Homework 3

Please answer question number 1.

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Select the right answer for each of the following: 1- The damped vibrational system is. system, (c) none of these, (d) false statement. (a) conservative system, (b) nonconservative (a) viscous (b) solid 2- When a system vibrates in a fluid medium, the damping is (c) Coulomb. 3- When an initial excitation exerted on a vibrational system results in - ippearance. (a) natural frequency. (b) resonance. (c) beating. (d) none of these. 4- Undamped vibration is characterized by no loss of 1) (a) speed, (b) acceleration, (c) mass, (d) energy. (a) mass, (b) damper, 5- Vibratory system should at least consists of a spring and - (c) mass and damper.

A strain gage system is mounted on an airplane wing to measure wing oscillations and strain during wind gusts. Thesystem is second order, having a 90 % rise time of 100 ms, a ringing frequency of 1200 Hz, and a damping ratio of0.8. Determine (a) the dynamic error when subjected to a 1 Hz oscillation and (b) the time lag (in seconds).

4. A convenient way to measure damping in a single-degree-of-freedom system is called logarith- mic decrement of free response. Consider the free response given by (4). Let's denote t₁ and t2 denote the times corresponding to two consecutive displacements ₁ and 2 measured one cycle apart, i.e., where t₂ = t₁ + T 2π Wd According to (4), show that the logarithmic decrement dis T = x1 8 = log = x2 where log has base e. Combine (5), (7), and (8) to obtain = (WnT 8 4π² +8² Will this method work for other initial conditions? why?

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Vibrations please help

An undamped simple harmonic oscillator has mass 2.0 kg and spring constant 50 N/m. The initial displacement from equilibrium (at time t = 0) is 0.30 m and the initial velocity is 2.0 m/s. %3D 1. Determine the angular frequency (in rad/s), the frequency (in Hz), and the period (in s). Determine numerical values (not in terms of .) Determine the displacement in the form x(t) = Acos(mot) + Bsin(@ot). That is, solve for A and B in meters, and write oo in radians per second.

QI: A portion of an automobile suspension system consists of an elastic spring and a viscous damper, as shown. If the natural frequency is 10 rad/sec, a) Determine the damping ratio so that any oscillations that occur will decay from 11-0.5 m to x2-D 0.025 m. b) If the system is displaced 0.2 m from its equilibrium position, and released from rest, determine the position of the mass after it has ascillated through 3 cycles.

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Q.4 Show that the time tm corresponding to the peak response of a damped single degree of freedom system due to an impulsive force is given by the equation /1 – §2 - tan watm where wg is the damped natural frequency and č is the damping factor.

Rayleigh's damping model has been used to evaluate the damping matrix, C=aM+ 3K, for a fixed-fixed bar, split in 3 elements, and subject to axial vibration. M and K are the mass and stiffness matrices respectively. Please select all the answers that apply: The coupled equations of motion can now be decoupled into single mode equations. The damping is stiffness dominated when ß is large. Following observations through experimental testing, a=ß in most cases. We will obtain large damping at high frequencies only if a is large. Following observations through experimental testing, ß is always null.

The graph shows the displacement versus time of a mass in a mass-damper-spring system. Which of the following could be the damping ratio of the system? x(t) in 0,5 O 0.65 O 0.38 O 1.22 0.12 1.5 t 20