Chapter 15 Intro - Chapter 15 Introduction The Kinematics...

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Click to edit Master subtitle style Chapter 15 Introduction The Kinematics of Rigid Bodies We will investigate the relations existing between the time, the positions, the velocities, and the accelerations of the various particles forming a rigid body.
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Applications Will involve primarily the analysis of mechanisms such as gears, connecting rods, and pin-connected linkages.
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5 types of motion Translation Rotation about a fixed axis General plane motion Motion about a fixed point General motion
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Click to edit Master subtitle style 1) Translation Includes rectilinear and curvilinear r B = r A + r B/A v B = v A + v B/A a B = a A + a B/A = 0 = 0 v B/A = 0 a B/A = 0 because rB/A is a constant in translation
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Translatio n
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Click to edit Master subtitle style 2) Rotation about a fixed axis Δs = (BP) Δθ = (r sin ) φ Δθ Δs/ Δt = (BP) Δθ/ Δt = (r sin φ) Δθ/ Δt Take limit as = Δt approaches 0 The velocity v of P is a vector perpendicular to the plane containing AA′ and r , and of magnitude v . θ φ v=ds/dt=r sin
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Click to edit Master subtitle style 2) Rotation about a fixed axis, cont. v = d r /dt = x r ω ω = ω k where ω = k is the axis of rotation. You may have to calculate the axis of rotation if it does not line up with the usual i, j, k axes. r is the position vector drawn from any point on the axis of rotation to point P. This is the same result we would get if we said: θ &
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Click to edit Master subtitle style 2) Rotation about a fixed axis, cont. a = α x r + ω x (ω x r) k is the axis of rotation. You may have to calculate the axis of rotation if it does not line up with the usual i, j, k axes. v = d r /dt = ω x r a = d v /dt = d( ω x r)/ dt =d ω /dt x r + ω x d r /dt let d ω /dt = α
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Click to edit Master subtitle style 2) Rotation about a fixed axis, cont. a = α x r + ω x (ω x r) = α k x r – ω2 r v = ω x r = ω k x r Rotation of a slab about a fixed axis a t = α k x r at = r α a n = - ω 2 r an = r ω2
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Example of rotation about a fixed axis
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Click to edit Master subtitle style 3) General Plane Motion May be considered as the sum of a translation and a rotation Any plane motion which is neither a rotation nor a translation is called general plane motion
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Click to edit Master subtitle style 3) General Plane Motion (cont)
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Click to edit Master subtitle style 3) General Plane Motion (cont) v B = v A + v B/A v B/A = ω k x r B/A
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Chapter 15 Intro - Chapter 15 Introduction The Kinematics...

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