Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2
Rigid-Body Kinematics Objects as sets of points. Relative distances between all points are invariant to rigid movement. Free body movement: around the center of mass (COM). Movement has two components: Linear trajectory of a central point. Relative rotation around the point. 3
Mass The measure of the amount of matter in the volume of an object: m = # ρ dv ( ρ : the density of each point the object volume V. dv: the volume element. Equivalently: a measure of resistance to motion or change in motion. 4
Mass For a 3D object, mass is the integral over its volume: m = # # # ρ(x, y, z) dx dy dz For uniform density (ρ constant): m = ρ V 5
Center of Mass The center of mass (COM) is the average point of the object, weighted by density: COM = 1 m p: point coordinates. # ρ 4 p dv ( Point of balance for the object. Uniform density: COM ó centroid. COM 6
Center of Mass of System Sets of bodies have a mutual center of mass: m 8 : mass of each body. 9 COM = 1 m 7 m 8 p 8 8:; p 8 : location of individual COM. m= 9 m 8 8:;. Example: two spheres in 1D x 2 x =>? = m ;x ; + m A x A m ; + m A x 1 m 1 x COM COM m 2 7
Center of Mass Quite easy to determine for primitive shapes What about complex surface based models? 8
Rotational Motion P a point on to the object. C is the center of rotation. Distance vector r = P C. u C θ r P v s P r = r : the distance. Object rotates ó P travels along a circular path. Unit-length axis of rotation: u. Here, u=z ( out from the screen). Rotation: counterclockwise. right-hand rule. 9
Angular Displacement Point p covers linear distance s. θ is the angular displacement of the object: u C θ r P v s P θ = s/r s: arc length. Unit is radian (rad) 1 radian = angle for arc length 1 at a distance 1. 10
Angular Velocity Angular speed: the rate of change of the angular displacement: unit is UVW XYZ. ω = dθ dt The angular velocity vector is collinear with the rotation axis: ω = ωu 11
Angular Acceleration Angular acceleration: the rate of change of the angular velocity: α = dω dt Paralleling definition of linear acceleration. Unit is rad/s A 12
Tangential and Angular Velocities Every point moves with the same angular velocity. Direction of vector: u. Tangential velocity vector: Or: v = ω r ω = r v r A u C θ r P v s P ω = ^_ (abs. values) U due to s = θ 4 r. V(t) C P(t + t) P(t) Only the tangential part matters! T(t) 13
Dynamics The centripetal force creates curved motion. In the direction of (negative) r Object is in orbit. Constant force ó circular rotation with constant tangential velocity. Why? 14
Tangential & Centripetal Accelerations Tangential acceleration α holds: a = α r cf. velocity equation v = ω r. The centripetal acceleration drives the rotational movement: a 9 = ^b U r = ωa r. What is the centrifugal force? 15
Angular Momentum Linear motion è linear momentum: p = mv. Rotational motion è angular momentum about any fixed relative point (to which r is measured): L = # r p ρdv unit is N 4 m 4 s ρdv = dm (mass element) Angular momentum is conserved! Just like the linear momentum. Caveat: conserved w.r.t. the same point. ( 16
Angular Momentum Plugging in angular velocity: r p dv = r v dm = r ω r dm Integrating, we get: L = # r ω r dm ( Note: The angular momentum and the angular velocity are not generally collinear! 17
Define: r = x y z Moment of Inertia and ω = ω f ω g ω h. For a single rotating body: the angular velocity is constant. We get: L = # r ω r dm ( = # y A + z A ω f xyω g xzω h yxω f + z A + x A ω g yzω h zxω f zyω g + (x A + y A )ω h dm = I ff I fg I fh I gf I gg I gh I hf I hg I hh ω f ω g ω h. Note: replacing integral with a (constant) matrix operating on a vector! 18
Momentum and Inertia r = x y z The inertia tensor I only depends on the geometry of the object and the relative fixed point (often, COM): I ff = # y A + z A dm I fg = I gf = # xy dm I gg = # z A + x A dm I fh = I hf = # xz dm I hh = # x A + y A dm I gh = I hg = # yz dm 19
The Inertia Tensor Compact form: # y A + z A dm # xy dm # xz dm I = # xy dm # z A + x A dm # yz dm # xz dm # yz dm # x A + y A dm The diagonal elements are called the (principal) moment of inertia. The off-diagonal elements are called products of inertia. 20
The Inertia Tensor Equivalently, we separate mass elements to density and volume elements: I = # ρ x, y, z ( y A + z A xy xz xy z A + x A yz dx dy dz xz yz x A + y A The diagonal elements: distances to the respective principal axes. The non-diagonal elements: products of the perpendicular distances to the respective planes. 21
Moment of Inertia The moment of inertia I j, with respect to a rotation axis u, measures how much the mass spreads out : I j = # r j A dm ( r j : perpendicular distance to axis. Through the central rotation origin point. Measures ability to resist change in rotational motion. The angular equivalent to mass! 22
Moment and Tensor We have: r j A = u r A = u k I(q)u for any point q. (Remember: r is distance to origin). Thus: I j = u k I(q)u dm=u k Iu? The scalar angular momentum around the axis is then L j = I j ω. Reducible to a planar problem (axis as Z axis). 23
For a mass point: I = m 4 r j A Moment of Inertia For a collection of mass points: I = m 8 r 8 A 8 r A r m m r ; ; m A For a continuous mass distribution on the plane: I = r A? j dm dm r o r m o 24
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 A + y A + z A = R A 25
Inertia of Primitive Shapes Distance to axis of rotation is the radius of the disc at the cross section along z: r A = x A + y A = R A z A. Summing moments of inertia of small cylinders of inertia I q along the z-axis: = Ub r A We get: v di q = 1 2 ra dm = 1 2 ra ρdv = 1 2 ra ρπr A dz v I q = ; ρπ A wv ru dz = ; ρπ A wv RA z A A dz = ; ρπ [R u z 2R A z o 3 A + z z 5] v wv = ρπ 1 2 3 + 1 5 R z. As m = ρ 4 3 πr o, we finally obtain: I q = A z mra. 26
Inertia of Primitive Shapes Solid sphere, radius r and mass m: I = 2 5 mra 0 0 0 2 5 mra 0 0 0 2 5 mra Hollow sphere, radius r and mass m: I = 2 3 mra 0 0 0 2 3 mra 0 0 0 2 3 mra z y x 27
Inertia of Primitive Shapes Solid ellipsoid, semi-axes a, b, c and mass m: I = 1 5 m(ba +c A ) 0 0 0 1 5 m(aa +c A ) 0 0 0 1 5 m(aa +b A ) z y x Solid box, width w, height h, depth d and mass m: h w d I = 1 12 m(ha +d A ) 0 0 1 0 12 m(wa +d A ) 0 0 0 1 12 m(wa +h A ) 28
Inertia of Primitive Shapes Solid cylinder, radius r, height h and mass m: h I = 1 12 m(3ra +h A ) 0 0 0 1 12 m(3ra +h A ) 0 0 0 1 2 mra Hollow cylinder, radius r, height h and mass m: h I = 1 12 m(6ra +h A ) 0 0 0 1 12 m(6ra +h A ) 0 0 0 mr A 29
Parallel-Axis Theorem The object does not necessarily rotate around the center of mass. Some point can be fixed! parallel axis theorem: I^: inertia around axis u. I^ = I =>? + md A I =>? inertia about a parallel axis through the COM. d is the distance between the axes. 30
Parallel-Axis Theorem More generally, for point displacements: d f, d g, d h I ff = # y A + z A dm + md f A I fg = # xy dm + md f d g I gg = # z A + x A dm + md g A I fh = # xz dm + md f d h I hh = # x A + y A dm + md h A I gh = # yz dm + md g d h 31
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 h = I f + I g for any planar object I h = 2I f = 2I g for symmetrical objects 32
Reference Frame The representation of the inertia tensor is coordinate dependent. The physical effect should be invariant to coordinates! If transformation R changes bases from body to world coordinate, the inertia tensor in world space is: I U W = R 4 I Wg 4 R k The moment of inertia is invariant! Why? 33
Torque A force F applied at a distance r from a (fixed) center of mass. Tangential part causes tangential acceleration: F = m 4 a The torque τ is defined as: τ = r F So we get τ = m r α r = m r A α unit is N m rotates an object about its axis of rotation. 34
Newton s Second Law The law F = m 4 a has an equivalent with the inertia tensor and torque: τ = Iα Force ó linear acceleration Torque ó angular acceleration 35
Torque and Angular Momentum Reminder: in the linear case: F = W (p is the linear WŒ momentum). Similarly with torque and angular momentum: dl dt = dr dp p + r dt dt = v mv + r F = 0 + τ Kinematics: dl dt = d(iω) = I dω = Iα = τ dt dt Force ó derivative of linear momentum. Torque ó derivative of angular momentum. 36
Rotational Kinetic Energy Translating energy formulas to rotational motion. The rotational kinetic energy is defined as: E ŽU = 1 2 ωk I ω 37
Conservation of Mechanical Energy Adding rotational kinetic energy E ŽŒ t + t + E t + t + E ŽU t + t = E ŽŒ t + E t + E ŽU (t) + E > E ŽŒ is the translational kinetic energy. E is the potential energy. E ŽU is the rotational kinetic energy. E > the lost energies (surface friction, air resistance etc.). 38
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 39
Rigid Body Forces A force can be applied anywhere on the object, producing also a rotational motion. F COM α a 40 40
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 41 41
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. 42 42