03-09-2012, 11:26 AM
Dynamic Optimization of Spur Gears
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Abstract
This paper presents a global optimization method focused on gear vibration reduction by means of
profile modifications. A nonlinear dynamic model is used to study the vibration behavior; such model is
validated using data available in literature. The optimization method considers different regimes and
torque levels, the objective function can be the static transmission error or the maximum amplitude of the
gear vibration in terms of dynamic transmission error. The procedure finds the optimal profile
modification that reduces the vibrations over a wide range of operating conditions. In order to reduce the
computational cost, a Random-Simplex optimization algorithm is developed; the optimum reliability is
estimated using a Monte Carlo simulation. The approach shows good performances for the computational
efficiency as well as the reliability of results. Finally, an application to High Contact Ratio (HCR) gears is
presented and an extremely good performance is obtained by combining optimization procedures and
HCR properties.
Introduction
The noise generated by spur gears is mainly due to the transmission error fluctuation, which is the
difference between the theoretical relative position of two unloaded gears (without manufacturing errors)
and the relative position of an actual loaded gear pair [1].
Introducing suitable profile modifications allows to change the phase and the ratio of the teeth load
exchange and to reduce the transmission error during a single meshing period [2]; this approach is valid
when only the teeth elastic deflections are considered.
In 1983 Sato et al. [3] studied analytically and experimentally the influence of profile modifications on
gear vibrations. This paper was one of the first works that obtained experimental confirmation regarding
gears optimization.
One of the most interesting studies on profile modifications, toward noise gear reduction, is due to
Tavakoli and Houser [4], 1986, which developed an original optimization algorithm based on the
modified Complex method, the steepest descent and the sequential one-variable search. The optimization
method was a local approach. An objective function based on the mean value of the transmission-errorharmonics
was used; out of design torques were analyzed on the optimal gears. Furthermore, the
dynamics was not studied and the static transmission error was evaluated by means of a cantilever beam
model.
In 1989, Simon [5] used a regressive analysis based on numerical computations in order to minimize
the fluctuation of the teeth load distribution. In 1990 Munro et al. [6] developed a simple optimization
approach, based on the Harris maps; the method was qualitative, because it did not consider the load
distribution due to the real teeth deformations.
Physical model
In order to model the dynamic behavior of a spur gear pair, some physical characteristics must be
considered. First of all, the parametric excitation due to the meshing stiffness fluctuation and transmission
error variations, due to the teeth elastic deflections, must be included in the model. Moreover, there is the
possibility of loss of contact due to the circumferential clearance between teeth. The present model has
been widely used in the literature [15–17], it allows to take into account the aforementioned phenomena;
conversely, the compliance of bearings and shafts is neglected as well as the effect of manufacturing
errors and misalignments.
Case study and dynamic analysis
The numerical analysis is carried out on the spur gear pair described in Table II, which is part of an
agricultural vehicle gearbox (courtesy of Case New Holland Italy S.p.A.); profile modifications reported
in Table II follow the Case New Holland standard; the gear pair with such modifications will be
considered the reference system (standard modifications). It is to note that the reference profile
modifications are used for comparison only; indeed, the optimization algorithm presented here does not
need any seed set.
Random-simplex optimization
The optimization strategy proposed in this paper is based on the combination of a Random search of
the optimum, followed by a refinement carried out using the Simplex method [13]; the latter one is very
robust and does not require derivatives evaluation of the objective function; however, it is not suitable for
global optimization.
Let us consider the minimization of a nonlinear function of n variables F[x1. . . xj, . . . , xn]. The first
step is to perform a Random sampling using a ―uniform distribution‖, and the optimum is the seed for the
Simplex Method.
The Simplex method is an iterative algorithm based on the concept of a polyhedron of n + 1 vertices
that is moved, iteration by iteration, toward the optimal point. The algorithm starts with an initial
polyhedron (P1, . . . , Pj, . . . , Pn) of points in the parameter space, each vertex is provided by the user or
randomly chosen; then the centroid of the Simplex is evaluated as ; the objective
function is computed for every vertex; then, such vertices are ordered. The vertex with the worst value of
F is moved. Three types of motions are possible: reflection (the vertex is reflected toward the opposite
face of the simplex), expansion (the vertex is expanded toward the minimum), contraction (the vertex is
moved toward the simplex face; for example, in Fig. 11 the point R would move toward the segment AD).
The single iteration starts with a reflection move; then, if the new vertex is better than the old one, the
new vertex is moved further with an expansion in the same direction, otherwise it is contracted. In Fig. 11
a simple reflection move of the worst point B is represented; after reflection, the new vertex R is
accepted. The algorithm stops when it cannot find an improved solution or when the maximum number of
iterations is reached.
Conclusions
A global optimization method, based on the Random plus Simplex approach, has been developed in
order to suggest the best geometry of spur gears versus vibration reduction; tip and root relief parameters
are considered for optimizing the gears. The optimization strategy considers the whole dynamic scenario,
i.e. the vibration level at different rotational speeds and different torques. This allows to find an optimized
gear pair having good dynamic behavior over a wide range of operating conditions.
The effectiveness of the dynamic optimization strategy is proved by means of comparisons with the
classical methods based on the peak to peak of the transmission error. The dynamic optimization produces
better results than the static optimization, it increases the computational cost of 15% only. The approach
includes a reliability analysis, which is limited to the parameter space; it allows to evaluate the robustness
of the optimum when design parameters cannot be controlled with extreme precision.