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APPENDIX

APPENDIX 60

Appendix A Deviation of the Logarithmic Decrement Calculating the logarithmic decrement δ is important for the determination of the damping ratio D of a mechanical system. The formulas presented in [7] allow to calculate D when the stationary amplitude x0 is known, to which ∆1 and ∆2 can be related. When the system cannot reach a steady state due to a periodical excitation, the amplitude of a third peak is required for calculating δ. In the following part, the required formulas will be derivated refering to figure A.1. With δ = ∆1 ∆2 and a simple geometrical consideration follows = ∆2 ∆3 ∆1∆3 = ∆ 2 2 ∆1 + ∆2 = x1 − x2 (A.1) (A.2) (A.3) x3 = x2 + ∆2 + ∆3. (A.4) When substituting (A.2) with (A.3), the quadratic equation results with the roots (A.4) can now be written as ∆ 2 2 + ∆2∆3 − ∆3(x1 − x2) = 0 (A.5) ∆2(1,2) = −∆3 ± ∆2 3 + 4∆3(x1 − x2) . (A.6) 2 2x3 = 2x2 + (−∆3 ± ∆ 2 3 + 4∆3(x1 − x2)) + 2∆2 (A.7) 61

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