Summary
The degree of influence of rotatory inertia and shear deformation on natural frequency determinations is discussed. It is found that for material such as wood, which has a relatively high ratio of bending modulus to shear modulus, these effects need to be taken into account even for supposedly ‘slender’ beams. The discussion covers the four most common types of support condition, namely simply supported, free-free, clamped-free and clamped-clamped ends. As a perfectly rigid clamped condition can never be achieved in practice, especially for relatively ‘soft’ material like wood, consideration is extended to also include partially-clamped beams.
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Abbreviations
- A:
-
cross sectional area
- E:
-
modulus of elasticity in bending
- Ec :
-
modulus of elasticity in compression perpendicular to the grain
- G:
-
shear modulus
- I:
-
second moment of area
- K′:
-
shear shape factor
- KH, KHO, KHL :
-
translational spring stiffness
- KT, KTO, KTL :
-
rotational spring stiffness
- L:
-
span
- b:
-
breadth
- d:
-
depth
- s:
-
clamp length
- m:
-
mass per unit length
- t:
-
time
- v:
-
transverse displacement
- x:
-
distance along the length of a beam
- y:
-
transverse displacement with time
- α:
-
total rotation
- ω:
-
angular natural frequency
- fT :
-
natural frequency of a Timoshenko beam
- fE :
-
natural frequency of an Euler beam
References
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This work was undertaken with financial support from Canadian Natural Sciences and Engineering Research Council under Special Support for Forestry Grant No. FRP0030800 and Operating Grant No. OGP0004417
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Chui, Y.H., Smith, I. Influence of rotatory inertia, shear deformation and support condition on natural frequencies of wooden beams. Wood Sci.Technol. 24, 233–245 (1990). https://doi.org/10.1007/BF01153557
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DOI: https://doi.org/10.1007/BF01153557