MEAM 535 Newton-Euler Equations From Dynamics of Systems of Particles to Rigid Body Dynamics University of Pennsylvania 1 MEAM 535 Outline 1. Newton’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of mass, F=ma for the system of particles 4. Angular momentum (relative to O or the center of mass) 5. Newton’s 2nd Law for Rigid Body Motion - F=ma - Rotational equation of motion 6. Principal axes and moments of inertia 7. Newton-Euler equations of motion University of Pennsylvania 2 MEAM 535 Newton’s Second Law for a Particle Newton’s Second Law for a particle P Position vector p in an inertial frame A F is the force acting on the particle, mass m P m a2 Linear momentum in A p Angular Momentum of the system S relative to O in A O a1 N A A P d p HO = ∑p × m i=1 dt a3 Rate of change of angular and linear momentum in A A € A d L d AHP =… O =… dt dt University of Pennsylvania 3 € € MEAM 535 Newton’s Second Law for a System of Particles Newton’s Second Law for a particle P Position vector p in an inertial frame A F is the force acting on the particle, mass m Pi mi a Linear momentum in A 2 pi O a Extend to many particles 1 Mass mi at Pi F is the external force acting on P i i a3 Fij is the force exerted by Pj on Pi University of Pennsylvania 4 MEAM 535 Center of Mass Define Newton’s Second Law j Fij k i Fik Fi P a i 2 mi pi O a1 a3 University of Pennsylvania 5 MEAM 535 Newton’s Second Law for a System of Particles The center of mass for a system of particles, S, accelerates in an inertial frame (A) as if it were a single particle with mass m (equal to the total mass of the system) acted upon by a force equal to the net external force. Center of mass Conservation of Linear Momentum The linear momentum of a system of particles stays constant in an inertial frame (A) if the net external force equals zero. University of Pennsylvania 6 MEAM 535 Outline 1. Newton’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of mass, F=ma for the system of particles 4. Angular momentum (relative to O or the center of mass) 5. Newton’s 2nd Law for Rigid Body Motion - F=ma - Rotational equation of motion 6. Principal axes and moments of inertia 7. Newton-Euler equations of motion University of Pennsylvania 7 MEAM 535 Angular Momentum of a System of Particles Angular Momentum of Pi relative to O in A Angular Momentum of Pi relative to C in A Note: C is not fixed in A P i Angular Momentum of the system mi S relative to O in A ri pi rC Angular Momentum of the system O a1 S relative to C in A a3 University of Pennsylvania 8 MEAM 535 Rate of Change of Angular Momentum relative to O Angular Momentum of the system S relative to O in A Resultant moment of all external forces acting on the system S about (relative to) O P i The rate of change of angular mi momentum of the system S relative to O in A is equal to the resultant moment of ri pi all external forces acting on the system relative to O rC O a1 Notes: (1) A is inertial; and Note (2) O is fixed in A a3 University of Pennsylvania 9 MEAM 535 Rate of Change of Angular Momentum relative to C Angular Momentum of the system S relative to C in A Resultant moment of all external forces acting on the system S about (relative to) C Pi mi 0 0 ri pi The rate of change of angular momentum of the system S relative to C rC in A is equal to the resultant moment of O a1 all external forces acting on the system relative to C a3 University of Pennsylvania 10 MEAM 535 Principles of Conservation of Angular Momentum for a System of Particles (1) The angular momentum of S relative to O is constant in A (2) The angular momentum of S relative to C is constant in A University of Pennsylvania 11 MEAM 535 Examples 1 Particles A and B, each of mass m, are attached to massless rods of length l which are pivoted at A and O in the horizontal plane. Solve for the velocities as a function of positions, assuming θ and φ are zero at t=0, and the rate of change of θ and φ are known to be zero and ω0 respectively at t=0. Answer 3 − 2cosφ φ˙ = ω 0 1+ sin2 φ (1− cosφ)ω 0 θ˙ = (1+ sin2 φ)(3 − 2sinφ) € University of Pennsylvania 12 € MEAM 535 Examples 2 Each of the three balls has a mass m and is welded to rigid angular frame of negligible mass. The assembly rests on a smooth horizontal surface. If a force F is applied to one bar as shown, determine (a) the acceleration of the point O and the (b) the angular acceleration of the frame. m Answer F r d b Fb = −3mr2ω O dt ( ) m r r m € University of Pennsylvania 13 MEAM 535 Outline 1. Newton’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of mass, F=ma for the system of particles 4. Angular momentum (relative to O or the center of mass) 5. Newton’s 2nd Law for Rigid Body Motion - F=ma - Rotational equation of motion 6. Principal axes and moments of inertia 7. Newton-Euler equations of motion University of Pennsylvania 14 MEAM 535 Angular Momentum for a System of Particles that are Rigidly Connected (relative to O) B is a reference frame that is attached to the rigid body of mass m N A S HO = ∑ri × mi p˙ i i=1 N A d = p × m r + r m ∑ i i ( C i ) i i=1 dt b2 N ri A C A B = rC × m v + ∑ri × mi ( ω × ri ) p i=1 i b b1 3 C € Angular momentum Angular momentum rC of a single particle of of the rigid body due O a1€ mass m (equal to the to its rotation sum of the N masses) translating as if it a3 were attached to the center of mass C. University of Pennsylvania 15 MEAM 535 Angular Momentum for a System of Particles that are Rigidly Connected (relative to C) B is a reference frame that is attached to N the rigid body of mass m A S HC = ∑ri × mi p˙ i i=1 N A d mi = r × m r + r b ∑ i i ( C i ) 2 i=1 dt ri p N i b1 A B b3 C = ∑ri × mi ( ω × ri ) i=1 rC O a1 a3 € University of Pennsylvania 16 MEAM 535 Inertia Dyadic N A S A B Definition HC = ∑ri × mi ( ω × ri ) i=1 € A B A B IC ⋅ ω = ∑mi [(ri ⋅ ri )U − riri ]⋅ ω i=1,N m r r A B r r A B = ∑ i [( i ⋅ i ) ω − i ( i ⋅ ω )] i=1,N A B = ∑miri × ( ω × ri ) i=1,N University of Pennsylvania 17 € MEAM 535 Rotational equations of motion The rate of change of angular momentum of the system S relative to C in A is equal to the resultant moment of all external forces acting on the system relative to C What is the difficulty? University of Pennsylvania 18 MEAM 535 Outline 1. Newton’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of mass, F=ma for the system of particles 4. Angular momentum (relative to O or the center of mass) 5. Newton’s 2nd Law for Rigid Body Motion - F=ma - Rotational equation of motion 6. Principal axes and moments of inertia 7. Newton-Euler equations of motion University of Pennsylvania 19 MEAM 535 Principal Axes and Principal Moments Principal axis of inertia u is a unit vector along a principal axis if I . u is parallel to u There are 3 independent principal axes! Principal moment of inertia The moment of inertia with respect to a principal axis, u . I . u, is called a principal moment of inertia. AωB Physical interpretation AωB B B O A B O HO and ω are not parallel! A B HO and ω are parallel! University of Pennsylvania 20 MEAM 535 Properties of the inertia matrix With respect to any origin, and any reference triad! 1. The inertia matrix is Symmetric 2. The inertia matrix is Positive Definite 3. Moments of inertia are always positive 4. Moments of inertia are the least when C is the reference point 5. It is always possible to find three distinct mutually orthogonal principal axes of inertia. [may not be unique] 6. If the reference triad is aligned with the principal axes, the products of inertia are zero University of Pennsylvania 21 MEAM 535 Outline 1. Newton’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of mass, F=ma for the system of particles 4.