Natural Vibration Characteristics of a Cantilever Beam Parametrically Excited by Pitch Motion

Natural vibration characteristics of a cantilever beam that is parametrically excited by pitch motion are theoretically investigated in this work. A partial differential equation governing the parametrically excited bending vibration of the beam is established by employing the generalized Hamiltonian principle. Normal modal functions of the bending vibration are obtained based on the method of separation of variables. It reveals that the dynamic pitch motion cannot change the modal shapes of the bending vibration of a uniform beam. Principal vibrations of the beam are studied by taking into account two cases of Formula: see text and Formula: see text, where Formula: see text and Formula: see text are the first and second principal bending stiffness, respectively. The bending stiffness is considered as the superposition of a constant stiffness with a small perturbation. With the aid of the Fourier series theory and the method of multiple scales, the principal vibrations of the parametrically excited system at non-resonant and resonant cases are analyzed by introducing a small natural parameter Formula: see text, while natural vibration characteristics of the beam are obtained and the orthogonal properties of modal functions are illustrated. Theoretical methods provided in this work are verified by using the finite element method, assumed modes method and direct numerical integrations to the governing equation based on a rectangular beam model. It reveals that theoretical results coincide with numerical results very well. Finally, the influences of design parameters including the beam height, beam width, beam length, slender ratio, aspect ratio, pitch amplitude, pitch frequency and pitch phase on natural frequencies of a harmonic pitching beam are discussed. It shows that the width, height, length, slender ratio, aspect ratio and pitch amplitude have dramatic influences on the natural frequencies of the pitching beam, while the pitch frequency and pitch phase have little effect on natural frequencies.