ABSTRACT
This chapter examines variational aspects of the dynamics of beams and plates. It presents Hamilton's principle. The chapter focuses on beams, first deriving the equations of motion from William Rowan Hamilton's principle and considers both exact and approximate solutions for free vibrations. It examines the Rayleigh quotient, in a more general manner and develops strong supporting arguments for some physically inspired assertions made earlier concerning the Rayleigh and the Rayleigh-Ritz methods. The chapter presents the very powerful maximum theorem and "min-max" theorem from the calculus of variations. Those who have studied dynamics of particles and rigid bodies at the intermediate level should recall Hamilton's principle as employed for discrete systems. The chapter provides to develop the equations of motion and boundary conditions for classical plate theory, using Hamilton's principle. It re-examines the Rayleigh-Ritz method itself in a more formal manner in order to prepare the groundwork for assessing its validity.
