Equation (2.57) indicates flexure induced torsion is a nonlinear phenomenon (Malatkar, 2003). Substituting (2.56) and (2.57) into (2.49) and (2.50) yields the order three equations of motion for the flexural-flexural-torsional vibration of a cantilever beam. s ⎡ s ⎤ mv&& + cv v& + Dζ v iv = Qv + {( Dη − Dζ ) ⎢ w' ' ∫ v" w" ds − w' ' ' ∫ v" w' ds ⎥ 0 ⎣ l ⎦ s s ( D − Dζ ) 2 − η ( w' ' ∫ ∫ v' ' w' ' dsds )'}'− Dζ {v' (v' v' '+ w' w' ' )'}' Dξ 0 l s
s
(2.58)
s
1 ∂2 − m{v' ∫ 2 [ ∫ (v'2 + w'2 )ds ]ds}'−(v' ∫ Qu ds )' 2 ∂t 0 l l s ⎤ ⎡ s && + cw w& + Dη w = Qw − {( Dη − Dζ ) ⎢v' ' ∫ v" w" ds − v' ' ' ∫ w" v' ds ⎥ mw 0 ⎦ ⎣ l 2 s s ( D − Dζ ) + η (v' ' ∫ ∫ v' ' w' ' dsds )'}'− Dη {w' (v' v' '+ w' w' ' )'}' Dξ 0 l iv
s
s
(2.59)
s
1 ∂2 − m{w' ∫ 2 [ ∫ (v'2 + w'2 )ds ]ds}'−( w' ∫ Qu ds )' 2 ∂t 0 l l
The boundary conditions for (2.58) and (2.59) are given by v(0, t ) = 0, v" (l , t ) = 0,
w(0, t ) = 0, w" (l , t ) = 0,
v' (0, t ) = 0, v' ' ' (l , t ) = 0,
w' (0, t ) = 0 w' ' ' (l , t ) = 0
(2.60)
The boundary conditions for the free end are derived from (2.53). The equation of motion and boundary conditions for the forced planar flexural vibration of the beam is obtained from equations (2.58) and (2.59). For planar motion, equation (2.59) is dropped along with the w terms in (2.58). With these substitutions equation (2.58) becomes
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