Rigid Body DynamicsCross ProductSlide 3Slide 4Slide 5Derivative of a Rotating VectorDerivative of Rotating MatrixProduct RuleSlide 9Dynamics of ParticlesKinematics of a ParticleMass, Momentum, and ForceMoment of MomentumSlide 14Slide 15Moment of Force (Torque)Slide 17Rotational InertiaSlide 19Slide 20Slide 21Slide 22Systems of ParticlesVelocity of Center of MassForce on a ParticleSlide 26Torque in a System of ParticlesSlide 28Internal ForcesKinematics of Rigid BodiesKinematics of a Rigid BodySlide 32Offset PositionOffset VelocityOffset AccelerationKinematics of an Offset PointDynamics of Rigid BodiesRigid BodiesRigid Body MassRigid Body Center of MassRigid Body Rotational InertiaDiagonalization of Rotational InertialDerivative of Rotational InertialDerivative of Angular MomentumNewton-Euler EquationsApplied Forces & TorquesProperties of Rigid BodiesRigid Body SimulationSlide 49Slide 50Rigid Body DynamicsCSE169: Computer AnimationInstructor: Steve RotenbergUCSD, Winter 2005Cross Product xyyxzxxzyzzyzyxzyxbababababababbbaaabakjibaCross Product zyxxzzzxyxzyzyyzxxxyyxzxxzyzzybbabacbabbacbababcbabababababa000bacbaCross Productzyxxyxzyzzyxzyxxzzzxyxzyzyyzxxbbbaaaaaacccbbabacbabbacbababc000000Cross Product000ˆˆ000xyxzyzzyxxyxzyzzyxaaaaaabbbaaaaaacccababaDerivative of a Rotating VectorLet’s say that vector r is rotating around the origin, maintaining a fixed distanceAt any instant, it has an angular velocity of ωrωrdtdrω rωDerivative of Rotating MatrixIf matrix A is a rigid 3x3 matrix rotating with angular velocity ωThis implies that the a, b, and c axes must be rotating around ωThe derivatives of each axis are ωxa, ωxb, and ωxc, and so the derivative of the entire matrix is:AωAωAˆdtdProduct Rule dtdcabcdtdbabcdtdadtabcddtdbabdtdadtabdThe product rule defines the derivative of productsProduct RuleIt can be extended to vector and matrix products as well dtddtddtddtddtddtddtddtddtdBABABAbabababababaDynamics of ParticlesKinematics of a Particleonaccelerati ity veloc position 22dtddtddtdxvaxvxMass, Momentum, and Forceforce momentum mass apfvpmdtdmmMoment of MomentumThe moment of momentum is a vectorAlso known as angular momentum (the two terms mean basically the same thing, but are used in slightly different situations)Angular momentum has parallel properties with linear momentumIn particular, like the linear momentum, angular momentum is conserved in a mechanical systemprL Moment of MomentumprL p1r2r3rppL is the same for all three of these particles•••Moment of MomentumprL p1r2r3rppL is different for all of these particles•••Moment of Force (Torque)The moment of force (or torque) about a point is the rate of change of the moment of momentum about that pointdtdLτ Moment of Force (Torque) frτfrvvτfrpvτprprLτprLmdtddtddtdRotational InertiaL=rxp is a general expression for the moment of momentum of a particleIn a case where we have a particle rotating around the origin while keeping a fixed distance, we can re-express the moment of momentum in terms of it’s angular velocity ωRotational Inertia rrIωILωrrLωrrrωrLvrvrLprLˆˆˆˆmmmmmmRotational Inertia222222000000ˆˆyxzyzxzyzxyxzxyxzyxyxzyzxyxzyzrrrrrrrrrrrrrrrrrrmrrrrrrrrrrrrmmIIrrIRotational Inertia ωILI222222yxzyzxzyzxyxzxyxzyrrmrmrrmrrmrrrmrmrrmrrmrrrmRotational InertiaThe rotational inertia matrix I is a 3x3 matrix that is essentially the rotational equivalent of massIt relates the angular momentum of a system to its angular velocity by the equationThis is similar to how mass relates linear momentum to linear velocity, but rotation adds additional complexityωIL vp mSystems of Particlesmomentum tal tomass ofcenter ofposition particles all of mass l tota1iiicmiiicmniitotalmmmmmvppxxVelocity of Center of MasscmtotalcmtotalcmcmiiiiiicmiiicmcmmmmmmdtdmmmdtddtdvppvvxvxxvForce on a ParticleThe change in momentum of the center of mass is equal to the sum of all of the forces on the individual particlesThis means that the resulting change in the total momentum is independent of the location of the applied forceiiicmicmdtddtddtdfpppppSystems of Particles icmicmiicmpxxLprLThe total moment of momentum around the center of mass is:Torque in a System of Particles iicmiicmiicmcmiicmdtddtddtdfrτprτprLτprLSystems of ParticlesWe can see that a system of particles behaves a lot like a particle itselfIt has a mass, position (center of mass), momentum, velocity, acceleration, and it responds to forcesWe can also define it’s angular momentum and relate a change in system angular momentum to a force applied to an individual particle iicmfrτicmffInternal ForcesIf forces are generated within the particle system (say from gravity, or springs connecting particles) they must obey Newton’s Third Law (every action has an equal and opposite reaction)This means that internal forces will balance out and have no net effect on the total momentum of the systemAs those opposite forces act along the same line of action, the torques on the center of mass cancel out as wellKinematics of Rigid BodiesKinematics of a Rigid BodyFor the center of mass of the rigid body:22dtddtddtdcmcmcmcmcmcmxvaxvxKinematics of a Rigid BodyFor the orientation of the rigid body:onacceleratiangular
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