Computational Methods in Rotor Dynamics: Transfer Matrix Approach

The transfer matrix method is a fundamental approach in rotor dynamics for analyzing vibration characteristics of multi-disk rotor systems. Its core concept involves decomposing complex rotor systems into simple elements and using transfer matrices to describe the propagation of state variables between adjacent elements.

Transfer matrix coefficient calculation typically follows these implementation steps: System Discretization: Divide the rotor system into characteristic elements (shaft segments, mass disks, bearings) based on physical properties. Each element's state vector contains displacement, rotation angle, bending moment, and shear force parameters. In code implementation, this involves creating element objects with properties like material type and geometric dimensions. Element Transfer Matrix Construction: Derive transfer matrices according to element types (e.g., uniform shaft segments, elastic bearings). For uniform shafts, the transfer matrix can be obtained from beam vibration differential equations, incorporating material parameters (Young's modulus, density) and geometric parameters (length, cross-sectional moment of inertia). Programming typically uses matrix exponential functions or analytical solutions of differential equations. Global Matrix Assembly: Multiply individual element transfer matrices in topological order to form the system's overall transfer matrix. Boundary conditions (free/fixed ends) affect initial/final matrix forms - this requires conditional matrix modifications in code implementation. Eigenequation Solution: Establish characteristic equations through the global transfer matrix and boundary conditions to solve for eigenvalues (natural frequencies) and eigenvectors (mode shapes). Numerical methods like the Riccati transfer matrix method can be implemented to handle ill-conditioned problems using matrix transformation techniques.

Implementation considerations: Asymmetric bearings or gyroscopic effects introduce complex terms, requiring matrix dimension expansion through complex number arithmetic in programming. Numerical instability in high-frequency calculations may be addressed using orthogonal transformations or segmented optimization strategies in algorithmic design.