# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 1 Introduction  
The gear train is the most widely-used power transmission system, providing torque from rotary motion to other devices. The gear train's most common use is in motor vehicles, such as automobiles, aeronautic vehicles, heavy equipment, and agricultural machines. The first step in the gearbox design process is to determine the macro-geometry of the gears, including the normal module, number of gear teeth, pressure angle, helix angle, face width, and center distance. The next step is to predict the gear mesh misalignment by considering the elastic stiffness of the shaft, bearing, and housing under a quasi-static condition [1; 2]. However, the gear train is fundamentally a rotating machine, and, therefore, gears in the general operating condition cause dynamic loadings that are transmitted from the bearings to the housing structure. This causes time-varying interactions among the gearbox components, as can be seen in Figure 1. Therefore, the dynamic response of a geared rotor-bearing system should be considered for precise gear train design and analysis. Such designs have been studied extensively [3; 4; 5; 6; 7; 8]. Recently, considering the housing stiffness has become recognized as important for controlling the gear train's noise and vibration due to the use of lightweight gearbox housings in various vehicle industries for improved fuel efficiency. In such cases, various design codes encourage dynamic analysis to support the gearbox design procedure; however, such analyses incur heavy computational costs. Therefore, the initial steps, such as determining the macro- and micro-geometries of the gearsand computing the gear mesh misalignment, generally continue to be performed under the static condition [9; 10; 11; 12; 13; 14; 15; 16; 17]. As a potential solution to this problem, we propose introducing a dynamic stiffness component that considers the relevant elastic and kinetic energies. This enables compensation of the dynamic features of the mechanical components in the quasi-static analysis. Under harmonic excitation, we can simply separate spatial and temporal variables in the dynamic equations of motion and compute the dynamic stiffness of the gearbox components as a quasi-static analysis. We can then compute the dynamic gear mesh misalignment and transmission error following the standard gear design procedure by simply replacing the static stiffness with the dynamic stiffness. Numerical investigations demonstrate that this simple modification of the typical gear design procedure can produce relevant differences in the gear mesh misalignment. We emphasize that the aim of this work is not a noise, vibration, and harshness (NVH) analysis, but rather a consideration of the dynamic stiffness effect in the conventional gear design process under a quasi-static condition. Among the mechanical components, modeling the housing structure with the finite element (FE) method generally requires several numerical treatments. First, the degrees of freedom (DOFs) of the housing mesh surfaces where the bearings are located must be connected to the center of gravity of the bearing using rigid body elements (RBEs), which are also known as multi-point constraints [18; 19; 20; 21]. In addition, simplifying the dynamic model of the FE housing, which has a tremendous number of DOFs, requires the use of reduced-order modeling (ROM) techniques [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Component mode synthesis (CMS) is currently the most popular ROM technique for dynamic systems [26; 27; 28; 29; 30; 31]. The typical CMS method uses a small number of important and constraint modes to construct a reduced model that mathematically projects the displacement fields of the gearbox components onto modal coordinates. Thus, the accuracy level of the reduced system can be adjusted by adding (or eliminating) the desired number of modes. However, such modal projections require additional numerical treatments for application to the typical gear design process. To address this problem most efficiently, we use a Guyan reduction.