| Through in-depth analysis of the harmonic balance method,we put forward a Newton iterative harmonic balance method,and at the same time,we comparatively analyze the similarities and differences between the Newton harmonic balance method and the Newton iterative harmonic balance method.Essentially,the modified harmonic balance method is mainly to overcome difficulties in solving nonlinear algebraic equations of the classical harmonic balance method in the process of constructing high order solutions.To linearize the governing equations or the high order algebraic equations can overcome these difficulties.The former is the Newton harmonic balance method,the latter is the Newton iterative harmonic balance method that proposed in this paper.Consider the following nonlinear vibration governing equation:Here,supposc f(u)is an odd function,namely,f(-u)=f(u),and when u?[-A,A],u?0,satisfy uf(u)>0.1.Single harmonic balance method Where2.N items harmonic balance methodTake the corresponding solutions are(?)where x3,x5,…x2N-1 are unknowns.Further,based on the assumption of odd function that f(-u)=-f(u),f(uNHB(?))can be expanded to Fourier series as follows Where(?)Substituting Equation(4-6)into Equation(1),after finishing,we make the coefficients of cos?,cos3?,...,cos(2N-1)? are zero,obtain nonlinear algebraic equations about ?,x3,x5,…,x2N-1Generally,it is difficult to establish the analytic solutions of nonlinear algebraic Equations(7).3.Newton Iterative harmonic balance method Write the Equation(7)as where,X=(?,x3,x5,…x2N-1)T,F=(f1,f3,…f2N-1)T,andWe can establish high precision analytical approximate solutions for Equation(8),with the help of numerical iteration method into numerical algebra,like Newton method and various Quasi-Newton Methods.Unlike the numerical iteration method,Newton Iterative harmonic balance method that is proposed by this paper,use the solution of single harmonic balance as iterative initial value,and the results of each iteration are all analytic expression.We can obtain the recurrence formula with the help of Newton iteration method as follows.where,F'(Xk)is a Jacobian matrix of F at Xk.It can be seen from the above derivation,that the main difference between the Newton harmonic balance method and the Newton iterative harmonic balance method is that the Newton method is used to approximate the nonlinear governing equations or to approximate the nonlinear algebraic equations,some specific examples can be used to prove that the two methods are equivalent when establish approximate solutions.Change iterative formula of Newton iterative harmonic balance method to Quasi-Newton method,we can obtain the following quasi Newton iterative solution equation.Generally speaking,the expression of the Quasi-Newton iteration harmonic solution is much simpler,but the accuracy of the solution is usually not as high as that of the Newton harmonic balance method and the Newton iterative harmonic balance method.4.Automatic program design of modified harmonic balance methodBased on Mathematica Software,three program modules are designed,including the initialization module,the solution module and the post-processing module,the automatic solution of above-mentioned various modified harmonic balance methods is realized.The program can not only give the approximation periods and the periodic solutions that are constructed by various methods,but also can give the comparison of these solutions and numerical solutions.Finally,the Cubic Duffing equation and the non-natural vibration system are studied by using the modified harmonic balance that was above-mentioned and the Mathematica program,the results show that the modified harmonic balance method can be used to establish the corresponding high-precision analytical approximation and periodic solutions.The program can be used to automatically establish the analytical approximate solutions of various nonlinear vibration systems and finish comparison between the numerical solution and the numerical solution. |
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