b^2 = - 57" [sS + c(l-cC)] Elq 



1^ 2, _ ,_ .., __ 2 



^21 = - ^T" [(1+2C )sS - c(l-cC)] Elq 



b = - -— 1/1+5^ (i- sC - — cS) Elq 

 22 Db ' ^ qi q2 



2 

 D = 1 + cC + £;sS + 2^ 



Here both q and E, increase a^ oj increases. ^ is a dimensionless quantity 

 and may serve as a relative measure of the effect of shear warping on the 

 vibrations of the beam. The increase of E, with increasing cj means that 

 the effect of shear warping increases with increasing frequency , as is 

 well known in other connections. 



Using a common terminology, the coefficients a , a p, a , and a 



may be called influence coefficients for forcing of the beam at one end. 



** 



Similarly b,., bio, b , and b might be called inertia coefficients. 

 11 12' 21 22 



Because of the factors sin q£ and cos q i in the formulas, the values of 



both sets of coefficients, as w increases from 0, range in roughly cyclic 



fashion over all values between -» and +«>. 



These coefficients may be of use in treating the vibrations of a 



system composed of a uniform beam attached at one end to another 

 structure. 



* See pp. 9 and I80 of Reference 11. 



** The a 's refer to the influence of forces in producing vibrational 



ij 



displacements; the b 's to the effects of beam mass in making F and G 



ij 

 necessary. 



