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Application of double-crowning allows to assign ahead that function of transmission errors is a parabolic one, and allows to assign as well the maximal value of transmission errors as of 6–8″. The expected magnitude of the predesign parabolic function of transmission errors and the magnitude of the parabolic plunge of the generating tool have to be correlated. Fig. 2 shows the case wherein due to a large magnitude of error of misalignment, the function of transmission errors is formed by two branches: of surface-to-surface contact and of surface-to-curve contact.
(31K)
Fig. 2. Results of TCA of a case of double-crowned helical gear drive with a large error Δγ = 10′: (a) function of transmission errors wherein corresponds to surface-to-surface tangency and correspond to surface-to-curve tangency; (b) path of contact on pinion tooth surface; (c) path of contact on gear tooth surface.
4. Transmission errors of a loaded gear drive
The contents of this section cover the procedure of determination of transmission errors of a loaded gear drive by application of a general purpose FEM computer program [3]. Transmission errors of an unloaded gear drive are directly determined by application of TCA. Comparison of transmission errors for unloaded and loaded gear drives is represented in Section 5.
4.1. Preliminary considerations
(i) Due to the effect of loading of the gear drive, the maximal transmission errors are reduced and the contact ratio is increased
(ii) The authors’ approach allows to reduce the time of preparation of the model by the automatic generation of the finite element model [1] for each configuration of the set of applied configurations.
(iii) Fig. 3 illustrates a configuration that is investigated under the load. TCA allows to determine point M of tangency of tooth surfaces Σ1 and Σ2, before the load will be applied (Fig. 3(a)), where N2 and N1 are the surface normals (Fig. 3(b) and (c)). The elastic deformations of tooth surfaces of the pinion and the gear are obtained as the result of applying the torque to the gear. The illustrations of Fig. 3(b) and (c) are based on discrete presentations of the contacting surfaces.
Fig. 3. Illustration of: (a) a single configuration; (b) and (c) discrete presentations of contacting surfaces and surface normals N1 and N2.
(iv) Fig. 4 shows schematically the set of configurations in 2D space. The location of each configuration (before the elastic deformation will be applied) is determined by TCA.
Fig. 4. Illustration of set of models for simulation of meshing of a loaded gear drive.
4.2. Application of finite element analysis for determination of function of transmission errors of a loaded gear drive
The described procedure is applicable for any type of a gear drive. The following is the description of the required steps:
(i) The machine-tool settings applied for generation are known ahead, and then the pinion and gear tooth surfaces (including the fillet) may be determined analytically.
(ii) Related angular positions are determined by (a) applying of TCA for Nf configurations (Nf = 8–16), and (b) observing the relation
(iii) A preprocessor is applied for generation of Nf models with the conditions: (a) the pinion is fully constrained to position , and (b) the gear has a rigid surface that can rotate about the gear’s axis (Fig. 5). Prescribed torque is applied to this surface.
(vi) The total function of transmission errors for a loaded gear drive is obtained considering: (i) the error caused due to the mismatched of generating surfaces, and (ii) the elastic approach .
5. Numerical examples
A helical gear drive with design parameters given in Table 1 is designed. The following conditions of meshing and contact of the drive are considered:
(1) The gear and pinion rack-cutters are provided with a straight-line and parabolic profiles as cross-section profiles, respectively, for generation of the gear and the pinion. Mismatched rack-cutter profiles yield the so-called profile crowning.