A boom is supported by two cables, as shown below. Cable BD is 15.6 ft long, and cable CD is 23.7 ft long. Determine the length of the boom and the angles cable BD and cable CD make with the horizontal

A boom is supported by two cables, as shown below. Cable BD is 15.6 ft long, and cable CD is 23.7 ft long. Determine the length of the boom and the angles cable BD and cable CD make with the horizontal

FSE 100 Extra Credit (20 points) Instructions: Read the description below and work through the design process to build an automated waste sorting system. Turn in the following deliverables in one document, typed: 1. Problem Statement – 1 point 2. Technical System Requirements (at least 3 complete sentences using “shall”) – 3 points 3. Judging Criteria (at least 3, explain why you chose them) – 2 points 4. AHP – 2 points 5. Summaries of your 3 design options (paragraph minimum for each option) – 3 points 6. Design Decision Matrix – 3 points 7. Orthographic Drawing of your final design (3 projections required) – 3 points 8. Activity Diagram of how your sorter functions – 3 points Description: The city of Tempe waste management has notified ASU that due to the exceptional effort the Sundevil students have made in the sustainability area, ASU has been contributing three times the amount of recyclable materials than what was predicted on a monthly basis. Unfortunately, due to the immense amount of materials being delivered, the city of Tempe waste management has asked for assistance from ASU prior to picking up the recyclable waste. They have requested that ASU implement an automated waste sorting system that would pre-filter all the materials so the city of Tempe can collect the materials based on one of three types and process the waste much faster. ASU has hired you to design an automated sorter, but due to the unexpected nature of this request, ASU prefers that this design be as simple and inexpensive to build as possible. The city of Tempe would like to have the waste categorized as either glass, plastic, or metal. Paper will not be considered in this design. Any glass that is sorted in your device needs to stay intact, and not break. Very few people will be able to monitor this device as it sorts, so it must be able to sort the items with no input from a user, as quickly as possible. This design cannot exceed 2m in length, width, or height, but the weight is unlimited. ASU is not giving any guidance as to the materials you can use, so you are free to shop for whatever you’d like, but keep in mind, the final cost of this device must be as inexpensive as possible. Submit through Blackboard or print out your document and turn it in to me no later than the date shown on Blackboard

FSE 100 Extra Credit (20 points) Instructions: Read the description below and work through the design process to build an automated waste sorting system. Turn in the following deliverables in one document, typed: 1. Problem Statement – 1 point 2. Technical System Requirements (at least 3 complete sentences using “shall”) – 3 points 3. Judging Criteria (at least 3, explain why you chose them) – 2 points 4. AHP – 2 points 5. Summaries of your 3 design options (paragraph minimum for each option) – 3 points 6. Design Decision Matrix – 3 points 7. Orthographic Drawing of your final design (3 projections required) – 3 points 8. Activity Diagram of how your sorter functions – 3 points Description: The city of Tempe waste management has notified ASU that due to the exceptional effort the Sundevil students have made in the sustainability area, ASU has been contributing three times the amount of recyclable materials than what was predicted on a monthly basis. Unfortunately, due to the immense amount of materials being delivered, the city of Tempe waste management has asked for assistance from ASU prior to picking up the recyclable waste. They have requested that ASU implement an automated waste sorting system that would pre-filter all the materials so the city of Tempe can collect the materials based on one of three types and process the waste much faster. ASU has hired you to design an automated sorter, but due to the unexpected nature of this request, ASU prefers that this design be as simple and inexpensive to build as possible. The city of Tempe would like to have the waste categorized as either glass, plastic, or metal. Paper will not be considered in this design. Any glass that is sorted in your device needs to stay intact, and not break. Very few people will be able to monitor this device as it sorts, so it must be able to sort the items with no input from a user, as quickly as possible. This design cannot exceed 2m in length, width, or height, but the weight is unlimited. ASU is not giving any guidance as to the materials you can use, so you are free to shop for whatever you’d like, but keep in mind, the final cost of this device must be as inexpensive as possible. Submit through Blackboard or print out your document and turn it in to me no later than the date shown on Blackboard

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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!

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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/The following graphs depict the motion of an object starting from rest and moving without friction. Describe how you would calculate the object’s acceleration, instantaneous speed, and distance at time “p” from each graph (slope, area-under-curve, etc.). 15. An object is launched at an angle of 450 from the ground of a mystery planet. The object hits the ground 20m away after a total flight time of 4.0s. Assume no air resistance. a. What are the initial vertical and horizontal velocities? b. Calculate the acceleration due to gravity. c. Draw graphs to quantitatively represent the vertical and horizontal velocities for the entire 4.0s of flight. Linear Dynamics 1. A block of mass 3 kg, initially at rest, is pulled along a frictionless, horizontal surface with a force shown as a function of time by the graph above. Calculate the acceleration and speed after 2s. Questions 2-4: Two blocks of masses M and m, with M > m, are connected by a light string. The string passes over a frictionless pulley of negligible mass so that the blocks hang vertically like Atwood’s machine. The blocks are then released from rest as shown above. 2. Draw a free-body diagram for each mass. Compare and contrast the tension on each. 3. Compare and contrast the net-force acting on each block. 4. Draw a free-body diagram for the string holding the pulley. Explain whether the force increases, decreases, or remains the same as the blocks accelerate. Questions 5-6: A ball is released from the top of a curved hill as shown above; the hill has sufficient friction so that the ball rolls as it moves down the hill. 5. What can be inferred about the ball’s linear acceleration and speed as the ball goes from the top to the bottom? (Increase, decrease, or remain the same) 6. Draw a free-body diagram for each location in the diagram to compare the weight, normal, and friction forces as it rolls down hill. Questions 7-8: Consider the above block sitting on a smooth tabletop. It is connected by a light string that passes over a frictionless and massless pulley to a pulling force of 30N downward. 7. Use Newton’s 2nd Law to determine what will happen to the net force, mass, and acceleration of the entire system if the pulling force of 30N is replaced with another block weighing 30N. 8. What will happen to the tension on each body?

/The following graphs depict the motion of an object starting from rest and moving without friction. Describe how you would calculate the object’s acceleration, instantaneous speed, and distance at time “p” from each graph (slope, area-under-curve, etc.). 15. An object is launched at an angle of 450 from the ground of a mystery planet. The object hits the ground 20m away after a total flight time of 4.0s. Assume no air resistance. a. What are the initial vertical and horizontal velocities? b. Calculate the acceleration due to gravity. c. Draw graphs to quantitatively represent the vertical and horizontal velocities for the entire 4.0s of flight. Linear Dynamics 1. A block of mass 3 kg, initially at rest, is pulled along a frictionless, horizontal surface with a force shown as a function of time by the graph above. Calculate the acceleration and speed after 2s. Questions 2-4: Two blocks of masses M and m, with M > m, are connected by a light string. The string passes over a frictionless pulley of negligible mass so that the blocks hang vertically like Atwood’s machine. The blocks are then released from rest as shown above. 2. Draw a free-body diagram for each mass. Compare and contrast the tension on each. 3. Compare and contrast the net-force acting on each block. 4. Draw a free-body diagram for the string holding the pulley. Explain whether the force increases, decreases, or remains the same as the blocks accelerate. Questions 5-6: A ball is released from the top of a curved hill as shown above; the hill has sufficient friction so that the ball rolls as it moves down the hill. 5. What can be inferred about the ball’s linear acceleration and speed as the ball goes from the top to the bottom? (Increase, decrease, or remain the same) 6. Draw a free-body diagram for each location in the diagram to compare the weight, normal, and friction forces as it rolls down hill. Questions 7-8: Consider the above block sitting on a smooth tabletop. It is connected by a light string that passes over a frictionless and massless pulley to a pulling force of 30N downward. 7. Use Newton’s 2nd Law to determine what will happen to the net force, mass, and acceleration of the entire system if the pulling force of 30N is replaced with another block weighing 30N. 8. What will happen to the tension on each body?

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Vermont Technical College Electronics I – Laboratory ELT-2051 Lab 07: Transistor Biasing Circuits and Q-point Stability Objectives: • To set an operating point for a transistor using three different bias techniques • To explore amplification of an AC signal • To use MultiSim to verify your experimental data General: In this laboratory, you will be supplied with two NPN transistors with varying ß’s. Prelab: Calculate values of Rb in Figures 1 and 2 assuming ß = 200, VCE = 6V . For Figure 3, calculate R1 and R2 so that their parallel resistance is about 20KΩ or 10% of (ß+1)RE. Also, calculate the critical frequency of the 1uF capacitor in Figure 4. Materials: • 2N3904, 2N4123 NPN TXs (1 high ß, 1 low ß) • (2) 1 k Ohm, 100 k Ohm, assorted resistors • 1uF, 10uF capacitors • Curve Tracer • DC Power Supply • Multimeter • Signal Generator • Oscilloscope • Breadboard Procedure: 1. Use the curve tracer to plot the curves for each of your transistors. From these curves, again using the curve tracer, determine the ßDC for each transistor at the IC currents of 1mA, 3mA, 6mA, and 10mA with VCE = 6V. Of course, be sure to keep track of which transistor goes with which curve. Verify that the ßDC values that you obtain are within the manufacturer’s specifications. Remember– ßDC = hFE ! 2. For each of the three circuits shown in Figures 1-3, using the R values calculated in your prelab, determine the operating points IC and VCE for each of the transistors. Be sure to table your data. In addition, plot ß vs IC for both transistors on a single graph so that the data is meaningful! What conclusions can be reached for the 3 biasing circuits? 3. Lastly – Build Figure 4 and determine the ratio (Gain) of Vout/Vin at 1KHz. Now vary the frequency of Vin to determine at what frequencies this ratio decreases to 0.707 of the value at 1KHz. 4. Use the Bode Plotter feature in MultiSim to verify your data of Part 3. Is the cut-off frequency the same as you measured in the lab? Base Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Emitter Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Voltage Divider Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Laboratory Report: This lab is a semi-formal lab. Be sure to collect all data necessary to make observations and answer questions before you leave the lab. Also, you and your lab partner should discuss the results and outcomes prior to leaving. Take notes, fill in tables and include diagrams as needed. Your report should include: • Data Table • Beta Plot • MultiSim Frequency Response • Comparison of biasing schemes • Comparison of measurements vs. simulations and expectations.

Vermont Technical College Electronics I – Laboratory ELT-2051 Lab 07: Transistor Biasing Circuits and Q-point Stability Objectives: • To set an operating point for a transistor using three different bias techniques • To explore amplification of an AC signal • To use MultiSim to verify your experimental data General: In this laboratory, you will be supplied with two NPN transistors with varying ß’s. Prelab: Calculate values of Rb in Figures 1 and 2 assuming ß = 200, VCE = 6V . For Figure 3, calculate R1 and R2 so that their parallel resistance is about 20KΩ or 10% of (ß+1)RE. Also, calculate the critical frequency of the 1uF capacitor in Figure 4. Materials: • 2N3904, 2N4123 NPN TXs (1 high ß, 1 low ß) • (2) 1 k Ohm, 100 k Ohm, assorted resistors • 1uF, 10uF capacitors • Curve Tracer • DC Power Supply • Multimeter • Signal Generator • Oscilloscope • Breadboard Procedure: 1. Use the curve tracer to plot the curves for each of your transistors. From these curves, again using the curve tracer, determine the ßDC for each transistor at the IC currents of 1mA, 3mA, 6mA, and 10mA with VCE = 6V. Of course, be sure to keep track of which transistor goes with which curve. Verify that the ßDC values that you obtain are within the manufacturer’s specifications. Remember– ßDC = hFE ! 2. For each of the three circuits shown in Figures 1-3, using the R values calculated in your prelab, determine the operating points IC and VCE for each of the transistors. Be sure to table your data. In addition, plot ß vs IC for both transistors on a single graph so that the data is meaningful! What conclusions can be reached for the 3 biasing circuits? 3. Lastly – Build Figure 4 and determine the ratio (Gain) of Vout/Vin at 1KHz. Now vary the frequency of Vin to determine at what frequencies this ratio decreases to 0.707 of the value at 1KHz. 4. Use the Bode Plotter feature in MultiSim to verify your data of Part 3. Is the cut-off frequency the same as you measured in the lab? Base Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Emitter Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Voltage Divider Bias: Parameter Calculated Value Simulated Value Measured Value VCE1 (high β) VCE2 (low β) n/a n/a |VCE1 – VCE2| 0 0 IC1 (high β) IC2 (low β) n/a n/a |IC1 – IC2| 0 0 Laboratory Report: This lab is a semi-formal lab. Be sure to collect all data necessary to make observations and answer questions before you leave the lab. Also, you and your lab partner should discuss the results and outcomes prior to leaving. Take notes, fill in tables and include diagrams as needed. Your report should include: • Data Table • Beta Plot • MultiSim Frequency Response • Comparison of biasing schemes • Comparison of measurements vs. simulations and expectations.

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Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

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A bar is supported by a ball-and-socket joint and two cables as shown below. a) Draw a clear and neat free body diagram (FBD) for the bar. This FBD must be a figure that is separately from the figure given below. b) Determine the reaction at support A (the ball-and-socket joint). Express the force in Cartesian vector form. c) Determine the magnitude of the tension in each cable.

A bar is supported by a ball-and-socket joint and two cables as shown below. a) Draw a clear and neat free body diagram (FBD) for the bar. This FBD must be a figure that is separately from the figure given below. b) Determine the reaction at support A (the ball-and-socket joint). Express the force in Cartesian vector form. c) Determine the magnitude of the tension in each cable.

 
20.) To the right of the monkey display, there is an exhibit about fossils once believed to have been of human ancestors. Summarize the story of one of the specimens shown. (Homo diluvia testis, Oreopithicus, “Piltdown man” and Gigantapithicus). My favorite is the one with the salamander 

20.) To the right of the monkey display, there is an exhibit about fossils once believed to have been of human ancestors. Summarize the story of one of the specimens shown. (Homo diluvia testis, Oreopithicus, “Piltdown man” and Gigantapithicus). My favorite is the one with the salamander 

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MCE 260 Fall 2015 Homework 9, due November 12, 2015. PRESENT CLEARLY HOW YOU DEVELOPED THE SOLUTION TO THE PROBLEMS Each problem is worth up to 5 points. Points are given as follows: 5 points: Work was complete and presented clearly, the answer is correct 4 points: Work was complete, but not clearly presented or some errors in calculation 3 points: Some errors or omissions in methods or presentation 2 points: Major errors or omissions in methods or presentation 1 point: Problem was understood but incorrect approach was used 1. The radial compressor shown above (with dimensions) has a crank that rotates at 120 RPM. What is the maximal acceleration of each piston? 2. Design a follower displacement profile for a double-dwell (RDFD) cam, with these performance specifications: Machine cycle is 5 seconds. Rise from 0 to 22 mm in 45 degrees. Dwell for 45 degrees. Fall from 22 mm back to zero in 90 degrees. Dwell for the remainder of the machine cycle. You may use any profile that satisfies the fundamental law of cam design. Write equations for follower displacement (s) as a function of cam angle (θ) for each of the four segments of the cam. 3. (10% extra credit) What is the maximal acceleration of the follower? At which point in the cycle does it occur? Page 1 of 1

MCE 260 Fall 2015 Homework 9, due November 12, 2015. PRESENT CLEARLY HOW YOU DEVELOPED THE SOLUTION TO THE PROBLEMS Each problem is worth up to 5 points. Points are given as follows: 5 points: Work was complete and presented clearly, the answer is correct 4 points: Work was complete, but not clearly presented or some errors in calculation 3 points: Some errors or omissions in methods or presentation 2 points: Major errors or omissions in methods or presentation 1 point: Problem was understood but incorrect approach was used 1. The radial compressor shown above (with dimensions) has a crank that rotates at 120 RPM. What is the maximal acceleration of each piston? 2. Design a follower displacement profile for a double-dwell (RDFD) cam, with these performance specifications: Machine cycle is 5 seconds. Rise from 0 to 22 mm in 45 degrees. Dwell for 45 degrees. Fall from 22 mm back to zero in 90 degrees. Dwell for the remainder of the machine cycle. You may use any profile that satisfies the fundamental law of cam design. Write equations for follower displacement (s) as a function of cam angle (θ) for each of the four segments of the cam. 3. (10% extra credit) What is the maximal acceleration of the follower? At which point in the cycle does it occur? Page 1 of 1

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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

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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