power flow of network in saudi arabia – network modules (load/demand) – types of generators – cost efficient – power flow after injecting renwable energy all using matlab and all about saudi arabia its for my project

power flow of network in saudi arabia – network modules (load/demand) – types of generators – cost efficient – power flow after injecting renwable energy all using matlab and all about saudi arabia its for my project

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The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

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ENGR 1120 – PROGRAMMING FOR ENGINEERS (MATLAB) Homework Program #2 Objectives: Demonstrate knowledge of data files, vector variables, intrinsic functions, subscript manipulation, for loops, and plotting in MATLAB. You have been given a set of ASCII data files that contain directions for laying out patterns in a field. The data files contain in the first column a distance to travel and in the second column a direction heading. Unfortunately, the person who created the data did not have a good understanding of orienteering and the direction headings are given as referenced to a clock face. The pattern begins at the origin of a Cartesian coordinate system with the person facing 12 o’clock, see the figure below. The figure shows an example of the first step in the pattern being a distance of 1.5 feet in the direction of 7 o’clock. All direction headings are given in terms of this clock orientation. The distance values given are in feet. There are 5 data files provided online for testing of the program. Write a script file that will allow the user to input from the keyboard the filename of the file that they wish to analyze. Load only that ONE data file and plot the resulting pattern. Once each point forming the pattern has been located, find and designate on the plot which of the resulting nodes was the farthest away from the origin. Also find and designate the center of the pattern as defined to occur at the coordinate location corresponding to (average x, average y). When plotting the resulting pattern on the Cartesian coordinate system, set the axes limits to appropriate values. HINT: Correlate the direction headings provided in the data files to a Cartesian coordinate system by using the following vector in your script file. This requires subscript manipulation. angle = [60; 30; 0; 330; 300; 270; 240; 210; 180; 150; 120; 90] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1 2 3 4 5 6 7 8 9 10 11 12 you are here

ENGR 1120 – PROGRAMMING FOR ENGINEERS (MATLAB) Homework Program #2 Objectives: Demonstrate knowledge of data files, vector variables, intrinsic functions, subscript manipulation, for loops, and plotting in MATLAB. You have been given a set of ASCII data files that contain directions for laying out patterns in a field. The data files contain in the first column a distance to travel and in the second column a direction heading. Unfortunately, the person who created the data did not have a good understanding of orienteering and the direction headings are given as referenced to a clock face. The pattern begins at the origin of a Cartesian coordinate system with the person facing 12 o’clock, see the figure below. The figure shows an example of the first step in the pattern being a distance of 1.5 feet in the direction of 7 o’clock. All direction headings are given in terms of this clock orientation. The distance values given are in feet. There are 5 data files provided online for testing of the program. Write a script file that will allow the user to input from the keyboard the filename of the file that they wish to analyze. Load only that ONE data file and plot the resulting pattern. Once each point forming the pattern has been located, find and designate on the plot which of the resulting nodes was the farthest away from the origin. Also find and designate the center of the pattern as defined to occur at the coordinate location corresponding to (average x, average y). When plotting the resulting pattern on the Cartesian coordinate system, set the axes limits to appropriate values. HINT: Correlate the direction headings provided in the data files to a Cartesian coordinate system by using the following vector in your script file. This requires subscript manipulation. angle = [60; 30; 0; 330; 300; 270; 240; 210; 180; 150; 120; 90] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1 2 3 4 5 6 7 8 9 10 11 12 you are here

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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)

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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EEGR 221 MATLAB Project 1 Basic Signals Fall 2015 Due date: 10/5/15 1. (a) Plot ?1(?) = ?(?+1)−?(?−5) where -7 < t < 7 seconds. Use millisecond units. (b) Plot ? = 5 ??? (??)[ ?(?+1)−?(?−5)] 2. (a) Plot x2(t) exactly as shown in this figure. Include the same titles and labels for the signal. Hint: Find the amplitude equations as function of time and insert those to your MATLAB script to create and plot this signal. (b) Decompose x2(t) into its even and odd components and plot x2e(t) and x2o(t). (c) Plot x2e(t) + x2o(t) and verify that x2e(t) + x2o(t) = x2(t). How to report the results?  For each plot you must label x and y axis and have a title for the plot. Following commands could be used. heaviside, plot, axis, ylabel, ylabel, title, fliplr, etc … At the command prompt of MATLAB you can type >> help [command name] to get help with any command.  Plot all of the signal for t between -7 and 7 seconds.  Save your commands in an m-file with your name in the name field. (e.g. John_Scott.m) and append the code to the end of your report.  Your report must be organized and your solution for each question mu st be clearly marked by the number of the question. Example 2.a or 2.b, … In each part the problem should be clearly identified. Type the problem statement in each section. Show the plots of input and output signals. Draw conclusions based on your plots and in problem 3 discuss why the property is not satisfied based on your plots.  Turn in a hard copy of your report in class. This report must include a cover page with the name of both student partners.

EEGR 221 MATLAB Project 1 Basic Signals Fall 2015 Due date: 10/5/15 1. (a) Plot ?1(?) = ?(?+1)−?(?−5) where -7 < t < 7 seconds. Use millisecond units. (b) Plot ? = 5 ??? (??)[ ?(?+1)−?(?−5)] 2. (a) Plot x2(t) exactly as shown in this figure. Include the same titles and labels for the signal. Hint: Find the amplitude equations as function of time and insert those to your MATLAB script to create and plot this signal. (b) Decompose x2(t) into its even and odd components and plot x2e(t) and x2o(t). (c) Plot x2e(t) + x2o(t) and verify that x2e(t) + x2o(t) = x2(t). How to report the results?  For each plot you must label x and y axis and have a title for the plot. Following commands could be used. heaviside, plot, axis, ylabel, ylabel, title, fliplr, etc … At the command prompt of MATLAB you can type >> help [command name] to get help with any command.  Plot all of the signal for t between -7 and 7 seconds.  Save your commands in an m-file with your name in the name field. (e.g. John_Scott.m) and append the code to the end of your report.  Your report must be organized and your solution for each question mu st be clearly marked by the number of the question. Example 2.a or 2.b, … In each part the problem should be clearly identified. Type the problem statement in each section. Show the plots of input and output signals. Draw conclusions based on your plots and in problem 3 discuss why the property is not satisfied based on your plots.  Turn in a hard copy of your report in class. This report must include a cover page with the name of both student partners.

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 a ect 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 simpli ed 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 (simpli ed). 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 di erence 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 de ne di erence 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 di erence (deg) between two sinusoidal signals.

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 a ect 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 simpli ed 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 (simpli ed). 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 di erence 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 de ne di erence 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 di erence (deg) between two sinusoidal signals.

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PlotCycloidArc(8.5) Math98 HW4 The cylcoid is the plane curve traced out by a point on a circle as the circle rolls without slipping on a straight line.1 In this exercise we will use MATLAB to create an animation of a circle rolling on a straight line, while a point on the circle traces the cycloid. a. Implement a MATLAB function of the form function PlotCycloidArc(ArcLength). This function takes in a positive number ArcLength and displays a snapshot of the cirle rolling (without slipping) on the x-axis while a point on the cirlce traces the cycloid. The circle is initially centered at (0,1) and has radius 1, and the initial tracing point is taken to be (0, 0). An example output is shown in the above ?gure. As in the ?gure, plot the cycloid arc black, the circle blue, and use a red dot for the tracing point. Hint: If the circle has rolled for a length of arc t = 0, the coordinates of the tracing point are (t-sin t, 1-cos t). b. Implement a MATLAB function of the form function CycloidMovie(NumHumps,NumIntervals). This function will output an animation of the circle rolling along a line while a point on the circle traces the cycloid. This function inputs two natural numbers NumHumps and NumIntervals representing the number of periods (or humps) of the cycloid and the number or frames per hump that will be used to make the animation, respectively. Use the getframe command to save frames outputted from PlotCycloidArc and the movie command to play them back together as a movie. Use the axis command to view the frames on the rectan- gle [0, 2pNumHumps] × [0, 5/2]. Also label the ticks 0, 2p, . . . , 2pNumHumps on the x axis with the strings 1See Wikipedia for more on the cycloid. 0, 2p, . . . , 2pNumHumps and do the same for 1, 2 on the y axis (see the ?gure above). Label the movie ’Cycloid Animation’. Submit MATLAB code for both parts a and b and a printout the ?gures obtained by the commands PlotCycloidArc(8.5), PlotCycloidArc(12), and CycloidMovie(3,10)

PlotCycloidArc(8.5) Math98 HW4 The cylcoid is the plane curve traced out by a point on a circle as the circle rolls without slipping on a straight line.1 In this exercise we will use MATLAB to create an animation of a circle rolling on a straight line, while a point on the circle traces the cycloid. a. Implement a MATLAB function of the form function PlotCycloidArc(ArcLength). This function takes in a positive number ArcLength and displays a snapshot of the cirle rolling (without slipping) on the x-axis while a point on the cirlce traces the cycloid. The circle is initially centered at (0,1) and has radius 1, and the initial tracing point is taken to be (0, 0). An example output is shown in the above ?gure. As in the ?gure, plot the cycloid arc black, the circle blue, and use a red dot for the tracing point. Hint: If the circle has rolled for a length of arc t = 0, the coordinates of the tracing point are (t-sin t, 1-cos t). b. Implement a MATLAB function of the form function CycloidMovie(NumHumps,NumIntervals). This function will output an animation of the circle rolling along a line while a point on the circle traces the cycloid. This function inputs two natural numbers NumHumps and NumIntervals representing the number of periods (or humps) of the cycloid and the number or frames per hump that will be used to make the animation, respectively. Use the getframe command to save frames outputted from PlotCycloidArc and the movie command to play them back together as a movie. Use the axis command to view the frames on the rectan- gle [0, 2pNumHumps] × [0, 5/2]. Also label the ticks 0, 2p, . . . , 2pNumHumps on the x axis with the strings 1See Wikipedia for more on the cycloid. 0, 2p, . . . , 2pNumHumps and do the same for 1, 2 on the y axis (see the ?gure above). Label the movie ’Cycloid Animation’. Submit MATLAB code for both parts a and b and a printout the ?gures obtained by the commands PlotCycloidArc(8.5), PlotCycloidArc(12), and CycloidMovie(3,10)

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Project 1: Particle Trajectory in Pleated Filters Due: 12:30 pm, Dec. 1, 2015, submission through blackboard Course: Numerical Methods Instructor: Dr. Hooman V. Tafreshi Most aerosol filters are made of pleated fibrous media. This is to accommodate as much filtration media as possible in a limited space available to an air filtration unit (e.g., the engine of a car). A variety of parameters contribute to the performance of a pleated filter. These parameters include, but are not limited to, geometry of the pleat (e.g., pleat height, width, and count), microscale properties of the fibrous media (e.g., fiber diameters, fiber orientation, and solid volume fraction), aerodynamic and thermal conditions of the flow (e.g., flow velocity, temperature, and operating pressure), and particle properties (e.g., diameter, density, and shape). Figure 1: Examples of pleated air filters [1‐2]. In this project you are asked to calculate the trajectory of aerosol particles as they travel inside a rectangular pleat channel. Due to the symmetry of the pleat geometry, you only need to simulate one half of the channel (see Figure 2). Figure 2: The simulation domain and boundary conditions (the figure’s aspect ratio is altered for illustration purposes). Trajectory of the aerosol particles can be calculated in a 2‐D domain by solving the Newton’s 2nd law written for the particles in the x‐ and y‐directions, v(h) inlet velocity fibrous media v(y) y tm l h x Ui u(l) u(x) 2 2 p 1 p 1 ( , ) d x dx u x y dt  dt    2 2 p 1 p 1 ( , ) d y dy v x y dt  dt    where 2 1/18 p p   d    is the particle relaxation time, 10 μm p d  is the particle diameter, 1000 kg/m3 p   is the particle density, and   1.85105 Pa.s is the air viscosity. Also, u(x, y) and v(x, y) represent the components of the air velocity in the x and y directions inside the pleat channel, respectively. The x and y positions of the particles are denoted by xp and yp, respectively. You may use the following expressions for u(x, y) and v(x, y) .     2 3 1 2 u x, y u x y h                  sin 2 v x,y v h π y h        where   i 1 u x U x l h          is the average air velocity inside the pleat channel in the x‐direction and Ui is the velocity at the pleat entrance (assume 1 m/s for this project). l = 0.0275 m and h =0.0011 m are the pleat length and height, respectively. Writing the conservation of mass for the air flowing into the channel, you can also obtain that   i v h U h l h         . These 2nd order ODEs can easily be solved using a 4th order Rung‐Kutta method. In order to obtain realistic particle trajectories, you also need to consider proper initial conditions for the velocity of the particles: x(t  0)  0 , ( 0) i p p y t   y , p ( 0) cos i i dx t U dt    , p ( 0) sin i i dy t U dt     . where i  is the angle with respect to the axial direction by which a particle enters the pleat channel (see Figure 3). The inlet angle can be obtained from the following equation: 2 75 0.78 +0.16 1.61St i i p p i y y e h h                    where   2 St 18 2 ρPdPUi μ h  is the particles Stokes number. Figure 3: An illustration of the required particle trajectory calculation inside a rectangular pleated filter. You are asked to calculate and plot the trajectories of particles released from the vertical positions of ?? ? ? 0.05?, ?? ? ? 0.25?, ?? ? ? 0.5?, ?? ? ? 0.75? , and ?? ? ? 0.95? in one single figure. To do so, you need to track the trajectories until they reach one of the channel walls (i.e., stop when xp  l or p y  h ). Use a time step of 0.00001 sec. For more information see Ref. [3]. For additional background information see Ref. [4] and references there. In submitting your project please stick to following guidelines: 1‐ In blackboard, submit all the Matlab files and report in one single zip file. For naming your zip file, adhere to the format as: Lastname_firstname_project1.zip For instance: Einstein_albert_project1.zip 2‐ The report should be in pdf format only with the name as Project1.pdf (NO word documents .docx or .doc will be graded). 3‐ Your zip file can contain as many Matlab files as you want to submit. Also please submit the main code which TA’s should run with the name as: Project1.m (You can name the function files as you desire). Summary of what you should submit: 1‐ Runge–Kutta 4th order implementation in MATLAB. 2‐ Plot 5 particle trajectories in one graph. 3‐ Report your output (the x‐y positions of the five particles at each time step) in the form of a table with 11 columns (one for time and two for the x and y of each particle). Make sure the units are second for time and meter for the x and y. 4‐ Write a short, but yet clean and professional report describing your work. Up to 25% of your grade will be based solely on the style and formatting of your report. Use proper heading for each section of your report. Be consistent in your font size. Use Times New Roman only. Make sure that figures have proper self‐explanatory captions and are cited in the body of the report. Make sure that your figures have legends as well as x and y labels with proper and consistent fonts. Don’t forget that any number presented in the report or on the figures has to have a proper unit. Equations and pages in your report should be numbered. Embed your figures in the text. Make sure they do not have unnecessary frames around them or are not plotted on a grey background (default setting of some software programs!). inlet angle Particle trajectory i p y i 0 p x  Important Note: It is possible to solve the above ODEs using built‐in solvers such as ode45 in MATLAB, and you are encouraged to consider that for validating your MATLAB program. However, the results that you submit for this project MUST be obtained from your own implementation of the 4th order Runge‐Kutta method. You will not receive full credit if your MATALB program does not work, even if your results are absolutely correct! References: 1. http://www.airexco.net/custom‐manufacturedbr12‐inch‐pleated‐filter‐c‐108_113_114/custommadebr12‐ inch‐pleated‐filter‐p‐786.html 2. http://www.ebay.com/itm/Air‐Compressor‐Air‐Filter‐Element‐CFE‐275‐Round‐Pleated‐Filter‐ /251081172328 3. A.M. Saleh and H.V. Tafreshi, A Simple Semi‐Analytical Model for Designing Pleated Air Filters under Loading, Separation and Purification Technology 137, 94 (2014) 4. A.M. Saleh, S. Fotovati, H.V. Tafreshi, and B. Pourdeyhimi, Modeling Service Life of Pleated Filters Exposed to Poly‐Dispersed Aerosols, Powder Technology 266, 79 (2014)

Project 1: Particle Trajectory in Pleated Filters Due: 12:30 pm, Dec. 1, 2015, submission through blackboard Course: Numerical Methods Instructor: Dr. Hooman V. Tafreshi Most aerosol filters are made of pleated fibrous media. This is to accommodate as much filtration media as possible in a limited space available to an air filtration unit (e.g., the engine of a car). A variety of parameters contribute to the performance of a pleated filter. These parameters include, but are not limited to, geometry of the pleat (e.g., pleat height, width, and count), microscale properties of the fibrous media (e.g., fiber diameters, fiber orientation, and solid volume fraction), aerodynamic and thermal conditions of the flow (e.g., flow velocity, temperature, and operating pressure), and particle properties (e.g., diameter, density, and shape). Figure 1: Examples of pleated air filters [1‐2]. In this project you are asked to calculate the trajectory of aerosol particles as they travel inside a rectangular pleat channel. Due to the symmetry of the pleat geometry, you only need to simulate one half of the channel (see Figure 2). Figure 2: The simulation domain and boundary conditions (the figure’s aspect ratio is altered for illustration purposes). Trajectory of the aerosol particles can be calculated in a 2‐D domain by solving the Newton’s 2nd law written for the particles in the x‐ and y‐directions, v(h) inlet velocity fibrous media v(y) y tm l h x Ui u(l) u(x) 2 2 p 1 p 1 ( , ) d x dx u x y dt  dt    2 2 p 1 p 1 ( , ) d y dy v x y dt  dt    where 2 1/18 p p   d    is the particle relaxation time, 10 μm p d  is the particle diameter, 1000 kg/m3 p   is the particle density, and   1.85105 Pa.s is the air viscosity. Also, u(x, y) and v(x, y) represent the components of the air velocity in the x and y directions inside the pleat channel, respectively. The x and y positions of the particles are denoted by xp and yp, respectively. You may use the following expressions for u(x, y) and v(x, y) .     2 3 1 2 u x, y u x y h                  sin 2 v x,y v h π y h        where   i 1 u x U x l h          is the average air velocity inside the pleat channel in the x‐direction and Ui is the velocity at the pleat entrance (assume 1 m/s for this project). l = 0.0275 m and h =0.0011 m are the pleat length and height, respectively. Writing the conservation of mass for the air flowing into the channel, you can also obtain that   i v h U h l h         . These 2nd order ODEs can easily be solved using a 4th order Rung‐Kutta method. In order to obtain realistic particle trajectories, you also need to consider proper initial conditions for the velocity of the particles: x(t  0)  0 , ( 0) i p p y t   y , p ( 0) cos i i dx t U dt    , p ( 0) sin i i dy t U dt     . where i  is the angle with respect to the axial direction by which a particle enters the pleat channel (see Figure 3). The inlet angle can be obtained from the following equation: 2 75 0.78 +0.16 1.61St i i p p i y y e h h                    where   2 St 18 2 ρPdPUi μ h  is the particles Stokes number. Figure 3: An illustration of the required particle trajectory calculation inside a rectangular pleated filter. You are asked to calculate and plot the trajectories of particles released from the vertical positions of ?? ? ? 0.05?, ?? ? ? 0.25?, ?? ? ? 0.5?, ?? ? ? 0.75? , and ?? ? ? 0.95? in one single figure. To do so, you need to track the trajectories until they reach one of the channel walls (i.e., stop when xp  l or p y  h ). Use a time step of 0.00001 sec. For more information see Ref. [3]. For additional background information see Ref. [4] and references there. In submitting your project please stick to following guidelines: 1‐ In blackboard, submit all the Matlab files and report in one single zip file. For naming your zip file, adhere to the format as: Lastname_firstname_project1.zip For instance: Einstein_albert_project1.zip 2‐ The report should be in pdf format only with the name as Project1.pdf (NO word documents .docx or .doc will be graded). 3‐ Your zip file can contain as many Matlab files as you want to submit. Also please submit the main code which TA’s should run with the name as: Project1.m (You can name the function files as you desire). Summary of what you should submit: 1‐ Runge–Kutta 4th order implementation in MATLAB. 2‐ Plot 5 particle trajectories in one graph. 3‐ Report your output (the x‐y positions of the five particles at each time step) in the form of a table with 11 columns (one for time and two for the x and y of each particle). Make sure the units are second for time and meter for the x and y. 4‐ Write a short, but yet clean and professional report describing your work. Up to 25% of your grade will be based solely on the style and formatting of your report. Use proper heading for each section of your report. Be consistent in your font size. Use Times New Roman only. Make sure that figures have proper self‐explanatory captions and are cited in the body of the report. Make sure that your figures have legends as well as x and y labels with proper and consistent fonts. Don’t forget that any number presented in the report or on the figures has to have a proper unit. Equations and pages in your report should be numbered. Embed your figures in the text. Make sure they do not have unnecessary frames around them or are not plotted on a grey background (default setting of some software programs!). inlet angle Particle trajectory i p y i 0 p x  Important Note: It is possible to solve the above ODEs using built‐in solvers such as ode45 in MATLAB, and you are encouraged to consider that for validating your MATLAB program. However, the results that you submit for this project MUST be obtained from your own implementation of the 4th order Runge‐Kutta method. You will not receive full credit if your MATALB program does not work, even if your results are absolutely correct! References: 1. http://www.airexco.net/custom‐manufacturedbr12‐inch‐pleated‐filter‐c‐108_113_114/custommadebr12‐ inch‐pleated‐filter‐p‐786.html 2. http://www.ebay.com/itm/Air‐Compressor‐Air‐Filter‐Element‐CFE‐275‐Round‐Pleated‐Filter‐ /251081172328 3. A.M. Saleh and H.V. Tafreshi, A Simple Semi‐Analytical Model for Designing Pleated Air Filters under Loading, Separation and Purification Technology 137, 94 (2014) 4. A.M. Saleh, S. Fotovati, H.V. Tafreshi, and B. Pourdeyhimi, Modeling Service Life of Pleated Filters Exposed to Poly‐Dispersed Aerosols, Powder Technology 266, 79 (2014)

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Lab Assignment-09 Note: Create and save m-files for each problem individually. Copy all the m-files into a ‘single’ folder and upload the folder to D2L. Read chapters 2 and chapter 3.1-3.3 of the textbook (Introduction to MATLAB 7 for Engineers), solve the following problems in MATLAB. Given A= [■(3&-2&1@6&8&-5@7&9&10)] ; B= [■(6&9&-4@7&5&3@-8&2&1)] ; C= [■(-7&-5&2@10&6&1@3&-9&8)] ; Find the following A+B+C Verify the associative law (A+B)+C=A+ (B+C) D=Transpose(AB) E=A4 + B2 – C3 Find F, given that F = E-1 * D-1 – (AT) -1 Use MATLAB to solve the following set of equations 5x+7y + 9z = 12 7x- 4y + 8z = 86 15x- 9y – 6z = -57 Write a function that accepts temperature in degrees F and computes the corresponding value in degree C. The relation between the two is Aluminum alloys are made by adding other elements to aluminum to improve its properties, such as hardness or tensile strength. The following table shows the composition of five commonly used alloys, which are known by their alloy numbers ( 2024, 6061, and so on) [Kutz, 1999]. Obtain a matrix algorithm to compute the amounts of raw materials needed to produce a given amount of each alloy. Use MATLAB to determine how much raw material each type is needed to produce 1000tons of each alloy. Composition of aluminum alloys Alloy % Cu % Mg % Mn % Si % Zn 2024 4.4 1.5 0.6 0 0 6061 0 1 0 0.6 0 7005 0 1.4 0 0 4.5 7075 1.6 2.5 0 0 5.6 356.0 0 0.3 0 7 0

Lab Assignment-09 Note: Create and save m-files for each problem individually. Copy all the m-files into a ‘single’ folder and upload the folder to D2L. Read chapters 2 and chapter 3.1-3.3 of the textbook (Introduction to MATLAB 7 for Engineers), solve the following problems in MATLAB. Given A= [■(3&-2&1@6&8&-5@7&9&10)] ; B= [■(6&9&-4@7&5&3@-8&2&1)] ; C= [■(-7&-5&2@10&6&1@3&-9&8)] ; Find the following A+B+C Verify the associative law (A+B)+C=A+ (B+C) D=Transpose(AB) E=A4 + B2 – C3 Find F, given that F = E-1 * D-1 – (AT) -1 Use MATLAB to solve the following set of equations 5x+7y + 9z = 12 7x- 4y + 8z = 86 15x- 9y – 6z = -57 Write a function that accepts temperature in degrees F and computes the corresponding value in degree C. The relation between the two is Aluminum alloys are made by adding other elements to aluminum to improve its properties, such as hardness or tensile strength. The following table shows the composition of five commonly used alloys, which are known by their alloy numbers ( 2024, 6061, and so on) [Kutz, 1999]. Obtain a matrix algorithm to compute the amounts of raw materials needed to produce a given amount of each alloy. Use MATLAB to determine how much raw material each type is needed to produce 1000tons of each alloy. Composition of aluminum alloys Alloy % Cu % Mg % Mn % Si % Zn 2024 4.4 1.5 0.6 0 0 6061 0 1 0 0.6 0 7005 0 1.4 0 0 4.5 7075 1.6 2.5 0 0 5.6 356.0 0 0.3 0 7 0

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