1. Two carts with negligible rolling friction are connected as shown in Figure 1.An input force u(t) is applied. The masses of the two carts are M₁ and M2 andtheir displacements are denoted as x(t)) and q(t), respectively. The twocarts are connected by springs, k₁, k2 and dampers b₁, b₂.x(t)g(t)k₁k,M₁M,u(t)Figure 1By using Newton's Second Law, derive the two equations that describe themotion of the two carts. Hence derive the transfer function that relates thedisplacement q(t) and the input u(t), i.e. G(s) : =Q(s)U(s)Support your answer with complete working steps and explanation.

Step 1: Step 2: Step 3: Step 4:

Answered: 1. Two carts with negligible rolling…

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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The following data is given for the spring-mass-damper system shown in Figure • K1 = 250 N/m, K2 = 350 N/m • C1 = 150 N/ (m/sec), C2 = 200 N/ (m/sec) m = (25 +6) kg Determine: i. The undamped natural frequency. ii. The damping ratio. iii. The damped natural frequency. x(t) k 1 k 2 m C1 C2 Figure
1- Two mass-spring-damper systems A & B. System A has = 0.1 and wn = 20 rad/sec, and system B has < = 0.3 and w₁ = 5 rad/sec. Which system will vibrate longer in time? Why?
Damped Oscillatory Motion The setup consists of a mass m and a mass ", connected by a massless string of length { via a massless pulley. There is no friction between the string and the pulley. Mass m/2 is connected to a spring (spring constant k) and to a damper, damping constant b. Write down the equation of motion for the mass m. Use the coordinate x, with x = 0 being the equilibrium position of the system. Question т m/2] k b
1- Two mass-spring-damper systems A & B. System A has = 0.1 and @n = 20 rad/sec, and system B has < = 0.3 and wn = 5 rad/sec. Which system will vibrate longer in time? Why?
Problem #1 Consider the system shown below. The two pulleys, small and large, are bolted together and move as a single piece. The parameter values are as follows: m = 5 kg m₂ = 1 kg m₂ = 2 kg R₁.m R₂m₂ SA R₂ = 0.200 m k₂ = 400 N/m k₂ = 300 N/m 772 x(t) Note: x(t =0) = 0.01 m *(t =0)=0 Jaise = mR² R₂ = 0.400 m C₁ = 1.5 N-s/m C₂= design parameter a. Determine the characteristic equation in terms of the position of the hanging mass, x(t). b. The parameter c₂ can be adjusted in the design phase. Determine the value of c, required to yield a damping coefficient of 0.2. c. In the post-design phase structure, determine the system response, x(t), in response to initial conditions of
Write down the differential equation(s) that model the vertical dynamics of the system below (assume ( that no appreciable horizontal or twisting motion occurs). Two masses are suspended from the ceiling. Mass mị is attached to the ceiling by a spring with spring constant k1. Mass m2 is attached to the ceiling by a spring with spring constant k2 and is attached to mass m1 by a damper with damping constant b. A force u(t) is applied to mass m1 in the downward direction. The variable x(t) denotes the vertical displacement of mass mị and y(t) denotes the vertical displacement of the mass m2. Ki M, Ke M2
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
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…
A machine is supported on four steel springs. By neglecting damping, the natural frequency of vertical vibration the machine-spring system runs at f=200 cycles per minute. The machine generates a vertical force p(t)=posinot. When the machine runs at f₁ =30 cycle per minute the amplitude of vertical vibration is measured as u₁=0.2cm, at f₂=170 cycle per minute is u₂=1.042cm, and at f3-400 cycle per minute is u3=0.0248cm. Calculate the amplitude of vertical motion of the machine if the steel springs are replaced by four rubber isolators which provide the same stiffness, but introduce damping equivalent to -0.25 for the system. Evaluate the response of the system for frequency ratios r, respectively.
Two carts with negligible rolling friction are connected as shown in Figure (1b). An input force u(t) is applied. The masses of the two carts are M, and M, and their displacements are x(t) and q(t), respectively. The carts are connected by a spring k and a damper b. Answer the following questions: (b) By using Newton's Second Law, derive two mathematical equations that describe the motion of the two carts. Hence derive the following two transfer functions: G,(s) = S) U(s) (c) x(1) 9(t) k M, M2 u(t) Figure (1b)
6. Figure 8 shows a damped spring-mass system that contains a fixed pulley, a slider and a mass. There is a string whose one side is hung vertically by a mass m, = 1 kg. the other side connect to a slider m, = 05 kg which can slide along the incline (A = 30°). The center of slider is pulled by a damper with damping coefficient e, as well as a spring with force constant k = 2 N/m. Moment of inertia of the pulley is / = 0.01 kg m? and the radius of the pulley is r = 0.1 m. The spring remains unstretched when mass m, is released from rest at the position x = 0, then it oscillates in vertical direction. Note that the cable do not slip, the air resistance against motion and the friction between the slider and the incline can be neglected. i) Draw the free-body diagrams of mass m, and pulley /; i) Formulate the oardinary differential equation (ODE) of this vibration system in terms of x; i) Determine the damping coefficient e such that the system is being subcritical damped; iv) Given the…
Figure 1 shows a system comprising a bar with mass m-12 kg and the length of the bar L=2 m, two springs with stiffness kt=1000 N-m/rad and k = 2000 N/m, one damper with damping coefficient c=50 N-s/m and two additive masses at the end of the bar, where each mass (M) is equal to 50 kg. The rotation about the hinge A, measured with respect to the static equilibrium position of the system is 0 (t). The system is excited by force (F) so find the state space representation of the system if the output y=ġ(t).

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