Rotational Dynamics

Paper Summaries

• Any takers? Dynamics II

Rotational Motion

Projects Projects

• Proposals • Question about presentations – Should all have received feedback via e-mail. – We have 21 projects – Grading – We can either: • Dedicate last 4 classes for presentation • ACCEPTED – 30 minute presentations • CONDITIONALLY ACCEPTED – 20 minute presentations w/leftovers • PLEASE RESUBMIT WITH MORE INFO • Dedicate last 3 classes for presentation – 20 minute presentations – Will run 20 minutes late – One extra class – READING / PROJECT Day

Projects Assignment 1

• Next class: • Any questions? – Project roll call • Short (1 min) summary of project. • Assignment 1 due April 9th

1 Plan for today A note for last time

• Continue Physics 101 – Last time we spoke about linear motion Supporting object – Today we consider rotation F Resting contact s F

Normal force FN

Static friction Fs = us * FN

Motivation Films Motivational Film

• Believe it or not… • Change Myself (1991) – There is an esoteric link between these two – Todd Rundgren short films. • Hello It’s Me / Bang on the Drum • Producer of Meatloaf’s Bat Out of Hell

– First fully computer-animated music video produced on a desktop system

Motivational Film Motivational Film

• Mathematica: The Theorem of Pythagoras • So what’s the link? (1988) – Jim Blinn played trombone on “The Want of a – Jim Blinn, Cal Tech (now at Microsoft Research) Nail” off of Todd’s 1989 album Nearly Human – This is how I wish I could teach Mathematics! – One of several visual Math lessons • http://www.projectmathematics.com/

2 Let’s get started Laws of Motion

• Physics for Rigid Body Dynamics •Law I – Every object in a state of uniform motion tends to – Note: Change in original schedule remain in that state of motion unless an external • Last Wednesday: Linear Motion force is applied to it. (Inertia) • Today: Rotational Motion • Law II: • Wednesday: Collisions – The acceleration of a body is proportional to the • Monday, April 7: Numerical Integration / resulting force acting on the body, and this Constraints acceleration is in the same direction as the force. • Law III: – For every action there is an equal and opposite reaction.

Linear Motion Linear Motion

• For Linear Physical Motion • One term we failed to define –Mass –Momentum • Measure of the amount of matter in a body • Defined as mass x velocity • From Law II: Measure of the a body’s resistance to motion • Another way of stating Law I – Velocity – Momentum is conserved • Change of motion with respect to time – Cannot change unless force is acted upon it – Acceleration • Change of velocity with respect to time –Force • In short, force is what makes objects accelerate

Terms Terms

• Mathematically defined • Let’s switch notation – Velocity v = ds / dt – Velocity v = s&(t) – Acceleration a = dv / dt = d 2s / dt – Acceleration a = v&(t) = &s&(t) –Force –Force F = ma F = ma = mv& = M& –Momentum M = mv –Momentum M = mv

3 Let’s Talk About Rotation Let’s talk about rotation

• Figures taken from • Two coordinate systems to consider – Watt & Policarpo, “3D Games: RealTime – World space Rendering and Software Technology” • Coordinate system of space where object resides • Known as inertial frame – Object space • Local coordinate system of the object • Origin is the center of mass of the object • Known as body frame

Let’s talk about rotation Transformation matrix Translation s(t)

xw  m11 m12 m13 m14  xo  y  m m m m  y   w  =  21 22 23 24  ⋅  o  zw  m31 m32 m33 m34  zo         1  m41 m42 m43 m44   1  { 142444 43444 { world transformation object

Rotation = R(t)

Rotation Matrix Rotation Matrix

• Reinterpreting the rotation matrix • Factoids about the rotation matrix – R is orthogonal rxx ryx rzx  • Each column vector is of unit length   • Determinant = 1 R(t) = rxy ryy rzy  • Inverse is equal to it’s transpose r r r  –R-1 = RT  xz yz zz  • Al the above must hold x y z – Each column gives the world coordinates of an axis in – Real reason why interpolation of this matrix directly the local coordinate system does not work!

4 Calculating position Angular velocity

• Taking into account rotation, the position (in • Amount of rotation with respect to time world space) of a point r (in object space) body – Measured w.r.t. the body frame r(t) = s(t) + R(t)rbody – Given by a 3D vector ω(t) • Direction of ω gives the axis of rotation • Which is a different way of saying • Magnitude of ω, | ω | gives the amount of rotation

Angular velocity Angular velocity

• To obtain velocity at a point rbody with respect to point in object space

r&body (t) = ω(t)× rbody

• Note that this is independent of linear velocity

•Link

Angular Velocity Angular Velocity

• Apply to each axis in object space • Apply to each axis in object space – Recall

 rxx  ryx  rzx         R&(t) = ω(t)× rxy ω(t)× ryy ω(t)× rzy  = ω(t)∗ R(t) rxx ryx rzx         r  r  r      xz   yz   zz  R(t) = rxy ryy rzy    rxz ryz rzz  x y z

5 Complete velocity equation Putting it all together

• For a point r in object space Object properties body Position, orientation Calculate forces • Position Linear and angular velocity translation Linear and angular momentum mass r(t) = s(t) + R(t)rbody rotation • Velocity

r&(t) = s&(t) + R&(t)rbody Update object properties Calculate accelerations Using mass, momenta r&(t) = v(t) +ω(t)∗ R(t)rbody

Putting it all together Putting it all together

• Step 1 • What is missing? – Calculate Forces, F(t) – How about rotational force / acceleration • Step 2 – Integrate position/rotation – This is known a Torque • s(t +∆t) = s(t) + v(t)∆t •R(t +∆t) = R(t) + (ω(t)*R(t)) ∆t /* CAREFUL HERE */

• r(t +∆t) = s(t +∆t) + rbodyR(t +∆t) – Update Momentum • More after break •M(t +∆t) = M(t) + F(t) ∆t • Step 3 – Calculate velocities •v(t +∆t) = M(t +∆t)/m

Rotational acceleration Rotational acceleration

• Consider • Observations – Rotation occurs around the center of mass Will result in (COM) of an object rotation – Force applied to point at a distance from COM will cause a rotation

Apply force

6 Torque Torque

• Force applied to an object that results in rotational • Now force can be seen as two separate acceleration. components: – Dependent upon the distance from the center of mass – Force (Linear) τ = (r − x)× F – Torque (Rotational) –Where • τ = Torque • x = center of mass • r = point where force is applied • F = force applied

Torque More about mass

• When considering rotation, we need to know 3 things about the mass of an object – Magnitude –Center of mass – Description of how the mass is distributed in the object

• This final item is given by the Inertia Tensor

Inertia tensor Inertia tensor

• Symmetric 3x3 matrix • Each element of the tensor is calculated by integrating the density of the object at points q=(x,y,z) I xx I xy I xz  I = (y 2 + z 2 )dm I = − xydm   xx ∫ xy ∫ I I I I 2 2 body = xy yy yz I = (x + z )dm I = − xzdm   yy ∫ xz ∫

  2 2 I I I I = (x + y )dm I yz = − yzdm  xz yz zz  zz ∫ ∫

7 Inertia Tensor Formulas for Inertia Tensor

• Like in the linear case, • For symmetrical bodies, I contains only – The inertia tensor can be interpreted as the non-zero components on the diagonal resistance to acceleration F = ma I 0 0 τ = Iω&  xx  I =  0 I 0  F −1 body  yy  a = ω& = α = I τ  0 0 I  m  zz 

Formulas for Inertia Tensor Formulas for Inertia Tensor

• For common symmetric objects: • For common symmetric objects: – Sphere with radius r and mass m – Rectangular box of sides a, b, c coinciding to x, 2 y, and z axes respectively I = I = I = mr 2 xx yy zz 5 1 2 2 I xx = m(b + c ) – Cylinder with radius r, height h, where z axis is 3

coincident with the long axis of cylinder 1 2 2 I yy = m(a + c ) 1 1 1 3 I = I = m(r 2 + h2 ) I = mr 2 xx yy 4 3 zz 2 1 I = m(a2 + b2 ) zz 3

Inertia Tensor Let’s recap

• Important: • For linear motion – With respect to time, in object space, I is – Velocity v = s&(t) constant – Acceleration a = v&(t) = &s&(t) • To convert to world space: –Momentum M = mv I(t) = R(t)I R(t)T –Force body F = ma = mv = M& •Furthermore & −1 −1 T I (t) = R(t)I body R(t)

8 Let’s Recap Putting it all together

• For rotational motion Object properties Position, orientation Calculate forces – Angular Velocity ω = r&(t) Linear and angular velocity – Angular Acceleration α = ω&(t) Linear and angular momentum mass – Angular Momentum L = I(t)ω(t) – Torque τ = I(t)α(t) = I(t)ω& (t) = L&(t) Update object properties Calculate accelerations Using mass, momenta

Putting it all together Putting it all together

• State of object at any given time • Derivative of object state  s(t)  position  s&(t)   v(t)   R(t)   R&(t)  ω(t)∗ R(t) S(t) =   rotation (in world coords) S&(t) =   =   M (t) momentum M& (t)  F(t)         L(t)  angular momentum  L&(t)   τ (t) 

Putting it all together Putting It all together

• Step 1 • Step 3 – Calculate Forces, F(t), τ(t) – Calculate velocities • Step 2 •v(t +∆t) = M(t +∆t)/m -1 -1 T – Integrate position/rotation •I (t +∆t) = R(t +∆t)I body R(t +∆t) -1 • s(t +∆t) = s(t) + v(t)∆t • ω(t +∆t) = + I (t +∆t)L(t +∆t) •R(t +∆t) = R(t) + (ω(t)*R(t)) ∆t /* CAREFUL HERE */ • Go to step 1 – Update Momentum •M(t +∆t) = M(t) + F(t) ∆t •L(t +∆t) = L(t) + τ(t) ∆t • Questions?

9 Next Time

• Collisions / Impulse Forces

• Questions?