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

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SUMMARY

The discussion focuses on solving the analytical solution for a single-degree-of-freedom (single-DOF) system characterized by the non-linear damping equation: mx'' + (B2)(x')² + (B1)x' + Kx = 0. Attempts to solve this equation using Maple were unsuccessful due to the non-linear term. It is established that, except for specific coefficient values, there is no known analytical method to solve this non-linear ordinary differential equation (ODE) for a general case. Numerical methods or approximate developments are recommended as viable alternatives for finding solutions.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with non-linear dynamics
  • Proficiency in using Maple software for mathematical computations
  • Knowledge of numerical methods for solving differential equations
NEXT STEPS
  • Explore numerical methods for solving non-linear ODEs
  • Learn about approximate solutions for non-linear systems
  • Investigate the use of Maple for numerical calculus
  • Study specific cases of single-DOF systems with known coefficients
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Engineers, mathematicians, and researchers working on dynamic systems, particularly those dealing with non-linear damping in mechanical systems, will benefit from this discussion.

actionman26
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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.
 
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actionman26 said:
mx'' + B2x'x' + B1x' + Kx = 0.
Writing x'x' looks odd, some mistake?
 
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.
 

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