Derive the state-space model (state equation and output equation) in vector form for thefollowing system. The system outputs are the displacements of each spring. Assume that theconnection between the spring and rope is massless and that the rope is inextensible. Assumethat gravity is an input as well as the applied forces Fi(t) and F2(t). Neglect friction forceson mass m₂.If q₁ is the state variable for the bottom spring connected to m₁, q2 is the state variable for themass m₁, q3 is the the top spring connected to the pulley, and q4 is the state variable connectedto the mass m2, then you should expect to get the following state-space representation:9243y0LaLa(L+L)04₂kL₂mi Li0m2013(L+L)0CA=- [8]9293 +9293 +L94m₂F₂(t)m₂TOLF₂(t)Figure 2: Diagram for problem 2X500X2[000]00U₁U12U13m₂.21142143

Given: To find: State-space model

Derive the state-space model (state equation and output equation) in vector form for the following system. The system outputs are the displacements of each spring. Assume that the connection between the spring and rope is massless and that the rope is inextensible. Assume that gravity is an input as well as the applied forces Fi(t) and F2(t). Neglect friction forces on mass m₂. If qi is the state variable for the bottom spring connected to m₁, q2 is the state variable for the mass m₁, q3 is the the top spring connected to the pulley, and q4 is the state variable connected to the mass m2, then you should expect to get the following state-space representation:

Derive the state-space model (state equation and output equation) in vector form for the following system. The system outputs are the displacements of each spring. Assume that the connection between the spring and rope is massless and that the rope is inextensible. Assume that gravity is an input as well as the applied forces Fi(t) and F2(t). Neglect friction forces on mass m₂. If qi is the state variable for the bottom spring connected to m₁, q2 is the state variable for the mass m₁, q3 is the the top spring connected to the pulley, and q4 is the state variable connected to the mass m2, then you should expect to get the following state-space representation:

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

2. Find the equation(s) of motion of the following system. How many DOF does this system have? The displacement of the mass is measured from the position where the spring is unstreched. Is this system linear or nonlinear? Define the state variables and express the equation(s) of motion in state form. Gravity should be included and the Coulomb friction coefficient of the rough ground is u. F,(0 x,(t) Friction, u
Find the state-space equation of the following mass-springdamper system. Assume that the bottom surface is frictionless. The input is external forcef(t)and the output is displacementx1(t)
For the system shown in the figure; the input is the torque T and the outputs are the displacements Q1 and Q2.a) Draw the free body diagrams and derive the equations of motions. (Assume Q1 >Q 2).b) Obtain the state-space representation by using the given state variables (vector)
3. Consider the system shown below. The outputs of the system are the angular displacement of the upper gear (positive about the x-axis) and the Contact force between the upper and lower gear. Assume that the initial conditions for all state variables are zero and that the gears are massless. There are two inputs Ti(t) acting on the top gear and T₂(t) acting on the rightmost disk. If you let • 9₁ denote the state variable for the spring 92 denote the state variable for the rightmost disk. u₁ denote T₁. u₂ denote T₂. You should expect to get the following state space representation and 9= KR + 0₁ 0 LIR -1. 7/2 Ti(t) Jun 0:0⁰ 40² T₂(t) 03 Figure 3: System for problem 3 21 (a) Derive the state-space model (state equation and output equation) in vector form. (b) For the system parameters I = 8 kg m², k = 1 N m,b=2 N s m/rad, R₁ = 1 m, and R₂ = 3 m: i. Use MATLAB to determine the transfer function matrix [G(s)]. ii. What is the ristic equation AS the system? iii. What are the values of the…
Consider the translational mechanical system as shown in Figure Q3 with two force inputs, F,(t) and F2(t) being applied on Ma and M3 respectively. Derive the equations of motion for each mass M,, M2 and M3. Then, determine the state space representation of the system if the state variables are as: X(t) = [x, (t) *(e) x2(t) i(t) x3(t) ž()]". The output of the system is taken as displacement at x, (t). X1(t) X3(t) K X2(t) F2(t) M1 M3 M2 F1(t) Figure Q3
a) Determine the state space representation for the translational mechanical system shown in Figure Q4(a), where force, f(t) and displacement, x(t) are the input and output of the system respectively. Use these state variables in your answer. oooo K M -x(1) -ƒ(1)
1. For the following mechanical translational system a. Write two differential equations of Order in s domain b. Change to time domain, and choose state variables c. Write the state equations, and the state matrix equation d. Write the output equation if x2 is the output Hint: the state variables will be x1, V1, X2, V2 fv, fit), K3 M K2 M2 0000
Question 3. Consider a mass-spring translational mechanical system in series. The spring has a nonlinear characteristics such that the relationship between the spring force and the spring displacement can be described mathematically f.(t) = 3x. Assume that the applied force is f(t) = 6+ 8f (t), where 8f(t) is a small force about the 6 Newton constant value. Assuming the output to be the displacement of the mass, obtain the state space representation of the system about the equilibrium displacement.
Taking the velocity v and the gravitational spring force F, as the state vari- ables (Note: This is the systematic and unified approach where we choose the across-variable v of the A-type element-mass, and the through-variable F, of the T-type element-spring, as the state variables) and F(t) as the input vari- able, develop a state-space model for the mass dynamics.
b m dy(t) +b. m d² y(t) dt² (5 points) Obtain the state space model of the system with input ƒ(t) and output y(t). Calculate the system matrices for m = 1, k = 4 and b = 1. dt ►y Hon - + ky(t) = f(t) f
Consider the following mechanical system: k m +f b d²y(t) +b- dy(t) + ky(t) = f (t) m %3D dt? dt Obtain the state space model of the system with input f (t) and output y(t). Calculate the system matrices for m = 1, k = 1 and b = 2. Check the stability by using the second method of Lyapunov. 3.
penaulum shown in the figure, the nonlinear equations of motion are given by where g is gravity, L is the length of the pendulum, m is the mass attached at the end of the pendulum (we - 0. assume the rod is massless), and k is the coefficient of sin e friction at the pivot point. a) Obtain a state-variable representation of the system. b) Linearize the dynamics and obtain a state-space representation of the system around the equilibrium position @ = 180 degrees, ở = 0. Pvor point Llength c) A constant torque, in counter-clockwise direction, is applied as an input to the pendulum. Angular position is the measured output. Derive the transfer function from the torque to angular position for the linearized pendulum system obtained in part b). Massles rod m, mass