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Home , Dynamics (mechanics), Inertia

MEAM 535

Newton-Euler Equations

From of Systems of Particles to Dynamics

University of Pennsylvania 1 MEAM 535 Outline 1. ’s 2nd Law for a particle - recap 2. Newton’s 2nd Law for a system of particles - recap 3. Center of , F=ma for the system of particles 4. Angular (relative to O or the center of mass) 5. Newton’s 2nd Law for Rigid Body - F=ma - Rotational equation of motion 6. Principal axes and moments of 7. Newton-Euler

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 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. (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 of all external 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 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 of the point O and the (b) the 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 sum of the N ) 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

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€ 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 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. 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 22 MEAM 535 Newton-Euler Equations

Newton’s second law

Note C can be replaced by any point that is fixed in A R b2 Pi If b1, b2, b3, are along principal axes and ρ b1 A B i ω = ω1 b1 + ω2 b2 + ω3b3 C

B b3

MC University of Pennsylvania 23 MEAM 535 Euler’s Equations

1

2 R b2 Pi Let b1, b2, b3, be along principal axes and b ρi 1 A B ω = ω1 b1 + ω2 b2 + C ω3b3

B b3

MC University of Pennsylvania 24 MEAM 535 Example a3

45 deg with reference to pi F1 , M1

p Problem Weld 2 a 2 The rectangular homogeneous plate is welded to p1 the shaft and the assembly rotates about a vertical O θ axis at uniform (ignore ). Find the 2b forces and moments that the weld at O must be a1 subject to. What are the forces and moments that P the bearings must support?

2a

F2 , M2

University of Pennsylvania 25 MEAM 535 Newton-Euler Equations (point O fixed in A)

Newton’s equations Euler’s equations 1

2 R b2 If b1, b2, b3, are along principal axes and b1 A B ω = ω1 b1 + ω2 b2 + b2 ω b C 3 3 b1 B O

b3 A

MC b3

University of Pennsylvania 26 MEAM 535 Rotation of an axisymmetric b f body about a principal axis 3 3

b2 OR f2

Ωt f b Suppose bi are along principal axes, and 1 1

Option 2 And, suppose the rigid body is axisymmetric With respect to fi with b3 as an axis of symmetry Option 1 Note: The angular is With respect to bi

Which expression is simpler?

University of Pennsylvania 27 MEAM 535 Example: Precession of a Spinning

Three simple angular velocities  Spin  Precession  Nutation

Three

 Rotation about a3 through φ

 Rotation about f1 through θ

 Rotation about f3 through ψ

University of Pennsylvania 28 MEAM 535 Example (continued)

B

University of Pennsylvania 29 MEAM 535 Example (continued)

Let |OG|=l

B

mgl sinθ

Transverse moments of inertia Axial moment of inertia

University of Pennsylvania 30 MEAM 535 Problem: Rate Gyro

A rate gyro produces a measurable nutation angle θ proportional to the precessional or yaw . Assume the precession angular velocity is constant and is directed along the vertical axis, the linear spring holds the angle θ to small values and the linear damper keeps the angular rate small. Derive the equation of motion and show when and velocities are small, the angle θ is proportional to the yaw velocity.

University of Pennsylvania 31 MEAM 535 Problem: Dynamics of a Thrown Football

b1 b A football is thrown horizontally at an 2 angle θ with respect to the horizontal axis with a spin about its axial FD symmetry axis. The air can be e assumed to act at a e from the θ center of mass in the horizontal direction. Because the ball is not vC thrown perfectly there are small rotational velocity components in the transverse directions.

Write down equations of motion for mg the football and show that a sufficiently large spin velocity stabilizes the motion for θ=0.

University of Pennsylvania 32 MEAM 535

Problem Problem Two particles of mass m are attached to a massless A thin homogeneous disk of mass 800 rod at an angle α as shown in the figure in such a grams and radius 100 mm rotates at a way that the center of mass is on the horizontal constant rate of ω2 = 20 rads/sec with shaft of length 2l. Calculate the bearing reactions respect to the arm ABC, which itself if the horizontal shaft rotates with a constant angular velocity ω. rotates at ω1 = 10 rads/sec in an inertial frame. For the position shown, determine the dynamic reactions at the bearings D and E.

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