Question

Use Hamilton's principle to derive the equation of motion and boundary conditions for a uniform 2. cantilever beam with a spring attached in the middle.

please use Hamilton's principle and keep attention a spring is attached in the middle

Answer 1

By using Hamilton's principle By balancing the force in Z . -v(v + dv)+ f(x,f)(partialdifferential x + v) middot p A(x) middot partialdifferential x partialdifferential^2 w(x,f_c)/ partialdifferential t^2 (M+ partialdifferential M) - (v+ partialdifferential v)+f(x,f) partialdifferential x/2 -M =0 partialdifferential v = partialdifferential v/ partialdifferential x & partialdifferential M = partialdifferential M/ partialdifferential x middot dx partialdifferential x(x,f)/ partialdifferential x + f(x,f)= rho A(x) middot partialdifferential^2 w(x,f)/ partialdifferential t^2 partialdifferential(M(x,t))/ partialdifferential x + f(x,t)= rho A(x) middot partialdifferential^2 W(x,t)/ partialdifferential t^2 from elementary theory of bending beam M(x,t)= sigma l(x) middot partialdifferential^2 W(x,f)/ partialdifferential x^2 where epsilon = young's modules l(x)= MOI eq. of motion for transverse form for non - uniform partialdifferential/ partialdifferential x^2(sigma l(t) middot partialdifferential^2 w(x,t)/ partialdifferential x^2) + rho A(x) partialdifferential^2 w(x,f)/ partialdifferential t^2 = F(x,t) Uniform beam middot [sigma l(x) partialdifferential^4 w(x,t)/ partialdifferential x^4] - rho A(x) middot partialdifferential^2 w(x,t)/ partialdifferential t^2 = f(x,t) f(x,t)=0 rightdoublearrow c^2 partialdifferential^2 w(x,t)/ partialdifferential x^2 + partialdifferential^2 w(x,t)/ partialdifferential x^2 =0 c = squareroot epsilon x/rho A