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| موضوع: محاضرة بعنوان Free Vibration of 1-DOF System السبت 01 أكتوبر 2022, 1:47 am | |
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2.0 Outline Free Response of Undamped System Free Response of Damped System Natural Frequency, Damping Ratio Ch. 2: Free Vibration of 1-DOF System 2.1 Free Response of Undamped System Free vibration is the vibration of a system in response to initial excitations, consisting of initial displacements/ velocities. To obtain the free response, we must solve system of homogeneous ODEs, i.e. ones with zero applied forces. The standard form of MBK EOM is 2.1 Free Response of Undamped System mx cx kx x x t + + = = 0, ( )Ch. 2: Free Vibration of 1-DOF System If the system is undamped, c = 0. The EOM becomes 2.1 Free Response of Undamped System subject to the initial conditions 0 , 0 The solutions of homogeneous ODE are in the form x t Ae A s ( ) = st , is the amplitude and is constant Subs. the solution into ODE, we get 0 0 ** characteristic equation ** ** characteristic roots, eigenvalues ** general solution: by superposition n n = +Ch. 2: Free Vibration of 1-DOF System We can now apply the given i.c. to solve for A1 and A2. However we will use some facts to arrange the solutions into a more appealing form. 2.1 Free Response of Undamped System Because is real, . Let . Therefore . The given i.c. are then used to solve for the amplitude C and the phase angle Φ . Note ω n, known as natural frequency, is the system parameter. The system is called harmonic oscillator because of its response to i.c. is the oscillation at harmonic frequency forever.Ch. 2: Free Vibration of 1-DOF System 2.1 Free Response of Undamped System If the initial conditions are 0 and 0 cos and sin and tan cos sin as the function of i.c. and system parameter Ch. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped System We normalize the standard MBK EOM by mass m: 2.2 Free Response of Damped System / natural frequency / 2 viscous damping factor subject to the initial conditions 0 and 0 The solutions of homogeneous ODE are in the form x t Ae A s ( ) = st , is the amplitude and is constant Subs. the solution into ODE, we getCh. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped System ( ) ( ) 1 2 2 2 2 2 2 12 1 2 2 0 2 0 **CHE** 1 ** characteristic roots, eigenvalue ** general solution: by superposition = +Ch. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped System 12 Response of 0, i.c. 0 and 0 i) 0 : cos harmonic oscillation with frequency entially decaying amplitude oscillation with damped frequency and envelope ωd Ce−ζωntCh. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped SystemCh. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped System Response of 0, i.c. 0 and 0 iii) 1: 1 aperiodic decay with peak more suppressed and decay further slow down as inc aperiodic decay with highest peak and fastest decay = +Ch. 2: Free Vibration of 1-DOF System 2.2 Free Response of Damped SystemCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio 2.3 Natural Frequency, Damping Ratio The response will partly be dictated by the roots s12, which depend on Root locus diagram gives a complete picture of the manner in which s12 change with The focus will be the left half s-plane where the system response is stable. and . ζ ωn and . ζ ωnCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System constant, vary i) 0 undamped , on the Im-axis far from origin by . The motion is with natural frequency . ii) 0 1 underdamped , 1 harmonic oscillation ir of symmetric points moving on a semicircle of radius . The motion is . iii) 1 critically damped , repeated roots. The motion is . n n oscillatory decay s aperiodic decay 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System iv) 1 overdamped , 1 ( ) 12 2 two negative real roots going to 0 and . The motion is . constant, vary The symmetric roots will be far from origin along the radius n making an angle cos with - axis. −1 ( ) ζ x 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Determination of M-B-K • Mass – directly measure the weight or deduced from frequency of oscillation • Spring – from measures of the force and deflection or deduced from frequency of oscillation • Damper – deduced from the decrementing responseCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ex. 1 A given system of unknown mass m and spring k was observed to oscillate harmonically in free vibration with T n = 2π x10-2 s. When a mass M = 0.9 kg was added to the system, the new period rose to 2.5π x10-2 s. Determine the system parameters m and k.Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ex. 2 A connecting rod of mass m = 3x10-3 kg and IC = 0.432x10-4 kgm2 is suspended on a knife edge about the upper inner surface of a wrist-pin bearing, as shown in the figure. When disturbed slightly, the rod was observed to oscillate harmonically with ω n = 6 rad/s. Determine the distance h between the support and the C.M.Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio 3 10 kg, 0.432 10 kgm , 6 rad/s 0.2, 0.072 m Radius of gyration 0.12 m must be greater than the longest length of the object. 0.072 m ∴ =Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ex. 3 A disk of mass m and radius R rolls w/o slip while restrained by a dashpot with coefficient of viscous damping c in parallel with a spring of stiffness k. Derive the differential equation for the displacement x(t) of the disk mass center C and determine the viscous damping factor ζ and the frequency ω n of undamped oscillation.Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ex. 4 Calculate the frequency of damped oscillation of the system for the values m = 1750 kg, c = 3500 Ns/m, k = 7x105 N/m, a = 1.25 m, and b = 2.5 m. Determine the value of critical damping.Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio Ex. 5 A projectile of mass m = 10 kg traveling with v = 50 m/s strikes and becomes embedded in a massless board supported by a spring stiffness k = 6.4x104 N/m in parallel with a dashpot of c = 400 Ns/m. Determine the time required for the board to reach the max displacement and the value of max displacement.Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio ( ) 7 0.4447 m =Ch. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping RatioCh. 2: Free Vibration of 1-DOF System 2.3 Natural Frequency, Damping Ratio
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