Mechanical Vibration Problem 3.33. The following data are given for a damped single degree of freedommass-spring system: mass m=6 kg, damping coefficient c= 25 N · s/m, andstiffness coefficient k = 5.5 x 10° N/m. The mass is subjected to a velocity-dependent force. The mass has zero initial displacement and initial velocityof 1 m/s. Determine the displacement and velocity of the mass after 0.1s inthe following two cases of the forcing function F(a) F = 50x N;(b) F = 400x N.

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Answered: 3.33. The following data are given for…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

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Consider the two-degrees-of-freedom mass-spring system below. Assume m₁ = 30 kg, m2 = 35 kg. If the second natural frequency of the system is 100 rad/s, the value of stiffness k is www. k₁=k k₂ = k k3= k 174.41 kN/m 107.38 kN/m 140.93 kN/m 39.24 kN/m 73.62 kN/m Next 000 00 Previous 4 x₁ (t) x₂ (t) m₁ m₂. mm
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?
An electric motor of mass 25 kg is supported on a three-spring system. Thesprings are arranged in parallel, with the three stiffnesses being 10 kN/m,5 kN/m and 25 kN/m respectively. Viscous damping is present within the systemand has a magnitude of 0.4 kNs/m.i) Produce a sketch of the system and state its equation of motion.ii) Determine the damping ratio and comment on its value.iii) Determine the amplitude of vibration of the motor 5 cycles afterit is released from an initial displacement of 15 mm.
2- Two mass-spring systems A & B both have stiffness of k-10 N/m. The mass of system A is twice the mass of system B (mA=2mg= 1 kg), which system will oscillate faster? Why? Support your answer with a plot.
A 3-kilogram mass hangs from a spring with a constant of 4 newtons per meter. The mass is set into motion by giving it a downward velocity of 3 meters per second. Damping in newtons equal to five times the velocity in meters per second acts on the mass during its motion. At time t = 6 seconds, it is struck upwards with a hammer imparting a unit impulse force. Set up the initial-value problem to compute the displacement of the mass as a function of time. Do not solve the equation.
4.8. A damped single degree of freedom mass-spring system has a mass m = 3 kg, a spring stiffness k = 2700 N/m, and a damping coefficient c = 18 N-s/m. The mass is subjected to a harmonic force which has an amplitude Fo = 20N and a frequency wf = 15 rad/s. The initial conditions are xo = 4 cm, and to = 0. Determine the displacement, velocity, and acceleration of the mass after time t = 0.5 s.
Consider the two-degrees-of-freedom mass-spring system below. Assume m₁ = 50 kg. m₂ = 55 kg. If the second natural frequency of the system is 100 rad/s, the value of stiffness kis k₁=k T x₂ (t) Ţ *₂(1) 107.38 kN/m 39.24 kN/m 73.62 kN/m 140.93 kN/m 174.41 kN/m 1₁ My kg = k
4. A mass-spring system exhibits a period of 5 s and a measured mass of 20 kg. Calculate the spring stiffness k.
QUESTION 3. Consider the mass-spring oscillator without friction: y" + y = 0. Suppose we add a force x(t) which corresponds to a push (to the left) of the mass as it oscillates. We will suppose the push is described by the function x(t) = -u(t – 2n) + u(t – (2n + a)) for some a > 2n which we are allowed to vary. (A small a will correspond to a short duration push and a large a to a long duration push). Here, u(t) is the unit step function. We are interested in solving the initial value problem y" + y = x(t), y(0) = 1, y'(0) = 0.
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…
5 rad/s A mass-spring system oscillates on a frictionless horizontal surface in simple harmonic motion with an amplitude A = 0.16 m. At what position (x = ?) would the kinetic energy of the system be equal to three times its elastic potential energy (K = 3U)? %3D At x = +0.1 m O At x = 10.05 m At x = +0.025 m OAt x = +0.04 m At x = +0.08 m
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…

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