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Mechanics

Mechanics. 8.3 Angular Momentum of a Rigid Body 刚体的角动量. Chapter 8. Planar Kinetics of Rigid Bodies. Section 17.3. G. A. 8.3 Angular Momentum of a Rigid Body. The angular momentum of a particle system about an arbitrary reference point A. L A =  L A i =  r i  m i v i.

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Mechanics

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  1. Mechanics 8.3 Angular Momentum of a Rigid Body 刚体的角动量 Chapter 8. Planar Kinetics of Rigid Bodies Section 17.3 Ch. 8 Planar kinetics of rigid bodies

  2. G A 8.3 Angular Momentum of a Rigid Body • The angular momentum of a particle system about an arbitrary reference point A LA= LAi=  ri  mivi ri: Position vector of mi relative to A vi: Velocity of mi relative to an inertial reference frame LG: angular momentum about the mass center G in CoM Ch. 8 Planar kinetics of rigid bodies

  3. 8.3 Angular Momentum of a Rigid Body • For rigid body with continuous mass distribution mi dm    • Choosing the mass center G of the body as the base point, the velocity of an arbitrary point B on the body is given by vB = vG +vB/G = vG + w×rB/G = vG +w×r´ v´=w×r´ : velocity of point B relative to mass center G = the velocity of B in CoM Ch. 8 Planar kinetics of rigid bodies

  4. p= vdm y´ v´=w  r´ r´ dm  x´ G r y x A 8.3 Angular Momentum of a Rigid Body For plane motion: • Assume the plane of motion is parallel to x-y plane • Angular velocity of the body: w = wk 1) Angular momentum about mass center r´ = x´i + y´j + z´k w = wk A (BC) = (A· C)B– (A ·B)C  Ch. 8 Planar kinetics of rigid bodies

  5. p= vdm y´ v´=w  r´ r´ dm  x´ G r y x A 8.3 Angular Momentum of a Rigid Body Rectangular components: Where The rotational inertia of the body about z´ axis Ch. 8 Planar kinetics of rigid bodies

  6. 8.3 Angular Momentum of a Rigid Body Ix´z´ and Iy´z´ are called the products of inertia(惯量积) Note: • Generally, LGxand LGyare not zero so that  and LGhave different directions • If the body is symmetric about the x´y´-plane, then the products of inertia vanish. In this case, LGxand LGyare zero, and the angular momentum of the body about its mass center reduces to LG = IGwk or LG = IGw In this chapter, we’ll assume that the body is symmetric!! Ch. 8 Planar kinetics of rigid bodies

  7. y´ mvG  G x´ rG  d A 8.3 Angular Momentum of a Rigid Body 2) Angular momentum about an arbitrary point A Consider a cross section of the body • Containing the mass center G • Parallel to the plane of motion • A lies in the same cross section LG = IGwk or LG = IGw d: the moment arm or Ch. 8 Planar kinetics of rigid bodies

  8. y´ mvG  G x´ rG A(fixed) 8.3 Angular Momentum of a Rigid Body 3) Rotation about a fixed axis The he body rotates about a fixed axis that passes through A and is perpendicular to the plane of motion. • Because G is a point on the body, mvGis perpendicular to rG and the magnitude of vG is rGw parallel axis theorem IA: The rotational inertia of the body about the fixed axis. IA is constant!! Ch. 8 Planar kinetics of rigid bodies

  9. 8.3 Angular Momentum of a Rigid Body Ch. 8 Planar kinetics of rigid bodies

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