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}}

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