Consider a mass m suspended by a spring of stiffness k and damper ofcoefficient c, arranged in parallel with each other as illustrated in Figure Q2.The mass is initially pulled downwards, in a direction termed positive-x, by aforcing function F = F, cos wt. After a short period of time steady-statebehaviour, characterised by a lag between force and displacement of 6 radians,comes to predominate. The maximum displacement described by this motionhas a magnitude of X. The system is in a resonant state.Q2kmF = F,cos(wt)Figure Q2Write equations for the displacement x, velocity x, and acceleration i of themass, using cosine terms.(a)Thus, given that the overall equation of motion is Fo cos(wt) – mä – ci -kx = 0, express the terms of the equation using a vector diagram. Identify theangles ot and o, and ensure that the angles you draw are labelled (whereappropriate) with values that are compatible with the state of the systemdescribed in the question above. This is to be a professional-quality engineeringsketch.(b)Rearrange this diagram into a vector polygon, again to a professional quality,and use this polygon to extract X/F, in terms of k, 5, and B. Characterise thetransmitted force you expect to see, given the state of the system.(c)(d)Imagine that Fo is set to zero, and the attachment point is instead displaced,sinusoidally, in a new vertical co-ordinate y. Using a value of X/F. derived fromyour vector polygon, show that adjustments to the value of B could make thedisplacement x a function of either y or y.Discuss the two devices you have modelled in part (d). What are their names,what do they measure, and what size are they?(e)

Answered: Image /qna-images/answer/e17165fc-97fd-4fc7-9220-59ee1eae94ad.jpg

Consider a mass m suspended by a spring of stiffness k and damper of coefficient c, arranged in parallel with each other as illustrated in Figure Q2. The mass is initially pulled downwards, in a direction termed positive-x, by a forcing function F F, cos wt. After a short period of time steady-state behaviour, characterised by a lag between force and displacement of o radians, comes to predominate. The maximum displacement described by this motion has a magnitude of X. The system is in a resonant state. Q2 k F = F,cos(wt) %3D Figure Q2 Write equations for the displacement x, velocity x, and acceleration i of the mass, using cosine terms. (a) Thus, given that the overall equation of motion is Fo cos(wt) - mä – cx – kx = (b) 0, express the terms of the equation using a vector diagram. Identify the angles ot and , and ensure that the angles you draw are labelled (where appropriate) with values that are compatible with the state of the system described in the question above. This is to be a professional-quality engineering sketch. E

Consider a mass m suspended by a spring of stiffness k and damper of coefficient c, arranged in parallel with each other as illustrated in Figure Q2. The mass is initially pulled downwards, in a direction termed positive-x, by a forcing function F F, cos wt. After a short period of time steady-state behaviour, characterised by a lag between force and displacement of o radians, comes to predominate. The maximum displacement described by this motion has a magnitude of X. The system is in a resonant state. Q2 k F = F,cos(wt) %3D Figure Q2 Write equations for the displacement x, velocity x, and acceleration i of the mass, using cosine terms. (a) Thus, given that the overall equation of motion is Fo cos(wt) - mä – cx – kx = (b) 0, express the terms of the equation using a vector diagram. Identify the angles ot and , and ensure that the angles you draw are labelled (where appropriate) with values that are compatible with the state of the system described in the question above. This is to be a professional-quality engineering sketch. E

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

Needs Complete solution with 100 % accuracy.
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…
MECHANICAL VIBRATIONS
A mass, m of 20 kg consider one vertical motion only with angular displacement, 0 = wt, with spring coefficient, k of 13 kN/m as shown in Figure Q1. If the system oscillates at free motion y = Y sin(@nt) with amplitude, Y of 13 mm: (a) (i) Determine the natural frequency of the system, o,. (ii) Solve the time taken to complete one cycle of motion. (iii) Analyze the velocity of the mass, ý at position A.
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…
%3! Q4: - For static system as shown in figure below, if k 50 N/mm, m =m2, u = 0.25 for the blocks m, and m2, tensions TI= 28.35N, spring elongation 5 mm and the force F1 m, cos 0 + m, sin 0. Find the inclination angle, Tension in other wire, and the mass of blocks. By using the Grammar method. T2 m2 T1 ml F1
For the mass spring system shown in figure spring constant (k 1 kg/m), mass= 8 kg and damping constant (C= 6 kg/sec). The steady state for this system is %3D k-IKg/m M-8Kg C-6 Kg/ sec
Mechanical Vibration Problem
Underdamped Mass-Spring System Find and graph the motion of a damped mass-spring system with mass m = 0.25 slugs, spring constant k = 4 lb/ft. and damping constanl b = I lb sec/fl. The mass is initially pulled to the right s1re1ching the spring by I ft. and th:n released.
For a mass-spring oscillator, Newton's second law implies that the position y(t) of the mass is governed by the second-order differential equation my''(t) +by' (t) + ky(t) = 0. (a) Find the equation of motion for the vibrating spring with damping if m= 20 kg, b = 80 kg/sec, k = 260 kg/sec², y(0) = 0.3 m, and y'(0) = -0.3 m/sec. (b) After how many seconds will the mass in part (a) first cross the equilibrium point? (c) Find the frequency of oscillation for the spring system of part (a). (d) The corresponding undamped system has a frequency of oscillation of approximately 0.574 cycles per second. What effect does the damping have on the frequency of oscillation? What other effects does it have on the solution? (a) y(t) = plz answer a-d
Forcing Function Spring Constant (2)4 Mass k Friction Constant b m Mass Displacement y(t) 3. (20 pts.) Consider the following spring-mass-damper mechanical system (it is placed sideways, so that you won't need to consider gravity). The input is given by f(t), and the output is y(t). Find an equation in frequency domain (s domain) that defines the relationship between the input F(s) and the output Y(s).
t. Design the following application case m is The mechanical vibration system is shown in Figure (a). When a step input with shown in Figure (b). Evaluate the values of mass m, damping constant c and spring constant k. B (a) yu P=2N X(t)/cm4 0.1095 0.1 0 2 (b) 3 4