Can a single-DOF system with non-linear damping be solved analytically?

[actionman26]

Hi I am trying to solve an analytical solution for a single-DOF system with a non-linear damping component:

mx'' + B2x'x' + B1x' + Kx = 0.

Tried in maple with no success with the non-linear term in the eqn.

Thanks in advance.

 

[epenguin]

mx'' + B2x'x' + B1x' + Kx = 0.

Writing x'x' looks odd, some mistake?

 

[JJacquelin]

The second order non-linear ODE :
mx'' + (B2)(x')² + (B1)x' + Kx = 0
can be reduced to a first order non-linear ODE :
Let dx/dt = p(x) which leads to :
m*(dp/dx)*p +(B2)*p² +(B1)*p +K*x = 0
Except for particular values of the coefficients, i.e. in the general case, there is no known analytical method to solve this non-linear ODE in odrer to obtain the result on the form of a combination of a finite number of standard functions.
You may solve it by numerical calculus or by approximate developments.