Theoretical analysis of the vibration inducing processes of the "workpiece-mandrel" assemblies during gear-tooth milling

. The theoretical study of the acoustic characteristics of the "workpiece-mandrel" technological system during gear-teeth milling has shown that for calculating the sound pressure levels, it is necessary to determine the vibration velocities of the workpieces and mandrels at their natural vibration frequencies. The paper presents the results of the theoretical studies of the vibration velocities of the processed gear wheels and mandrels during gear-tooth milling by the rolling and copying method.


Introduction
The tooth-wheel cutting is carried out by copying and rolling methods. Both operational methods rotate the workpiece by one tooth. For cutting teeth process, the "workpiece-cutting tool" system is dispatched by movements: the main movement is the rotation of the cutter; supply motion is the relative linear movement of the cutting tool along the length of the tooth can be improved as a machine platen using a dividing head installed on it with a workpiece being processed or a spindle unit. The processed tooth-wheels are mounted on mandrels with a cylindrical shape [1][2][3][4][5].

Article
With setting the force action, the natural vibration frequencies, the dependence of the maximum value of the real part of the vibration velocity is reduced to the following form: the tooth-wheel material, kg / m 2 ; h is the length of the tooth, m; k is a coefficient that determines the natural frequencies of vibrations; M is the mass of the tooth-wheel, kg. The components of the cutting forces during gear-tooth milling are determined according to the standards of cutting conditions = 10 * * * * * * where t* is the cut depth, mm; Sz is a supply per cutter tooth, mm / tooth; B is the geartooth milling width, mm; Ср, х*, у*, n*, g*, w* are coefficients given according to [6]; K_mp is a correction factor that takes into account the quality of the processed material.
The ratio between the components of the cutting forces as Рz, Ру, Рх is determined from the data [6,7] as follows: where φ is the main angle in the cutter plan.
Using the paper works [6][7][8][9], the differential equations of the mandrel vibrations of a geartooth milling machine as a cantilever-fixed rod of a cylindrical shape for the conditions of constancy of the application cutting force coordinate and the ratio of the corresponding Рz and Ру are determined as follows: The vibrations differential equations in the direction of the coordinate axes are completely identical and they differ by a factor with a cutting force (Ру = 0,4 Pz). Therefore, the solutions of the equations with respect to the real part of the modulus of the vibration velocities are given for the OZ axis. The rms-root mean square value of the vibration velocity, which the sound pressure levels are calculated, it is defined as The vibration calculation speed of the processed tooth-wheel with the ratio D/l > 2 is based on its representation as a plate, and the dependence of the vibration speed in a complex form [9,10] = ( ) where w is the circular frequency of the power disturbance, p / s; m is the distributed mass, kg / m 2 ; is a coefficient of the vibrational energy losses.
The differential equations of the tooth-wheels oscillation with the single-sided support are obtained for two options for gear-teeth milling depending on the ratio of the tooth length and the pitch circle diameter. For the ratio l/D > 2, it is used the cutting force moves along the part of the workpiece with the supply rate . Then the differential equation of the workpiece vibrations and their solutions due to the modulus of the real part of the vibration velocities with the coefficient of vibrational energy loss, are obtained in the following form:  For cutting process using the copying method, the mandrels are circular two supported bars. The vibrations` equations of the wheel mandrels and disk cutters for cutting teeth use the copying method in Figure 1. In this cutting method, the options for the mandrel supports should be considered as articulated, rigid and elastic-dissipative. In addition, the specificity of the force action should be given in the differential equations. For the cutter mandrel, the coordinate of the force action is constant. Similarly, the force action on the gear-tooth wheel mandrel is set for a tooth length of up to 50 mm. For tooth lengths of 80, 125, 180, 220, 300, 350, 560 and 1350 mm, the force action is specified as moving along the workpiece with the supply rate.
Then the differential equation of the mandrel vibrations of the cutting tool and the geartooth wheel is determined as follows using the conditions of the force action with a constant application coordinate: It is given for the hinge conditions For the conditions of the technological load moving along the processed tooth, the differential equations of the mandrel vibrations are determined as follows: It is given for the hinged fastening