310 S.D. Yavuz et al. T Torque t Time U Solution vector u Gear mesh displacement harmonic amplitude Z Number of teeth z Perturbation state vector ˛ Phase of static transmission error harmonic Discrete time interval Discontinuous separation function Rotational displacement … Phase difference between meshes Dimensionless frequency ! Characteristic frequency Damping ratio Angle between the lines connecting the centers of the gears Constant angle in mesh phasing calculation Subscripts a Alternating component i Mesh index m Mean component Superscripts i Mesh index rms Root-mean-square value T Matrix transpose . Derivative with respect to time ’ Derivative with respect to dimensionless time Dimensional quantities 30.1 Introduction Gear vibration is an important consideration in drive-train systems due to noise and durability problems. Under dynamic conditions, gear systems produce much higher gear mesh forces than static forces transmitted. These high frequency dynamic forces must be supported by the bearings and are eventually transmitted to the housing to act as the main excitations for gear related noise. Furthermore, alternating forces induced by the vibration reduce fatigue life of the driveline components. Therefore, a better understanding of the gear system dynamics is vital in order to design more silent and durable transmission. There are a large number of gear dynamics related studies in the literature and in the vast majority of these studies, a single gear pair is considered. Numerous mathematical models are constructed and analytical and numerical solution methods are developed in those studies. The models including a spur gear pair are mostly nonlinear (piecewise-linear) due to backlash but differ in incorporating time variation of mesh stiffness. Some of these models are nonlinear time-invariant (NTI) [1], whereas the others are nonlinear time-varying (NTV) [2–6]. However, published experimental data [3, 4, 7] show that the dynamic behavior of a spur gear pair can only be described by a NTV model. These single-degree-of-freedom (DOF) models are extended to multiple DOFs nonlinear models of geared rotor-bearing systems [8–11]. Moreover, linear and time-invariant characteristics of helical gears are studied in [12–14]. The studies on multi-mesh gear systems are fewer than the ones on single gear pair systems even though most practical systems use multi-mesh, multi-stage gear trains. Nonlinear time-varying dynamic models of multi-mesh spur gear trains are
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