generalized velocities

Course AE4-T40
Lecture 2:
2D Models
Of Kite and Cable
Overview
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2D system model
3D system model
Kite steering
3D kite models
2D System Model
Tether: lumped masses with rigid links. External Forces applied at the lumped masses
Kite is controlled via angle of attack, attitude and flexible dynamics are ignored
Generalized coordinates are the angular rotation of each link θj, Φj, with j= 1…n,
and n the number of masses
Length of the links lj(t) are a function of time and thus no generalized coordinates
When the line length is changed, only line length of segment ln is changed
2D System Model
For 2D case assume out of plane angle Φj = 0 for all cable elements
For illustration purposes now consider a system model with n = 3
With unit vectors i, j, k in x, y and z (using only I and j) we now define the inertial
positions of the three point masses with respect to the reference axes in
terms of the generalized coordinates
Position Vectors of line lump masses
Corresponding velocities and accelerations are determined by
differentiation of the position vectors
Velocity vectors of line lump masses
In general we have:
Acceleration vectors of
line lump masses
Kanes equations of motion
Lagrange’s equations are second order differential equations in the generalized
coordinates qi (i = 1,…,n). These may be converted to first-order differential
equations or into state-space form in the standard way, by defining an additional
set of variables, called motion variables.
To convert Lagrange’s equations, one defines the motion variables simply as
configuration variable derivatives, sometimes called generalized velocities. Then
the state vector is made up of the configuration and motion variables:
the generalized coordinates and generalized velocities.
In Kane’s method, generalized coordinates are also used as configuration
variables. However, the motion variables in Kane’s equations are defined as
functions that are linear in the configuration variable derivatives and in general
nonlinear in the configuration variables. The use of such functions can lead to
significantly more compact equations. The name given to these new motion
variables is generalized speeds and the symbol commonly used is u.
Kanes Equations
Kanes equations
Generalized Inertia Forces for n = 3
Generalized Inertia Forces for n = 3
Generalized Inertia Forces for n = 3
Generalized Inertia Forces for n = n
Generalized External Forces
Kane’s equations of motion
Where for our system the external forces are composed of:
-Tether drag
-Kite Lift and Drag
-Gravity
Tether Drag
Kite Lift and Drag
Gravity
Model results
Control results
Control Results
Control Results