M.Sc Thesis

M.Sc StudentRosenstock David
SubjectNon-Linear Vibrations of Circular Ring
DepartmentDepartment of Mechanical Engineering
Supervisor PROF. David Elata
Full Thesis textFull thesis text - English Version


In this work we rigorously analyze the in-plane nonlinear flexural vibration of a thin circular ring, and focus on the primary elliptic mode. We derive the kinematics of a thin circular ring using a polar coordinate system to describe the displacement field. Then we consider the Lagrangian function of the system, and derive the equations of motion using standard procedures of variational calculus (i.e. the Euler-Lagrange equations). We solve the equations of motions using the Galerkin method, and derive the mode and frequency of the free-vibrations of the ring. The nonlinear analysis shows that the free-vibrations frequency of the ring is affected by the amplitude of vibrations. Initially, as the amplitude of vibration is increased from zero, the free-vibrations frequency decreases. However, at some point, a further increase of vibration amplitude may cause an increase in the frequency of free-vibrations. Our investigation also considers the effect of the ring thickness, and we show that the initial reduction in the frequency of free-vibrations is more dominant for thicker rings. We conclude that the nonlinear mode of vibrations of the ring includes the linear mode, and is augmented by two additional components at double the frequency of the linear mode. The amplitude of these two components if proportional to the square of the amplitude of the linear components. We use several methods of numerical calculations to verify the validity of our results. Our insight is that the initial decrease in free-vibrations frequency is due to an increase in inertia in the system and not due to a ‘softening’ effect. Specifically, this added inertia is directly caused by the two additional components of the nonlinear mode of vibration.