Complex Exponential Method for Modal Parameter Identification

The Complex Exponential Method operates on the principle that free vibration responses or impulse response functions can be expressed as summations of complex exponential terms, utilizing linear estimation techniques to determine unknown parameters. The fundamental concept originates from the modal superposition principle in vibration differential equations, establishing analytical relationships between dynamic responses and modal parameters. By fitting impulse response functions, complete modal parameters including natural frequencies, damping ratios, and mode shapes can be obtained with excellent fitting precision. The core algorithm involves constructing Prony polynomials where the Z-transform factors contain the frequencies to be identified. The zeros of these polynomials are designed to equal the values of the Z-transform factors, thereby transforming the problem of solving for Z-transform factors into determining Prony polynomial coefficients. This mathematical transformation enables efficient numerical implementation through linear algebra solutions. Below is a MATLAB implementation for solving Prony polynomial coefficients using matrix operations: The code typically involves: 1. Sampling the impulse response data and constructing a Hankel matrix from measurement data 2. Performing singular value decomposition (SVD) to determine system order 3. Solving the characteristic equation to obtain polynomial coefficients 4. Calculating complex roots to extract frequency and damping parameters 5. Reconstructing mode shapes through least-squares fitting Key MATLAB functions employed include: - hankel() for matrix construction - svd() for system order identification - roots() or eig() for solving characteristic equations - polyval() for polynomial evaluation - backslash operator () for linear system solutions This implementation provides robust numerical stability and computational efficiency for practical modal parameter identification applications in structural dynamics.