THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

Then the energy dissipated in a cycle is obtained from∫ 2π20.6 Techniques for Vibration Control 607∫ 2πCẋ 2 (t) d(ωt) = CA 2 ω cos 2 (ωt + φ) d(ωt) = πCA 2 ωω 00The energy dissipated per radian is CA 2 ω/2, and the peak potential energy iskA 2 /2. Therefore, the system loss factor isη s = CA2 ω/2kA 2 /2= CωkIntroducing Equations (20.8) and (20.9), Equation (20.45) becomes(20.45)η s = 2ξ ω ω n(20.46)At resonance, η s assumes the value of 2ξ. This indicates that systems withlarge loss factors are highly damped. For small values of damping, applyingEquations (20.18), (20.28), and (20.29):η S = 2ξ = δ π = 1(MF) resonanceIn the analysis of structural damping, complex stiffness and complex moduliare principal parameters. We recall that the steady-state oscillation is described bythe following expression:which can be rewritten aswhere(−ω 2 m + iωC + k)X = F 0(−ω 2 m + k ∗ )X = F 0k ∗ = k + iωCThe imaginary stiffness is the damping. Employing Equation (20.45), we canexpress the complex stiffness ask ∗ = k(1 + iη s )In an analogous manner, the complex Young’s modulus E for damping materialcan be expressed asE ∗ = E(1 + iη M )where E is the real part of the Young’s modulus of the damping material and η Mis the loss factor of the damping material. 1The loss factors and complex moduli vary with frequency and temperature. It hasbeen assumed that the damping force is proportional to velocity and independentof amplitude, but there are situations where the damping of the materials depends1 η s is the loss factor of the system structure and damping material.