Lecture 3: Kinematics of Particles

Transcription

1 Lecture 3: Kinematics of Particles Recap: Numerical integration [Appendix A, 2.1] 2 and 3-Dimensional Motion Resolving vectors in 3-Dimensions Differentiation of vectors leading into a review of Polar coordinates [2.3] Path coordinates [2.4] State vector, state space: Extensions to 2 and 3 dimensions Reducing higher order differential equations to first order differential equations

2 2-Dimensional motion Positions require two coordinates (Cartesian coordinates) x(t) y(t) Need position vectors r(t) = x(t) i + y(t) Need a reference frame Fixed reference frames Frame fixed to i, orthogonal, unit vectors Classroom, Earth, Center of the Earth, Center of the Sun, Center of the Universe, Moving reference frames Frames fixed to moving bodies Need to be able to differentiate vectors v(t), a(t), (t), s(t), i

3 Differentiation of Vectors B e 3 e 2 e 1 u = u x i + u y i + u z k u Differentiate with respect to time: z y k u = u 1 e 1 + u 2 e 2 + u 3 e 3 O i Differentiate with respect to time: x

4 Transformations between unit vectors Understanding the relationship between sets of unit vectors is very important Visualize Write down the dot products e 1 e 2 e 3 i k i.e 1 i.e 2 i.e 3.e 1.e 2.e 3 k.e 1 k.e 2 k.e 3 Dot product of unit vectors = Cosine of angle between vectors

5 Polar coordinates e θ e r i e r cosθ sinθ e θ sinθ cosθ y P r O θ i x

6 Polar coordinates y O θ i r e θ x P e r d r e dt de dt θ i = = = = e r cosθ sinθ sin θ & θ i + e & θ θ cosθθ& e r θ & d v re r& P / O = ( r ) = er + dt i e θ sinθ + cosθ cosθθ& sinθθ& r & θ e θ 2 a = (& r r & θ ) e + ( r && θ 2r& & θ ) P / O r + e θ

7 Typhoon Imbudo, July 2003 Hurricanes, anticyclones (later) Gafilo, March 2004

8 Path coordinates i e n cos β sin β e t sin β cos β e t P y O r i r C e n β x v a P p = = r v c e t = n r c β & & 2 β e + v 2 r C Normal acceleration r e c t & β e v& Tangential acceleration t

9 State Space

10 State MEAM 211 Actual (internal) state of the system Geometry of Behavior, Abraham 1981 Mathematical model requires an idealized state The idealized state must be observable (in order for results to be practical)

11 Modeling: State Fang Exposure q = q q 1 2 Ear Attitude Geometry of Behavior, Abraham 1981

12 Modeling: State Geometry of Behavior, Abraham 1981 Ear Attitude Fang Exposure

13 State Space and Time Geometry of Behavior, Abraham 1981

14 Integration in State Space ODE in one scalar variable We know how to integrate or

15 State Space Notation Motivation We know how to integrate or Can we do 2 nd or higher order differential equations?

16 State Space Notation Motivation We know how to integrate or Can we do 2 nd or higher order differential equations? Yes - n th order ODEs can be converted into n first order ODEs

17 Converting 2nd order ODEs to 2 1st order ODEs Example

18 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector

19 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector Write state equations

20 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector Write state equations With initial conditions

21 Solving Equations of Motions for Particles Example Particle in 3-D subect to thrust, gravity and drag State vector is 6x1 6 State Equations

22 MATLAB Example A ball is thrown upward against gravitational attraction and air resistance with an initial velocity of 30 meters/second. The air resistance opposes the velocity and is proportional to the square of the velocity. The acceleration is: a = - g cv 2 sign(v) where g = 9.81 meter/sec 2 and c = /meter. Solve for the position and velocity of the particle as a function of time through a six second time interval. main.m timeinterval=[0, 6]; % interval for integration x0=[0; 30]; % initial position = 0, velocity = 30 intfn ('vertical', timeinterval, x0); function xdot=vertical(t, x); vertical.m xdot=[xdot1; xdot2]; c=0.001; % c= /m g=9.81; % g=9.81 m/sec/sec xdot1 = x(2); % derivative of position = velocity xdot2 = -g - c*x(2)*abs(x(2)); % acceleration

23 Example on particle kinematics Path and Polar Coordinates

24 Example y MEAM 211 B r 3 m A 4 m θ x O Carousel i The carousel rotates about an axis perpendicular to the plane of this paper passing through O at a constant rate of 3 rotations per minute. Your friend is standing (feet fixed to the carousel and not moving relative to the carousel) on the carousel at point B. You are standing at A with your feet on firm ground and at rest with respect to the Earth.

25 Example y e t MEAM 211 e θ θ B B e r θ r e n 3 m θ x A 4 m O Carousel i e t e r cosθ e θ sinθ e n sinθ cosθ