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}


Equivalent stiffness when spring are connected in series


Equivalent stiffness when spring are connected in parallel


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 )



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