fundamentals of engineering supplied-reference handbook - Ventech!

It may also be shown that the undamped natural frequency
may be
expressed in terms of the static deflection of the system as
ω = g / δ
n st
The undamped natural period of vibration may now be
written as
τ = 2 π/ ω = 2 π m/ k = 2 π δ / g
n n st
Torsional Vibration
I
kt
θ
For torsional free vibrations it may be shown that the
differential
equation of motion is
�� θ+ ( kt/ I)
θ= 0,
where
θ= the angular displacement of the system
kt
=
the torsional stiffness of the massless rod
I = the mass moment of inertia of the end mass
36
DYNAMICS (continued)
The solution may now be written in terms of the initial
conditions
θ (0) =θ 0 and θ �(0) =θ�
0 as
θ ( t) =θ cos( ω t) + ( θ� / ω )sin( ω t)
0 n 0 n n
where the undamped natural circular frequency is given by
ω =
n t
k / I
The torsional stiffness of a solid round rod with associated
polar
moment-of-inertia J, length L , and shear modulus of
elasticity G is given by
k = GJ / L
t
Thus the undamped circular natural frequency for a system
with a solid
round supporting rod may be written as
ω =
n
GJ / I L
Similar to the linear vibration problem, the undamped
natural period may be written as
τn
= 2π / ωn
= 2π
I / kt
= 2π
I L/
GJ