The harmonic balance method uses a description function to approximate the nonlinearity caused by the gap and is widely used in PGSs (Acri et al., 2019). The excitation and response parameters are assumed to be harmonic functions and are substituted in the nonlinear equation. The approximate expressions of the response and phase parameters can thus be obtained using the condition of equal power coefficients. Since this method is not limited by the degree of nonlinearity, all the frequency response values can be obtained. However, owing to the limitations of the assumed excitation and response form, the accuracy of this method is not satisfactory, particularly if the first harmonic is considered. It results in the artificial loss of the superharmonic, subharmonic, or chaotic responses. Lau and Cheung proposed the incremental harmonic balance (IHB) method in 1981 to improve the accuracy of the existing harmonic balance method. A Taylor series expansion was performed on the nonlinear differential equations while ignoring the higher-order derivatives to obtain the differential equations in an incremental form. The Fourier series and the Galerkin method were then used to obtain nonlinear algebraic equations. The entire process is divided into an incremental component (Newton–Raphson method) and a harmonic balance component (Galerkin method). This method has the advantage of free control algorithm convergence accuracy, among many others, and is thus an effective method for solving complex nonlinear problems (Daneshjou et al., 2017). Rohan and Lukeš (2019) used the incremental harmonic balance method for a two-stage star gear train with multiple degrees of freedom; however, it is only used the response as an incremental parameter. If a singular point is encountered along the path of the solution branch, the quasi-arc length parameter is introduced. The original variables and parameters are assumed to be functions of the arc length. This condition is added to the original equation to smoothly track the path through the singular point (Guo et al., 2014). The harmonic balance method, based on the continuation of the arc-length, has been applied to the pure torsion model of a fixed shaft and a single-stage planetary transmission (Tomilina, 2015). The incremental harmonic balance method, based on the continuation of the arc length, is used to calculate the dynamic characteristic equation of the system used in this study. The two incremental parameters (response and fundamental frequency) are expressed in terms of the arc length to smoothly overcome the singular points along the path. The formula to calculate the steady-state response of a two-stage herringbone PGS is presented in this study. This method has not yet been applied to planetary gear transmission systems.

A new time variable τh=Ωτ is introduced in this method. The expression of y, initially written in terms of τ, is rewritten in terms of τh. As a result, Eq. (4) is written as 5Ω2My¨+ΩCy˙,τhy˙+Ky,ωmiΩτhy=F(τh), where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the TsSCS respectively. ωmi is the natural angular frequency. The incremental process is the first component of the IHB method. If yj0 and Ω0 are the solutions of Eq. (5), their neighboring points can be expressed as 6yj=yj0+Δyj,Ω=Ω0+ΔΩ,j=1,2,…,M+N, where Δyi and ΔΩ are the incremental parameters.

By substituting Eq. (6) into Eq. (5) and omitting high-order small quantities, the incremental equation matrix can be obtained with Δyi and ΔΩ as the unknown quantities. 7Ω02MΔy¨+Ω0CΔy˙+KΔy=R-2Ω0My¨0+Cy˙0ΔΩ8R=Fτh-Ω02My¨0+Ω0Cy˙0+Ky0, where R is the unbalanced force vector (also referred to as the residual correction term in some studies). If yj0 and Ω0 are exact solutions, then R=0.

The second component of the IHB method involves the harmonic balance process. The steady-state response of the system is described by a Fourier series. The response contains only odd harmonics and is given as follows. 9yj0=aj0+∑k=1Ncajkcos⁡kτh+bjksin⁡kτh=CsAj10Δyj0=Δaj0+∑k=1NcΔajkcos⁡kτh+Δbjksin⁡kτh=CsΔAj

The response of the system and its increment can be written in the following matrix form. 11y0=SA,Δy=SΔA

After substituting Eq. (11) into the incremental Eq. (7) and the unbalanced force Eq. (8), the Galerkin averaging process is applied to obtain the equations for the unknown quantities ΔA and ΔΩ. 12KjcΔA=R-RjcΔΩ, where Kjc=∫02πST (Ω02MS′′+Ω0CS′+KS)dτh, R=∫02πSTF(τh)-∫02πST(Ω02MS′′+Ω0CS′+KS)dτhA, and Rjc=∫02πST(2Ω0MS′′+CS′)dτhA.

The arc length parameter equation, corresponding to Eq. (5), can be expressed as 13g(p)-s=0.

Assuming g(p)=pTp, p=AT,ΩT and substituting the increments of A, Ω, and S in Eq. (13), the increment equation can be obtained, as shown below. 14∂g∂AT{ΔA}+∂g∂ΩΔΩ-Δs+g-s=0

Figure 1 shows a part of the analytical balance path of the arc-length extension method. And Eq. (13) can be rewritten as 15g(p)-s=p′Tp-pc=016p′=pc-pcc/pc-pcc.

The initial values of the upper and lower points pu of the balance path are determined by the values of the previous two points, pc and Pcc.

The complete incremental equation can be obtained by combining Eqs. (12) and (14). 17Jp{Δp}=KjcRjc{∂g/∂A}T∂g/∂ΩΔAΔΩ=R-g+Δs+s=[R̃], where Jp is the Jacobian matrix relative to {p}.

The above equation is expressed in an iterative form that can be easily calculated, as shown below. 18pi+1=pi+Jp-1piR̃pi

Equation (18) represents the Newton–Raphson iterative equation that is obtained after introducing the arc length parameter. The arc length parameter is used to predict the value of the next solution from the current solution and is used as the initial value in the next iteration.

The transmission diagram of the double-wide helical planetary composite system, with a two-level power branch, for high speed and heavy duty applications, is shown in Fig. 2. The system is composed of a star gear train I (sun gear zsI, planet gear zm, and ring gear zrI) that is connected to a planet gear train II (sun gear zsII, planet gear zn, ring gear zrII, and planet carrier H). The superscripts I and II correspond to the series of the component. The input power is transmitted to the load L by the sun gear zsI. The input speed of the second-stage planetary gear train is reduced according to the deceleration of the first-stage planetary gear train, thereby increasing the stability and smoothness of the transmission. The system parameters are listed in Tables 1 and 2. The calculated nonlinear response characteristics of the system are shown in Figs. 3 to 5, which correspond to the time domain response history, phase diagram, and Poincaré mapping of the system, respectively. Owing to the space constraints in this paper, we have only mentioned the torsional response of the representative components. However, this does not imply that the translational response of the components can be neglected.

Since the number of unknowns exceeds the number of equations, the expected increment must be specified before performing the actual calculation. The increment specified in this article is equal to ΔΩ. Based on the structural flowchart of the improved incremental harmonic balance method depicted in Fig. 3 and the dimensionless parameters listed in Tables 1 and 2, the specific iterative process of the improved method is given as follows.

The amplitude–frequency characteristic curves of the system operating at the approximate working speed are shown in Fig. 4. The figure compares the results obtained by the incremental harmonic balance method and those obtained by the numerical integration (NI) method. The latter is based on the variable step size of the fourth and fifth steps, which is the Runge–Kutta method.

The meshing frequency lies in the range of 3.3–3.7 kHz, as shown in Fig. 4b and c. Amplitude jumps are observed in both the sun gear and star gear of the star gear system. The two methods are in this area. The amplitude values obtained from both the methods do not match as well in this region as they do in the others. In addition, the results of the numerical integration method indicate the presence of a resonance peak in other regions; however, the IHB method does not seem to indicate the presence of a resonance peak. This is because the number of harmonic response terms is insufficient.

The ring gear of the star gear system, shown in Fig. 4a, and the sun gear of the planetary gear system, shown in Fig. 4e, produce minimal amplitude resonance. The results obtained by the numerical method and IHB method follow a similar trend; however, a significant difference exists between the amplitude values. The variations in the amplitude of the planetary carrier, shown in Fig. 4d, and the planetary gears, shown in Fig. 4f, are relatively gentle. The curves obtained from the two methods gradually tend to become consistent with increasing mesh frequency, resulting in a significant amplitude variation of the planetary gears. This is consistent at a 4.0 kHz bit frequency. The above analysis demonstrates that the improved method is in agreement with the law of amplitude–frequency change for each part of the system, thereby illustrating the feasibility of this method.

The incremental harmonic balance method, which is based on the arc-length continuation technique, is a semi-analytical and semi-numerical method. It is necessary to perform a purely analytical study of the dynamic characteristic equation of the system. Current studies adopt the multi-scale perturbation analysis method to obtain the analytical solutions of parametric excitation and gap nonlinear system equations (Li et al., 2019). The multi-scale perturbation (MsP) method can obtain the analytical frequency response functions of a system, including the fundamental, subharmonic, and superharmonic resonance responses (Tittus et al., 2020). Thus, this technique demonstrates the impact of important parameters on the response of the nonlinear dynamic characteristics, unlike conventional numerical methods.

The small parameter ε=|c^sn(I)|/kspII, c^sn(I) is introduced in the first-order Fourier coefficient of the meshing stiffness of the sun gear and planetary gears in the planetary gear system.

kspII refers to the average meshing stiffness and is written in its dimensionless form for each gear pair, as shown below. 19ksmI(τ)=kspI1+ε∑l=1∞bsmlejlωm1τ+cc,krmI(τ)=krpI1+ε∑l=1∞brmlejlωm1τ+cc20ksnII(τ)=kspII1+ε∑l=1∞csnlejlωm2τ+cc,krnII(τ)=krpII1+ε∑l=1∞crnlejlωm2τ+cc, where csn(l)=c^sn(l)|c^sn(l)|=O(1), crn(l)=c^rn(l)|c^sn(l)|kspIIkrpII=O(1), bsm(l)=kspIIkspIb^sm(l)c^sn(l)=O(1), brm(l)=kspIIkrpIb^rm(l)c^sn(l)=O(1).

The time required for the contact gear pair to disengage is assumed to be negligible with respect to the response period, i.e., ξT=O(ε), where ξ is the disengagement time, and T=2πΩ is the response period. The Fourier expansion of the non-meshed function of the contact gear pair is expressed in terms of the fundamental frequency Ω, as shown below. 21ΘδsmI=1+ε∑h=0∞g^sm(h)ejhΩτ+cc,ΘδrmI=1+ε∑h=0∞g^rm(h)ejhΩτ+cc22ΘδsnII=1+ε∑h=0∞θ^sn(h)ejhΩτ+cc,ΘδrnII=1+ε∑h=0∞θ^rn(h)ejhΩτ+cc

The corresponding eigenvalue of Eq. (11) is expressed as 23Kb+K0Vi=ci2MVi, where K0 is the linear time-invariant average meshing stiffness matrix, and K(y,t)=Ke+Kb+K0+Kd(y,t). Kd represents the change range matrix of the average meshing stiffness K0, with its mean value equal to zero. Kb is the support torsional stiffness matrix (including the support stiffness of the star and planet gears). Ke is the additional stiffness matrix generated due the transient phase angle of the planetary gear, Ke=CpnII+CsnII+CrnII, where CpnII, CsnII, and CrnII are the coefficient matrices related to CpnII, CpsnII, and CrnII, respectively. The vibration mode is given by V=V1,…,V3(M+N+4), and it satisfies the relation VTMV=I. The average stiffness matrix K is given below. 24K=K0+Kb=∑m=1MkspIKsmI+krpIKrmI+∑n=1NkspIIKsnII+krpIIKrnII, where KsmI, KrmI, KsnII, and KrnII are the coefficient matrices related to kspI, krpI, kspII, and krpII.

Substituting Eqs. (25)–(28) into Eq. (11), we obtain 25My˙+Cy˙+Key+Kb+K0y+ε∑n=1NkspIIQsIIKsnIIy+ε∑n=1NkrpIIQrIIKrnIIy+ε∑m=1MkspIQsIKsmIy+ε∑m=1MkrpIQrIKrmIy+O(ε)=F, where QsI=∑l=1∞bsm(l)ejlωm1τ+∑h=0∞g^sm(h)ejhωτ+cc, QrI=∑l=1∞brm(l)ejlωm1τ+∑h=0∞g^rm(h)ejhωτ+cc, QsII=∑l=1∞csn(l)ejlωm2τ+∑h=0∞θ^sn(h)ejhωτ+cc, QrII=∑l=1∞csr(l)ejlωm2τ+∑h=0∞θ^rn(h)ejhωτ+cc.

The modal coordinate transformation y=Vz can be used to write the additional stiffness matrix coefficients in terms of the small parameters. The resulting modal coordinate form, obtained after the transformation of Eq. (31), is given as 26z¨q+ελqz˙q+cq2zq+εncII∑w=13(M+N+4)∑n=1NCpnIIEpnqw+∑n=1NCsnIIEsnqw+∑n=1NCrnIIErnqwzw+ε∑w=13(M+N+4)[∑m=1MkspIQsIGsmqw+krpIQrIGrmqw+∑n=1NkspIIQsIIGsnqw+krpIIQrIIGrnqw]zw=VqTF=fq, where Gsm=VTKsmIV, Grm=VTKrmIV, Gsn=VTKsnIIV, Grn=VTKrnIIV, Epn=VTCpnIIV, Esn=VTCsnIIV, and Ern=VTCrnIIV, Epnqw, Esnqw, and Ernqw represent the elements in the qth row and ωth column of the matrices Epm, Esn, and Ern, respectively. Gsmqw, Grmqw, Gsnqw, and Grnqw represent the elements in the qth row and the ωth column of the matrices Gsm, Grm, Gsn, and Grm, respectively. The modal damping factor 2ζqcq has been introduced and rewritten in terms of the small parameter ε as ελq=2ζqcq. εncII=ωcII, where ωcII is the speed of the second stage planet carrier. The small parameter ε is related to the time change of the phase angle of the planet gear in the second stage. A multi-scale method is applied by introducing multi-scale variables such as τn=εnτ and zq(τ0,τ1,…)=zq0(τ0,τ1,…)+εzq1(τ0,τ1,…)+O(ε2). Based on the abovementioned variables, the perturbation equation with the first approximate solution is proposed, as shown below. 27∂2∂τ02zq0+cq2zq0=fq28∂2∂τ02zq1+cq2zq1=-2∂2zq0∂τ0∂τ1-λq∂zq0∂τ0-ncII∑w=13(M+N+4)(∑n=1NCpnIIEpnqw+∑n=1NCsnIIEsnqw+∑n=1NCrnIIErnqw)zw0-∑w=13(M+N+4)[∑m=1M(kspIQsIGsmqw+krpIQrIGrmqw)+∑n=1NkspIIQsIIGsnqw+krpIIQrIIGrnqw]zw0

Equation (34) is the perturbation equation used to calculate the closed solution. The frequency response characteristics of the system under different excitations can be studied using this equation.

The amplitude–frequency characteristics of the system are obtained and studied after solving the frequency response equations under different resonance conditions according to the multi-scale perturbation analysis method. A natural frequency of 973.1 Hz is selected to study the frequency response characteristics of the system during resonance in the vibration mode of the planetary gear system. The amplitude–frequency characteristics of the system are analyzed and compared with the results obtained using the numerical integration method. The calculated amplitude–frequency characteristic curve is shown in Fig. 5. Figure 5a shows that the response amplitude of the ring gear of the star gear train calculated by the multi-scale method differs significantly from that of the numerical method. No amplitude jump was observed, as shown in Fig. 5b and c. The variation trends of the planetary carrier and gears of the planetary gear train, as shown in Fig. 5d and f, are identical. The trends followed by the variation of the responses in both methods are identical, as shown in Fig. 5e. The larger difference is also the magnitude of the amplitude. The difference between the values of the amplitude obtained through both methods is relatively large for the planet carrier and small for the planet gear.