<P> For example, metal structures (e.g., airplane fuselages, engine crankshafts) have damping factors less than 0.05, while automotive suspensions are in the range of 0.2--0.3 . The solution to the underdamped system for the mass - spring - damper model is the following: </P> <Dl> <Dd> x (t) = X e − ζ ω n t cos ⁡ (1 − ζ 2 ω n t − φ), ω n = 2 π f n . (\ displaystyle x (t) = Xe ^ (- \ zeta \ omega _ (n) t) \ cos (((\ sqrt (1 - \ zeta ^ (2))) \ omega _ (n) t - \ phi)), \ qquad \ omega _ (n) = 2 \ pi f_ (n).) </Dd> </Dl> <Dd> x (t) = X e − ζ ω n t cos ⁡ (1 − ζ 2 ω n t − φ), ω n = 2 π f n . (\ displaystyle x (t) = Xe ^ (- \ zeta \ omega _ (n) t) \ cos (((\ sqrt (1 - \ zeta ^ (2))) \ omega _ (n) t - \ phi)), \ qquad \ omega _ (n) = 2 \ pi f_ (n).) </Dd> <P> The value of X, the initial magnitude, and φ, (\ displaystyle \ phi,) the phase shift, are determined by the amount the spring is stretched . The formulas for these values can be found in the references . </P>

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