MT-1620 al.2002 Can recast equation(23-6) to be w ma W (237) El The solution to this homogeneous equation is of the form Putting this into (23-7)yields 0 E m10 El So this is an eigenvalue problem (spatially). The four roots are p=+λ,-λ,+i,-i Where m10 E This yields Paul A. Lagace @2001 Unit 23-6
MIT - 16.20 Fall, 2002 Can recast equation (23-6) to be: 4 d w mω2 dx 4 − w = 0 (23-7) EI The solution to this homogeneous equation is of the form: w x() = e px Putting this into (23-7) yields 4 px mω2 pe − e px = 0 EI 4 mω2 ⇒ p = EI So this is an eigenvalue problem (spatially). The four roots are: p = + λ, -λ, + iλ, - iλ where: 14 mω2 / λ = EI This yields: Paul A. Lagace © 2001 Unit 23 - 6
MT-1620 Fall 2002 w(x)= Ae+ Be Ce/+ De or w(x)=Csinh/x C, coshnx+ C3 sin/x+ C4 cosx(23-8 The constants are found by applying the boundary conditions (4 constants=>4 boundary conditions EXample: Simply-supported beam Figure 23.2 Representation of simply-supported beam 2 EI. m= constant with x d w E m Paul A Lagace @2001 Unit 23-7
MIT - 16.20 Fall, 2002 λ w x() = Ae λ x + Be −λ x + Ce i λ x + De −i x or: w x() = C1 sinhλx + C2 coshλx + C3 sinλx + C4 cosλx (23-8) The constants are found by applying the boundary conditions (4 constants ⇒ 4 boundary conditions) Example: Simply-supported beam Figure 23.2 Representation of simply-supported beam EI, m = constant with x 4 2 EI dw + m dw = pz dx 4 dt 2 Paul A. Lagace © 2001 Unit 23 - 7