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An inverted pendulum in which the pivot is oscillated rapidly up and down can be stable in the inverted position. This is called Kapitza's pendulum, after Russian physicist Pyotr Kapitza who first analysed it. The equation of motion for a pendulum connected to a massless, oscillating base is derived the same way as with the pendulum on the cart.
In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics. [41] A classic example along these lines is the inverted pendulum. [42] Other examples are
Kapitza noted that a pendulum clock with a vibrating pendulum suspension always goes faster than a clock with a fixed suspension. [11] Walking is defined by an 'inverted pendulum' gait in which the body vaults over the stiff limb or limbs with each step. Increased stability during walking might be related to stability of Kapitza's pendulum.
A pendulum is a body suspended from a fixed support such that it freely swings back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position.
Pendulum. Inverted pendulum; Double pendulum; Foucault pendulum; Spherical pendulum; Kinematics; Equation of motion; Dynamics (mechanics) Classical mechanics; Isolated physical system. Lagrangian mechanics; Hamiltonian mechanics; Routhian mechanics; Hamilton-Jacobi theory; Appell's equation of motion; Udwadia–Kalaba equation; Celestial ...
Rotational Inverted Pendulum: Classic pedagogical example of application of control theory. The Furuta pendulum, or rotational inverted pendulum, consists of a driven arm which rotates in the horizontal plane and a pendulum attached to that arm which is free to rotate in the vertical plane.
Since the system is invariant under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards: [1] = The region close to the pivot is singular, since is close to zero and the equations of motion require dividing by . As such, special techniques must be used to rigorously analyze these cases.
The phase portrait of the pendulum equation x ″ + sin x = 0.The highlighted curve shows the heteroclinic orbit from (x, x′) = (–π, 0) to (x, x′) = (π, 0).This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.