4 Pre-LabEquations Governing the Cart DynamicsDerive the following equation of motion for the cart system shown in Figure 5:(mer² Rm+RmKJm) ï + (K₁KmK?) i = (rK,Kg) VTable 1 lists the parameters that appear in (5).Parameter UnitVoltkgmeterΩN-m/AVs/radVmeTRmK₁KmKgJmDescriptioninput voltagemass of the carradius of the motor gearsresistance of the motor windingstorque motor constantback EMF constantgearbox ratiomoment of inertia of the motorkg m²Table 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 inputvoltage 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 ofthe cart. The encoder is used to keep track of the position of the cart on the track.CartEncoderMaterMarch 7, 2023TrackXFigure 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 DynamicsThe input to your system is actually a voltage to the cart's motor. Thus, you need to derive the dynamicsof the system that converts the input voltage to the foren exertal on the cart. These are the dynamicsof the motor.Figure 6 shows a diagram of the electrical components of the motor.Figure 6: Clasic armature circuit of a standard DC motorFor 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 inertinTa-Kila JulHere K, is the motor torque constant, I is the current flowing through the coil, Jobs the moment ofinertia of the motor and is the angular acceleration of the motor.ME C134/KE C128 Spring 2023 Lab 3The current flowing through the motor can be related to the motor voltage input by:VIR+R-IR + Ku4 of 10UC Berkeleywhere is the angular velocity of the motor, R, is the resistance of the motor windings and K. is theback KMP constant (in).The torque is related to the applial force viawhere is the radius of the motor gear and K, is the gearbox gear ratio. The motor's angular velocityis related to the eart's linear velocity viaK-0¹ Kž – 0 ¹1

Given: To find: Derive equation

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

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Similar questions

The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas--pressure, temperature, and density (or quantity per volume): PV = NkBT (or pV = nRT), Figure L₂ Lx 1 of 1 Part A Find the magnitude of the average force (F) in the x direction that the particle exerts on the right-hand wall of the container as it bounces back and forth. Assume that collisions between the wall and particle are elastic and that the position of the container is fixed. Be careful of the sign of your answer. Express the magnitude of the average force in terms of m, vr, and L₂. ► View Available Hint(s) Submit Part B IVE ΑΣΦ ? Imagine that the container from the problem introduction is now filled with N identical gas particles of mass m. The particles each have different x velocities. but their average x velocity squared. denoted
Consider the system shown below. Answer Questions 6-11 based on this figure. How many degrees-of-freedom does the system shown below has? The two pulleys have the identical mass, radius, and the moment of inertia. т, 1, R k т, 1, R Variables x: vertical position of the moving pulley 8: rotation of the moving pulley p: rotation of the fixed pulley m: pulley mass R: pulley radius I: mass moment of the pulley about its own center of mass. k: spring stiffness T: cable tension Assumptions 1. x = 0, 0 = 0, and o = 0 when the spring is undeformed. 2. The cables and springs have negligible mass. 3. There is no slip between the pulley and the cable. 4. Only two-dimensional planar motion is allowed.
Question 9: Figure 3 shows a mechanical system. The rod (with moment of inertia J) rotates about the pivot at only small rotation angles. As pictured, theta is positive clockwise. Attached is mass m, which moves positive to the right. When stationary in the position shown, all springs are undeflected. Using BOBODDY, find the mathematical model of this system assuming small rotation angle 0. Link, moment of inertia J k3 L₁ 12 5 Ꮎ wwww Figure 3: Mechanical System m k₂ b
ll b) Obtain the mathematical model of the system shown in Figure Q2b using Newton's second law of motion, F=ma. k₁ w 3- 777777 C1 7771 k₂ D 7777 Figure: Q2b Page 2 of 7 A C2 1112
Fundamental.8 deals with the equation of motion. The use of the equation of motion to solve this problem is mandatory. Solution using other approaches (conservation of energy...) will be automatically considered false. For the problem related to Fundamental.8 sketches of the system showing: • the respective velocity and acceleration and the frame of reference considered • the forces acting on the system of considered, in other words, a free body diagram (FBD) are mandatory. Their absences will automatically make the problem false. A force P is applied at an angle =39 to a 311-kg cart. 0.4 m 0.3 m 0.08 m The acceleration of the cart is 2 m.s-². G B 0.2 m 0.3 m 1. What is the magnitude of the force P (answer on your hand-written work and in the cell below)? 2. What are the reaction at A and B?
Fundamental.8 deals with the equation of motion. The use of the equation of motion to solve this problem is mandatory. Solution using other approaches (conservation of energy...) will be automatically considered false. For the problem related to Fundamental.8 sketches of the system showing: • the respective velocity and acceleration and the frame of reference considered • the forces acting on the system of considered, in other words, a free body diagram (FBD) are mandatory. Their absences will automatically make the problem false. A force P is applied at an angle 0 =51 to a 869-kg cart. The kinetic friction coefficient on wheels is 0. P 0 0.4 m 0.3 m 0.08 m The acceleration of the cart is 1.2 m.s2. G B 0.2 m 0.3 m 1. What is the magnitude of the force P (answer on your hand-written work and in the cell below)? 2. What are the reaction at A and B?
A solid cylinder (mass 1.2kg, radius 5.0cm) on a horizontal table is connected to a block (mass 200 g) via a pulley (mass 0.80kg, radius 10.0cm) via a light, rigid string. Draw free body diagram for each of the object 1) list the dynamic and kinematic equations that related to all three objects. 2) find the acceleration of the falling block 3) fnd the angular acceleration of the cylinder and the friction force on it
Fundamental.9 deals with the equation of motion. The use of the equation of motion to solve this problem is mandatory. Solution using other approaches (conservation of energy...) will be automatically considered false. For the problem related to Fundamental.9 sketches of the system showing: • the respective velocity and acceleration and the frame of reference considered • the forces acting on the system of considered, in other words, a free body diagram (FBD) are mandatory. Their absences will automatically make the problem false. The uniform 19 lb beam is initially at rest when the forces FA applied at A (on its left hand-side) is vertical and FB applied at B (on its right hand-side) forms an angle 0=56° with the horizontal. FA FB A В. B 0 12 ft Determine FA (answer in the cell bellow) and FB to make sure that: • the beam has no angular acceleration • its horizontal horizontal acceleration is 1.6-times smaller than its vertical acceleration
Problem 5 Consider the two-mass system connected by two springs. Let coordinates x, and x, be such that when they are both zero, the springs are at their natural lengths. Write down the equations of motion. Use the Matlab command ode45 to solve the initial value problem with parameters m = m2 = k, = 1, k2 = 10 and initial conditions x,(t = 0) = *,(t = 0) = *,(t = 0) = 0, x2(t = 0) = 0.1 for te [0,67]. Plot x, and x2 as functions of t. Submit your plot as well as codes.
For the rotational mechanical system below, write, but do not solve, the equations of motion. T 大 Ji cele DI
3) Here is a simplified model of vehicle suspension system. The vehicle is being driven on the road below. The profile of the road and the velocity of the car in the horizontal direction are given. A road with such a profile will create an oscillation in the body of the car (the oscillation is denoted by y(t)). 1 y 1) Please obtain the magnitude of y(t) as a function of k and c. 2) Assume the mass of the car as 1 kg, k=10 N/m and c=2 Ns/m. What should the velocity of the car be to make ly(t)| less Vx = 3 m/s than 0.001 m? X : horizontal distance in meters sin(4x) the profile of the road in meters
Fundamental.7 deals with the equation of motion. The use of the equation of motion to solve this problem is mandatory. Solution using other approaches (conservation of energy ...) will be automatically considered false. For the problem related to Fundamental.7 sketches of the system showing: • the respective velocity and acceleration and the frame of reference considered • the forces acting on the system of considered, in other words, a free body diagram (FBD) are mandatory. Their absences will automatically make the problem false. B r C The pendulum bob B has a weight of 5lb and is released from rest in the position shown, 0 =0°. Determine the tension in string BC of length r = 5 ft at the instant the bob reaches point D, =7º.