Lumped System Analysis:

Sound is caused by vibrating objects or media. In this section, we study somewhat overly simplified, ideal structures to gain an understanding of the fundamental concepts of vibrating systems. The analysis of systems in terms of finite collections of masses, springs, and friction sources is referred to as lumped modeling and the resulting approximate system responses are called ``lumped characterizations''.

Lumped Elements:

  1. The Ideal Dashpot:
    Figure 1: An ideal dashpot or damper.
    \begin{figure}\begin{center}
\epsfig{file=figures/dashpot.eps, width=1.5in}
\end{center}
\end{figure}

  2. The Ideal Mass:
    Figure 2: An ideal mass.
    \begin{figure}\begin{center}
\epsfig{file=figures/mass.eps, width=2.5in}
\end{center}
\end{figure}

  3. The Ideal Spring:
    Figure 3: An ideal spring.
    \begin{figure}\begin{center}
\epsfig{file=figures/spring.eps, width=2.0in}
\end{center}
\end{figure}

Free Vibrations of Ideal Systems:

In this section, we analyze the behaviour of particular combinations of masses, springs, and dashpots when given an initial displacement and/or velocity. We will consider forced oscillations in a subsequent section.

A Mass-Spring Oscillator

Figure 4: An ideal mass-spring system.
\begin{figure}\begin{center}
\epsfig{file=figures/mass-spring.eps, width=2.3in}
\end{center}
\end{figure}

A Damped Mass-Spring Oscillator

A physical system without losses is rare (if not impossible). The introduction of a mechanical dashpot in the mass-spring system provides damping and causes the resulting vibrations to decay over time.

Figure 5: An ideal mass-spring-damper system.
\begin{figure}\begin{center}
\epsfig{file=figures/msd.eps, width=3in}
\end{center}
\end{figure}

A One-Mass, Two-Spring System

  1. Longitudinal Motion (along x-axis):
    Figure 7: A one-mass, two-spring system: Longitudinal motion.
    \begin{figure}\begin{center}
\epsfig{file=figures/mass-twospring.eps, width=3in}
\end{center}
\end{figure}

A Two-Mass, Three-Spring System

  1. Longitudinal Motion (along x-axis):
    Figure 8: A two-mass, three-spring system: Longitudinal motion.
    \begin{figure}\begin{center}
\epsfig{file=figures/two-mass.eps, width=4in}
\end{center}
\end{figure}

    \begin{displaymath}
m \frac{d^{2}x_{1}}{dt^{2}} + k x_{1} + k (x_{1} - x_{2}) = ...
... m \frac{d^{2}x_{2}}{dt^{2}} + k x_{2} + k (x_{2} - x_{1}) = 0
\end{displaymath}

Multiple Mass Systems

Figure 9: An infinite mass-spring system.
\begin{figure}\begin{center}
\epsfig{file=figures/mass-spring-etc.eps, width=7in}
\end{center}
\end{figure}

Forced Vibrations:

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

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