Free Vibration Analysis of Cantilever Beams with Variable Cross-Sections: Experimental Measurement and Analytical Comparison

Abstract

Free vibration of cantilever beams with non-uniform cross-sections is a fundamental problem in structural dynamics with direct relevance to engineering applications including turbine blades, robotic arm links, aircraft wings, crane booms, bridge girders, and micro-electromechanical systems (MEMS). The uniform Euler-Bernoulli cantilever beam possesses well-known closed-form natural frequency solutions; however, beams with linearly or parabolically varying cross-sections require series solutions or numerical and approximate analytical methods for accurate natural frequency prediction. This study experimentally measures the first three natural frequencies of four cantilever beam configurations  (A) uniform rectangular, (B) linearly width-tapered, (C) linearly depth-tapered, and (D) doubly-tapered (width and depth simultaneously)  machined from mild steel (IS 2062) and aluminium alloy Al 6061 to identical root dimensions. An accelerometer-based impact hammer test provided Frequency Response Functions (FRFs) from which natural frequencies were extracted by peak-picking with phase verification. Results are compared with analytical predictions obtained using the Rayleigh-Ritz method employing polynomial trial functions satisfying cantilever boundary conditions. Experimental and analytical first natural frequencies agreed within 3.9 percent for all configurations. The depth-tapered beam (Configuration C) exhibited the highest fundamental frequency-to-mass ratio among all configurations (0.066 Hz/g for steel), while the doubly-tapered beam (D) achieved the highest absolute fundamental frequency (72.3 Hz steel) and frequency-to-mass ratio (0.115 Hz/g). These results establish tapered cantilever beams as mass-efficient design solutions for resonance-avoidance in lightweight structural applications.