Page 407 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

394 Classical Methods Chap. 12

sections, we have

(12.6-3)
®, + i - + ^ í+i[ e i ), ^ + ‘(2 £ '/),.
^i+i =Vi + (12.6-4)

where {l/EI)¿ = slope at / + 1 measured from a tangent at i due to a unit moment
at i + 1;
(l^/2EI)¿ = slope at / + 1 measured from a tangent at i due to a unit shear at
/ + 1 = deflection at / + 1 measured from a tangent at i due to a
unit moment at / + 1;
(l^/3EI)¿ = deflection at / + 1 measured from a tangent at i due to a unit
shear at / + 1.
Thus, Eqs. (12.6-1) through (12.6-4) in the sequence given enable the calculations
to proceed from / to / + 1.
Boundary conditions. Of the four boundary conditions at each end, two
are generally known. For example, a cantilever beam with / = 1 at the free end
would have = 0. Because the amplitude is arbitrary, we can choose
yj = 1.0. Having done so, the slope 0^ is fixed to a value that is yet to be
determined. Because of the linear character of the problem, the four quantities at
the far end will be in the form
K = ^i+ ^1^1

T

M n — Cl2 ¿^2^1
= ^3 + ^3^1
y„ = «4 +