Abstract: This paper addresses the issue of extensive matrix calculations involved in the numerical solution of the differential equations for torsional vibration dynamics of a transmission system. A method for developing a solver using hybrid programming with the Matcom environment integrated under VC++ is proposed. This solver is then used to calculate the eigenvalues ​​and eigenvectors of the torsional vibration of a vehicle's transmission system. The calculation results, along with test results and comparisons with Amesim, demonstrate the correctness and reliability of the conclusions. Compared to solving using VC++ programming, introducing Matcom improves program execution efficiency and shortens software development time. Keywords: Transmission system, torsional vibration, eigenvalues, Matcom Introduction Due to work requirements, the author developed an analysis software for calculating the torsional vibration characteristics of vehicles—the "Vehicle Torsional Vibration Analysis and Calculation System." This software is a visual transmission system torsional vibration modeling and analysis tool. Starting from the actual needs of vehicles and considering the characteristics of the transmission system, it utilizes object-oriented programming technology to provide a basis for calculating torsional vibration, stiffness and strength verification, coupling selection, and overall vehicle performance matching. The vehicle transmission system is actually a complex gear transmission system. The aforementioned vehicle transmission torsional vibration analysis system uses a lumped mass model to analyze the torsional vibration of the shaft system. According to the characteristics of torsional vibration, it is decomposed into several common subsystems in the transmission system, such as the engine module, the gearbox module (including the fixed-axis gear train and planetary gearbox module), the coupling module, and the coupler module. A complete vehicle transmission system torsional vibration analysis model can be built through a simple drag-and-drop operation. It also automatically assembles the constant coefficient terms in the dynamic differential equation and solves its natural frequencies and principal vibration modes in the computer, thereby completing the torsional vibration analysis of the vehicle transmission system. As the solver part of the software, its solution stability and reliability are the most important aspects. As is well known, when analyzing the modal problems of undamped vibration of discrete systems, the torsional vibration dynamic differential equation can be uniformly expressed as: where [J] is the moment of inertia matrix. The moment of inertia coefficient matrix generated when using the lumped mass model is generally a positive definite diagonal matrix. [K] is the stiffness coefficient matrix. The stiffness coefficient matrix generated by the simplified model of the branch shaft system of the typical transmission system is generally a positive definite or semi-positive definite sparse tridiagonal matrix. {0} — Generalized coordinate vector, which is the number of independent coordinates in the system. The stiffness matrix and moment of inertia matrix are square matrices of the same dimension. Numerical methods for solving time-invariant linear differential equations with constant coefficients have been extensively covered in relevant books, and will not be explained in detail here. It is conceivable that the numerical solution of this differential equation involves a large number of matrix operations, such as matrix arithmetic operations, extracting a column or row to form a new vector, etc. These operations do not have default corresponding functions in VC++. To implement these operations, a class library must be written to overload the operators. Furthermore, considering the sparsity of the stiffness matrix, memory management also requires corresponding code, all of which increase the programming burden. In fact, the Matlab software, familiar to engineers, can easily solve these problems. As a dedicated numerical computation software, Matlab has considerable advantages over C and other languages ​​in matrix operations. However, whether it is a secondary development program written in Matlab or an application written in VC++ that calls the Matlab engine, it cannot be separated from the Matlab environment. That is to say, users must install Matlab software, which is very uneconomical and completely unnecessary. This approach integrates Matcom with VC++ for hybrid programming, leveraging Matlab's powerful matrix computation capabilities to solve matrix problems in numerical calculations while also allowing for the deployment of standalone applications. Users no longer need to rely on a platform with Matlab installed , achieving complete independence from Matlab's limitations. Matcom, a component of MathWorks' Mathtools suite, is used for the automatic C++ conversion of Matlab M-files. In numerical computations, applications using C code can significantly reduce simulation time and memory requirements compared to those using Matlab M-files. Matcom can generate MEX files, creating dynamic link libraries for Excel and Visual Basic or for standalone C++ applications. Matcom also has a dedicated matrix algorithm library (Mattix). This library contains over 600 commonly used functions from Matlab, encapsulated in a separate DLL. It can be called by Windows applications or custom-developed programs, and the resulting executable file is quite small with high execution efficiency. It can be applied to engineering calculations in linear algebra, polynomials, signal processing, etc., and also features file input/output streams, graphical visualization capabilities, and powerful post-processing functions. The matrix library in Matcom is included in Matrix... The three libraries are MatrixXL and MatrixVB. MatrixXL is a matrix library for Microsoft Excel, and MatrixVB is a matrix library for Microsoft Visual Basic. These will not be discussed here. Below is a brief introduction to Matrix. The process of integrating into VC++, taking Microsoft Visual C++ 6.0 as an example: [Step 1] Add the compiled "v4501v.ib" file to the project, usually located in the Lib folder under the Matcom installation directory. [Step 2] Include the "matlib.h" header file, such as "#include <matlib.h>". "It is recommended to place the function in the header file of the class that calls Matcom, otherwise it may conflict with other libraries during use. [Step 3] Set the library function compilation directory, such as: Project Settings/C, C++ Preprocessor/Additional include directories. Add the directory ... \matcom45\lih in the box. [Step 4] Add the library initialization statement at the call location, such as int main() { initM(MATCOM.VERSION); // Enter your code here exitM(); return 0; } After the above steps, Matcom can be integrated into VC++, and matrix operations will become very simple. For example, define the following matrices A and B: Mm A, B; To add matrices A and B, simply use "A+B". For basic functions and operations, you can refer to Matcom's help. It is recommended to directly refer to Matlab's help, as the two have many similarities. It can be said that in matrix operations, Matcom can completely replace Matlab's functions. 2 Engineering Example Applications " After establishing the Matcom environment according to the above steps, you can write your own solution program. The following is a dynamic solution of the torsional vibration equivalent system of the engine and transmission system of the CA1150PK2L2T heavy-duty truck in reference [2]. The simplified diagram of the transmission system, the equivalent moment of inertia of the system, and the equivalent stiffness value are listed in detail in reference [2]. Due to space limitations, they will not be repeated here. As a comparison of the calculation results and accuracy, in order to reflect the reliability and accuracy of matrix calculation in the Matcom integrated environment, this paper only compares the results of reference [2]. [2] A comparison is made between the analysis results of the torsional vibration of the transmission system under the 3rd gear shift condition. The transmission system can be modeled in the "Vehicle Torsional Vibration Analysis and Calculation System". It is simplified in the paper as a multi-branch equivalent torsional vibration model consisting of 24 concentrated rotational inertia units and 23 shaft equivalent elastic elements. J1 to J10 are the engine parts, such as the crankshaft, piston, shock absorber, etc., which are simplified into 10 equivalent rotational inertia units. When modeling in the "Vehicle Torsional Vibration Analysis and Calculation System", the engine module and shaft segment torsional unit are mainly used. The model is shown in Figure 1. In addition, in order to verify the correctness of the calculation results, in addition to comparing the calculation results with those in the original article, the author also calculated the torsional vibration natural frequency of the vehicle under the third gear condition in the large-scale simulation software Amesim, which integrates mechanical, electrical and hydraulic systems. The results are listed in Table 1. 3 Comparison of Calculation Results and Error Analysis From the comparison in Table 1, it can be seen that the results are in good agreement with those in reference [2], but the second, third and fourth natural frequencies are not listed in reference [2]. In addition, it can be seen that the results are quite accurate compared with those calculated by Arnesim. The conclusions of the result comparison and error analysis are summarized as follows: ① For the first natural frequency, the three results are quite close. Reference [2] mentions that under the natural frequency of the single-node torsional vibration of the third gear transmission system of the vehicle, the node is on the through shaft. Now, the first principal vibration mode of the calculation results in this paper is plotted as shown in Figure 2. It can be seen that the node occurs between J17 and J19. According to reference [2], the node is indeed on the through shaft, indicating that the calculation of the first natural frequency and principal vibration mode is effective. ② The second and third natural frequencies are not listed in reference [2], but compared with Amesim, it can be found that the natural frequencies exist, and the second and third natural frequencies should be equal, that is, frequency repetition occurs. In this case, there are two modes of vibration for the same frequency. ③ In order to further verify the possibility of frequency repetition, the original model was modified by deleting the rotational inertia of J14, J19, and J20 to form two completely symmetrical branch shaft systems, that is, the influence of the front and rear axle drive shafts was removed, and only the symmetrical wheel assembly was directly connected. Due to its symmetry, the branch shaft system formed in this way should exhibit frequency repetition, as shown in Figure 3 after modeling in the "Vehicle Torsional Vibration Analysis and Calculation System". Similarly, calculations were performed in Amesim and the "Vehicle Torsional Vibration Analysis and Calculation System" software, and the calculation results are listed in Table 2. As can be seen from Table 2, the repeated frequencies still occur at the second and third natural frequencies. This is mainly because J14, J19, and J20 are smaller than the wheel parts J17, J18, J23, and J24, so they have little impact on their lower-order vibration modes. Compared with the original transmission system natural frequencies, they are basically only missing the 6th natural frequency. Therefore, it can be judged that the above-mentioned repeated frequency phenomenon occurs because the overall wheel part of the vehicle is the main factor influencing the natural frequency. ① As can be seen from Figure 3, the number of nodes considered in the reference [2] mainly includes the number of nodes on the main through shaft (i.e., the section from J17 to J19) before the through shaft of J19. The nodes of the first four vibration modes all occur between J17 and J19, indicating that the first four vibration modes are all single nodes. The reference [2] may have considered that the natural frequency values ​​are not much different and the nodes are basically located in the same area, so it did not list the natural frequency values ​​of the other single nodes. ⑤ The "Vehicle Torsional Vibration Analysis and Calculation System" uses the matrix iteration method to calculate the natural frequencies of this transmission system. When there are equal natural frequencies in the system, the matrix iteration method calculates these equal natural frequencies sequentially, and the principal coordinates corresponding to these natural frequencies are mutually orthogonal. As shown in Figure 3, when the second principal mode reaches its maximum value, the third principal mode remains essentially at its minimum value, i.e., at the lowest point. Discarding the errors, the result is essentially a pair of orthogonal mode shapes, further verifying the conclusion that the vehicle's transmission shaft system has equal natural frequencies. ⑥ The matrix iteration method is a numerical solution that sets an initial iteration value and repeats the iteration to obtain an accurate value within the allowable error range. The error of each iteration will be accumulated in the next result. Comparing the 6th node in reference [2] with the 9th natural frequency value calculated in Amesim, it can be seen that the error is relatively large. Experienced users can reduce this error caused by iteration by setting an initial iteration matrix and setting a correction value for the positive semi-definite matrix in the iteration matrix method, or by choosing other numerical calculation methods, such as the IAanczos method commonly used in large software. 4 Conclusion In summary, the solver for the "Vehicle Torsional Vibration Analysis and Calculation System" was developed using a hybrid programming approach integrating Matcom with VC++. It solved the eigenvalues ​​and eigenvectors of the differential equation for the torsional vibration dynamics of a vehicle's transmission system. The method is feasible and the results are accurate. In fact, it can be extended to solve other problems such as responses under external forces; the only difference lies in the algorithm. Furthermore, utilizing Matcom's powerful plotting functions, it is easy to draw mode shapes and other parameters. The solver for the "Vehicle Torsional Vibration Analysis and Calculation System," developed using hybrid programming with integrated Matcom technology, provides correct calculation results and meets the computational requirements of the system.