93
Natural Frequency of Free Transverse Vibrations for a Shaft Subjected
to several Point Loads
Dunkerley’s method
Let W1, W2, W3, ……. be the concentrated loads on the shaft due to masses m1, m2, m3…... and δ1, δ2, δ3, …... the
static deflections of this shaft under each load when each load acts alone on the shaft. Let the shaft carry a
uniformly distributed mass of m per unit length over its whole span and the static deflection at mid-span due to
the load of this mass be δs.
fn = frequency of transverse vibration of the whole system
fns = frequency with the distributed load acting along
fn1, fn2, fn3, …... = frequency of transverse vibrations when each of W1, W2, W3, ……. Act alone.
Thus, according to Dunkerley’s empirical formula,
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1.27
Energy Method
Consider a shaft of negligible mass m, carrying point loads W1, W2, W3, ……. due to masses m1, m2, m3…... and y1,
y2, y3, …... the deflections of these loads.
In the extreme positions of the shaft, it possesses maximum potential energy and zero Kinetic energy and at
mean positions it possesses maximum Kinetic energy and zero potential energy.
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ω is the circular frequency of vibration.
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