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4 Pre-Lab 4.1 Equations Governing the Cart Dynamics Derive the following equation of motion for the cart system shown in Figure 5: (mer² Rm + RmK²Jm) ï + (K₁KmK²) à = (rK₂Kg) V Table 1 lists the parameters that appear in (5). Parameter Unit Volt kg meter Ω N-m/A Vs/rad V me T Rm K₁ Km Ka Jm Description input voltage mass of the car radius of the motor gears resistance of the motor windings torque motor constant back EMF constant gearbox ratio moment of inertia of the motor kg m² Table 1: Parameters of the cart system (5) In order to derive the equation (5), follow the steps below: 1. Using the free body diagram in Figure 5, apply Newton's second law to the cart. 2. Combine the motor dynamics, equations (1) - (4), to obtain the relationship between the input voltage V and the applied force Fa. Substitute this relationship into your equation from Step 1. This is the final model of your plant. 3. Is this system linear? If not, linearize the system. If so, leave as is.

4 Pre-Lab 4.1 Equations Governing the Cart Dynamics Derive the following equation of motion for the cart system shown in Figure 5: (mer² Rm + RmK²Jm) ï + (K₁KmK²) à = (rK₂Kg) V Table 1 lists the parameters that appear in (5). Parameter Unit Volt kg meter Ω N-m/A Vs/rad V me T Rm K₁ Km Ka Jm Description input voltage mass of the car radius of the motor gears resistance of the motor windings torque motor constant back EMF constant gearbox ratio moment of inertia of the motor kg m² Table 1: Parameters of the cart system (5) In order to derive the equation (5), follow the steps below: 1. Using the free body diagram in Figure 5, apply Newton's second law to the cart. 2. Combine the motor dynamics, equations (1) - (4), to obtain the relationship between the input voltage V and the applied force Fa. Substitute this relationship into your equation from Step 1. This is the final model of your plant. 3. Is this system linear? If not, linearize the system. If so, leave as is.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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4 Pre-Lab Equations Governing the Cart Dynamics Derive the following equation of motion for the cart system shown in Figure 5: (mer² Rm+RmKJm) ï + (K₁KmK?) i = (rK,Kg) V Table 1 lists the parameters that appear in (5). Parameter Unit Volt kg meter Ω N-m/A Vs/rad V me T Rm K₁ Km Kg Jm Description input voltage mass of the car radius of the motor gears resistance of the motor windings torque motor constant back EMF constant gearbox ratio moment of inertia of the motor kg m² Table 1: Parameters of the cart system (5) In order to derive the equation (5), follow the steps below: 1. Using the free body diagram in Figure 5, apply Newton's second law to the cart. 2. Combine the motor dynamics, equations (1) - (4), to obtain the relationship between the input voltage V and the applied force Fa. Substitute this relationship into your equation from Step 1. This is the final model of your plant. 3. Is this system linear? If not, linearize the system. If so, leave as is.
Transcribed Image Text:4 Pre-Lab Equations Governing the Cart Dynamics Derive the following equation of motion for the cart system shown in Figure 5: (mer² Rm+RmKJm) ï + (K₁KmK?) i = (rK,Kg) V Table 1 lists the parameters that appear in (5). Parameter Unit Volt kg meter Ω N-m/A Vs/rad V me T Rm K₁ Km Kg Jm Description input voltage mass of the car radius of the motor gears resistance of the motor windings torque motor constant back EMF constant gearbox ratio moment of inertia of the motor kg m² Table 1: Parameters of the cart system (5) In order to derive the equation (5), follow the steps below: 1. Using the free body diagram in Figure 5, apply Newton's second law to the cart. 2. Combine the motor dynamics, equations (1) - (4), to obtain the relationship between the input voltage V and the applied force Fa. Substitute this relationship into your equation from Step 1. This is the final model of your plant. 3. Is this system linear? If not, linearize the system. If so, leave as is.
diagram, F, is the input force exerted on the cart by the voltage applied to the motor, m, is the mass of the cart. The encoder is used to keep track of the position of the cart on the track. Cart Encoder Mater March 7, 2023 Track X Figure 5: Free body diagram of the cart (ignoring friction) Using Figure 5 and basic Newtonian dynamics you can derive the equations governing the system. 3.3 Motor Dynamics The input to your system is actually a voltage to the cart's motor. Thus, you need to derive the dynamics of the system that converts the input voltage to the foren exertal on the cart. These are the dynamics of the motor. Figure 6 shows a diagram of the electrical components of the motor. Figure 6: Clasic armature circuit of a standard DC motor For this derivation of the motor dynamics we assume the following: • We disregard the motor inductance: IR, so we can use the approximation 0. • Perfect efficiency of the motor and gearbox: -- 1. The torque generated by the motor is proportional to the current flowing through the motor windings, but is lessenal due to the moment of inertin Ta-Kila Jul Here K, is the motor torque constant, I is the current flowing through the coil, Jobs the moment of inertia of the motor and is the angular acceleration of the motor. ME C134/KE C128 Spring 2023 Lab 3 The current flowing through the motor can be related to the motor voltage input by: VIR+R-IR + Ku 4 of 10 UC Berkeley where is the angular velocity of the motor, R, is the resistance of the motor windings and K. is the back KMP constant (in). The torque is related to the applial force via where is the radius of the motor gear and K, is the gearbox gear ratio. The motor's angular velocity is related to the eart's linear velocity via K-0¹ Kž – 0 ¹1
Transcribed Image Text:diagram, F, is the input force exerted on the cart by the voltage applied to the motor, m, is the mass of the cart. The encoder is used to keep track of the position of the cart on the track. Cart Encoder Mater March 7, 2023 Track X Figure 5: Free body diagram of the cart (ignoring friction) Using Figure 5 and basic Newtonian dynamics you can derive the equations governing the system. 3.3 Motor Dynamics The input to your system is actually a voltage to the cart's motor. Thus, you need to derive the dynamics of the system that converts the input voltage to the foren exertal on the cart. These are the dynamics of the motor. Figure 6 shows a diagram of the electrical components of the motor. Figure 6: Clasic armature circuit of a standard DC motor For this derivation of the motor dynamics we assume the following: • We disregard the motor inductance: IR, so we can use the approximation 0. • Perfect efficiency of the motor and gearbox: -- 1. The torque generated by the motor is proportional to the current flowing through the motor windings, but is lessenal due to the moment of inertin Ta-Kila Jul Here K, is the motor torque constant, I is the current flowing through the coil, Jobs the moment of inertia of the motor and is the angular acceleration of the motor. ME C134/KE C128 Spring 2023 Lab 3 The current flowing through the motor can be related to the motor voltage input by: VIR+R-IR + Ku 4 of 10 UC Berkeley where is the angular velocity of the motor, R, is the resistance of the motor windings and K. is the back KMP constant (in). The torque is related to the applial force via where is the radius of the motor gear and K, is the gearbox gear ratio. The motor's angular velocity is related to the eart's linear velocity via K-0¹ Kž – 0 ¹1
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