Finding natural frequency of a simple pendulum by energy method?

Answer 1

Energy methodIt is possible to calculate the frequency of different mode shapes of system manually by the energy method. For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system. For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method. Rayleigh's principle states:

"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than - or equal to - the fundamental frequency ωn."

For an assumed mode shape , of a structural system with mass, M; stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x):

then, as above:

Answer 2

The natural frequency of a simple pendulum can be found using the equation: ( f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} ), where ( f ) is the frequency, ( g ) is the acceleration due to gravity, and ( L ) is the length of the pendulum. This equation is derived from the conservation of energy principle, where the potential energy at the maximum height is equal to the kinetic energy at the equilibrium position.