For the simple mechanical system model shown in fig.2 with (M=2, B=0.7, k=1). Write the differential equation f(t) M AB y(t) Fig.2 Simple mechanical system
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- Figure Q3 shows one cart with a mass that is separated from two walls by two springs and a dashpot, where kı, k2 and ka are the first, second spring and dashpot coefficients, respectively. The mass, m could represent an automobile system. An external force is also shown as F(t). Only horizontal motion and forces are considered. F(t) is input and x2(t) is output. (a) Derive all equations related to the system (b) Construct the block diagram from equation in (a) (c) Obtain the transfer function of the systemFor the simple mechanical system model shown in fig.2 with (M=2, B=0.7, k=1). Write the differential equation and obtain the simulation diagram. f(t) K M y(t) AB Fig.2 Simple mechanical systemP3) Find the transfer function (T.F.) of the systems as in the following fig.p3, 9 m K₂ C₂ fig.p3 ٤/٢ ... ...
- A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): 1. What is the order of this system?A spring system is shown here: k₁ 3 Ę k3 2 K₂ www ma 4 Ę₂ Part 1: For this specific system, develop the: a. Global stiffness matrix . b. Boundary condition vector • c. Load vector • d. Reduced system of equations • e. Reaction force equations (i.e., the equations eliminated by the boundary conditions) Part 2: Given: k1 = 70 N/mm, k2 = 110 N/mm, k3 = 165 N/mm, F1 = 150 N, F2 = 100 N, and nodes 1 and 3 are fixed; calculate the: a. Global stiffness matrix b. Displacements of nodes 2 and 4 c. Reaction forces at nodes 1 and 3 d. Spring force in each of the springs
- You are requested to design an automotive suspension or shock absorber system. In order to simplify the problem to one dimensional multiple mass-spring-damper system, a quarter vehicle model is used. The system parameters and free-body diagram of such system is shown below. M₁: Automobile body mass M₂: Wheel and suspension mass K₁: Spring constant of suspension system K2: Spring constant of wheel and tire B: Damping constant of shock absorber (a) Obtain the transfer function of X₁ (s) F(s) T₁(s) = = 2500 kg = 320 kg and T₂(s) = = 80,000 N/m = 500,000 N/m = 350 N-s/m Automobile- Suspension system Wheel- M₁ M₂ X₁ (s) - X₂ (S) F(s) in terms of the parameters of mass, damper and elastance (M, B and K). (b) Express the T₁ (s) and T₂ (s) with numerical values. c) Plot the x₁ (t) and x₁ (t) = x₂(t) outputs of this passive suspension system for the input torque f(t) = 2,000 N. fit) K₂ x₂(t) -Tire1. A spring mass system serving as a shock absorber under a car's suspension, supports the M 1000 kg mass of the car. For this shock absorber, k = 1 × 10°N /m and c = 2 × 10° N s/m. The car drives over a corrugated road with force %3| F = 2× 10° sin(@t) N . Use your notes to model the second order differential equation suited to this application. Simplify the equation with the coefficient of x'" as one. Solve x (the general solution) in terms of w using the complimentary and particular solution method. In determining the coefficients of your particular solution, it will be required that you assume w – 1z w or 1 – o z -w. Do not use Matlab as its solution will not be identifiable in the solution entry. Do not determine the value of w. You must indicate in your solution: 1. The simplified differential equation in terms of the displacement x you will be solving 2. The m equation and complimentary solution xe 3. The choice for the particular solution and the actual particular solution x,…The oven shown in the figure has a heating element with appreciable capacitance C. The other capacitance is that of the oven air C. The corresponding temperatures are T and T, and the outside temperature is To The thermal resistance of the heater-air interface is R; that of the oven wall is R. Identify a model for T1 and T2, with input q;, the heat flow rate delivered to the heater mass. Oven air T2 C2 T. R| T C 19i R2 Heating element a. Develop the dynamic equations for this system. (submit in Connect) b. Develop the state variable Matrix form of the model in terms of the system parameter variables (not numerical values). c. Using the numerical values for the system parameters provided, use MATLAB to calculate the transfer functions 1 and 2 T and describe the expected behavior based on both of these transfer functions. d. Plot the dynamic behavior of the actual temperature of the oven chamber if the heating element is turned on with a constant heat rate of qj. Assume that the entire…
- P3) Find the transfer function (T.F.) of the systems as in the following fig.p3, www kg CE TTTTTT fig.p3Consider a mass of 10 kg. attached to a spring with (K=40 N/m) and to a dashpot with damping constant 40 Ns/m. The spring is initially stretched by 2 meters then let go. 1) Setup the model (Organize info into Differential equation) 2) Convert model into normal form. 3) Solve1 An object of mass 125 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of times the weight of the object is pushing the object up (weight = mg). If we assume that water 40 resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 90 m/sec? Assume that the acceleration due to gravity is 9.81 m/ sec2. Find the equation of motion of the object. X(t) = %3D