Decay constant

Representation of the time course of the variable with a free damped oscillation. ${\displaystyle x(t)}$

In linear oscillation systems with one degree of freedom, the decay constant is the product of the undamped natural angular frequency and Lehr's damping . ${\displaystyle \omega _{0}}$ ${\displaystyle D}$

${\displaystyle \delta =\omega _{0}\cdot D,\,\left|D\right|<1}$

The time course of a linear oscillation can be given by the equation:

${\displaystyle x(t)=x_{0}\,e^{-\delta t}\sin(\omega _{d}\,t+\varphi _{0})\,}$, With ${\displaystyle \omega _{d}=\omega _{0}\,{\sqrt {1-D^{2}}}}$

to be discribed. With a positive sign of the decay constant, the oscillation decays ; with a negative sign, the amplitude of the oscillation increases exponentially .

In the case of a damped oscillation ( ), the amplitude has decayed to less than 5% of the initial amplitude after approximately time . ${\displaystyle \delta >0}$ ${\displaystyle t_{\mathrm {\mbox{ü}} }={\frac {3}{\delta }}}$

In the case of measured step responses of any vibration system, the decay constant can be calculated approximately from the logarithmic decrement and the period of vibration . ${\displaystyle \Lambda }$ ${\displaystyle T_{\mathrm {d} }}$

${\displaystyle \delta ={\frac {\Lambda }{T_{\mathrm {d} }}}}$

The logarithmic decrement is calculated from two amplitudes that are separated by the period of oscillation. For linear systems, two amplitudes are sufficient. For weakly non-linear systems, several logarithmic decrements should be averaged. In the case of strongly non-linear systems, it is better to determine the time until the amplitude has entered a strip around ± 5 percent of the steady-state value.

${\displaystyle \Lambda =\ln {\frac {A(t)}{A(t+T_{\mathrm {d} })}}}$

Systems with PT1 behavior , e.g. B. the series connection of a spring and a damper are given by the differential equation

${\displaystyle T_{1}\cdot {\dot {y}}(t)+y(t)=x(t)}$

described. The time constant is the reciprocal of the decay constant. ${\displaystyle T_{1}}$

See also

• Cooldown
• Transient process

literature

• Hans Dresig , Franz Holzweißig : machine dynamics . 10th edition. Springer, Berlin 2011, ISBN 978-3-642-16009-7 , pp. 44 ( limited preview in Google Book search).

Individual evidence

1. ^ Otto Föllinger : Control engineering . 6th improved edition. Hüthig Verlag 1985. ISBN 3-7785-1137-8