A torsional model of multi-stage gears – inﬂuence of external excitations and tooth shape modiﬁcations

– In this paper, a torsional dynamic model of multi-stage idler spur and helical gears is presented which combines mesh internal excitations and external forcing terms such as time-varying external torque. Each contact line in the various base planes is discretized in elemental cells which are all attributed a time-varying mesh stiﬀness element and initial separation to account for tooth shape deviations from ideal involute ﬂanks. The mesh stiﬀness functions are estimated from the formulae of Weber & Banaschek and their relative phasing is determined based on gear geometry and relative positioning. The corresponding non-linear diﬀerential system is solved by combining a Newmark’s numerical time-step integration scheme and a normal contact algorithm. A number of simulation results are presented on the combined inﬂuence on dynamic tooth loads of errors and shape deviations along with external excitations.


Introduction
In a number of mechanisms, multi-stage geared systems need to be employed to reduce speed and/or split the total power into several paths in order to operate pumps, generators, motors, etc. In such conditions, load machines control the load sharing between the various meshes to a large extent and can also introduce additional excitations via time-varying resisting torque for example. It is therefore interesting to analyse the potential influence of power circulation on the dynamic behaviour of multi-stage geared systems along with the contributions of mesh and external excitations. Most of the literature on gear dynamics deals with individual pinion-gear pairs and only a limited number of papers on multi-stage gears can be found. Choy et al. [1,2] investigated the modal response of double-stage gears for steady-state and transient conditions. Velex and Saada [3], Velex and Raclot [4], Kubur et al. [5] introduced finite element models of two-stage units and investigated the influence of key design parameters on the system dynamics. More recently, Carbonelli et al. [6,7] presented a robust optimization method for tooth corrections in truck timing gear cascades based on a swarm particle algorithm to minimize static transmission error time-variations. Yu et al. [8] proposed a dynamic model with several pinions and studied the contribution of mesh frequency, bearing stiffness to the load sharing properties. Yang [9] studied the vibrational response in torsion of a two-stage gear submitted to deterministic and random excitations in the presence of backlash.
The objective of this paper is to present a simplified torsional model that can simulate dynamic tooth loads on several idler gears in the presence of time-varying mesh stiffness functions and tooth shape modifications. The main contribution is the analysis of the influence of power circulation when several constant or time-varying torques are applied which have been rarely tackled in the literature. Unlike the majority of the models, transmission errors are one of the results of the simulations and dynamic responses are derived by the simultaneous solution of the instant contact conditions on the teeth and the equations of motion.

Model of N-1 stage gear systems
The model is composed of N idler spur or helical pinions simulated as rigid discs and connected by time-varying mesh stiffness functions [5,10] (Fig. 1). Each member has one single torsional degree-of-freedom (θ 1 , θ 2 , . . . , θ N ) which represents the angular perturbation superimposed on rigid-body rotation. All the contact lines in the base planes are discretized into thin slices which are all attributed (i) an elemental time-varying mesh stiffness, and (ii) a normal deviation to account for mesh stiffness variations (including their relative phasing) and tooth shape deviations with respect to perfect involute tooth flanks. Constant or time-varying external loads can be introduced at any node of the model and, in this paper, periodic torques of the form Ct(t) = C 0 + C s sin(f t Ωt) will be considered where Ω is the rotational frequency of the member.

Rigid body motions -Tooth shape modifications
For multi-stage idler gears, every mesh can alter the speed ratio because of its individual errors and shape deviations. Rigid-body kinematics can be characterised by extending the formulation in reference [11] to system comprising N meshes which leads to the following expression for the ith member angular speed: (1) Tooth shape modifications are accounted for by distributions of equivalent normal deviations with respect to the ideal flanks for both the pinion and the gear. At any potential point of contact in the base plane, the corresponding initial gaps (before deformation) are updated based on the relative rigid motion of the teeth. By adding the contributions from the mating pinion and gear flanks, initial separations δe m (M m ) are defined for each mesh whose relative time-phasing is determined as explained in the following section.

Mesh phasing
Mesh phasing in multi-stage idler gears is a key parameter since it controls the combinations of mesh excitations in terms of mesh stiffness functions and tooth shape modifications. Two general situations can be identified depending on the position of the power input and the angular positions of the pinions. Considering first the case of three members such that the driving gear is either the first or the third one in the kinematic chain (Fig. 2), the meshing conditions on the two base planes are linked and the phase difference on the second base plane (with respect to the first one) can be expressed as a distance Δl such that [12] where κ l = |T 11 T 12|

P ba
with : remainder of the natural division.
Distance |T 1 1 T 1 2 | depends on angle φ (Fig. 2) hence on the relative positions of the three pinions and reads: where φ is the angle between the two lines of actions (dependent on the sign of the driving torque Ct 1 ): with κ e = 0 and The second case of interest corresponds to three members where the driving pinion is the intermediate one ( Fig. 3) for which the phase difference between the two meshes defined as a distance on the base plane becomes: with |T 1 2 M | the distance between the first point of contact T 1 2 and point M: Fig. 3 for the parameter definition) and nP ba is an integral multiple of P ba.

Equation of motion in the presence of shape deviations/errors
At any potential point M m on a base plane, the mesh deflection is the normal approach relative to rigid-body positions from which the initial normal separation due to errors or/and shape modifications is subtracted: Based on the gear mesh interface model in Figure 1, and, after applying the resulting total mesh forces and moment at the centres of the pinion and gear of each stage [13], the inter-force wrench associated with mesh m can be deduced in a compact form as:

See equations (10) and (11) next page.
In what follows, the equations of motion are solved by using a Newmark's time-step integration scheme combined with a normal contact algorithm which verifies that all the contact forces between the tooth flanks are compressive.

Applications
The numerical applications are conducted on the 3stage idler spur gear system defined in Figure 4 and Table 1, the pinion/gear centres are all aligned such that γ 1 = γ 2 = 180 • .

Oscillating torque
In a number of applications, the total power is split between several auxiliary components such as pumps, generators, motors, etc., and the influence of the additional intermediate resisting torques on the overall dynamic behaviour needs to be investigated. In [ The response curves in Figures 5a and 5c represent the maximum dynamic mesh forces for constant loads and clearly exhibit response peaks which correspond to a natural frequency being excited by the mesh frequency and its harmonics: 5250 rpm ∼ = (W 2 /z 1 ) rad.s −1 , 3250 rpm ∼ = (W 4 /z 1 ) rad.s −1 , 1600 rpm ∼ = (W 4 /z 1 )/2 rad.s −1 , 500 rpm ∼ = (W 3 /z 1 ) rad.s −1 . The results in Figures 5b and 5d correspond to the cases with time-varying torques (of frequency 10Ω 1 ) for which the numerical simulations were performed over at least 5 mesh periods in order to cover the system period (T sys = LCM (z 1 = 50, f t = 10) Tm xft ; T sys = 5T m ). The addition of a periodic torque introduces an additional response peak at 2600 rpm ∼ = (W 3 /f t ) rad.s −1 on the three stages showing that there exist significant inter-mesh couplings. Moreover, the response curves for (c) and (d) highlight the power circulation influence on dynamic tooth forces since, for the same gear geometry and arrangement, stages 1 and 2 experience higher dynamic loads whereas stage 3 is less loaded compared with the previous examples (a) and (b). Some slight frequency shifts can also be noticed which are probably caused by the changes in nominal tooth loading hence on the mesh stiffness functions (the contact compliance being load dependent). Finally, it should be noticed that the introduction of a periodic component on torque Ct 1 or Ct 3 leads to modified response spectra (not shown here) which exhibit significant modulation side-bands.

Linear tooth modifications
In this section, tip relief was applied on all teeth in order to compensate for the elastic deflections at engagement/end of recess and also improve the tooth load distribution along the path of contact. Tooth reliefs are characterised by an amplitude at tooth tips (depth of modification) and by an extent of modification measured along the path of contact on the base plane. For each stage m, the depth of modification depends on the maximum mesh deflection and the extent of relief is expressed as a fraction of the length of the active contact line ε αm P ba [11].
Based on the results in Figure 6, it can be observed that the system exhibits contact losses between the teeth for some of the reduction stages at some critical speeds mostly at 1600 rpm, 2600 rpm and 5250 rpm. The introduction of short tip reliefs (Table 1) leads to a substantial improvement over the whole speed range (Figs. 6a  and 6b). However, the presence of a periodic torque slightly downgrades the dynamic performance mainly at the additional peaks caused by the time-varying torque excitation around 2600 rpm (Figs. 6c and 6d) for the treated example. This observation confirms that profile modifications can be effective in reducing mesh excitations whereas they hardly modify the contributions from external forcing terms.

Conclusion
A simplified torsional dynamic model of multi-stage gears has been presented which can account for N stages with any relative orientation of their lines of action. The relative phasing between the various mesh excitations is determined analytically and realistic mesh stiffness functions are employed. The results highlight the influence of the power circulation (several power outputs or inputs, etc.) in multi-mesh gear units along with the contributions of external excitations such as periodic torques on dynamic tooth loads and critical speeds. Periodic torques generate additional response peaks and introduce strong modulations in the dynamic response. It has also been shown that short tip profile modifications improve dynamic behaviour over all the range of rotational speeds but have limited influence on external excitation sources. The proposed global (load boundary conditions, power circulation) and local (profile modifications) approach is original as far as the authors know and shed light on multi-mesh gear dynamics. However, torsional models are known to be inaccurate in a number of realistic situations when bearing/casing deflections cannot be ignored. Further research is therefore needed in order to incorporate all three-dimensional displacements particularly when flexible components (casing) can strongly influence the loading conditions on the teeth. Finally, research is currently under way to investigate optimum profile modifications and critically assess the concept of quasistatic transmission errors in multi-stage geared systems.