A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass Mis suspended from the collar by a frictionless pivot. L, M 2.1 Select a complete and independent set of generalized coordinates for this system. 2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix differential equation. 2.3 Develop the state-space model. å 2.4 Derive an eigenvalue problem of the form [K]{a}=¹[M]{a}

A collar of mass m slides without friction on a horizontal rigid rod and is restrained by a pair of identical springs with spring constant k. A pendulum consisting of a uniform rigid bar of length L and mass Mis suspended from the collar by a frictionless pivot. L, M 2.1 Select a complete and independent set of generalized coordinates for this system. 2.2 Derive using Lagrange's principle, the equations of motion of the system in the form of a matrix differential equation. 2.3 Develop the state-space model. å 2.4 Derive an eigenvalue problem of the form [K]{a}=¹[M]{a}

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 Rod with mass m and length L can rotate freely about one of its ends. A point mass with the same mass as the rod (m) can move freely along the length of the rod (ignore friction). The point mass is attached to a spring with a spring constant K and a negligible initial length. Gravity acts. Use Newton's Second Law (this includes the torque equation if you need it) and derive the Equation(s) of motion for this system. L mu K m m
A circular solid disc of uniform thickness 20 mm, radius 200 mm and mass 20 kg, is used as flywheel. If it rotates at 600 rpm, the kinetic energy of the flywheel, in joules is
Q4. A two-degree-of-freedom model consisting of two masses connected in series by two springs is shown in the figure below. The physical parameters have the values m, = 8 kg, m, = 2 kg, k, = 20 N/m, and k2 = 30 N/m. X1 X2 m1 m2 k1 k2 (A) Write down the equation of motion for mass m, (B) Write down the equation of motion for mass m, Calculate the first (larger) natural frequency of the system (D) Calculate the second (smaller) natural frequency of the system
A uniform disk of radius 0.455 m and unknown mass is constrained to rotate about a perpendicular axis through its center. A ring with the same mass as the disk is attached around the disk's rim. A tangential force of 0.227 N applied at the rim causes an angular acceleration of 0.131 rad/s^2. Find the mass of the disk.
4.61. Consider the double pendulum shown in Fig. P4.61 The two point masses of equal sn are attached to two rigid links of equal length L. The links are assumed massless Kance their masses are small compared to the point masses. The hinged joints constrain mesystem in planar motion. The friction is negligible, so the system can be assumed frictionless. A Using Lagrange's equations, derive the general nonlinear differential equations of motion. Obtain the linearized differential equations for small motions, and then express the Yesults in the second-order matrix form. 7. FIGURE PM
Asap
Consider a system of three masses, each with mass m, joined by ideal, massless springs, each with spring's constant k, as shown in figure 1. k m x1 k m x2 k m X3 k Figure 1: 3 identical masses attached with identical ideal, massless springs. The coordi- nates 1, 2, 3 represent the position in time of each of the masses. Neglect any effect due to gravity. (a) Show the spring forces acting upon each mass when T1, T2, T3 > 0. (b) Find the equations of motion of each mass. That is, a system of second order, coupled differential equations for x1, x2, x3. (c) Find the normal modes of the system and their associated angular frequencies.
4.13. Derive the differential equation of motion of the system shown in Fig. P4.1. Obtain the steady state solution of the absolute motion of the mass. Also obtain the displacement of the mass with respect to the moving base. For this system, let m=3 kg, k1 =k2 = 1350 N/m, c= 40 N s/m, and y=0.04 sin 15t. The initial conditions are such that xo =5 mm and io =0. Determine the displacement, velocity, and acceleration of the mass after time t =1s. y = Y, sin @,1 k1 in k2 m Fig. P4.1
2. A spring is hung vertically and an object of mass m attached to the lower end is then slowly lowered a distance d to the equilibrium point a. find the value of the spring constant if the magnitude of the displacement d is 2.0 cm and the mass is .55 kg b. if a second identical spring is attached to the object in parallel with the first spring, where is the new equilibrium point of the system? c. What is the effective spring constant of the two springs acting as one?
1. The figure below shows a one-dimensional, horizontal beam of uniform-density in static equilibrium. The beam is supported by a frictionless support at a point 1/3 of the way down its length (L). A rope running through a frictionless pulley is attached to the far end of the beam. The mass of the beam per-unit-meter (m) is known, and the magnitude of the acceleration due to gravity is g. y | 9 13 a) What is the total force due to the weight of the beam and where does it act? b) Draw the free body diagram (FBD) of this beam. c) Derive expressions for the vertical reaction force at the support (R) and the force applied to the end of the rope (F). Your expressions should be in terms of L, m,, and g; that is, you should find Ry = f(L, mr, g) and F = f(L, mz.g).
A uniform disk of mass 2M and radius a can roll along a rough horizontal rail. A particleof mass m/2 is suspended from the centre C of the disk by a light inextensible string oflength b. The whole system moves in the vertical plane through the rail. Take as generalisedcoordinates x, and θ the angle between the string and the downward vertical.(a) Write the Lagrangian for the system.(b) Obtain Euler-Lagrange equations.(c) Identify the cyclic coordinate and find the corresponding conserved momentum (say px).(d) Is px the horizontal linear momentum of the system?
9. Two masses connected by a massless string of length I slide without friction on two inclined planes as shown in the figure above. The motion is restricted to the x-y plane indicated in the figure. Gravity is acting in the negative y-direction. (a). Explain the term "virtual displacement". (b). State d'Alembert's principle of virtual work. (c). Write down d'Alemberts equation for this system in cartesian coordinates. Need detailed and step by step answer of all subparts with good handwriting Please I want to learn