Second, the maximum developed torque is independent of the rotor resistance.First, the slip at maximum developed torque, s mt, is directly proportionate to the rotor resistance, R r.Several observations can be made from equations 8 and 9. Referring to the equivalent circuit of the stator and applying voltage division, the Thevenin’s voltage can be calculated as: The Thevenin’s voltage, V 1a, is equal to E 1 in figure 2.įIGURE 2 Induction motor stator equivalent circuit.įIGURE 3 Thevenin equivalent of the induction motor stator. Figure 3 shows a Thevenin’s equivalent of the stator. We can simplify our calculations by replacing it with a Thevenin’s equivalent so that we do not have parallel branches in the equivalent circuit. S2CID 121743087 – via IoP journals.FIGURE 1 Induction motor torque-speed characteristic curves.įigure 2 showed the equivalent circuit for the stator. "Simple 'log formulae' for pendulum motion valid for any amplitude". "A New and Wonderful Pendulum Period Equation" (PDF). Suppes, Patrick (December 2008), "Approximations for the period of the simple pendulum based on the arithmetic-geometric mean" (PDF), Am. "The pendulum - Rich physics from a simple system". , Part 4, Definition 3, translated July 2007 by Ian Bruce ^ defined by Christiaan Huygens: Huygens, Christian (1673).The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian mechanics. Paul Appell pointed out a physical interpretation of the imaginary period: if θ 0 is the maximum angle of one pendulum and 180° − θ 0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.Ĭoupled pendula Two identical simple pendulums coupled via a spring connecting the bobs.Ĭoupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in a supporting structure (such as a tabletop). The real period is, of course, the time it takes the pendulum to go through one full cycle. The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. Physical interpretation of the imaginary period Where θ 0 is the maximum angle of oscillation (with respect to the vertical) and K( k) is the complete elliptic integral of the first kind. The differential equation which represents the motion of a simple pendulum isĭ d t d θ d t = d d t 2 g ℓ ( cos θ − cos θ 0 ), d 2 θ d t 2 = 1 2 − 2 g ℓ sin θ 2 g ℓ ( cos θ − cos θ 0 ) d θ d t = 1 2 − 2 g ℓ sin θ 2 g ℓ ( cos θ − cos θ 0 ) 2 g ℓ ( cos θ − cos θ 0 ) = − g ℓ sin θ, d 2 θ d t 2 + g ℓ sin θ = 0, The motion does not lose energy to friction or air resistance.the bob does not trace an ellipse but an arc. Motion occurs only in two dimensions, i.e.The rod or cord on which the bob swings is massless, inextensible and always remains taut.Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.Ī simple gravity pendulum is an idealized mathematical model of a real pendulum. The mathematics of pendulums are in general quite complicated. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. 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 toward the equilibrium position. A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity.
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