Natural Frequencies and Modal Analysis of Cantilever Beams - Wang Deya - Experiment 3

Natural frequency and modal analysis of cantilever beams constitute fundamental experiments in structural dynamics. Natural frequency refers to the inherent oscillation frequency of a structure during free vibration, while modes describe the specific vibration patterns at particular frequencies.

Natural frequency measurement typically involves exciting the cantilever beam and recording its response signals. By applying Fast Fourier Transform (FFT) analysis to the response data, the first several natural frequencies of the cantilever beam can be extracted. In code implementation, this would involve using FFT algorithms (e.g., numpy.fft.fft() in Python or fft() in MATLAB) to convert time-domain signals to frequency-domain spectra, followed by peak detection algorithms to identify dominant frequencies.

Modal analysis further investigates the vibration patterns at different frequencies. Mode shape diagrams visually represent deformation patterns under various modal orders. For instance, the first mode typically exhibits single bending vibration at the free end, while higher-order modes may display multiple nodes and anti-nodes. Computational methods like the Finite Element Method (FEM) can be implemented using eigenvalue solvers (e.g., scipy.linalg.eig() in Python) to calculate mode shapes from mass and stiffness matrices.

During experimentation, careful attention must be paid to boundary condition simulation to ensure the fixed-end constraints approximate ideal conditions. Additionally, excitation methods (such as impact hammer testing or shaker excitation) and sensor placement significantly affect measurement accuracy. Code implementations often include signal preprocessing routines (filtering, windowing functions) to improve signal-to-noise ratio before modal parameter extraction.

Mode shape visualization can be achieved through multi-point measurements or optical methods to obtain displacement data, combined with modal analysis software for graphical representation. This process not only validates theoretical calculations but also helps engineers optimize structural designs to avoid resonance issues. Modern implementations may involve Python libraries like Matplotlib or Plotly for dynamic visualization of mode shapes, incorporating animation functions to demonstrate vibration patterns over time.