Using variable modulus to modify a rack cutter and generate a gear pair

In this paper, two normal imaginary helical rack cutters were first established. One of these cutters is a skewed-rack cutter with an asymmetrical straight edge. The other is a rack cutter with an asymmetric parabolic profile. Second, the gear’s tooth surface of the asymmetric parabolic rack cutter is modified to be barrel-shaped based on a variable modulus. The tooth thickness of the gear is gradually reduced along the face width of the tooth from the middle of the tooth surface. Then the coordinate relationship between the gears’ blanks and the imaginary helical rack cutters was established. Through the differential geometry, crowned and uncrowned helical gear pairs were generated. Because of human factors, when the gear pair is installed, it is easy to cause the gear pair edge contact. It is necessary to add artificial assembly error settings through the tooth contact analysis to investigate the kinematic errors and contact conditions of the crowned and uncrowned helical gear pair. The mathematical models and analysis methods proposed for the crowned imaginary rack cutter using variable modulus should be useful for the design and production of double crowned helical gears with asymmetric parabolic teeth.

Using a planar rack cutter with variable modulus to generate a spherical gear pair

In this study, an imaginary planar rack cutter with variable modulus and discrete conical teeth was used to generate a spherical gear pair with double degree of freedom. First, a geometric method was used to design the mathematical model of the imaginary planar rack cutter with variable modulus and discrete conical teeth. Next, the relationship of coordinate systems between the generating and generated surfaces was established. Then, the family of the imaginary planar rack cutter surfaces was obtained by homogeneous coordinate transformation matrix. Further, two equations of meshing between the generating and generated surfaces were determined by the two-parameter envelope theory. The mathematical model of spherical gear pair with variable modulus and discrete ring-involute teeth can be created by using the two equations of meshing and the family of the imaginary planar rack cutter surfaces. The mathematical models of the spherical gear pair with double degree of freedom and tooth contact analysis method were used to investigate the assembly errors of the gear pair in four cases.

Complex anti-synchronization of two indistinguishable chaotic complex nonlinear models

The principal target of this work is to introduce and examine a novel kind of complex synchronization. This sort may be called complex anti-synchronization. There are surprising properties of complex anti-synchronization that do not exist in the writing, for example, (1) this sort of synchronization can dissect just for complex nonlinear frameworks. (2) The complex anti-synchronization contains or connects two sorts of synchronizations (anti-synchronization and complete synchronization). Anti-synchronization happens between the real part of main framework and the imaginary part of the slave framework, although complete synchronization accomplishes between the real part of slave framework and the imaginary part of the main framework. (3) In complex anti-synchronization, the attractors of the essential and slave structures are moving symmetrical to each other with a similar structure. (4) The state variable of the standard framework synchronizes with an other state variable of the slave structure. An explanation of complex anti-synchronization is presented for two indistinguishable chaotic complex nonlinear frameworks. In view of the Lyapunov function, a plan is intended to accomplish complex anti-synchronization of disordered or chaotic attractors of these frameworks. The effectiveness of the obtained results is outlined by a reenactment illustration. Numerical outcomes are plotted to show state variable, modulus errors, phase errors and the development of the attractors of these chaotic frameworks after synchronization to demonstrate that complex anti-synchronization is accomplished.