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\section{Mechanical Oscillations\label{Mechanical oscillations}}

A spring with constant $C$ and damping $k$ is connected to a mass $M$. If a
periodic force $F(t)$ of frequency$\ \omega $ is applied, this equation of
motion holds: 
\[
m\ddot{x}=F(t)-k\dot{x}-Cx 
\]
With $F(t)=e^{i\omega t}\ $and$\ x(t)=xe^{i\omega t}$, this becomes 
\[
-m\omega ^{2}x=F-Cx-ik\omega x 
\]
Setting $\omega _{0}^{2}=\dfrac{C}{m}$ it follows that 
\[
x=\frac{F}{m(\omega _{0}^{2}-\omega ^{2})+ik\omega } 
\]
and 
\[
~\dot{x}=\frac{F}{i\sqrt{Cm}\delta +k} 
\]
where $\delta =\dfrac{\omega }{\omega _{0}}-\dfrac{\omega _{0}}{\omega }$.
The quantity $Z=\dfrac{F}{\dot{x}}$ is called the \textit{impedance} of the
system. The \textit{quality} of the system is given by 
\[
Q=\frac{\sqrt{Cm}}{k} 
\]

The frequency with minimal $|Z|$ is called \textit{velocity resonance
frequency}. This is equal to $\omega _{0}$. In the \textit{resonance curve,} 
$\dfrac{|Z|}{\sqrt{Cm}}$ is plotted against $\dfrac{\omega }{\omega _{0}}$.
The width of this curve is characterized by the points where $|Z(\omega
)|=|Z(\omega _{0})|\sqrt{2}$. At these points $R=X$ and $\delta =\pm Q^{-1}$%
, and the width is $2\Delta \omega _{B}=\dfrac{\omega _{0}}{Q}$.

The\label{Stiffness} \textit{stiffness} of an oscillating system is given by 
$\dfrac{F}{x}$. The\label{Amplitude resonance frequency} \textit{amplitude
resonance frequency} $\omega _{A}$ is the frequency for which $i\omega Z$ is
minimal. This is the case for 
\[
\omega _{A}=\omega _{0}\sqrt{1-\frac{1}{2}Q^{2}} 
\]

\label{Damping frequency}The \textit{damping frequency} $\omega _{D}$ is a
measure for the time in which an oscillating system comes to rest. It is
given by 
\[
\omega _{D}=\omega _{0}\sqrt{1-\frac{1}{4Q^{2}}} 
\]
\label{Damped oscillation}A \textit{weakly damped oscillation} $(k^{2}<4mC)$
dies out after $T_{D}=2\pi /\omega _{D}$. $\omega _{\mathrm{D}}=0$ for a 
\textit{critically damped} oscillation $(k^{2}=4mC)$. If $k^{2}\gg 4mC$, a 
\textit{strongly damped oscillation} $(k^{2}>4mC)$ drops like 
\[
x(t)\approx x_{0}\exp (-t/\tau ) 
\]
.

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/CAL_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Table of Integrals}{}{}{%
../Mathematics/TAB_integrals.tex#table of integrals}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Basic Techniques for Doing Mathematics}{}{}{%
../../Help/DM1.tex}}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Algebra}{}{}{../../Help/DM3.tex}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

{\small \hyperref{Function Definitions}{}{}{../../Help/DM5.tex}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{Differential Equations}{}{}{../../Help/DM10.tex}}\hfill

\noindent \hrulefill

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