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 rearranged 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.
Both Wilson-θ and Newmark-β methods are generally used approaches for solving numerical solutions of dynamical systems [
For some nonlinear problems, sometimes the two methods would provide false results, especially when there are quasi-periodic or chaotic responses [
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.
The nonlinear dynamical system is expressed as the following form
P [ x ¨ ( t ) , x ˙ ( t ) , x ( t ) ] = Q ( t ) (1)
where x ¨ ( t ) , x ˙ ( t ) , x ( t ) represent the generalized acceleration, velocity and displacement, respectively. After a step length Δ t , the following equation will be obtained as
P [ x ¨ ( t + Δ t ) , x ˙ ( t + Δ t ) , x ( t + Δ t ) ] = Q ( t + Δ t ) (2)
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
∂ P ( x ¨ , x ˙ , x ) ∂ x ¨ | t Δ x ¨ + ∂ P ( x ¨ , x ˙ , x ) ∂ x ˙ | t Δ x ˙ + ∂ P ( x ¨ , x ˙ , x ) ∂ x | t Δ x = Q ( t + Δ t ) − Q ( t ) (3)
which can be expressed in matrix form as
M ˜ Δ x ¨ + C ˜ Δ x ˙ + K ˜ Δ x = Δ Q (4)
with coefficient matrixes as
M ˜ = ∂ P ( x ¨ , x ˙ , x ) ∂ x ¨ | t , C ˜ = ∂ P ( x ¨ , x ˙ , x ) ∂ x ˙ | t , K ˜ = ∂ P ( x ¨ , x ˙ , x ) ∂ x | t , Δ P = Q ( t + Δ t ) − Q (t)
At the beginning of each time step, one should calculate the coefficient matrixes. One possible way is to substitute into Equation (4) the velocity, acceleration and displacement of the previous time step, i.e., x ¨ , x ˙ ( t ) , x ( t ) . It is worth noting that, however, this treatment will bring serious numerical errors as the higher-order terms are all removed from the Taylor expansion. On the other hand, Equation (1) will be unbalanced if we substitute x ¨ ( t + Δ t ) , x ˙ ( t + Δ t ) , x ( t + Δ t ) into the coefficient matrixes. Denote the unbalanced term as
R = P [ x ¨ ( t + Δ t ) , x ˙ ( t + Δ t ) , x ( t + Δ t ) ] − Q ( t + Δ t ) (5)
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 [
A basic assumption of the Wilson-θ method is that the acceleration varies linearly between time [ t , t + θ Δ t ] , so that the velocity and displacement can be expressed as
{ x ˙ ( t + Δ t ) = x ˙ ( t ) + θ Δ t 2 [ x ¨ ( t + Δ t ) + x ¨ ( t ) ] x ( t + Δ t ) = x ( t ) + θ Δ t x ˙ ( t ) + ( θ Δ t ) 2 6 [ x ¨ ( t + θ Δ t ) + 2 x ¨ ( t ) ] (6)
Rewrite Equation (6) equivalently in the incremental form and substitute it into (4), we can deduce the incremental displacement equation at [ t , t + θ Δ t ]
K ¯ Δ x θ t = Δ Q ¯ (7)
in which
K ¯ = K ˜ + 3 θ Δ t C ˜ + 6 ( θ Δ t ) 2 M ˜ , Δ Q ¯ = Δ Q ˜ + [ θ Δ t 2 x ¨ ( t ) + 3 x ˙ ( t ) ] C ˜ + [ 6 θ Δ t x ˙ ( t ) + 3 x ¨ ( t ) ] M ˜ (8)
With Δ x θ t being obtained, we can update the velocity, acceleration, and displacement as
Δ x ¨ = a 0 Δ x θ t + a 1 x ˙ + a 2 x ¨ Δ x ˙ = a 3 Δ x θ t + a 4 x ˙ + a 5 x ¨ Δ x = a 6 Δ x θ t + a 7 x ˙ + a 8 x ¨ (9)
where the coefficients a i , i = 0 , 1 , ⋯ , 8 are all listed in Appendix. And, the system response at time t + Δ t could be finally obtained as
x ¨ ( t + Δ t ) = x ¨ ( t ) + Δ x ¨ x ˙ ( t + Δ t ) = x ˙ ( t ) + Δ x ˙ x ( t + Δ t ) = x ( t ) + Δ x (10)
Similar to Equation (6), when the Newmark-β method is employed we can obtain
{ x ¨ ( t + Δ t ) = 1 α Δ t 2 [ x ( t + Δ t ) − x ( t ) ] − 1 α Δ t x ˙ ( t ) + ( 1 − 1 2 α x ¨ ( t ) ) x ˙ ( t + Δ t ) = x ˙ ( t ) + ( 1 − δ ) Δ t x ¨ ( t ) + δ Δ t x ¨ ( t + Δ t ) (11)
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
Δ x ¨ = b 0 Δ x + b 1 x ˙ + b 2 x ¨ Δ x ˙ = b 3 Δ x + b 4 x ˙ + b 5 x ¨ (12)
with the expressions of b i , i = 0 , 1 , ⋯ , 5 in Appendix. Substituting Equation (12) into (10) yields the response at time t + Δ t .
To elucidate the improved methods, we consider the following dynamic equation
M x ¨ ( t ) + C x ˙ ( t ) + K ( x ) x ( t ) = F ( x , x ˙ , t ) (13)
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 ( x ¨ 0 , x ˙ 0 and x 0 ), time step Δ t as well as θ ≥ 1.37 , calculate the integral constants c 0 ~ c 8 shown in Appendix.
Step 2a: Compute the payload, F ˜ t + θ Δ t , at time t + Δ t
F ˜ t + θ Δ t = F t + θ ( F t + Δ t − F t ) + M ( c 0 x t + c 2 x ˙ t + 2 x ¨ t ) + C ( c 1 x t + 2 x ˙ t + c 3 x ¨ t ) (14)
Step 3a: Solve the displacement at time t + θ Δ t according to the following equation
K * x t + θ Δ t = F ˜ t + θ Δ t (15)
where K * = K + c 0 M + c 1 C .
Step 4a: Finally, calculate the displacement, velocity and acceleration at time t + Δ t according to the rules
{ x ¨ ( t + Δ t ) = c 4 [ x ( t + θ Δ t ) − x ( t ) ] + c 5 x ˙ ( t ) + c 6 x ¨ ( t ) x ˙ ( t + Δ t ) = x ˙ ( t ) + c 7 [ x ¨ ( t + Δ t ) + x ¨ ( t ) ] x ( t + Δ t ) = x ( t ) + Δ t x ˙ ( t ) + c 8 [ x ¨ ( t + Δ t ) + 2 x ¨ ( t ) ] (16)
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 ( x ¨ 0 , x ˙ 0 and x 0 ), time step Δ t as well as δ ≥ 0.50 , α ≥ 0.25 ( 0.5 + δ ) 2 , compute the constants d 0 ~ d 7 as shown in Appendix.
Step 2b: Compute the payload, F ˜ t + θ Δ t , at time t + Δ t
F ˜ t + Δ t = F t + Δ t + M ( d 0 x t + d 2 x ˙ t + d 3 x ¨ t ) + C ( d 1 x t + d 4 x ˙ t + d 5 x ¨ t ) (17)
Step 3b: Solve the displacement at time t + Δ t according to the following equation
K * x t + Δ t = F ˜ t + Δ t (18)
where K * = K + d 0 M + d 1 C .
Step 4b: Finally, calculate the displacement, velocity and acceleration at time t + Δ t according to the rules
{ x ¨ ( t + Δ t ) = d 0 [ x ( t + Δ t ) − x ( t ) ] − d 2 x ˙ ( t ) − d 3 x ¨ ( t ) x ˙ ( t + Δ t ) = x ˙ ( t ) + d 6 ⋅ x ¨ ( t ) + d 7 ⋅ x ¨ ( t + Δ t ) x ( t + Δ t ) = F ˜ t + Δ t / K * (19)
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 [
The key procedure of the improved methods is to calculate the responses by a predictor-corrector algorithm, without transforming the nonlinearities into linear ones.
Denote the state responses of the system at time t as
{ u } t = { x ¨ ( t ) , x ˙ ( t ) , x ( t ) } (20)
After the time Δ t , the state responses are rewritten as
{ u } t + Δ t = { x ¨ ( t + Δ t ) , x ˙ ( t + Δ t ) , x ( t + Δ t ) } (21)
In both the improved Wilson-θ and Newmark-β methods, specifically, the second steps (Step 2a and Step 2b) which aim at calculating the payload are replaced by the following predictor-corrector algorithm
Step A: Predict the solution { u ′ } t + Δ t at t + Δ t according to priorly given responses { u } t with a small increment Δ u .
Step B: Compute the payload F ˜ ′ t + θ Δ t by applying the predicted solution { u ′ } t + Δ t to Equation (14) or (17).
Step C: Obtain a new solution { u ″ } t + Δ t at t + Δ t by applying F ˜ ′ t + θ Δ t to Eqs. (15-16) or (18-19);
Step D: Correct { u ″ } t + Δ t by proceeding Steps B-C-D if necessary, until the residual Re = | M x ¨ ( t ) + C x ˙ ( t ) + K ( x ) x ( t ) − F ( x , x ˙ , t ) | is smaller than a given tolerance T o l .
For more details, Step 2a and Step 2b will be realized by the procedures displayed in
In real practice, there are two key variables to be elaborately chosen such as the initial increment Δ u and the tolerance T o l . In principle, a small value of Δ u is helpful to the convergence of the predictor-corrector algorithm. In our
numerical study, Δ u is chosen randomly chosen to be at the order of 10-6. Differently, the value of T o l should be chosen to be relatively large to guarantee the convergence. In this paper, T o l is chosen as T o l = O ( Δ t 2 ) because both the Wilson-θ and Newmark-β methods generally have a precision of the second order.
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. [
Considering the forced system is subjected to external excitations varying over time domain such as [
M x ¨ + C x ˙ + ( K 1 + K 3 ( x ) ) x = F (22)
where x = [ x 1 x 2 ] , M = [ 1 0 0 1 ] , C = [ c 0 0 c ] , K 1 = [ 1 δ − δ 1 ] , K ( 3 ) ( x ) = [ a 1 x 1 2 + a 2 x 1 x 2 , a 3 x 1 x 2 + a 4 x 2 2 b 1 x 1 2 + b 2 x 1 x 2 , b 3 x 1 x 2 + b 4 x 2 2 ] , F = [ f 1 cos ω t f 2 sin ω t ] with the constants ω = 2.3112 , c = 0.12 , δ = 0.1299 , f 1 = f 2 = 0.1587 , a 2 = a 4 = 0 , b 1 = b 3 = 0 , a 1 = a 3 = b 2 = b 4 = 1 .
Also, the time step is chosen as Δ t = 0.01 in this example.
To further check the computaion accuracy, the error curves provided by the obtained results are shown in
classical method requires more iterations during Step 2a for convergence, that is designed to approximate the preload associated with nonlinearities. Interestingly, in most cases only two iterations are needed for the convergence of the second steps of the improved methods.
Two improved methods have been proposed based on Wilson-θ and Newmark-β methods, respectively, for solving quasi-periodic responses of nonlinear dynamical 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.
This work is supported by the National Natural Science Foundation of China (11572356, 11672337).
The authors declare no conflicts of interest regarding the publication of this paper.
Liu, G., Lv, Z.R. and Chen, Y.M. (2018) Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems. Journal of Applied Mathematics and Physics, 6, 1625-1635. https://doi.org/10.4236/jamp.2018.68138
Coefficients for Wilson-θ method based on the incremental equation
a 0 = 6 θ 3 Δ t 2 , a 1 = − 6 θ 2 Δ t , a 2 = − 3 θ , a 3 = 3 θ 2 Δ t , a 4 = − 3 θ 2 , a 5 = ( 1 − 3 2 θ ) Δ t , a 6 = 1 θ 3 , a 7 = ( 1 − 3 θ 2 ) Δ t , a 8 = 1 2 ( 1 − 1 θ ) Δ t 2
Coefficients for Newmark-β method based on the incremental equation
b 0 = 1 α Δ t 2 , b 1 = − 1 α Δ t , b 2 = − 1 2 α , b 3 = δ α Δ t , b 4 = − δ α , b 5 = 1 + δ 2 α
Integral constants in classical Wilson-θ method
c 0 = 6 ( θ Δ t ) 2 , c 1 = 3 θ Δ t , c 2 = 2 ⋅ c 1 , c 3 = Δ t 2 ⋅ θ , c 4 = c 0 θ , c 5 = − c 2 θ , c 6 = 1 − 3 θ , c 7 = Δ t 2 , c 8 = Δ t 2 6
Integral constants in classical Newmark-β method
d 0 = 1 α Δ t 2 , d 1 = δ α Δ t , d 2 = 1 α Δ t , d 3 = 1 2 α − 1 , d 4 = δ α − 1 , d 5 = Δ t 2 ( δ α − 2 ) , d 6 = Δ t ( 1 − δ ) , d 7 = Δ t ⋅ δ