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The definition of a unit includes determining node coordinates and deformation coordinates. The node coordinates are used to describe the position of the element in the generalized coordinate space, consisting of Cartesian coordinates and parameters representing the direction. Deformation coordinates are used to describe the elastic deformation of a unit or to describe large relative displacements or rotations between units, which should have clear physical meaning. The deformation coordinates are nonlinear functions of the node coordinates. The internal force of the unit is generally determined by the deformation coordinates. A pair of meshing gears is defined as a gear unit. The center of the gear is taken as the two nodes p, q of the unit. The gear unit defines eight node coordinates, namely: xkgear=[xp,yp,p,xq,yq,q,rp,rq]T(1) where p and q are the corners of the driving gear and the driven gear; xp, Yp, xq, yq are Cartesian coordinates of nodes p and q; rp and rq are the base circle radii of the driving gear and the driven gear.
For parallel shaft gearing, define 3 deformation coordinates. The first deformation coordinate is a deformation in the direction of the two nodes, including an error e1 due to gear manufacturing and installation. The second deformation coordinate is the rotation angle of the gear center line. The third deformation coordinate is a torsional deformation in the direction of the meshing lines of the two gears. That is: 1 = D1 = (xp-xq) 2 (yp-yq) 2-l0 e1 (2) 2 = D2 = arctan [(yq-yp) / (xq-xp)] - 0 (3) 3 = D3 =rpp-rqq e(p)(4) where l0 is the standard center distance of the gear; 0 is the initial rotation angle of the gear center connection; e(p) is the gear transmission error.
Gear corresponding to the internal force is warped coordinates: fm1 = K11 C11 (5) fm2 = K22 C22 (6) fm3 = K33 C33 (7) where, K1, C1 node along two directions stiffness and damping.
K2 and C2 are the stiffness and damping of the rotational deformation of the driven gear around the driving gear; K3 and C3 are the stiffness and damping along the meshing line of the two gears. Topology Description of Multibody Systems Gear multibody systems can be divided into a series of different units, the number of units depending on the complexity of the system and the required analytical accuracy.
Joining between different units by sharing all or part of the node coordinates. For example, when two units are rigidly coupled, the common nodes of the unit have the same movement coordinates. When the two units are hinged, the common nodes of the unit only have the same movement coordinates. The topological space Xk of the multi-body system node coordinates is the sum (and) sum of the coordinate subspaces of each unit node. k represents the kth unit body. The topological space Ek of the deformation coordinate of the multi-body system is the direct sum of the topological subspaces of the deformation coordinates of each unit. That is: X=Xk, E=Ek(8) by superimposing the deformation coordinates of each unit to obtain the deformation coordinate vector of the whole system, namely: XE,=D(x)(9) according to the constraint of the multi-body system selection generalized coordinates, topology multibody system node coordinate space and topological deformation space coordinates may be further divided into the following three separately sub-space: X = XoXcXm, E = Eo Em Ec (10) wherein the subscript o refers to a constant Node coordinates or deformation coordinates; c refers to independent node coordinates or deformation coordinates; m refers to independent node coordinates or deformation coordinates. The input motion of the system is determined by independent node coordinates and deformation coordinates.
Motion Analysis If independent coordinates (xm,m) are given, any node coordinates and deformation coordinates of the multibody system can be determined. This means that any node coordinates and deformation coordinates of a multibody system are functions of independent coordinates, ie: x=Fx(xm,m),=F(xm,m)(11) explicit functions, Fx and Fz, called geometry Transfer Function. This function cannot be obtained directly due to nonlinearity. However, the partial differential of the function for independent coordinates can be obtained by the partial differential of the deformation coordinates D (Equation 9) for the node coordinates, and then the velocity vector and acceleration vector under different independent coordinates can be derived from equation (11): x = DFxq, = DFq (12) x = qTD2Fxq DFxq, = qTD2Fq DFq (13) wherein DFx = [FxmFxm], DF = [FxmFm], q = xm, D2Fx = 2Fx2xm2Fxmxm2Fxm2Fx2m, D2F = 2F2xm2Fmxm2Fxm2F2m, q = xmm
The generalized mass matrix M, the generalized external force vector fx and the generalized internal force vector fin of the dynamic analysis multi-body system are formed by the mass matrix, the external force vector and the internal force vector of each unit. According to the principle of virtual work, the equation of motion of the system can be written as: [TxT]M(x,)x -fxfin=0(14) If the motion independent coordinate vector is divided into a dynamic independent coordinate vector qd and a constrained coordinate vector qr, Then qT=(qd, qr)T. In the above formula, the virtual velocity (xT, T) is derived from equation (11) as: x = dFxqd, = dFqd (15) where dFx = Fxqd, and dF = Fqd substituting equations (13) and (15) into equation (14) , simplified system equations of motion can be derived as: [(dF) TMdF] qd = (dF) T [-M (qTD2Fq drFqr) f] (16) where fT = (fxT, -fTin), dF = [FxqdFqd ]TdrF=[FxqrFqr]T (16) can be further abbreviated as: Moqd=Q(17) where Mo=(dF)TMdFQ=(dF)T[-M(qTD2Fq drFqr) f]
Computational examples This study has developed the finite element theory and new gear units into computer software. This software can be used for motion and dynamic analysis of multi-body systems. The user only needs to enter keywords describing the multi-body system, and does not need to be familiar with the software structure and the finite element theory. In order to verify the accuracy and reliability of the defined gear unit and finite element software, the simulation results of the single pair of gears in the direction of rotation were compared (see Figure 2). The pair of gears have the same number of teeth 56, a modulus of 3 mm, a tooth width of 8 mm, and a pressure angle of 20. a. The test gear has high precision, which can neglect the influence of manufacturing error on the measurement of the vibration energy peak energy and the applied acceleration. b shows the simulation results. It can be found that the acceleration frequency and amplitude of the simulation and test are very close. As a calculation example, the gear planetary mechanism shown is analyzed using a developed method. The mechanism consists of gears and rods. The rod 1 is a drive element. The organization is divided into 3 units: 1 gear unit and 2 beam units. It becomes engaged 2.6 108N / m and 4.4 108N / m stiffness variation between the gears 1,2.
The change in stiffness is one of the important causes of gear vibration and noise. Through the motion analysis of the illustrated planetary mechanism of the gear, the displacement, velocity, acceleration and nodal force of each node, internal force and deformation of each unit can be obtained. The effect of the stiffness of the rod 1 on the acceleration of the rotational vibration is shown.
Conclusion The new gear unit based on a special finite element theory can consider the time-varying stiffness of the gear, the gear manufacturing and assembly error, and the quality imbalance. The finite element theory and new gear units have evolved into computer software. By comparing the simulation results with the experimental results, the accuracy and reliability of the defined gear unit and finite element software are verified. A multi-body system consisting of gears, rods, shafts and bearings is modeled by the development of gear units and previously developed units. The software can be used to perform motion and dynamic analysis on a multi-body system with gears, and it is convenient to obtain the displacement, velocity, acceleration and nodal force of each node, internal force and deformation of each unit.