Generally, the gear meshing stiffness is regarded as the segment constant, and the variable coefficient differential equation is transformed into the constant coefficient differential equation, and then solved by numerical iterative. Another method is to simplify the gear meshing stiffness into a sine or cosine function. The system vibration equation is regarded as a eu-type equation, and then numerical solution is performed. This simplification does not reflect the variation of the meshing stiffness of the gear well.
In this paper, the piecewise cubic polynomial is used to represent the meshing stiffness function of the double circular arc gear. Therefore, when solving the differential equation of motion of the gear, a new solving method, namely the power series method, is adopted. The solution of the differential equation of the gear motion is given in the form of a power series, with its finite term as the approximate solution. At this point, as long as the geometric parameters of the gear pair are calculated, the corresponding vibration solution can be obtained.
When the degree of coincidence is constant, the increase in the knives narrows the tooth width, and the induced mass of the gear decreases, thereby increasing the natural frequency of the gear. In addition, when the gear diameter is constant, the increase in the helix angle reduces the number of teeth. Therefore, at a certain rotational speed, the meshing frequency of the gear decreases. Thus, at the same critical speed ratio, different helix angles correspond to different speeds. At a certain speed, the dynamic load coefficient varies with the helix angle. At this time, the increase of the helix angle causes the resonance speed of the gear to rise greatly. Therefore, increasing the helix angle at a lower speed can greatly reduce the dynamic load factor of the gear.
From the perspective of reducing the dynamic load coefficient, the appropriate gear helix angle should be selected according to the working speed of the gear to avoid the resonance region of the gear and achieve the best effect of reducing the dynamic load of the gear.
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