Combining the harmonic balance method with the incremental harmonic balance approach, an improved incremental harmonic balance method is presented to obtain the higher-order approximate steady-state solutions for strongly nonlinear systems, which can simplify the calculation process for high-order nonlinear terms. Taking a strongly nonlinear Duffing oscillator with cubic nonlinearity and a strongly nonlinear Duffing oscillator with quintic nonlinearity as examples, the forced vibrations under harmonic excitation are investigated. Based on the first-order approximate analytical solutions obtained by the harmonic balance method, the higher-order approximate solutions are obtained by the improved incremental harmonic balance method. The correctness of the approximate analytical results is verified by the numerical results. The comparison results show that the approximations obtained by the improved incremental harmonic balance method agree with the numerical solutions well, and the improved method is effective to analyze the dynamical response for strongly nonlinear systems.
ChenYMLiuJK (2007) A new method based on the harmonic balance method for nonlinear oscillators. Physics Letters A368(5): 371–378.
2.
ChungKWChanCLXuZet al. (2002) A perturbation-incremental method for strongly nonlinear autonomous oscillators with many degrees of freedom. Nonlinear Dynamics28(3): 243–259.
3.
DasSLChatterjeeA (2003) Multiple scales via galerkin projections: approximate asymptotics for strongly nonlinear oscillations. Nonlinear Dynamics32(32): 161–186.
4.
DonescuPVirginLNWuJJ (1996) Periodic solutions of an unsymmetric oscillator including a comprehensive study of their stability characteristics. Journal of Sound and Vibration192(5): 959–976.
5.
FarzanehYTootoonchiAA (2010) Global error minimization method for solving strongly nonlinear oscillator differential equations. Computers & Mathematics with Applications59(8): 2887–2895.
6.
GendelmanOV (2004) Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment. Nonlinear Dynamics37(2): 115–128.
7.
GuoZJZhangW (2016) The spreading residue harmonic balance study on the vibration frequencies of tapered beams. Applied Mathematical Modelling40(15–16): 7195–7203.
8.
HallKCThomasJPClarkWS (2002) Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA Journal40(5): 879–886.
9.
HuYDHuPZhangJZ (2015) Strongly nonlinear subharmonic resonance and chaotic motion of axially moving thin plate in magnetic field. Journal of Computational and Nonlinear Dynamics10(2): 021010.
10.
JuPXueX (2014) Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators. Applied Mathematical Modelling38(24): 6144–6152.
11.
KahramanASinghR (1990) Non-linear dynamics of a spur gear pair. Journal of Sound and Vibration142(1): 49–75.
12.
LabryerAAttarPJ (2009) High dimensional harmonic balance dealiasing techniques for a Duffing oscillator. Journal of Sound and Vibration324: 1016–1038.
13.
LaiSKLimCWWuBSet al. (2009) Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Mathematical Modelling33(2): 852–866.
14.
LauSLCheungYK (1981) Amplitude incremental variational principle for nonlinear vibration of elastic systems. Journal of Applied Mechanics48(4): 959–964.
15.
LeungAYTChuiSK (1995) Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. Journal of Sound and Vibration181(4): 619–633.
16.
LeungAYTGuoZJ (2011) Residue harmonic balance approach to limit cycles of non-linear jerk equations. International Journal of Non-Linear Mechanics46(6): 898–906.
17.
LeungAYGuoZJ (2012) Resonance response of a simply supported rotor-magnetic bearing system by harmonic balance. International Journal of Bifurcation and Chaos22(6): 349–361.
18.
LiaoS (2010) An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science & Numerical Simulation15(8): 2003–2016.
19.
LiuLDowellEHHallKC (2007) A novel harmonic balance analysis for the Van Der Pol oscillator. International Journal of Non-Linear Mechanics42(1): 2–12.
20.
LoghmaniGBHosseiniSM (2015) An improved piecewise variational iteration method for solving strongly nonlinear oscillators. Computational and Applied Mathematics34(1): 215–249.
21.
MahmoudGM (1997) Periodic solutions of strongly non-linear Mathieu oscillators. International Journal of Non-Linear Mechanics32(32): 1177–1185.
22.
OdibatZBatainehAS (2014) An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials. Mathematical Methods in the Applied Sciences38(5): 991–1000.
23.
PauzinMCMensahSCochelinBet al. (2011) High order harmonic balance formulation of free and encapsulated microbubbles. Journal of Sound & Vibration330(5): 987–1004.
24.
ShenYJWenSFLiXHet al. (2016) Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method. Nonlinear Dynamics85(3): 1457–1467.
25.
ShenYJYangSPLiuXD (2006) Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method. International Journal of Mechanical Sciences48(11): 1256–1263.
26.
StarosvetskyYGendelmanOV (2008) Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system. Journal of Sound and Vibration312(1–2): 234–256.
27.
SzeKYChenSHHuangJL (2005) The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration281(3–5): 611–626.
28.
ThomasJPDowellEHHallKC (2002) Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA Journal40(4): 638–646.
29.
WuBSSunWPLimCW (2006) An analytical approximate technique for a class of strongly non-linear oscillators. International Journal of Non-Linear Mechanics41(6–7): 766–774.
30.
XuLLuMWCaoQ (2002) Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method. Physics Letters A301(1–2): 65–73.
31.
XuLLuWMCaoQJ (2003) Analysis of dynamic properties of forced van der Pol equation with IHB method. Mechanics & Engineering25(1): 50–53.
