Particle Motion

1
Particle Motion

• Refresher on translational kinematics and
  kinetics.
• Linear momentum, Newtons laws, coordinate
  frames.
• Examples from aircraft performance, orbital
  mechanics, and launch vehicle trajectories.

2
Newtons Second Law

• Applicable to mass particles with constant mass
  and zero volume extensible to rigid bodies
• Not as simple as it looks
• Determination of the force usually requires
  free-body diagram can be extremely complicated
• Determination of the acceleration sometimes
  involves complicated kinematics
• Acceleration is second derivative of position
  vector with respect to time

3
Reference Frame Position Vector

• Position vector must be from inertial origin to
  mass particle

If expressed in terms of inertial frame
components, then differentiation is easy
4
Inertial Origin

• An inertial origin is a point that is not
  accelerating with respect to any other inertial
  origin
• Alternatively, an inertial origin is a point for
  which Newtons laws are applicable
• There is no known inertial origin, but for most
  problems an origin can be found that is inertial
  enough
• For some problems, an Earth-fixed reference point
  is sufficient, whereas for others, the rotation
  of the Earth must be considered

5
Inertial Reference Frame

• An inertial reference frame is a set of three
  unit vectors that are mutually perpendicular,
  with their origins at a single inertial origin,
  and whose directions remain fixed with respect to
  inertial space
• Alternatively, an inertial reference frame is a
  frame for which Newtons laws are applicable
• Usually the dc analyst must determine the
  simplest frame that is inertial enough
• What about these vehicles?
• Paper airplane, Cessna 152, B2, container ship,
  Space Shuttle, Apollo 11, Cassini, Rama

6
Reference Frames

• A reference frame is a set of three mutually
  perpendicular (orthogonal) unit vectors
• Typical notations include
• Typical reference frames of interest for vehicles
  include
• ECI (Earth-centered inertial)
• Perifocal (Earth-centered, orbit-based inertial)
• ECEF (Earth-centered, Earth-fixed, rotating)
• Orbital (Earth-centered, orbit-based, rotating)
• Wind (vehicle-centered, rotating)
• Body (vehicle-fixed, rotating)

7
Earth-Centered Inertial (ECI)

• Also called Celestial Coordinates
• The I-axis is in vernal equinox direction
• The K-axis is Earths rotation axis,
  perpendicular to equatorial plane
• The J-axis is in the equatorial plane and
  finishes the triad of unit vectors
• The IJ-plane is the equatorial plane

Towards Sun at vernal equinox
8
Perifocal Frame

• Earth-centered, orbit-based, inertial
• The P-axis is in periapsis direction
• The W-axis is perpendicular to orbital plane
  (direction of orbit angular momentum vector,
  )
• The Q-axis is in the orbital plane and finishes
  the triad of unit vectors

9
Orbital Frame

• Similar to roll-pitch-yaw frame, for spacecraft
• The o3 axis is in the nadir direction
• The o2 axis is in the negative orbit normal
  direction
• The o1 axis completes the triad, and is in the
  velocity vector direction for circular orbits

10
Body-Fixed Frame

• Applicable to any type of vehicle
• Typically denoted using b unit vectors
• For spacecraft
• The b3 axis is in the nadir direction
• The b2 axis is in the negative orbit normal
  direction
• The b1 axis completes the triad, and is in the
  velocity vector direction for circular orbits

11
Vector, Frame, and Matrix Notation
12
Position, Velocity and Acceleration Vectors

• Derivatives are simple because unit vectors are
  constant, in direction and magnitude

13
Application of

• Need to know the components of the force vector

Three second-order ordinary differential
equations. Linearity and coupling depend on
nature of forces.
14
Simple Feedback Control Forces

• Suppose the forces take the following form

Here the k terms are feedback gains, whose
values are selected by the dc analyst. The
terms are proportional to the position errors,
and the terms are proportional to the
velocity errors, or to the derivatives of the
position errors, hence the controller is a PD
controller
15
Equations of Motion with PD Control

• First, define some new variables, termed the
  states

x is the state vector
16
Linear System of Equations (PD Control)

• Because of the unique form of these equations,
  they can be written in matrix form
• Each right-hand side appears as a summation of
  constants multiplying states, with the states
  always appearing linearly
• Thus, the equations comprise a system of linear,
  constant-coefficient ordinary differential
  equations

17
Linear System (PD) (continued)

• Simple rearrangement of the equations of motion
  leads to the block matrix form

Because A is constant, the system is easily
solved
18
PD Example Plots

• The nature of the PD controller is that it causes
  all of the states to approach zero
  asymptotically thus it is a stable controller

Note poor quality of these graphs. Not
acceptable for technical presentations. How to
improve?
19
PD Example Exercise

• The lecture notes include the Matlab code for
  numerical integration of this system of equations
• Implement the Matlab code and carry out some
  simple experimentation with varying gains and
  initial conditions
• Compare the numerically integrated solution to
  the exact solution on the previous slide
• We will return to the closed form solution later
  in the course

20
Polar Coordinates, Rotating Reference Frame

• Convenient to visualize frame origin at particle,
  but position vector must be from inertial origin.
• Unit vectors of rotating reference frame are
  constant in length, but not in direction
• Unit vectors are orthogonal, in spite of
  distortion that appears in figure!

21
Rate of Change of the Unit Vectors

• The unit vectors change because the angle ??
  changes
• Denote small change in time t and corresponding
  change in angle ? with ?t and ??
• Consequently, the required derivatives can be
  written as

Exercise convince yourself this limit is correct
22
Orbital Mechanics Application

• When the net force is easily decomposed into
  radial and transverse components, polar
  coordinates are quite useful
• The most common example is the simple two-body
  problem, where the force is the inverse-square
  universal gravitational law

23
Simple Orbit Example

• Suppose we select units such that GM1 then the
  force is
• Application of Newtons 2nd law leads to the two
  scalar nonlinear ordinary differential equations
• Why are these equations nonlinear?

Exercise derive these equations
24
Rewrite as

• Define new state variables

Note the nonlinearities, as well as the
singularity for zero radius
25
Orbital Mechanics Example (cont)

• We can identify a special solution to the
  equations of motion
• Suppose x1constant (circular orbit), then x30,
  so x4constant also (constant angular rate)
• From the d.e. for x3 (which must be zero), we can
  deduce that
• This result is a form of Keplers Third Law,
  which relates the orbital period to the orbital
  radius the 1 is a result of our choice of
  units such that GM1

Exercise verify the equivalence between this
result and Keplers Third Law
26
Some Numerical Results

• Plot generated using Matlab code from lecture
  notes.

27
Particle Motion so far

• fma
• Reference frames, inertial origins and frames
• Position, velocity, acceleration
• State vector form of EOM
• PD control example
• Rotating reference frame, polar coordinates
• Simple orbital motion

28
Normal-Tangential Coordinates

• For a circular orbit, the radial-transverse frame
  and coordinates are also normal and tangential to
  the spacecraft flight path, respectively
• We can also define a frame that has these
  properties for non-circular orbits, but the
  gravitational force is more complicated in such a
  frame
• This normal-tangential frame is well-suited for
  problems involving aircraft trajectories, since
  lift and drag are aerodynamic forces defined in
  the normal and tangential directions, respectively

29
Normal-Tangential Coordinates
For aircraft, the velocity appears in the
aerodynamic forces, and these forces are defined
in the normal and tangential directions Unit
vectors of rotating reference frame are constant
in length, but not in direction
Flight path
30
Rate of Change of the Unit Vectors

• The unit vectors change because the angle ??
  changes
• Denote small change in time t and corresponding
  change in angle ? with ?t and ??
• Consequently, the various derivatives can be
  written as

Note that if ? is increasing, then the radius of
curvature of the flight path emanates from the
positive direction, whereas if ? is
decreasing, then the radius of curvature emanates
from the negative direction. The normal
acceleration (and hence the net normal force) is
in the same sense.
Exercise convince yourself
31
Normal-Tangential Coordinates

• The speed can generally be written in terms of
  the angular rate and the radius of curvature as
• Hence the acceleration vector can be written as
• This acceleration vector can be used in a
  straightforward way in developing the equations
  of motion for an aircraft modeled as a point mass
  moving in a vertical plane

32
Simplest Aircraft Translational Motion

• The four forces
• Lift L, perpendicular to flight path
• Drag D, parallel to flight path
• Weight W, toward center of Earth
• Thrust T, generally inclined wrt flight path

L
chord line
a
T
aT
q
D
W
33
Application of Newtons 2nd Law

• The simplest problem is the case of straight,
  level, non-accelerating flight, for ??T0
• Straight
• Level
• Equations simplify

34
Thrust Required for Straight, Level Flight

• Solve the thrustdrag, liftweight equations to
  obtain required thrust
• Obvious conclusion required thrust is minimized
  when L/D is maximized
• Further analysis requires stating lift and drag
  in terms of velocity and dynamic pressure
• Dynamic pressure where ?? is atmospheric
  density

35
Lift and Drag

• The lift force is written in terms of dynamic
  pressure, a characteristic area, and a
  dimensionless variable called the lift
  coefficient
• Similarly, the drag force is
• Keep in mind that dynamic pressure is

36
Drag Coefficient
37
Drag Polar

• The drag coefficient can be written as
• The CD,0 term is the zero-lift drag term, or
  parasite drag coefficient
• The term involving the lift coefficient is called
  the induced drag
• The term AR is the aspect ratio, AR b2/S
• The term 0 lt e lt 1 is the Oswald efficiency factor

38
Thrust Required

• Putting it all together, we can write

Why are lift and drag coefficients higher at low
speeds and lower at high speeds?
39
Lagranges Equations

• Brief treatment applicable only to systems of n
  particles subject only to conservative forces
• Basic idea
• Write down the position vectors of the n
  particles in terms of independent generalized
  coordinates
• Differentiate the position vectors to get
  velocity vectors in terms of generalized
  velocities
• Write down the kinetic energy, T
• Write down the potential energy, V
• Form the Lagrangian, LT-V
• Take derivatives of the Lagrangian to get the
  differential equations of motion

40
Lagranges Equations (2)

• The position vectors must be written in terms of
  independent generalized coordinates, denoted qk,
  k1,,N
• For example, for a particle with unconstrained
  motion in three dimensions, one might use q1x,
  q2y, q3z
• For a particle constrained to move on a circle in
  a plane (e.g., a pendulum), there is only one
  degree of freedom, so one might use q1q
• In general, the position vectors are written as
• Consequently, the velocity vectors are

41
Lagranges Equations (3)

• The potential energy depends only on the
  positions of the particle
• The kinetic energy depends on velocities
• Thus the Lagrangian depends on generalized
  coordinates and velocities

42
Lagranges Equations (4)

• Take the derivatives of the Lagrangian
• This step gives N second-order ordinary
  differential equations describing the motion of
  the n particles
• If there were non-conservative forces acting on
  the system of particles, then the right-hand
  sides of these equations would have non-zero
  terms, called generalized forces.

43
Lagranges Equations Example

• Consider a single particle of mass m connected to
  a linear spring with spring constant k, suspended
  in a constant gravitational field with
  acceleration g, and constrained to move in a
  vertical plane, i.e., an elastic planar pendulum.
  Assume, for simplicity, that the unstretched
  length of the spring is zero.
• Use polar coordinates, measured from the fixed
  end of the spring. The position of the particle
  with respect to this inertial origin is
• The potential energy associated with the spring
  is
• The potential energy associated with gravity is
• where is the angle between and the
  downward direction

44
Lagranges Equations Example (2)