Time It Takes to Approach Steady State
By Eric E. Ungar
Acentech, Inc., Cambridge, MA
In dealing with a structure or a mechanical system that is subjected to a vibratory force, such as that due to a rotating machine, it is common in practice to consider only how the system responds in the steady state—that is, after the excitation has been applied for a considerable time.
In theory, reaching the steady state takes infinite time—the final steady state is never reached. But how long does it take to reach a given fraction of the steady state response?
Consider the classical simple mass-spring-damper system consisting of a mass m that is mounted to a rigid support via a parallel arrangement of a spring with stiffness k and a damper with viscous damping coefficient c, and assume that a sinusoidal force of amplitude F and radian frequency ω acts on the mass. For such a system, the displacement x of the mass m from its equilibrium position in terms of time t obeys the well-known differential equation
the complete solution of which consists of the sum of a steady-state component xs(t) and a transient component xT(t). One may obtain expressions for these components by following the usual analytical procedures. The mathematical steps are not presented here, but the results are summarized below.
The magnitude XS of the steady-state component may be found to obey
with Xst = F / k denoting the static deflection of the mass due to a steady force of magnitude F and
Here β = ω/ωn is the ratio of the radian frequency ω of the applied force to the system’s radian natural frequency:
with the critical damping coefficient cc given by
For the commonly considered case where the system is at rest and undeflected from equilibrium before the force is applied, one finds that the magnitude XT of the transient component obeys
Thus, the ratio N of the magnitude of the transient component to the magnitude of the steady-state component is given by
where the latter form is an approximation for the often considered practical case of small damping. Since the total motion consists of the sum of the steady-state and transient components, the total motion nears the steady state as the transient decreases—that is, as N becomes smaller.
Results and Observations
As evident from equation 4, the ratio N depends on the system’s natural frequency but not on the frequency of the excitation. Further, this equation implies that the time TN it takes for the magnitude of the transient to decay to a given fraction Nc of the magnitude of the steady-state component obeys
and that the number of cycles nN it takes to reach the fraction Nc is
Note that nN depends only on the value of the damping ratio.
For example, consider how long it takes for the response of a system with 3% of critical damping and a natural frequency of 15 Hz to reach 99% of its steady-state amplitude. This is the time it takes for the transient amplitude to decay to 1% of the steady-state amplitude and may be found here to amount to
This corresponds to 1.63(15) = 24.4 cycles.
As apparent from equation 5, the time it takes for the transient to reach a given small fraction of the steady-state magnitude is shorter for (a) greater damping and (b) greater natural frequencies. This agrees with what one would expect since the transient motion is that of a system that is disturbed initially and then vibrates freely with decaying amplitude due to its damping.