this time, you should have done the following: Have code that correctly populates the coefficient matrix and ‘b’ vector by: Iterating through every node Determining which energy balance equation should be applied for each node Determining the appropriate columns for the coefficients of the energy balance equation (this comes from knowing the node numbers for all neighboring nodes) Have code that calculates the temperature distribution from the coefficient matrix and ‘b’ vector o Deadline 4: 11/20/2015. FINAL SUBMISSION. This week should be dedicated to: Creating the required plots Calculating the required heat flows Validating/verifying your simulation Plagiarism Note that this is an INDIVIDUAL project. While you may collaborate, you may NOT share code. Two people independently writing code will NEVER end up with the same code at the end of the day. Merely changing variable names or comments is not sufficient. I have been continuously impressed at how good the plagiarism checking software is at picking out copied code, both from other students and from internet sources. Plagiarized code from any class, past or present, will result in a severely reduced score (potentially a zero) and a report of academic dishonesty. Moreover, presenting results that were not obtained by your code will result in a zero grade and a report of academic dishonesty. Please do not test this – I derive no joy from catching people in plagiarism. Grading The grade breakdown will be as follows: Report: 30 pts o 20 pts: Justification of the validity of your code o 5 pts: required plots shown in report o 5 pts: required heat transfer values shown in table Code: 70 pts (50 pts max if you choose the easier geometry) o 45 pts (35 for easier geometry): coefficient matrix generation o 15 pts: (10 for easier geometry): calculation of heat transfer on edges o 10 pts (5 for easier geometry): temperature matrix and correct surface plot Subtractions: o Up to -10 points for grammar and writing clarity o -5 points for using imaginary nodes (for the harder geometry) o -5 points for incorrect file name and file format o -5 points/day late (no more than 6 days allowed) o Up to 100% deduction for plagiarized code o 100% deduction for reporting results not generated by your submitted code o 100% deduction for submitting code that cannot handle a variable mesh size Skeleton Pseudocode The code listed below is intended to help you organize your thought process on this project. Feel free to use it or not as you see fit. However, if you do not follow this general approach, I will be unable to help you. ****** %Choose mesh size (e.g. n=10 for 10×10, 20 for 20×20, etc.) %Calculate number of nodes (probably something related to n2) %Run a for loop to define all the nodetypes and put them in their proper location for i=1:allnodes if i == xxxxx %use if statements (may need more than one) to determine where node ‘i’ is %e.g. left boundary, bottom right corner, etc. Should be able to determine %using only i and n %Must be node type 1 (for example) Nodetype(i) = 1 elseif i==xxxxx %use if statements to determine where node ‘i’ is %must be node type 2 (for example) Nodetype(i) = 2 elseif i ==xxxx %continue for all node types end end %Run a single for loop to populate the coefficient matrix one row at a time for i =1:allnodes if nodetype (i) == 1 %use energy balance equation for node type 1 %put appropriate coefficients into the correct columns of row i for coefficient %matrix and b vector elseif nodetype(i) ==2 %use energy balance equation for node type 2 %etc., etc. end end %coefficient matrix and b vector should now be populated. Solve for T values %rearrange T values into matrix to make surface plot %verify sum of qin to each node adds to 0 %calculate heat flows on edges, verify that it matches generation %make edge plots Project Geometries Easier Geometry (Max Score is 80/100) The overall size is 1mx1m. The geometry is designed to work with any mesh size that is a multiple of 10×10. Harder Geometry The overall size is 1mx1m. The geometry is designed to work with any mesh size that is a multiple of 10×10.

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## Computer Project Overview You are to perform a 2D heat transfer simulation of the thermal system shown in the schematic below. All items, including justification, plots, and code must be put WITHIN A SINGLE WORD DOCUMENT, USING THE FILENAME “lastname.firstname”. Your document must be submitted into Safeassign in bblearn (in the computer project folder) by the date/time listed above. You will have the option of doing either of 2 different geometries for your simulation. The first geometry is easier, but will have a maximum possible score of 80%. The other geometry is more complex and is the only way to get a full 100% on the project. Your code must be able to handle any mesh size that is a multiple of 10 (e.g. 10×10, 20×20, etc). Code that does not allow a variable mesh size will result in a zero grade for the project. This project may or may not be painful, but I assign it because this is the direction that engineering (not just heat transfer) is going. Additionally, many students have expressed appreciation for the project, although for most of them that was after the fact. I include a couple quotes below from previous students who emailed me after graduating: “My mentor, who is the Lead Engineer for the testing group at Orbital, was SUPER impressed when I told him about the MATLAB project I did for heat transfer. My tears and extra gray hairs were worth it. Tell that to the new seniors when they whine about the project next year. “ “side note, i learned A LOT. i knew very little about matlab prior to this. thank you” “I gotta say though, I think the main thing that pushed me into the realm of being qualified for this job is my capability in Matlab. Your HT project you had all of us do was paramount in kicking me into gear.” Deliverables Your submitted document must contain all of the following: A one paragraph justification of how you know that your simulation is valid. Your justification must address EACH of the following tests: o Heat flows on each individual node should add up to 0 on every node (note: if done correctly, your sums should be something like 10-10) o Net heat flow summed over the boundaries of your simulation should match the heat generated in the system (again, if correct these sums should match within 10-10) o Your temperature profile should match what is physically expected given EACH of the prescribed boundary conditions (describe SPECIFICALLY what EACH boundary condition should do to the temperature profile, and verify that this is reflected in your 3D temperature map) o Doubling your mesh density should not affect your solution (i.e. your calculated heat flows) by more than 0.5% o If you do not get a reasonable answer, show how your solution is NOT valid and discuss HOW you would use these to check whether your code is valid A table listing the quantity of heat transfer per unit depth along each of the non-insulated edges, using a positive value for heat transfer into the solid and negative for heat transfer out of the solid. This should be the TOTAL (summed) heat transfer along each edge. A 3D projection of the temperature distribution (the ‘surf’ command may prove useful), arranged in a way that clearly shows the full temperature distribution, in the correct orientation. Plots of the temperature profiles along each of the edges. Put these together into a single figure with a 3×2 subplot (not a single plot) using the subplot command. Begin with the bottom edge, then work your way clockwise around the geometry. The code you wrote to complete the project Deadlines While there are no deliverables associated with the deadlines below other than the final submission, I will not provide assistance to you on the listed subtasks after their associated deadlines, unless it is during my normal office hours AND there is no one else waiting to ask questions. Also make use of the Finite Difference Overview, MATLAB primer, and tips document which are posted in bblearn. Deadlines o Deadline 1: 11/11/2015. By this time, you should have done the following: Make sure you understand how FD simulations work, by a) reviewing your lecture notes; b) reading Chapter 4 of your textbook; and c) reviewing the Finite Difference Overview posted in bblearn Chosen a numbering scheme for your nodes (where is node ‘1’, how do the nodes increment, etc.) Identified each of the ‘types’ of nodes present and determined the appropriate energy balance relation for each of them Determined the LOGIC to identify the node ‘type’ for an arbitrary node number ‘i’ (based on your numbering scheme and the size of the mesh ‘n’) For a given arbitrary node ‘i’, have a way of identifying the node numbers of all neighboring nodes using only ‘i’ and ‘n’. Written pseudocode that shows the logic of your program. If you do not have pseudocode, I cannot help you with your coding. o Deadline 2: 11/18/2015. By this time, you should have done the following: Have code that can identify the node numbers for all neighboring nodes for an arbitrary node ‘i’ (this will probably involve a series of ‘if’ statements) Have code that can identify which energy balance equation should be applied for any arbitrary node ‘i’ (again, this will probably involve a series of ‘if’ statements) o Deadline 3: 11/20/2015. By this time, you should have done the following: Have code that correctly populates the coefficient matrix and ‘b’ vector by: Iterating through every node Determining which energy balance equation should be applied for each node Determining the appropriate columns for the coefficients of the energy balance equation (this comes from knowing the node numbers for all neighboring nodes) Have code that calculates the temperature distribution from the coefficient matrix and ‘b’ vector o Deadline 4: 11/20/2015. FINAL SUBMISSION. This week should be dedicated to: Creating the required plots Calculating the required heat flows Validating/verifying your simulation Plagiarism Note that this is an INDIVIDUAL project. While you may collaborate, you may NOT share code. Two people independently writing code will NEVER end up with the same code at the end of the day. Merely changing variable names or comments is not sufficient. I have been continuously impressed at how good the plagiarism checking software is at picking out copied code, both from other students and from internet sources. Plagiarized code from any class, past or present, will result in a severely reduced score (potentially a zero) and a report of academic dishonesty. Moreover, presenting results that were not obtained by your code will result in a zero grade and a report of academic dishonesty. Please do not test this – I derive no joy from catching people in plagiarism. Grading The grade breakdown will be as follows: Report: 30 pts o 20 pts: Justification of the validity of your code o 5 pts: required plots shown in report o 5 pts: required heat transfer values shown in table Code: 70 pts (50 pts max if you choose the easier geometry) o 45 pts (35 for easier geometry): coefficient matrix generation o 15 pts: (10 for easier geometry): calculation of heat transfer on edges o 10 pts (5 for easier geometry): temperature matrix and correct surface plot Subtractions: o Up to -10 points for grammar and writing clarity o -5 points for using imaginary nodes (for the harder geometry) o -5 points for incorrect file name and file format o -5 points/day late (no more than 6 days allowed) o Up to 100% deduction for plagiarized code o 100% deduction for reporting results not generated by your submitted code o 100% deduction for submitting code that cannot handle a variable mesh size Skeleton Pseudocode The code listed below is intended to help you organize your thought process on this project. Feel free to use it or not as you see fit. However, if you do not follow this general approach, I will be unable to help you. ****** %Choose mesh size (e.g. n=10 for 10×10, 20 for 20×20, etc.) %Calculate number of nodes (probably something related to n2) %Run a for loop to define all the nodetypes and put them in their proper location for i=1:allnodes if i == xxxxx %use if statements (may need more than one) to determine where node ‘i’ is %e.g. left boundary, bottom right corner, etc. Should be able to determine %using only i and n %Must be node type 1 (for example) Nodetype(i) = 1 elseif i==xxxxx %use if statements to determine where node ‘i’ is %must be node type 2 (for example) Nodetype(i) = 2 elseif i ==xxxx %continue for all node types end end %Run a single for loop to populate the coefficient matrix one row at a time for i =1:allnodes if nodetype (i) == 1 %use energy balance equation for node type 1 %put appropriate coefficients into the correct columns of row i for coefficient %matrix and b vector elseif nodetype(i) ==2 %use energy balance equation for node type 2 %etc., etc. end end %coefficient matrix and b vector should now be populated. Solve for T values %rearrange T values into matrix to make surface plot %verify sum of qin to each node adds to 0 %calculate heat flows on edges, verify that it matches generation %make edge plots Project Geometries Easier Geometry (Max Score is 80/100) The overall size is 1mx1m. The geometry is designed to work with any mesh size that is a multiple of 10×10. Harder Geometry The overall size is 1mx1m. The geometry is designed to work with any mesh size that is a multiple of 10×10.

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## PHSX 220 Homework 13 Paper – Due Online April 28 – 5:00 pm SHM and Wave the Equation Problem 1: A hanging mass system with a mass of 85 kg, spring constant of k= 490 N/m is realeased from rest from a distance of 10 meters below the systems equilibrium position (similar values to the bottom of a bungee jump). Calculate the following quantities in regards to this system after being released at t=0: a) The angular frequency of the system (radians/sec) b) The frequency of oscillations for the system (Hz) c) The period of oscillations for the system (seconds) d) The time it takes to get back to the equilibrium position of the system for the rst time Problem 2: A horizontal spring-mass system (mass of 2:21×1025 kg) with no friction has an ocsillation frequency of 9,192,631,770 cycles per second. (a second is dened by 9,192,631,770 cycles of a Cs-133 atom)). Calculate the eective spring constant of the system Problem 3: A swinging person, such as Tarzan, can be modeled after a simple pendulum with a mass of 85 kg and a length of 10 m. Consider the mass being released from rest at t=0 at an angle of +15 degrees from the vertical. Calculate the following quantities in regards to this system. You need to be in radians mode for this problem a) The angular frequency of the system (radians/sec) b) The frequency of oscillations for the system in (Hz) c) The period of oscillations of the system (seconds) d) Sketch plots of the angular position, angular velocity and angular acceleration of the system as a function of time. Hint: These will always help you with these time to it takes to a certain point in it’s cycle questions. e) The time it takes for the mass to get half way through its rst cycle (or to the other side of the swing if you were interested in timing say a rescue eort or something along those lines) . f) The maximum angular velocity of the mass g) The maximum angular accleration of the mass h) The magnitude of the angular momentum of the mass at 3 seconds i) The magnitude of the torque acting on the mass at 3 seconds Problem 4: A wave has a wavenumber of 1 m-1, and an angular frequency of 2 radians per second, travels in the +x direction and has a maximum transverse amplitude of 0.1 m. At t=0, and x =0 the y position is equal to 0.0 m (y(0,0) = 0.0 m). a) Calculate the wavelength of the wave b) Calculate the period of oscillations for the wave c) Calculate the wave speed along the x axis d) Calculate the magnitude and direction of the transverse position of the wave at x=0.5 m and t = 8s e) Calculate the magnitude and direction of the transverse velocity of the wave at x=0.5 m and t = 8s f) Calculate the magnitude and direction of the transverse acceleration of the wave at x=0.5 m and t = 8s Problem 5-6: Chapter 16 Problem 10, 22 Additional Suggested Problems with Solutions Provided: Chapter 16 Problems 5, 9, 15, 45

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## Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

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