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- Q2: For mass-spring system shown. The mass is given an initial displacement x(0)= 0.1 m, and released from rest. Find: 1- The position of the mass after 2 seconds. 2- The velocity of the mass after 2 seconds. 3- Plot the response for three cycles and label the result from 1 & 2. 4- What is the period of oscillation? 5- What is the acceleration of mass (m) after 5 second? Į x(t) k=100 N/m m = 4 kg5. A mass on a spring is shown below. Let y(t) be the displacement of the spring as a function of time t. Position A is the equilibrium position corresponding to y = 0. As was done in class, we are taking the positive direction to be downward. The graph of the solution to an initial value problem for this spring is shown. Answer the following questions about this system. A m B A: y=0 C (a) The spring's starting position (at t = 0) could be at i. Position A (Equilibrium) ii. Position B (Above Equilibrium) iii. Position C (Below Equilibrium) iv. Not enough information to tell (b) The spring's starting velocity (at t = 0) is i. positive ii. zero iii. negative iv. Not enough information to tell (c) Immediately after being set in motion at t = 0) the springThank you in advance! A certain mass-spring-damper system has the following equation of motion. x''+ cx' + 100x = f(t) Suppose that the initial conditions are zero and that the applied force f(t) is a step function of magnitude 100. Solve for x(t) for the following three cases: (a) c=10 and (b) c=20, (c) c=50.
- 6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless Support3 Problem Consider the following model of a mechanical system: The system has a mass m, a linear spring with stiffness k, and two identical dampers with damping constant b. The left wall generates an input motion xin (t) that causes the mass to undergo a displacement x(t) from its equilibrium position. The initial position and velocity are zero. • Find the ODE describing the motion of the system by drawing a free-body diagram and applying Newton's 2nd Law. Your answer should be in terms of the following variables: Xin Xin, X, X, x, b,k, m Show that the transfer function is G(s) = - X(s) Xin(s) = 2b m k s+ m k s²+ + m When taking the Laplace transform L[xin(t)] you may assume the initial (input) condition Xin(0) = 0.A spring with mass 1 kg is attached to one side of a spring with spring constant k = 12 kg/sec2. The other end of the spring is connected to the side of a steel beam so that the spring may oscillate in the horizontal direction. Suppose that friction is proportional to velocity with constant of proportionality f = 7 kg/sec. Write a differential equation for y(t), the position of the mass at time t, with the initial conditions y(0) = 0 and y(0) =1 m/sec. Solve your resulting IVP.
- The figure that is attached illustrates a system in which a force F is applied to a mass m2 that is connected to another mass m1 via a spring and a damper, and mass m1 is connected to a wall via a damper. The equations of motion that govern the time evolution of the mass displacements, y1(t)and y2(t), are given below. A) Define 4 state variables of the system (xi, i = 1,…,4) as phase variables, and define the control input u. Convert the two equations of motion above into four 1st-order ODEs that are functions of the state variables and the control input.(b) Write the four 1st -order ODEs from part (a) in state-space form,x = Ax+Bu.Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05mA 25 kg mass is attached to a spring with spring constant 100 N/m. The mass is driven by an external force equal tof(t) = 5 cos(2t). The mass is initially released from rest from a point 1 m below the equilibrium position. (Use theconvention that displacements measured below the equilibrium position are positive.) Write the initial-value problem which describes the position of the mass.
- 2. A kilogram mass is attached to the end of the spring with spring constant 2 N/m. Find the equation of motion if the mass is initially released (set in motion) from rest from a point 1 meter above equilibrium position. (Use the convention that displacements measured below the equilibrium position are positive.) (a) Write the initial-value problem which describes the position of the mass. (b) Find the solution to your initial-value problem from part (a). (c) Graph the solution found in (b) on (0For the following, use the spring equation: e dy k + m dt m dt² + with c = 4, m ues of k: 1. k=5 (b) (c) -y 0 1, and the initial conditions: y(0) = 2 and y'(0) = 0. For each of the following val- Find the general solution. Find the solution for these initial values. Sketch the timeplot graph of the solution y(t). In terms of the spring, is it underdamped, critically damped, or overdamped?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 metersSEE MORE QUESTIONS