An aluminum rod of length L=1m has mass density p = 2700 kg andYoung's modulus E = 70 GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are wexactE=√ρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency w₁ of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = w² MU, and find the fundamentalfrequency w₁. Compute the relative error e₁ = (w1 - wexact) /w exactSketch the fundamental mode of vibration.2. Use COMSOL to solve the same problem. Show the steps necessaryto find the fundamental frequency and mode of the rod. What is therelative error using linear elements and a normal mesh?

Answered: Image /qna-images/answer/2abfe09b-215e-46c7-bdf0-b172a077bafe.jpg

Answered: An aluminum rod of length L = 1m has…

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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Find the system's natural frequency shown in the figure with references to x. The block moves downward and the disk has radius r and moment inertia (J.) rotates about the fixed point. Assume the cable is rigid with negligible mass. 24 ww Keg 23kme zm
Q2: Find the steady- state response of the system in Fig. below for the following data: kı=1000N/m, k=500N/m, c=500N.s/m, m=10kg, r=5cm, Jo=1kg.m², Fo=50N and 0=20rad/s Pulley, mass moment of inertia Jo k2 Fo sin wt m x(1) k
H6.
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…
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 o. 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 loge (n(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…
Sub- Mechanical vibrations
Derive the Lagrange's Equations of motion for the system shown in Fig. 1 using the displacement x as the generalized coordinate of the mass m. Assume that the pulley and the mass m are rigidly coupled: Find the steady-state response of the system for the following data: k = 1000N / m,k, = 500N / m, c = 500NS / m, m =10kg,r = 5cm,J, = 1kgm² , F, = 50N, @= 20rad / s. %3D %3D %3D %3D %3D Pulley, mass moment of inertia J 2r Fosin ast Fig. 1
Please answer: How many degrees of freedom are in the system Write out the equations of morion in matrix form & Find the natural frequencies Given: k = 10 N/m; r = 0.5 m; I1 = 1.25 kg*m2 ; I2 = 1.2 kg*m2; m = 1 kg
For the mechanical network in figure 2,(a) Draw block diagram. (b) Draw free-body diagram of mass, spring, and damper system. (c) Draw transformed free-body diagram.
6. A weight of mass M = 8 kg is suspended by a spring with k = 10 N/ m and c = 1.6 N s/m. Determine the general vibration response to this homogeneous system by converting the equation to a state-space equation and solving it using eigenvalues and eigenvectors.
3. Suppose the model of a vehicle suspension is given below. Write the equation of motion in matrix form (not in State Space form). Calculate the natural frequencies for k₁ k2 10 N/m, m2 = 50kg, and m₁ = 2000kg. -= - - 103 N/m, 1 x1(c) m1 www m1 Car mass k₁ Car spring k1 k₂ Tire stiffness www x2(1) m2 Tire mass м2 k₂ ↑
Q1: The system shown has two masses. Beam of mass (Jo#m L² kg.m²) rotates about fixed point (O) and its free end is connected to disk rotates about fixed point (O₂). Consider all connecting links are massless and rigid. Find 1- The displacements of points A, B, and C in addition to the rotations of masses, all in terms of 0. 2- Find the equation of motion (EOM) in terms of 0. 3- What is the natural frequency of the system? 0 L/2 8 Energy methods A Jo=m L²2 L/2 Joz-m R² R C B C 128

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