Forced Vibrations:

• 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 $s = j \omega$) 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
  $$Z(s) = \frac{1}{\frac{1}{Z_{1}(s)} + \frac{1}{Z_{2}(s)}} = \frac{Z_{1}(s) Z_{2}(s)}{Z_{1}(s) + Z_{2}(s)}.$$

A Forced, or Driven, Mass-Spring-Damper System

• When a mass-spring-damper system is driven by an external force, the system equation is
  $$m \ddot{x} + \mu \dot{x} + k x = f(t)$$

• The driven mass-spring-damper system can be described using the Laplace Transform as (assuming v(0) = 0 and x(0) = 0):
  $$m \ddot{x} + \mu \dot{x} + k x = f(t) \hspace{0.1in} \Longleftrightarrow \hspace{0.1in} V(s) \left[m s + \mu + k/s \right] = F(s),$$
  and its impedance determined as
  $$Z(s) \ensuremath{\stackrel{\Delta}{=}}F(s) / V(s) = m s + \mu + k/s.$$
  This impedance expression, Z(s), can be evaluating for $s = j \omega$, 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):
  $$m \ddot{x} + \mu \dot{x} + k x = f(t) \hspace{0.1in} \Longleftrightarrow \hspace{0.1in} X(s) \left[m s^2 + \mu s + k \right] = F(s)$$
  The steady-state force-to-displacement response is then given by:
  $$X(s) / F(s) = \frac{1}{m s^2 + \mu s + k},$$
  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.