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

A uniform slender rod of mass M and length L is pinned at one end and attached to a point mass, m through a linear spring of stiffness k as shown in Figure 5. The pinned end of the rod is constrained by a torsional spring of stiffness Kr. Given the following parameters: M=5 kg, L = 1.2 m, K, = 300 Nm/radian, m = 5 kg, k = 1000 N/m, (i) determine the equations of motion of the system
C Fill the correct answer in the blank space 1 The center of mass fem for three unit mass particles system equal: ... 2 The vector bi + 3j - k is perpendicular to the vector i + 2j-3k, then value of (b) is *** 3 If the damping frequency is wo/2 and the applied frequency is wo/4. Then tan of the phase angle equals; .... 4 Lagrangian approach is based on 5 Inertial term due to ............. of coordinate system. 6 A planet become faster in orbit around the sun when the sun.
a) A vehicle circulates on a road as shown in Figure Q2a. The road profile can be modelled as the input u(t). The vehicle is modelled as a quarter car of mass m, and the suspension has a spring stiffness coefficient k = 2 Nm and a damper of coefficient c = 2 Nm s. Find the position of the mass, y(t) and any time t if the road profile is a unit step. m 一 Road Figure Q2a b) In Figure Q2b, a disk flywheel J of mass m 32 kg and radius r = 0.5 m is driven by an electric motor that when it is working produces an oscillating torque of Tin = 10sin(wt) N- m. The shaft bearings may be modelled as viscous rotary dampers with a damping coefficient of BR = 0.4 N-m-s/rad and stiffness coefficient KR = 2 Nm If the flywheel is at rest at t 0 and the power is suddenly applied to the motor, do the following [Hint:J = mr²/2]: (i) Find the natural frequency of oscillation of the disk expressed in Hz. (ii) Find the damping ratio for this system. (iii) Describe in no more than 3-4 lines the behaviour of the…
A mass of weight 1 kg is attached to a spring, causing of a displacement of 20 cm in a liquid with damping constant is 14 kg/s. Then a second mass of 1 kg placed on top carefully so that the combined mass is in equilibrium. Then the second mass is removed abruptly, and the first mass start to move. Construct and solve the initial value problem representing the position of the first mass. Use the approximation g=9.8 m2/s. You may assume that no external force acts on the system.
5. Consider an underdamped system subjected to the force f(t) as shown. Assuming that all initial conditions are zero, find the system response using the convolution integral. x(1) meg Static equilibrium keg Cea a ww
2. Consider a car suspension, modeled as a mass/spring/damper system (mass m, stiffness k, damping b). Suppose the height of the chassis is lo at rest, the height of the terrain below the driver varies as h(t), and the height of the chassis is denoted lo + y(t). (i.e., spring deflection away from rest is y(t) – h(t)). 2 (a) Give the transfer function G(s) = H(s) · = (b) Suppose the ground follows an oscillatory profile h(t) A cos(wx (t)) with magnitude A (in meters) and frequency w (measured in radians per meter). Suppose the car is traveling at a constant forward speed v. Using a frequency response analysis strategy, give the amplitude of oscillations experienced by the driver at steady state as a function of m, k, b, A, w, and v. Hint: You can't simply consider |G(iw)| to get the amplification in this case. (c) Suppose the ground varies by A = 5cm, w = 2 rad/m, and you are driving at v = 15 m/s. Using your answer to part (b), what amplitude of oscillation is felt by the driver when m…
A mass is suspended from a spring and is pulled down 3 cm from equilibrium and released. It completes a period in 2 seconds. Ignoring damping factors, find the function for the displacement from equilibrium h(t) in centimeters t seconds after the mass is released.
Between a solid mass of 10 kg and floor, two slab of isolator, natural rubber and feltare kept in series as shown in Figure Q1(b). The natural rubber slab has a stiffnessof 3000 N/m and an equivalent viscous damping coefficient of 110 Ns/m. The feltslab has a stiffness of 12,200 N/m and an equivalent viscous damping coefficient of170 Ns/m. (i) Sketch the single degree of freedom spring-mass-damper system(ii) Determine the undamped and damped natural frequencies of the system inthe vertical direction. Neglect the mass of the isolator
In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t=0 no external torques act on the wheel. ✈0(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity 22o. Solve the differential equation for response (1). c) Normally, we plot the response as (t) versus t. In- stead, suppose we plot the response as log, (n(t)/Ng) versus t. What type of curve is this plot? If you mea- sured experimentally logg ((t)/no) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and…