9/9/20142.3 Energy in Free Vibration The energy input to an SDOF by imparting to it an initial displacement and initial velocity is given by Recall he displacement response of a free vibration with initial displacement and velocity: This is the same as the input energy in Eq. 2.3.1. Therefore the total energy is independent of time and is dependent only on the input energy. Hence, there is conservation of energy during free vibration of a system without damping. 1 9/9/2014 __________________________________________________________________ The energy dissipated is proportional to the square of the amplitude of motion. It is not a constant value for any given amount of damping and amplitude since the energy dissipated increases linearly with the excitation frequency. 2 9/9/2014 In steady-state vibration, the energy input to the system due to the applied force is dissipated in viscous damping. The external force p(t) inputs energy to the system which for each cycle is Using Eq. 3.2.12: Eq. 3.8.2 can be written as (see Derivation 3.6) Hence, from Eq. 3.8.1 and 3.8.3, ED = EI The preceding energy concepts help explain the growth of the displacement amplitude caused by harmonic force with ω = ωn until steady-state is attained as shown in Fig 3.2.2. For ω = ωn , ϕ = 90o and Eq. 3.8.2 gives EI = π po uo (3.8.4) 3 9/9/2014 The input energy and the dissipated energy vary linearly and quadratically, respectively with the displacement amplitude. Before steady-state is reached, the EI per cycle exceeds the ED during the cycle by damping, leading to a larger amplitude of displacement in the next cycle. With growing displacement amplitude, the dissipated energy increases more rapidly than does the input energy. Eventually, the EI and the ED will match at the steady-state displacement uo, which will be bounded no matter how small the damping. This energy balance provides an alternative means of finding uo due to harmonic force with ω = ωn . Equating Eq 3.8.1 and 3.8.4 gives πpo uo = πcωnuo2 → uo = po. / cωn This result agrees with previously derived results. 4 9/9/2014 (a) Fig. 3.8.2 Hysteresis Loop for (a) viscous damper; (b) spring and viscous damper in parallel 5 9/9/2014 6 9/9/2014 Equivalent Viscous Damping All internal damping mechanisms, e.g., internal friction, fluid resistance, etc., → grouped as Viscous Damping with parameter c or , Energy is lost in the system as evidenced by the amplitude decay. The loss is due to damping 7 9/9/2014 8 9/9/2014 Problem from Clough and Penzien (2e) damping ratio - and damping coefficient c. 9 9/9/2014 Solution: Only the column provides lateral stiffness and damping. (a) k = f s max /1 = 390 lbs/ 0.15 in = 2600 lb/in (b) From Eq. 3.9.2 -eq = (1/4:)(ED/Eso) E so = ½ k12 and ED = 26 lb-in Hence, -eq = 0.071 or 7.1% - = c/ 2m9 → - eq = c eq / 2m9 (at resonance) c eq = - eq (2m 9) = 2 - eq k/ 9 = 2(0.071)(2600)/10 = 36.92 lb-sec/in 10
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