12.9 Periodic Structures
481
Fig. 12.18 Moments & forces on beam elements
Slopes θ B and θ B must be equal for the continuity of displacement and slope
across the support.
Hence equating θ B and θ B from Eqs. (12.168) and (12.169), yields
M r − 1 + 2
β B B
β B A
+ M r + 1 = 0
(12.170)
An identical equation applies to moments at each support between free bays of
the beam. Now,
M r + 1 = e
μ M r
M r = e
μ M r − 1
(12.171)
Substituting Eqs. (12.171) into (12.169), yields
e
μ
+ e
− μ
= − 2
β B B
β B A
or,
cosh μ = −
β B B
β B A
(12.172)
Equation (12.172) gives the equation for propagation constant μ. The right hand
side of the equation is strongly frequency dependent and will be complex if the
damping exists in the beam.
The receptance functions β B B and β B A are transcendental functions of frequency,
the mass of the beam, the stiffness and damping Eq. (12.172) can be expressed as