Zipu Yan^1,2, Honghua Dai^1,2,*, Qisi Wang^1,2, Satya N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.137, No.2, pp. 1419-1459, 2023, DOI:10.32604/cmes.2023.028198

Abstract The harmonic balance (HB) method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems. However, it is confined to low-order approximations due to complex symbolic operations. Many variants have been developed to improve the HB method, among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation. So far, there are two problems remaining to be addressed. i) A dozen of different versions of HB-like methods, in frequency domain or time domain or in hybrid, have been developed; unfortunately, misclassification… More >

Haitao Liao¹

CMES-Computer Modeling in Engineering & Sciences, Vol.115, No.2, pp. 233-262, 2018, DOI: 10.3970/cmes.2018.01004

Abstract A hybrid method combined the reduced Sequential Quadratic Programming (SQP) method with the harmonic balance method has been developed to analyze the characteristics of mode localization and internal resonance of nonlinear bladed disks. With the aid of harmonic balance method, the nonlinear equality constraints for the constrained optimization problem are constructed. The reduced SQP method is then utilized to deal with the original constrained optimization problem. Applying the null space decomposition technique to the harmonic balance algebraic equations results in the vanishing of the nonlinear equality constraints and a simple optimization problem involving only upper and lower bound constraints on… More >

Haitao Liao

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.3, pp. 155-187, 2014, DOI:10.3970/cmes.2014.103.155

Abstract The improved constrained optimization harmonic balance method(COHBM) is presented to solve the Duffing oscillator with time delayed term. Within the framework of the proposed method, the analytical gradients of the objective function and nonlinear quality constraints with respect to optimization variables are formulated and the sensitivity information of the Fourier coefficients can also obtained. The general formulas of the geometrically nonlinear and time delayed terms are analytically derived, which makes the calculations of nonlinear differential equations in the frequency domain easily. A stability analysis method based on the analytical formulation of the nonlinear equality constraints is presented for the nonlinear… More >

Haitao Liao¹

CMES-Computer Modeling in Engineering & Sciences, Vol.95, No.3, pp. 207-234, 2013, DOI:10.3970/cmes.2013.095.207

Abstract The constrained optimization multi-dimensional harmonic balance method for calculating the quasi-periodic solutions of nonlinear systems is presented. The problem of determining the worst quasi-periodic response is transformed into a nonlinear optimization problem with nonlinear equality constraints. The general nonlinear equality constraints are built using a set of nonlinear algebraic equations which is derived using the multi-dimensional harmonic balance method. The Multi- Start algorithm is adopted to solve the resulting constrained maximization problem. Finally, the validity of the proposed method is demonstrated with a Duffing oscillator and numerical case studies for problems with uncertainties are performed on a nonlinear two-degree of… More >

Hong-Hua Dai^1,2, Matt Schnoor², Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.5, pp. 459-498, 2012, DOI:10.3970/cmes.2012.084.459

Abstract In this study, the harmonic and 1/3 subharmonic oscillations of a single degree of freedom Duffing oscillator with large nonlinearity and large damping are investigated by using a simple point collocation method applied in the time domain over a period of the periodic solution. The relationship between the proposed collocation method and the high dimensional harmonic balance method (HDHB), proposed earlier by Thomas, Dowell, and Hall (2002), is explored. We demonstrate that the HDHB is not a kind of "harmonic balance method" but essentially a cumbersome version of the collocation method. In using the collocation method, the collocation-resulting nonlinear algebraic… More >

Chein-Shan Liu^1,2, Hong-Hua Dai¹, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.2, pp. 195-228, 2011, DOI:10.3970/cmes.2011.081.195

Abstract In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x^· = λ[αR + βP], where P = F − R(F·R) / ||R||² such that P·R = 0; or x^· = λ[αF + βP^∗], where P^∗ = R − F(F·R) / ||F||² such that P^*·F =… More >