Advanced Computer Graphics Spring 2014 K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology

2 Today’s Topics Basic Concepts from Physics  Rigid Body Classification  Rigid Body Kinematics  Newton’s Laws  Forces  Momenta  Energy

3 Basic Concepts from Physics We will review some of the basic concepts of physics that are relevant to the analysis of motion and interaction of rigid bodies. An appropriate choice of coordinate system is important in many applications.  Cartesian, polar, cylindrical and spherical coordinate systems.

4 Rigid Body Classification Characterized by the region that its mass lives in.  Single particle  Particle system  Continuous material Curve mass Surface mass Volume mass

5 Rigid Body Classification Discrete Material  Single particle Mass m that occupies a single location x.  Particle system A collection of a finite number of particles  The I th particle having mass m i and located at x i. Q i : some physical quantity associated with the ith particle.

6 Rigid Body Classification Continuous Material  It consists of infinitely many particles that lie in a bounded region of space, R. -> a continuum of mass Curve mass: The region R is a bounded segment of a curve in one, two or three dimensions. Surface mass: The region R is a bounded region in the plane or a bounded portion of a surface in space. Volume mass: R can be a solid occupying a bounded region of space.

7 Rigid Body Classification Continuous material  Curve mass: the integration is computed as a line integral.  Surface mass: the integration is computed as a double integral.  Volume mass: the integration is computed as a triple integral.

8 Rigid Body Kinematics Kinematics  It is a study of motion of objects without considering the influence of external forces.

9 Rigid Body Kinematics Planar Motion of Single Particle in Cartesian Coordinates.  The position of a particle at time t is r(t) = x(t) i + y(t) j.  The velocity of the particle at time t is v(t) = x’(t) i + y’(t) j.  The acceleration of the particle at time t is a(t) = x’’(t) i + y’’(t) j.  The unit length tangent vector is given by T(t) = v/|v| = (cos(φ(t)),sin(φ(t))).  The unit length normal vector is chosen as N(t) = (- sin(φ(t)),cos(φ(t))).

10 Rigid Body Kinematics Planar Motion of Single Particle in Cartesian Coordinates.  A coordinate system at a point on the curve is defined by origin r(t) and coordinate axis directions T(t) and N(t). Moving frame: {r(t); T(t), N(t)}.  Velocity: v = |v|T = s’T.: s is the arc length measured along the curve. s’ is the speed.

11 Rigid Body Kinematics Planar Motion of Single Particle in Cartesian Coordinates.  Curvature of the curve at arc length s  Acceleration

12 Rigid Body Kinematics Planar Motion of Single Particle in Cartesian Coordinates.  Acceleration  The rate of change of the tangent vector with respect to arc length is related to the normal vector Tangent acceleration, the acceleration in the direction of motion normal acceleration, centripetal acceleration

13 Rigid Body Kinematics Planar Motion of Single Particle in Cartesian Coordinates.  Summarizing the s-derivatives in a format matrix notation. The coefficient matrix is skew-symmetric.

14 Rigid Body Kinematics Planar Motion of Single Particle in Polar Coordinates  Position vector: R = r/|r| and r = |r| from the origin -> r = rR.  A unit vector perpendicular to R is P. Rotated by п/2 counterclockwise.  The moving frame {r(t); R(t), P(t)} provides an alternate coordinate system to the tangent-normal one.

15 Rigid Body Kinematics Planar Motion of Single Particle in Polar Coordinates  Velocity  Acceleration

16 Rigid Body Kinematics Spatial Motion in Cartesian Coordinates  Position of a particle at time t  Velocity at time t  Acceleration at time t  Unit-length tangent vector

17 Rigid Body Kinematics Spatial Motion in Cartesian Coordinates  Normal vector The choice of normal should be made in an attempt to maintain a continuous function N(s)  Binormal vector

18 Rigid Body Kinematics Spatial Motion in Cartesian Coordinates  Normal vector  Frenet-Serret equation

19 Rigid Body Kinematics Spatial Motion in Cylindrical Coordinates  Position  Velocity  Acceleration

20 Rigid Body Kinematics Spatial Motion in Spherical Coordinates  Position  Velocity  Acceleration

21 Rigid Body Kinematics Motion About a Fixed Axis: the position, velocity and acceleration of a particle that is rotating about a fixed axis and is a constant distance from that axis.  R = (cosθ)ξ + (sinθ)η on the plane  Angular speed: σ(t) = θ’(t)  Angular velocity: w(t) = σ(t)D  Angular acceleration: α(t) = σ’(t)D

22 Rigid Body Kinematics Motion About a Fixed Axis  The position of a particle in motion about the axis r(t) = r 0 R(t) + h 0 D  The velocity v(t) = w ⅹ r  The acceleration a(t) = -r 0 σ 2 R + α ⅹ r -r 0 σ 2 R: the centripetal acceleration of the particle α ⅹ r: the tangential acceleration of the particle.-> tangent to the circle of motion.

23 Rigid Body Kinematics Motion About a Moving Axis  The concept of angular velocity for a time-varying axis with unit-length direction vector D(t) can be established by studying motion about a fixed axis.  Particle position: r(t) = r 0 (cosθ(t)ξ + sinθ(t)η)+h 0 D. Initial position: r 0 =r(0)=r 0 ξ Position at later times: r(t) = R(t)r 0 R(t) = I + (sin θ(t))Skew(D) + (1- cosθ(t))Skew(D) 2

24 Rigid Body Kinematics Motion About a Moving Axis  Linear velocity: r’(t) = w(t) ⅹ r(t) = Skew(w(t))r(t)  R’(t)=Skew(w(t))R(t)  If the unit direction vector D varies with time, i.e. D(t) R(t) = I + (sin θ(t))Skew(D(t)) + (1- cosθ(t))Skew(D(t)) 2  Angular velocity w = θ’D + (sin θ)D + (cos θ-1)D ⅹ D

25 Rigid Body Kinematics Particle Systems and Continuous Materials  World Coordinates Position at time t in world coordinates : x(t;P)  Body Coordinates Given the origin of body coordinate, x(t;C), the position x(t;P) with respect to body coordinate is given by b(t;P).  x(t;P) = x(t;C) + R(t)b(t;P), R(t) = [U 0 (t),U 1 (t),U 2 (t)], the body axis directions with respect to world coordinates. Here R(t) is a rotation matrix.

26 Rigid Body Kinematics Particle Systems and Continuous Materials  Consider a time-varying vector written in the body coordinate system ξ(t) = R(t)s(t), where s(t) is the body coordinates varying with time. What the world observer sees What the body observer sees

27 Rigid Body Kinematics Particle Systems and Continuous Materials  The body origin has world velocity v cen =dX(t;C)/dt and world acceleration a cen = dv cen /dt.  Then the world velocity v wor V cen : the drag velocity, the velocity of the body origin relative to the world coordinates Dr/Dt, the velocity of P measured relative to the body coordinates w ⅹ r : the velocity due to rotation of the frame

28 Rigid Body Kinematics Particle Systems and Continuous Materials  Then the world velocity a wor =dv wor /dt 1.a cen : the translational acceleration of the body origin relative to the world coordinates 2.w ⅹ (w ⅹ r): the centripetal acceleration due to rotation of the frame 3.(Dw/dt) ⅹ r: the tangential acceleration due to angular acceleration 4.2w ⅹ (Dr/Dt): the Coriolis acceleration 5.D 2 r/Dt 2 : the acceleration of P relative to the body

29 Rigid Body Kinematics Newton’s Law of Physics  In the absence of external forces, an object at rest will remain at rest. If the object is in motion and no external forces act on it, the object remains in motion with constant velocity  For an object of constant mass over time, its acceleration a is proportional to the force F and inversely proportional to the mass m of the object. a = F/m. If the mass changes over time, the more general statement of the law is F = d(mv)/dt = ma + (dm/dt)v. mv is the linear momentum of the object. The application of an external force on an object causes a change in the object’s momentum over time.

30 Newton’s Law If a force is exerted on one object, there is a force of equal magnitude but opposite direction on some other body that interacts with it. (Action/reaction) Inertial frame: position, velocity and acceleration are measured with respect to some coordinate system. -> Inertial frame.  The inertial frame can be fixed or have a constant linear velocity and no rotation.  Kinetic energy must be measured in an inertial system. Noninertial frame: any other frame of reference.

31 Forces Gravitational Forces  Approximation of the gravitational force on the earth.(near the flat surface of the earth) F = -mgU. Spring Forces  F = -cΔU: c>0 is the spring constant. (Hooke’s law)

32 Forces Friction and Other Dissipative Forces  A dissipative force is one for which energy of the system decreases when motion takes place.  F dissipative = c|v| n. c is mostly a constant but can vary with time.

33 Forces Friction and Other Dissipative Forces  Friction A friction force between two objects in contact opposes the sliding of one object over the surface of the adjacent one. It is tangent to the surface of the adjacent object and opposite in direction to the velocity of the moving object. The magnitude of the frictional force is assumed to be proportional to the magnitude of the normal force between surfaces. It is also assumed to be independent of the area of contact and independent of the speed of the object once that object starts to move.

34 Forces Friction and Other Dissipative Forces  Viscosity A viscous force has magnitude with n=1. Typically, when an object is dragged through a thick fluid, a viscous force can be experienced. The force is modeled to have direction opposite to that of the moving object.

35 Forces Torque  Let F be the applied force and r the position of the particle relative to the origin τ=r ⅹ F.  For a system of p particles located at position r i.  For a continuum of mass that occupies a region R,

36 Forces Equilibrium  An object is in equilibrium if The sum of all external forces acting on the object must be zero. The torques on the object must sum to zero.  An object in equilibrium is not necessarily stationary. It is possible that the inertial frame in which the object is measured is moving with constant velocity.

37 Momenta Linear and Angular Momentum Mass of an object  Center of mass of an object A finite number of point masses (discrete) A solid body (continuous) Moments and products of inertia. These concepts are particularly important when discussing motion of a rigid body.

38 Linear Momentum Single particle: p = mv  The applied force and momentum are related by F=dp/dt. A system of particle: A continuum of mass: Conservation of momentum  If the net external force on a system of objects is zero, the linear momentum is a constant.

39 Angular Momentum Single particle: L=r ⅹ p=r ⅹ mv  Torque is the time derivative of angular momentum A system of particle: A continuum of mass: Conservation of momentum  If the net external force on a system of objects is zero, the angular momentum is a constant.

40 Center of Mass Discrete Mass in One Dimension Continuous Mass in One Dimension

41 Center of Mass Discrete Mass in Two Dimension Continuous Mass in One Dimension

42 Center of Mass Discrete Mass in Two Dimension Continuous Mass in One Dimension

43 Center of Mass Discrete Mass in Three Dimension Continuous Mass in Three Dimension

44 Moments and Products of Inertia Moment of inertia is another important quantity in physics. This is a measure of the rotational inertia of a body about an axis. The more difficult it is to set the object into rotation, the larger the moment of inertia about that axis.

45 Moments and Products of Inertia Moment of Inertia in One Dimension  A system of discrete particles With respect to the origin of the real line With respect to the center of mass  A continuous mass With respect to the origin of the real line With respect to the center of mass

46 Moments and Products of Inertia Moment of Inertia in Two Dimensions  A system of discrete particles With respect to the origin of the real line With respect to the center of mass  A continuous mass With respect to the origin of the real line With respect to the center of mass

47 Moments and Products of Inertia Moment of Inertia in Three Dimensions  A system of discrete particles Moments of inertia about the x,y and z axes. Moments of inertia about a line L through origin O. given parametrically as O+tD with unit-length direction vector D=(d 1,d 2,d 3 )

48 Moments and Products of Inertia Moment of Inertia in Three Dimensions  A system of discrete particles Inertia Tensor Moments and products of inertia for a continuum of mass occupying a region R

49 Moments and Products of Inertia The angular momentum  L = r ⅹ mv = mr ⅹ (w ⅹ r) = m(|r| 2 I -rr T )w = Jw The linear momentum equation p = mv. The torque equation  τ=Jα: α is the angular acceleration Newton’s second law: F = ma. Euler’s equations of motion M: the diagonal matrix of principal moments

50 Moments and Products of Inertia Mass and Inertia Tensor of a Solid Polyhedron  Mirtich’s method Reduction from 3D -> 2D  The construction uses the Divergence theorem from calculus for reducing volume integrals to surface integrals.  It is a reduction from three-dimensional integrals to two- dimensional integrals. Reduction from 2D -> 1D  The polyhedron surfaces are planar faces.  Projection of them onto coordinate planes are used to set up yet another reduction in dimension.  Green’s Theorem is used to reduce the planar integrals to lin e integrals around the boundaries of the projected faces.

51 Energy Work and Kinetic Energy  Total work done by the force on the particle as it travels over a time interval [t 0,t 1 ] with the velocity of the particle v(t)

52 Energy Conservative Force  The work is independent of the path.  It only depends on the starting and ending points.  The total energy of the system is conserved. Nonconservative Force  Friction, viscous drag when moving through a fluid and many forces that have explicit dependence on time and velocity.

53 Energy Potential Energy  The work done by the forces in transferring the system is referred to as the potential energy. For having a conservative force