# Rotational & Rigid-Body Mechanics. Lectures 3+4

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1 Rotational & Rigid-Body Mechanics Lectures 3+4

2 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2

3 Circular Motion - Definitions C is the center of rotation. P a point on to the object. r is the distance vector P C r = r the distance between C and P. Object rotates P travels along a circular path after t, P covered distance s, and angle θ. Unit-length axis of rotation: u. In this example, the Z axis going out from the screen. Rotation: counterclockwise (right-hand rule). C θ r P s P 3

4 Angular Displacement This angle θ represents the rotation of the object: θ = s/r where s is the arc length and r the radius unit is radian (rad) 1 radian=angle for arc length 1 at a distance 1. ω C θ r P v s P 4

5 Angular Velocity Angular velocity: the rate of change of the angular displacement: ω = θ t = θ t + t θ(t) t unit is rad/s. The angular velocity vector is collinear with the rotation axis: ω = ωu 5

6 Angular Acceleration Angular acceleration: the rate of change of the angular velocity: α = ω ω t + t ω(t) = t t Exactly like acceleration is to velocity in a trajectory. unit is rad/s 2 6

7 Equations of Motion Deferring velocity from position and acceleration: ω(t + t) = ω(t) + α t ω = ω(t + t) + ω(t) 2 θ = 1 2 ω(t + t) + ω(t) t θ = ω(t) t α t2 ω(t + t) 2 = ω(t) 2 + 2α θ 7

8 Kinetics of Rotational Motion The centripetal force creates curved motion. Orthogonal to the velocity of the object. Object is in orbit. Constant force circular rotation with constant tangential velocity. Why? 8

9 Tangential and Angular Velocities Every point on a rigid body moves with the same angular velocity. Different points on a rigid body can have different tangential velocities. Different radii V(t) C P(t + t) P(t) T(t) 9

10 Tangential Velocity Angular velocity vector: direction of rotation axis Tangential velocity vector: direction of movement direction Relation: v = ω r ω Or: r v ω = r 2 C θ r P v s P Note that v = ω r (abs. values), due to s = θ r. Only the tangential part of any velocity matters for rotation! 10

11 Tangential Acceleration Tangential acceleration defined the same way a = α r where α is the angular acceleration. Similarly to the velocity equation v = ω r 11

12 Centripetal Acceleration The centripetal acceleration, orthogonal to the velocity (=towards the axis of rotation), drives the rotational movement: What is the centrifugal force? a n = v t 2 r = rω2 12

13 Rigid-Body Kinematics Rigid bodies have dimensions, as opposed to single points. Relative distances between all points are invariant. Movement can be decomposed into two components: Linear trajectory of any single points Relative rotation around the point Free body movement: around the center of mass (COM).

14 Mass The measure of the amount of matter in the volume of an object: m = ρ dv V where ρ is the density at each location in the object volume V. Equivalently: a measure of resistance to motion or change in motion. 14

15 Mass For a 3D object, mass is the integral over its volume: m = ρ(x, y, z) dx dy dz For uniform density objects (simple rigid bodies): m = ρ V where ρ is the density of the object and V is its volume. 15

16 Center of Mass The center of mass (COM) is the point at which all the mass can be considered to be concentrated obtained from the first moment, i.e. mass times distance. point of balance of the object if uniform density, COM is also the centroid COM 16

17 Center of Mass Coordinate of the COM: COM = 1 m V ρ p p dv where p is the position at each location in V. For a set of bodies: n COM = 1 m i=1 m i p i where m i is the mass of each COM p i in every body. 17

18 Center of Mass Example for a body made of two spheres in 1D x 2 x 1 x COM m 1 COM m 2 x COM = m 1x 1 + m 2 x 2 m 1 + m 2 18

19 Center of Mass Quite easy to determine for primitive shapes But what about complex surface based models? 19

20 Angular Momentum Remember linear momentum: p = m v. Rotational motion also produces angular momentum of a any point on the object about the center of mass (or any relative point): L = V r p dv unit is N m s Angular momentum is a conserved quantity! Just like the linear momentum. Remember: measured around a point. 20

21 Angular Momentum Plugging in angular velocity: r p = r v dm = r ω r dm Integrating, we get: L = r ω r dm M Note: The angular momentum and the angular velocity are not generally collinear! 21

22 Defining: r = x y z and ω = Moment of Inertia ω x ω y ω z. Remember that the angular velocity is constant throughout the body. We get: L = y 2 + z 2 ω x xyω y xzω z yxω x + z 2 + x 2 ω y yzω z zxω x zyω y + (x 2 + y 2 )ω z dm = I xx I xy I xz I yx I yy I yz I zx I zy I zz ω x ω y ω z.

23 Momentum and Inertia The inertia tensor only depends on the geometry of the object and the relative point (often, COM): I xx = y 2 + z 2 dm I xy = I yx = xy dm I yy = z 2 + x 2 dm I xz = I zx = xz dm I zz = x 2 + y 2 dm I yz = I zy = yz dm 23

24 The Inertia Tensor Finally the inertia can be expressed as the matrix y 2 + z 2 dm xy dm xz dm I = xy dm z 2 + x 2 dm yz dm xz dm yz dm x 2 + y 2 dm The diagonal elements are called the (principal) moment of inertia The off-diagonal elements are called products of inertia 24

25 The Inertia Tensor Equivalently, we separate mass elements to density and volume elements: I = V ρ x, y, z y 2 + z 2 xy xz xy z 2 + x 2 yz dx dy dz xz yz x 2 + y 2 The diagonal elements: distances to the respective principal axes. The non-diagonal elements: products of the perpendicular distances to the respective planes. 25

26 Moment of Inertia The moment of inertia of a rigid body is a measure of how much the mass of the body is spread out. A measure of the ability to resist change in rotational motion Defined with respect to a specific rotation axis u. Through the central rotation origin point. We have that: I u = V r u 2 dm 26

27 Moment and Tensor We have: r u 2 = u r 2 = u T I(q)u for any point q. (Remember: r is distance to origin). Thus: I u = M u T I(q)u dm=u T Iu The scalar angular momentum around the axis is then L u = I u ω. Reducible to a planar problem (axis is the new z axis).

28 Moment of Inertia For a mass point: I = m r u 2 r m For a collection of mass points: I = i m i r i 2 r 2 m 2 r 1 m 1 r 3 m 3 For a continuous mass distribution on the plane: I = M r 2 u dm dm r 28

29 Inertia of Primitive Shapes For primitive shapes, the inertia can be expressed with the parameters of the shape Illustration on a solid sphere Calculating inertia by integration of thin discs along one axis (e.g. z). Surface equation: x 2 + y 2 + z 2 = R 2 29

30 Inertia of Primitive Shapes Distance to axis of rotation is the radius of the disc at the cross section along z: r 2 = x 2 + y 2 = R 2 z 2. Summing moments of inertia of small cylinders of inertia I Z = r2 m We get: di Z = 1 2 r2 dm = 1 2 r2 ρdv = 1 2 r2 ρπr 2 dz 2 along the z-axis: I Z = 1 ρπ R 2 R r 4 dz = 1 ρπ R 2 R R 2 z 2 2 dz = 1 ρπ 2 R4 z 2R 2 z z 5 R 5 R as m = ρ = ρπ R πr 3, we finally obtain: I Z = 2 5 mr2. 30

31 Inertia of Primitive Shapes Solid sphere, radius r and mass m: I = 2 5 mr mr mr2 y Hollow sphere, radius r and mass m: 31 I = 2 3 mr mr mr2 z x

32 Inertia of Primitive Shapes Solid ellipsoid, semi-axes a, b, c and mass m: I = 1 5 m(b2 +c 2 ) m(a2 +c 2 ) m(a2 +b 2 ) z y x Solid box, width w, height h, depth d and mass m: 32 h w d I = 1 12 m(h2 +d 2 ) m(w2 +d 2 ) m(w2 +h 2 )

33 Inertia of Primitive Shapes Solid cylinder, radius r, height h and mass m: h I = 1 12 m(3r2 +h 2 ) m(3r2 +h 2 ) Hollow cylinder, radius r, height h and mass m: 1 2 mr2 h 33 I = 1 12 m(6r2 +h 2 ) m(6r2 +h 2 ) mr 2

34 Parallel-Axis Theorem The object does not necessarily rotate around the center of mass. Some point can be fixed! parallel axis theorem: I v = I COM + md 2 Where: I v : inertia around axis u. I COM inertia about a parallel axis through the COM. d is the distance between the axes. 34

35 Parallel-Axis Theorem More generally, for point displacements: d x, d y, d z I xx = y 2 + z 2 dm + md x 2 I xy = xy dm + md x d y I yy = z 2 + x 2 dm + md y 2 I xz = xz dm + md x d z I zz = x 2 + y 2 dm + md z 2 I yz = yz dm + md y d z 35

36 Perpendicular-Axis Theorem For a planar 2D object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane: z y z y x x I z = I x + I y for any planar object 36 I z = 2I x = 2I y for symmetrical objects

37 Reference Frame The inertia tensor is coordinate dependent. If R changes bases from body to world coordinate, the inertia tensor in world space is: I world = R I body R T 37

38 Rigid-Body Dynamics

39 Torque The torque is a force F applied at a distance r from a (held) center of mass. Tangential part causes tangential acceleration: F t = m a t Multiplying by the distance, the torque is: τ = F t r = m a t r We know that a t = r α So we have τ = m r α r = m r 2 α unit is N m rotates an object about its axis of rotation through the center of mass. 39

40 Torque A force in general is not applied in the direction of the tangent. The torque τ is then defined as: τ = r F The direction of the torque is perpendicular to both F and r. 40

41 Newton s Second Law The law F = m a has an equivalent with inertia tensor and torque: τ = I α Force linear acceleration Torque angular acceleration 41

42 Rotational Kinetic Energy Translating energy formulas to rotational motion. The rotational kinetic energy is defined as: E Kr = 1 2 ωt I ω 42

43 Conservation of Mechanical Energy Adding rotational kinetic energy E Kt t + t + E P t + t + E Kr t + t = E Kt t + E P t + E Kr (t) + E O E Kt is the translational kinetic energy. E P is the potential energy. E Kr is the rotational kinetic energy. E O the lost energies (surface friction, air resistance etc.). 43

44 Torque and Angular Momentum Remember, in the linear case: F = d p dt ( p is the linear momentum). Similarly with torque and angular momentum: dl d r d p = p + r dt dt dt = v m v + r F = 0 + τ Force derivative of linear momentum. Torque derivative of angular momentum.

45 Impulse We may apply off-center forces for a very short amount of time. Such angular impulse results in a change in angular momentum, i.e. in angular velocity: τ t = L 45

46 Rigid Body Forces A force can be applied anywhere on the object, producing also a rotational motion. F COM α a 46

47 Position of An Object Remember: the object moves linearly as the COM moves. Rotation: the movement for all points relatively to the COM. Total motion: sum of the two motions. F 47

48 Particle System Most simple instance of a physics system Each object (body) is a particle Each particle have forces acting upon it Constant, e.g. gravity Position dependent, e.g. force fields Velocity dependent, e.g. drag forces Event based, e.g. collision forces Restrictive, e.g. joint constraint So net force is a function F p o, v, a, m, t, Discretization: e.g., V f q dm becomes a sum: i=1 n f q i m i 48

49 Particle System Use the equations of motion to find the position of each particle at each frame. At the start of each frame: Sum up all of the forces for each particle. From these forces compute the acceleration. Integrate into velocity and position. Rigid body: all particles receive the same rotation and translation. 49

50 Complex Objects When an object consists of multiple primitive shapes: Calculate the individual inertia of each shape. Use parallel axis theorem to transform to inertia about an axis through the COM of the object. Add the inertia matrices together. 50

51 Motion Constraints A rigid body may not be free to move on its own. We wish to constrain its movement: wheels on a chair human body parts trigger of a gun opening door actually almost anything you can think of in a game... 51

52 Degree of freedom To describe how a body can move in space, specify its degrees of freedom (DOF): Translational Rotational 52

53 Kinematic pair A kinematic pair is a connection between two bodies that imposes constraints on their relative movement Lower pair, constraint on a point, line or plane: Revolute pair, or hinged joint: 1 rotational DOF. Prismatic joint, or slider: 1 translational DOF. Screw pair: 1 coordinated rotation/translation DOF. Cylindrical pair: 1 translational + 1 rotational DOF. Spherical pair, or ball-and-socket joint: 3 rotational DOF. Planar pair: 3 translational DOF. Higher pair, constraint on a curve or surface. 53

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