Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems

Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are re-arranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.


Introduction
Both Wilson-θ and Newmark-β methods are generally used approaches for solving numerical solutions of dynamical systems [1]- [6].As we know, a fundamental assumption of the Wilson-θ method lies in that, the acceleration changes linearly in a single time step [7].For this reason, the Wilson-θ is considered to be one of linear acceleration methods.One of the best merits of this method is that its result is unconditionally stable when the internal parameter θ is chosen to be larger than 1.37 [8].The Newmark-β method is in essence an extension of linear acceleration method [9].When solving linear problems, both the two methods can provide accurate results by adjusting the internal parameters.
For some nonlinear problems, sometimes the two methods would provide false results, especially when there are quasi-periodic or chaotic responses [10].
Even worse, the methods may even cease to be valid for some cases, for instance leading to numerical divergence as shown later.Fu et al. [11] proposed an approximate method to linearize a quasi-periodic nonlinear equation, and suggested a gradual integration process based on the Wilson-θ method.To the best of our knowledge, it is tough to make such a linearization to complex quasi-periodic systems especially when strongly nonlinearities are taken into account.Even though the linearization is done successfully, it will pose restriction to the computational accuracy [12] [13].
In this paper, both the Wilson-θ and Newmark-β methods will be improved with the help of a predictor-corrector algorithm.This algorithm is based on the incremental process which makes Wilson-θ and Newmark-β methods free of repeated linearizations of the nonlinear terms.The initial solution is predicted at the previous time point and corrected recursively to be the true one at the next point.Numerical solutions have been successfully obtained for quasi-periodic responses of both autonomous and non-autonomous nonlinear dynamical systems.Interestingly, the results show that the iterative correction of the predicted solution can converge within two iterations, which ensures the improved methods with high computational efficiency.Most importantly, the improved methods can directly deal with a variety of quasi-periodic dynamical systems with even strong nonlinearities, as there is no requirement of linearizing the considered equations.

Linearization
The nonlinear dynamical system is expressed as the following form where ( ) ( ) ( ) , , x t x t x t   represent the generalized acceleration, velocity and displacement, respectively.After a step length t ∆ , the following equation will be obtained as The incremental balance equation can be obtained by subtracting Equations ((1) from ( 2)).In practice, the linearization equation can also be obtained by expanding the incremental equilibrium equation as Taylor series at time t with the higher order terms being neglected , , , , , , which can be expressed in matrix form as with coefficient matrixes as where R is also the truncation error for each calculation.The precision of the computing result can be controlled by adjusting R to a relatively low magnitude [14].

Solving Incremental Equation
A basic assumption of the Wilson-θ method is that the acceleration varies linearly between time [ ] , t t t θ + ∆ , so that the velocity and displacement can be expressed as Rewrite Equation ( 6) equivalently in the incremental form and substitute it into (4), we can deduce the incremental displacement equation at [ ] where the coefficients , 0, Similar to Equation ( 6), when the Newmark-β method is employed we can obtain with α and δ are priorly given parameters.Also, rewrite Equation (11) in the incremental form and substitute it into (4), then we can obtain with the expressions of , 0,1, , 5 into (10) yields the response at time t t + ∆ .

Improving Wilson-θ and Newmark-β Methods
To elucidate the improved methods, we consider the following dynamic equation If Equation ( 13) is linear such that ( ) , , F x x t  is independent upon either displacement x or velocity x  , the Wilson-θ method can also be straightforwardly implemented by carrying out the following procedures Step 1a: Given any initial conditions ( 0 0 , x x   and 0 x ), time step t ∆ as well as , calculate the integral constants 0 8 c c shown in Appendix.
Step 2a: Compute the payload, Step 3a: Solve the displacement at time t t θ + ∆ according to the following equation where Step 4a: Finally, calculate the displacement, velocity and acceleration at time t t + ∆ according to the rules Then go to Step 1a to calculate the responses at the next time step.
Also, the Newmark-β method is capable of solving Equation ( 13) as long as it is linear, by employing the following procedures Step 1b: Given any initial conditions ( 0 0 , x x   and 0 x ), time step t ∆ as well as Step Step 3b: Solve the displacement at time t t + ∆ according to the following equation where * 0 1 Step 4b: Finally, calculate the displacement, velocity and acceleration at time Then go to Step 1b to calculate the responses at the next time step.
Note that, the above two methods are designed to solve linear systems.For nonlinear systems such as a cubic stiffness being included [15], the payload cannot be directly determined using either Step 2a or Step 2b, because the term t t F +∆ in both Equations ( 14) and ( 17) is a function of the responses (i.e., , , is smaller than a given tolerance Tol .
For more details, Step 2a and Step 2b will be realized by the procedures displayed in Figure 1.
In real practice, there are two key variables to be elaborately chosen such as the initial increment u ∆ and the tolerance Tol .In principle, a small value of u ∆ is helpful to the convergence of the predictor-corrector algorithm.In our numerical study, u ∆ is chosen randomly chosen to be at the order of 10-6.Differently, the value of Tol should be chosen to be relatively large to guarantee the convergence.In this paper, Tol is chosen as

Numerical Examples
In this section, a nonlinear dynamical system will be presented as numerical examples to validate the presented methods for solving quasi-periodic solutions.
It should be pointed out that, both the classical Wilson-θ and Newmark-β methods are incapable of solving nonlinear dynamical systems.As mentioned above, Fu et al. [11] suggested an approximate approach to enable Wilson-θ be capable of doing a nonlinear problem.The Wilson-θ modified by Fu et al. [11] will also be employed in all the following numerical studies, for the purpose of comparing the effectiveness and precision of the presented improved methods.
Considering the forced system is subjected to external excitations varying over time domain such as [16] ( ) ( ) where Also, the time step is chosen as 0.01 t ∆ = in this example.Figure 2 shows the comparisons of the displacement obtained by RK and the Wilson-θ methods, respectively.There is a slight discrepancy even at the early stage of the solution domain.As t increases further, the errors accumulate gradually resulting into noticeable differences.For quasi-periodic solutions, the coefficient matrix of the incremental balance equation should be corrected at every iteration step during calculation [17], which is possibly responsible for the rapid growing errors.The quasi-periodic solutions obtained by the improved methods are shown in Figure 3, also compared with the RK results.Nice agreement between the obtained soltuions domenstrate that both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions.
To further check the computaion accuracy, the error curves provided by the obtained results are shown in Figure 4.The error of the classical Wilson-θ is nearly of the same magnitue of the quasi-periodic responses, whereas the errors of the improved methods are at a much lower level.
Figure 5 shows the CPU running times for different methods, versus the number of time steps.The improved methods are more computational efficient than the classical Wilson-θ method.The reason consists in that the iteration of

Conclusions
Two improved methods have been proposed based on Wilson-θ and Newmark-β methods, respectively, for solving quasi-periodic responses of nonlinear dynam-ical systems.In order to avoid the same linearization procedure of Wilson-θ and Newmark-β methods, an improved fast algorithm is proposed in this paper, which enables both the two methods be capable of solving the quasi-periodic responses of nonlinear dynamical systems.In this algorithm, the solution at the next time step is first predicted according to the known one at the present time.
And the predicted solution is corrected by iterative manner, which together provides us with a predictor-corrector algorithm.Numerical examples have showed that, compared with the classical method, the improved methods can provide much more accurate quasi-periodic responses with less computational resources.Moreover, the improved methods are sometimes valid for systems for which the classical method is not.In addition, the convergence criterion is simple and effective for the presented methods, which guarantees both the computation accuracy and efficiency.c θ = − ,

Figure 1 .
Figure 1.A predictor-corrector algorithm for realizing the second steps in the improved Wilson-θ and Newmark-β methods, respectively.
-θ and Newmark-β methods generally have a precision of the second order.

Figure 4 .
Figure 4. Error curves of the obtained results for Equation (23) with RK results as the references.

Figure 5 .
Figure 5. CPU running time versus the number of time steps for the classical Wilson-θ, improved Wilson-θ and improved Newmark-β methods, respectively.