# Longitudinal vibrations

Notations and units:

$m$ = mass, $k$ = stiffness, $g$ = acceleration due to gravity, $E$ = modulus of elasticity, $W$ = load, $m$ = mass, $x$ = amplitude, $n$ = number of cycles

Units of the notations are in SI units.

Frequency = 1/ Time period

Natural frequency  $f_{n}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi}\sqrt{\frac{g}{\delta}}=\frac{0.4985}{\sqrt{\delta}}$

Natural frequency considering inertia  $f_{n}=\frac{1}{2\pi}\sqrt{\frac{k}{m+\frac{m_{s}}{3}}}$

$m_{s}$  = mass of spring

Circular frequency  $\omega_{n}=\sqrt{\frac{s}{m}}=\sqrt{\frac{g}{\delta}}$

$\omega_{n}=2\pi f_{n}$

Deflection of longitudinal vibrations  $\delta=\frac{WL}{AE}$

$\delta=\frac{mg}{k}$

Equivalent stiffness when spring are connected in series

$\frac{1}{k_{equivalent}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}+\frac{1}{k_{3}}+...$

Equivalent stiffness when spring are connected in parallel

$k_{equivalent}=k_{1}+k_{2}+k_{3}+...$

When spring is cut into “n” parts, stiffness of each cut spring becomes “nk”, where k= stiffness of spring before cutting.

Damping coefficient $c$ = Damping force / velocity

Damping factor or damping ratio  $\zeta=\frac{c}{c_{c}}$

Critical damping coefficient  $c_{c}=2m\omega_{n}$

Circular frequency of damped vibration  $\omega_{d}=2\pi f_{d}$

$\omega_{d}=\sqrt{1-\zeta^{2}} \times \omega_{n}$

$\zeta$ = 0 – Critically damped

$\zeta$ < 1 – Under damped

$\zeta$ > 1 – Over damped

Logarithmic decrement  $\delta_{L}=\frac{2\pi\zeta}{\sqrt{1-\zeta^{2}}}=\frac{1}{n}\times\ln\left ( \frac{x_{0}}{x_{n}} \right )$

$\frac{x_{0}}{x_{n+1}}=e^{\delta_{L}}$