Free and forced vibrations of elastically restrained cantilever with lumped oscillator

Abstract: The efficiency assessment of cantilever-based energy harvesters relies on vibrational analysis, which necessitates modifications aimed at enhancing efficiency. These modifications involve manipulating the fundamental frequency to lower values and encompassing a wider range of resonances within a specified bandwidth. Consequently, this paper introduces an original analytical-numerical exploration into the vibratory response of a cantilever with a novel boundary condition involving an elastically restrained oscillator-spring arrangement. At the microbeam's tip, an oscillator is elastically confined by a linear spring, resulting in a novel set of coupled governing equations and a distinct shearing boundary condition. Microbeam equations is derived from the modified couple stress theory to capture size dependency. During free vibration analysis, a previously unreported characteristic equation is derived. This nonlinear transcendental equation is numerically solved utilizing root-solver algorithms, such as those available in MATLAB. Significantly, it is discovered that the inclusion of a lumped oscillator with an elastic support induces a minimal (new) natural frequency. Applying the extended Hamilton's principle, the effect of the lumped oscillator emerges both on the governing equations of motion and boundary conditions of the microbeam. Novelty of the paper focuses on the both characteristic equation and transmissibility by adopting the Galerkin’s modal decomposition technique. This finding carries vital implications as the efficiency of cantilever-based energy harvesters is directly contingent upon the resonance frequency. Notably, the oscillator mass and spring constant are two parameters that directly influence the vibratory response of the microbeam. In the context of forced vibrations, harmonic base excitation is considered as the input excitation, and the mechanical frequency response function is provided. The proposed system offers two distinct advantages for energy harvester systems: the creation of minimal resonance at lower values and the potential to manipulate the system's resonance toward a desired frequency spectrum. Article Highlights: Modifying the boundary conditions of a cantilever beam with lumped-parameter system, can significantly change the behavior of the vibratory response. The boundary condition directly impact the resonance frequencies; which influences the maximum amount of harvestable voltage in vibration-based energy harvesters. Spring constant and mass of the lumped oscillator, are the key factors to alter the vibratory behavior and bandwidth of frequencies. Optimizing such mentioned parameters can help reaching to the maximum harvesting of energy.