• A Driven Mass-Spring-Damper Vibrator
• Standing Waves on a String
• Standing Waves in Pipes
• Impedance

Resonance

In this section, we consider the behavior of vibrating systems when they are driven by an external source.

A Driven Mass-Spring-Damper Vibrator

• When a mass-spring-damper system is driven by an external force, the system equation becomes $m \frac{d^{2}x}{dt^{2}} + R \frac{dx}{dt} + k x = f(t)$.

• For a sinusoidal driving force, the resulting solution has a transient component and a steady-state term.

• The steady-state displacement of the mass is dependent on the driving frequency and the damping in the system.

  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.

Standing Waves on a String

• A string of length $L$ fixed at $x=0$ and $x=L$ will have sinusoidal solutions of the form
  
  $$y_{n}(x,t) = (A_{n} \sin\omega_{n}t + B_{n} \cos\omega_{n}t) \sin\frac{\omega_{n}x}{c},$$
  
  
  where $\omega_{n} = n\pi c / L$.

• Only specific vibration frequencies are possible and these frequencies are related by integer multiples of the fundamental ($\omega_{1}$).

• Standing waves are the ``physical'' result of superposed traveling waves (Matlab example).

Standing Waves in Pipes

• One-dimensional sound waves propagate well in cylindrical pipes, just as mechanical waves travel along a string.

• Discontinuities in a pipe, like open and closed ends, cause wave reflections that result in the formation of standing waves patterns. These boundary conditions impose limitations on the particular frequency components that can be made to ``resonate''.

• Open end: acoustic pressure = ambient room pressure = 0.

• Closed end: particle velocity (and volume velocity) = 0.

• The acoustic length of a cylindrical pipe is slightly greater than its physical length by an additional $0.61 r$, where $r$ is the radius of the pipe.

Impedance

• Assume linear sinusoidal solutions.

• Impedance is a frequency-domain quantity (e.g. ratio of force to velocity at a specific frequency).

• Mechanical impedance: $Z = F/V$, where $F$ is force and $V$ is the resulting velocity.

  Figure 12: Impedance of a mass-spring-damper system.

  

• Acoustic impedance: $Z = P/U$, where $P$ is pressure and $U$ is volume flow.

  Figure 13: Input impedance of a cylindrical pipe (length 0.3 meters).