Physical pendulum

physical pendulum - motion equation

Physical pendulum is built with rigid body. One of rigid body ends is fixed to the ceiling. Rigid body is able to rotate around axis which is placed exactly in place where rigid’s end is fixed. Rotation axis is perpendicular to the plane of drawing. Rigid body has mass m and inertia I. Rigid body length is 2∙l. Note that inertia I is known for axis of rotations. If physical pendulum is in equilibrium position then it is not moving. In equilibrium position gravity force is balanced by rigid body’s reaction force. In a certain moment pendulum was deflected from its equilibrium and was inclined from vertical position by angle α. Pendulum is under gravity field, so gravity force works on it. Remember that physical pendulum is rigid body so gravity force is placed on rigid body’s gravity center C.

Physical pendulum – motion equation

Superposition method – currents calculation in electrical circuit

Electrical DC circuit - superposition method example 1.

Application of superposition method for simple electrical circuit. Electrical circuit is built from one voltage source and one current source. Main circuit will be divided into two sub-circuits because there are two sources. In the first sub-circuit current source will be the only extortion. In the second sub-circuit voltage source will be the only extortion.

Superposition method in electrical circuit – example 1

Mathematical pendulum

mathematical pendulum - motion equation

Mathematical pendulum is built with massless rope with length l. One of rope’s ends is fixed to the ceiling. To the second end of rope is fixed to the material point with mass m. Material point is then hanged to the rope. When mathematical pendulum is in equilibrium material point is not moving. In equilibrium position gravity force is balanced by rope’s tension force. In a certain moment mathematical pendulum was deflected from its equilibrium and was inclined from vertical position by angle α. Mathematical pendulum is under gravity field, so gravity force works on it.

Mathematical pendulum – motion equation

State space representation

state space representation - RLC circuit example 1.
In control theory and automation engineering dynamic models of system which are often created. It is a lot of cheaper and faster to simulate something than prepare real experiment. To simulate an object behaviour the state space representation is applied. State space representation allows to examine object features when it is under influence of various extortions. Set of examples about mentioned method are placed below.

State space representation for various systems