MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.W , T� Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 141 Reading: • Class Handout: Modeling Part 1: Energy and Power Flow in Linear Systems Sec. 3. • Class Handout: Modeling Part 2: Summary of One-Port Primitive Elements 1 The Modeling of Rotational Systems. With the the modeling framework as we defined it in Lecture 13, we have seen that in each energy domain we need to define (a) Two power variables, an across variable, and a through variable. the product of these variables is power. (b) Two ideal sources, and across variable source, and a through variable source. (c) Three ideal modeling elements, two energy storage elements (a T-type element, and a A-Type element), and a dissipative (D-Type) element.) (d) A pair of interconnection laws. We now address modeling of rotational mechanical systems. (a) Definition of Power variables: In a rotational system we consider the motion of a system around an axis of rotation: Consider the rotary motion resulting from a force F applied at a radius r from the rotational axis 1copyright cD.Rowell 2008 14–1W , TW , T12A n g u l a r v e l o c i t i e s W a n d W c a n b ed i f f e r e n t a c r o s s a n e l e m e n t .T h e t o r q u e T t r a n s m i t t e d t h r o u g h a ne l e m e n t i s c o n s t a n t .12FrDxqWThe work done by the force F in moving an infinitesimal distance Δx is ΔW = F Δx = F rθ and the power P is dΔW dθ P = = F r = T Ω dt dt where T = F r is the applied torque (N.m), and Ω = dθ/ dt is the angular velocity (rad/s). We note that if T and Ω have the same sign, then P > 0 and power is flowing into the system or element that is being rotated. Similarly, if T and Ω have the opposite signs, then P < 0 and power is flowing from the system or element, in other words the system is doing work on the source. Note that the angular velo city Ω can be different across an element, but that torque T is transmitted through an element: We therefore define our power variables as torque T and angular velocity Ω, where T is chosen as the through variable • Ω is chosen as the across variable. • (b) Ideal Sources: With the choice of modeling variables we can define our pair of ideal sources The Angular Velocity Source: Ωs(t) By definition the angular velocity source is an across variable source.The ideal angular velocity source will maintain the rotational speed regardless of the torque it must generate to do so: 14–2TW0Tt o r q u eT o r q u e T i s i n d e p e n d e n to f a n g u l a r v e l o c i t y W a r r o w s h o w s d i r e c t i o no f t o r q u e .rWmWW0Tt o r q u ea n g u l a r v e l o c i t y W i s i n d e p e n d e n t o f t o r q u e T a r r o w s h o w s d i r e c t i o no f a s s u m e d a n g . v e l .d r o p .WWabW > WabThe Torque Source: Ts(t) By definition the torque source is a through variable source. The ideal torque source will maintain the applied torque regardless of the angular velocity it must generate to do so: (c) Ideal Modeling Elements: 1 The Moment of Inertia: Consider a mass element m rotating at a fixed radius R about the axis of rotation. The stored energy is 1 1 E = m(rΩ)2 = JΩ2 2 2 where J = mr2 is defined to be the mo-ment of inertia of the particle. For a collection of n mass particles mi at radii ri, i = 1, . . . , n, the moment of inertia is n2J = � miri . i=1 14–3rm a s s d m a tr a d i u s rWLWJ = m L1 22w h e r e m i s t h e m a s so f t h e r o drw h e r e m i s t h e m a s so f t h e d i s kJ = 122m rFor a continuous distribution of mass about the axis of rotation, the moment of inertia is J = � R r 2dm 0 Examples: A uniform rod of length L rotating about its center. A uniform disc with radius r rotating about its center. The elemental equation for the moment of inertia J is dΩJTJ = J dt We note that the energy stored in a rotating mass is E = JΩ2/2, that is it is a function of the across variable, defining the moment of inertia as an A-type element. As in the case of a translational mass element, the angular velocity drop associated with a rotary inertia J is always measured with respect to a non-accelerating reference frame. Elemental Impedance: By definition ΩJ (s) 1 ZJ = = TJ (s) Js from the elemental equation. 14–4WWs h a f t c a n t w i s t w i t hd i f f e r e n t a n g u l a rv e l o c i t i e s a t t h e t w o e n d s12JJ12WTc o i l s p r i n g(2) The Torsional Spring: W , qaKTabbTW , qW = W - WbaLet θa and θb be the angular displacements of the two ends from their rest positions. Hooke’s law for a torsional spring is T = K(θa − θb). where K is defined to be the torsional stiffness. Differentiation gives dT d(θa − θb) = K dt dt dT = KΩ dt where Ω = (θ˙ a − θ˙ b) is the angular velocity drop across the spring. Torsional stiffness may result from the material properties of a “long” shaft or may be intentional, for example in a coil (“hair”) spring in a mechanical watch. 14–5WaTbTW v i s c o u s f r i c t i o n a lc o n t a c tW = W - WbaaTW e x t e r n …
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