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? s Mechanisms perform useful tasks s Velocity affects the time required to perform a given task b machining of a part b deploy or retract a classroom projector s Power is the product of force and velocity s Velocity analysis is required as a precursor to acceleration analysis
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Definition of Velocity s In Chapter 2, we defined Displacement of a moving point P as Δ R P = R P ’ – R P s The average velocity of this point over a period Δ T is Δ R P / Δ T V P = s This is instantaneous absolute velocity
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Rotation of a Rigid Body s In chapter 2 b we saw that when a body translates, every point moves the same distance b When a body rotates, two arbitrarily chosen points do not undergo the same motion other words, the displacement difference, is b In other words, the displacement difference, Δ R PQ , is 0 for translation and not equal to 0 for rotation. b Magnitude of angular velocity vector, ω,
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Angular Velocity s Is a vector s Magnitude given by the above equation s Direction is the instantaneous axis of rotation s Sense of direction is obtained by the right-hand rule s 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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s 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 s V P = V Q + V PQ (velocity difference equation) b Or, V PQ = V P - V Q b 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 s V PQ = ω x R PQ is valid only if both points are attached to the same rigid body herefore, customary to write the rigid s Therefore, customary to write the rigid body (e.g., 2) in the post-script, i.e., s V P2Q2 = ω 2 x R P2Q2
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Relative velocities of points on a rigid body Relative velocity vectors: s V P2Q2 = ω 2 x R P2Q2 s V P2O2 = ω 2 x R P2O2 s V A2B2 = ω 2 x R A2B2 Corresponding magnitudes: s V P2Q2 = R P2Q2 ω 2 OR ω 2 = V P2Q2 /R P2Q2 s V P2O2 = R P2O2 ω 2 OR ω 2 = V P2O2 /R P2O2 s V A2B2 = R A2B2 ω 2 OR ω 2 = V A2B2 /R A2B2
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Instantaneous Center of Velocity s Concept of instantaneous velocity axis for a pair of rigid bodies that move with respect to each other s An axis exists which is common to both bodies and about which either body can be considered rotating with respect to the other s 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 s A.k.a. Velocity Pole (in the text book), Centro (elsewhere) s Definition: The instantaneous location of a pair of coincident points of two different bodies for which
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This note was uploaded on 08/08/2011 for the course MAE 162A taught by Professor Glenn during the Spring '08 term at UCLA.

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

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