When the gear and the shaft are connected by spline, or the analysis object is the gear shaft, it can be regarded as the rigid connection between the gear node and the shaft node. Because the two nodes or degrees of freedom of the rigid connection have the same displacement, velocity, acceleration and external load, and do not care about the internal load of the rigid connection, the two nodes or degrees of freedom can be regarded as one, and the corresponding elements of the mass matrix, damping matrix, stiffness matrix and load vector are superimposed on each other. Taking the first and third degrees of freedom, the second and fifth degrees of freedom as examples (assuming that the number of degrees of freedom of the system is 5), the matrix operation of adding rigid connections between substructures is explained.
In the formula, P1 changes the elements of (1,3) and (2,5) to 1 on the basis of 5 × 5 identity matrix; P2 cuts out the third and fifth lines on the basis of 5 × 5 identity matrix. P1m means to add the third row of M matrix to the first row, and the fifth row to the second row, which is the row transformation; MPT1 means the corresponding column transformation. P2m means to delete the third and fifth rows of M matrix, which is row transformation; t2mp means corresponding column transformation. Therefore, MPPT represents the coefficient matrix of the first and third degrees of freedom, the second and fifth degrees of freedom with rigid connections. The third, fifth row and column are deleted because the state values of the third and fifth degrees of freedom are equal to the state values of the first and second degrees of freedom respectively after rigid connection. In order to save hard disk storage space, the rows and columns corresponding to the repeated degrees of freedom and their coefficient matrix are deleted.
Before adding rigid connection, the total degree of freedom of cutting transmission system is 386. After adding rigid connection, the total degree of freedom of cutting transmission system is 301. The nodes that are rigidly connected include: 4 pairs of gear nodes and shaft nodes (6 degrees of freedom for each node), 10 pairs of shaft nodes and box interface nodes (6 degrees of freedom for each node), 1 pair of planetary carrier nodes of the first stage planetary gear train and sun gear nodes of the second stage planetary gear train (1 degree of freedom for each node), and a total of 85 repeated degrees of freedom are deleted. It can also be seen that the work of merging 170 (deleting 85) degrees of freedom is rather tedious, and it will be very difficult to operate manually without matrix operation.