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382 G. F. HUTCHINS [J. S. M. p. E.
subject is ably discussed in detail in Horace Lamb's well-known work on Hydrodynamics (Cambridge Press).
A very interesting example of the type of problem to be encountered in cases where stresses in structures accompany motion is to be had in the cantilever beam. The expression for the end deflection of a cantilever beam with concentrated load of/7 pounds and length s' is
where e' is the modulus of elasticity of the material and i' is the moment of inertia of the beam with respect to the axis of bending. Dimensionally this expression may be written:
/ „ f'1'3 ~ e' i'
and in the miniature
* Ee'Ii'
If the densities are kept constant, F = L3 and a combination of equations gives:
El = L5.
If we use the same material in our miniature beam that we would use in the large beam, the modulus of elasticity factor, E, is unity, and we must reduce our moment of inertia in proportion to L5. If we merely reduce the linear dimensions of the beam by L and keep its structure the same, its moment of inertia will be reduced by only L4, so that it will be necessary to change the structure of the beam in the miniature to secure good results. A properly designed hollow beam would give the desired effect. The same end might also be obtained by designing the beam with a reduction in moment of inertia of L4 (reproduction of the structure in the large beam) and a reduction in the modulus of elasticity of L. This calls for a beam of different material, however, and it would therefore be necessary to also see that the density of the new material was right, and would probably not be as good a solution of the problem in most cases as the first alternative.
It may be noted that this problem is merely a special application of the resilience force case already covered. In our present problem
3ei