Problem 2. FConsider the Mass-Spring-Damper system shown in Fig. 2 (left),where m is the mass of the cart, k is the spring constant, and b is the damper constant.The input of the system is u(t), which represents the external force acting on the cart. Thedisplacement of the cart is denoted as x(t).=x(t).1) (10 points) Consider the velocity (t) as the output of the system, i.e., y(t)Assuming the zero initial conditions, write the transfer function G(s) from the inputu(t) to the output y(t).2) (15 points) If a proportional controller is used to control the system, then the blockdiagram of the closed-loop system is given by Fig. 2 (right). Assume that m =1 kg,b = 4 N.s/m, and k = 3N/m. Sketch the Bode plot of G(s) for K = 1.3) (15 points) Sketch the Nyquist plot based on the Bode plot. Determine the range ofK for which the system is stable. Consider both positive and negative values of K.bx(t)Ju(t)mRKG(s)YFigure 2: Mass-Spring-Damper system in Problem 2.

the image. The question asks you to find the transfer function of the system with a proportional…

Answered: Problem 2. F Consider the…

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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Find the differential equation of the mechanical system in Figure 1(a) To obtain the differential equation of motion of the mass and spring system given in Fig. 1. (a) one may utilize the Newton's law for mass and spring relations defined as shown in Fig. 1. (b) and (c) use f = cv for viscous friction, where v is the velocity of the motion and c is a constant. Z/////// k M F. F, F F F, F, k EF=ma F = k(x, - x,) = kx (b) (c) Figure 1: Mass-spring system (a), Force relations of mass (b) and spring (c)
4. Consider a mass of 500 g placed at the end of a spring with stiffness constant 100 N/m and hanging at rest. The mass is displaced downward by 1.0 cm and released from rest. When the motion sets in, a time-varying force F(t) = cos³ 2t is applied to the spring-mass system. Solve the nonhomogeneous 2nd order differential equation representing the motion. That is, solve for x(t). [Note: Take the downward direction as the negative x-axis.]
4) Consider the following mass spring damper system: m2 u(t) Assuming that y, > y1, equation of motion (EOM) for this system can be obtained by applying the Newton's second law: = mỹ (i. e. RHS is positive) Then, equations are written as: m,y, = k2(y2 – Yı) – kıyı – cỷı (1) m2yz = u(t) - k2(y2 - Yı) (2) where u(t) is the input function to the system. Take m, = 1 kg, m2 = 2 kg, c = 10 Ns/m and k = 10 N/m, k2 = 20 N/m. Then, answer the following questions. ¥½(s) a) Find the transfer functions G, (s) = 1 and G2(s) = U(s) U(s) b) Use Laplace transform method and obtain the displacements in Laplace space (i.e. s domain) Y,(s) and Y2(s) utilizing initial conditions y, (0) = 0, y2(0) = 1, ỷ, (0) = 1 and y2(0) = 0. Use Cramer's rule for the solutions of Y, (s) and Y2(s). c) Using Inverse Laplace transformation, obtain displacements in time space (i.e. time domain) Y1(t) and y2(t) if, c.l. u(t) = 8(t) (Impulse function) (Ramp function starts from t=2) c.2. u(t) = {"2, 2st<∞ 0, 0st< 2 u(t) t-2 O…
6. A 1kg mass is attached to a spring (with spring constant k = 4 N/m), and the spring itself is attached to the ceiling. If you pull the mass down to stretch the spring past its equilibrium position, when you release the mass and observe its (vertical) position, it's said to undergo simple harmonic motion. AT REST MASS PULLED DOWN wwww Under certain initial conditions, the mass's vertical position (in metres) relative to its equilibrium position at time t, y(t), can be modelled by the equation y(t) = cos(2t) – sin(2t), (Note that y measures how much the spring has been stretched, so y = 1 indicates the mass is Im below its equilibrium position, whereas y = -1 indicates it is 1m above its equilibrium position.) (a) Find expressions for the mass's (vertical) velocity v(t) (relative to its equilibrium position) and the mass's (vertical) acceleration a(t) (relative to its equilibrium position). (b) Is the mass moving toward the ceiling or toward the floor at t= T? Justify your answer with…
A 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.
Giving that the input to the shown system is ft) = sin(wt) and the output is the displacement y(t), determine Y(s). Hint: Start by getting the transfer function Y(s)/F(s). %3D k k y(t) m frictionless surface Select one: Y(s) = O a. (2ms2 + 2bs + k)(s2 + w²) 2k2w Y(s) = O b. (ms2 + bs + 2k)(s² + w²) Y(s) = O c. (ms2 + 2bs + 2k)(s2 + w2) 2ko Y(s) d. (ms2 + bs + 2lk2)(s2 + w²) ka Y (s) = O e. (2ms2 +2bs + k)(s2 + w²) In order to change the behavior of a 2nd order underdamped system, we need to move its closed-loop poles on the s- plane as shown, With this move, what happened to the damping ratio? it
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.
EHide blocks This course Clear my choice Giving that the input to the shown system is f(t) = sin(wt) and the output is the displacement y(t), determine Y(s). Hint: Start by getting the transfer function Y(s)/F(s). Y(t) frictionless surface 2b b Select one: Y(s) = O a. (3mbs2 + 3kbs + k)(s² + w²) Y(s) = O b. (ms² + 3bs + k)(s² + w²) 3w Y(s) c. (3ms² + 2bs + 3k)(s² + w²) 3bo Y(s) = O d. (mks2 + 3bs + k)(s² + w²) 3kbo Y(s) (mks2 + 3kbs + 1)(s² + w²) Oe. Clear my choice Activate Windows Mith the shown closed-loop poles on the s-plane has the shown response c(t) to a unit-step input. (True or False)
damping, c Sliding Crate with Bumper Stop A crate of mass m enters a lubricated slide at a loading dock with a velocity v, to the right and an initial position x(0) = x, in the diagram shown above. The sliding motion is lubricated, characterized by a linear damping coefficient C. At a distance x = L the crate contacts a bumper stop characterized by a spring constant k and a damping coefficient c. Numerical values for the system parameters are: C = 8 N-s/m k = 1000 N/m m = 50 kg X = 0 Vo = 5 m/s L= 20 m Ca is to be experimentally determined in completing this assignment Bumper: Assignment 1) Given these parameters, determine the governing dynamic equation for the crate position. 2) Convert the governing equation to state variable form. This requires two first-order equations: a. dv/dt f:(x,v) b. dx/dt = f.(x,v) where f, is a function of the states, x and v. where f, is a different function of the states, x and v. Make two sets of the above equations, one each depending on whether there…
Answer the following problem and write your complete solution and graph. 1. Write the equation of motion for the system given in the figure for the case that F(t) = F cos wt and the surface is friction free. Does the angle affect the magnitude of oscillation? Diagram: wwwm mass F(0) Fig. 4.4 Mass tied on the spring and Dashpot
Q6. For a point at distance 50 m and angle 450 to the axis which of the following are correct? Consider an infinite baffled piston of radius 5 cm driven at 2 kHz in air with velocity 10 m/s. You may choose multiple options. A. k = 18.32 and J1(ka sin(q))/ka sin(q) = 0.17 B. k = 36.64 and J1(ka sin(q))/ka sin(q) = 0.40 C.k=5.83 and J1(ka sin(q))/ka sin(q) = 3.16 D. None of the above Submit E
Find: X2(s)/ F(s) Given: M1 = 2 Kg M2 = 4Kg K1 = 2N.s/m K2 = 1 N.s/m K3 = 4 N.s/m B1 = B2 = B3 = 2 N/m x1(t) fv, x2(t) f(t) K1 K3 M1 K2 M2 fv, (a) F(s) (Fr,s + K,) X,(s) Δ (b)

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