162A 3 - Velocity

162A 3 - Velocity - 162A Introduction to Mechanisms and...

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162A. Introduction to Mechanisms and Mechanical Systems Chapter 3: Velocity
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Why is velocity important? square6 Mechanisms perform useful tasks square6 Velocity affects the time required to perform a given task machining of a part box2 box2 deploy or retract a classroom projector square6 Power is the product of force and velocity square6 Velocity analysis is required as a precursor to acceleration analysis
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Definition of Velocity square6 In Chapter 2, we defined Displacement of a moving point P as Δ R P = R P ’ – R P The average velocity of square6 this point over a period Δ T is Δ R P / Δ T V P = square6 This is instantaneous absolute velocity
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Rotation of a Rigid Body square6 In chapter 2 box2 we saw that when a body translates, every point moves the same distance box2 When a body rotates, two arbitrarily chosen points do not undergo the same motion box2 In other words, the displacement difference, Δ R PQ , is 0 for translation and not equal to 0 for rotation. box2 Magnitude of angular velocity vector, ω,
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Angular Velocity square6 Is a vector square6 Magnitude given by the above equation square6 Direction is the instantaneous axis of rotation square6 Sense of direction is obtained by the right-hand rule square6 One rigid body has only one angular velocity. In other words, if one line on the body rotates by an angle Δθ , any other line on the body also rotates the same amount Δθ. Therefore, one can select any line on the rigid body as the reference line to find the angular velocity of the body
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Pictorially,
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square6 V PQ = ω x R PQ (V PQ is called the relative velocity) From , dividing both sides by Δ T and taking the limit, we get Two velocity equations which becomes square6 V P = V Q + V PQ (velocity difference equation) box2 Or, V PQ = V P - V Q box2 Relative velocity of point P w.r.t. point Q is “the absolute velocity of P minus absolute velocity of Q”
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More on relative velocity square6 V PQ = ω x R PQ is valid only if both points are attached to the same rigid body square6 Therefore, customary to write the rigid body (e.g., 2) in the post-script, i.e., square6 V P2Q2 = ω 2 x R P2Q2
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Relative velocities of points on a rigid body Relative velocity vectors: square6 V P2Q2 = ω 2 x R P2Q2 square6 V P2O2 = ω 2 x R P2O2 square6 V A2B2 = ω 2 x R A2B2 Corresponding magnitudes: square6 V P2Q2 = R P2Q2 ω 2 OR ω 2 = V P2Q2 /R P2Q2 square6 V P2O2 = R P2O2 ω 2 OR ω 2 = V P2O2 /R P2O2 square6 V A2B2 = R A2B2 ω 2 OR ω 2 = V A2B2 /R A2B2
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Instantaneous Center of Velocity square6 Concept of instantaneous velocity axis for a pair of rigid bodies that move with respect to each other square6 An axis exists which is common to both bodies and about which either body can be considered rotating with respect to the other square6 In this class, we are restricted to planar motion. Each axis is perpendicular the plane of motion….so the velocity axis reduces to a center of velocity
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Instantaneous Center of Velocity square6 A.k.a. Velocity Pole (in the text book), Centro (elsewhere) square6
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