We are typically interested in the behavior of systems when driven by an external force f(t).
For a sinusoidal driving force, the resulting response has a transient component and a steady-state term. The transient component, which involves motion at the natural frequency of the system, decays away at a rate proportional to the damping in the system.
The mechanical impedance,
Z(s) = F(s) / V(s), (evaluated at ) of the system characterizes its steady-state response, after its initial transient behavior has decayed away.
Components connected in series share a common velocity and their impedances add (i.e.,
Z(s) = Z1(s) + Z2(s)).
Components connected in parallel share a common driving force but can develop different velocities. Admittances add in parallel, while a parallel combination of impedances is given by
When a mass-spring-damper system is driven by an external force, the system equation is
The driven mass-spring-damper system can be described using the Laplace Transform as (assuming v(0) = 0 and x(0) = 0):
and its impedance determined as
This impedance expression, Z(s), can be evaluating for , as shown in Fig. 10 for three different damping constants.
Figure 10:
Impedance of a mass-spring-damper system.
Mechanical driving-point impedance minima indicate the frequencies at which a sinusoidal driving force of constant amplitude will produce the greatest mass velocity (and displacement).
The Laplace Transform for the mass-spring-damper system can also be expressed in terms of its displacement (assuming x(0) = 0 and v(0) = 0):
The steady-state force-to-displacement response is then given by:
and is plotted in Fig. 11 for three damping constants. Resonance occurs when a system is driven near its natural frequency.
Figure 11:
Mass displacement vs. driving frequency for mass-spring-damper system.
Below resonance, the system is said to be “stiffness dominated”.
Above resonance, the system is said to be “mass dominated” and the resulting displacement approaches zero with increasing frequency.