DSolve[{E1*A1*W1''[Z] + \[Rho]*A1*\[Omega]^2*W1[Z]^2 == 0,
E1*J*U1''''[Z] - \[Rho]*A1*\[Omega]^2*U1 [Z] == 0, W1[0] == 0,
U1[0] == 0, U1''[0] == 0, U1''[L] == 0, U1[L] - \[Beta]*W1[L] == 0,
E1*A1*W1'[L] - E1*J*\[Beta]*U1'''[L] == 0}, {W1[Z], U1[Z]}, Z]

应该没别的问题，就是难解罢了
好像是梁方程？