Equations of motion of the typical section

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samcam
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Berichten: 4
Lid geworden op: 24 feb 2015, 11:06

Equations of motion of the typical section

Bericht door samcam » 18 mar 2015, 12:42

beste,

momenteel ben ik bezig met een taak, ik loop hier vast op een onderdeel en vroeg me af of iemand mij kan helpen bij deze taak. De afbeelding voor deze opdracht kunt u vinden bij de volgende link: http://nl.tinypic.com/r/29yjgjt/8. de bedoeling is dat er vergelijkingen worden opgesteld voor de pitch en de plunge.

Met Vriendelijke Groet,
samcam


General
A typical wing section (75% of half span) moves at airspeed U under an angle of attack α_0. The section is assumed to obey thin airfoil theory. The structural part of a wing section is modelled using a linear spring (stiffness K_h) and a torsional spring (stiffness K_α): in that way the plunging (the vertical motion) and pitching motion of the airfoil are described. Furthermore, small deviations from the mean equilibrium position are assumed. The distance between the elastic axis (EA) and the aerodynamic centre (AC) is designated the non-dimensional eccentricity e. The wing chord is frequently written in terms of two semi chords b, so c=2b. In this problem we assume that the angle of attack variations are small (≤10°) (what is the consequence for the dimensions of the airfoil given in figure 1 ?).
Since α and h are perturbation variables, the static gravitational force can be neglected for this problem. Assume that α_0=0°.

PART B: Equations of motion of the typical section
Since the airfoil is subjected to two motions, plunge and pitch, you will need two equations:
One equation for the vertical force;
One moment equation, with respect to the elastic axis;
Assume uniformly distributed mass of the airfoil section (so the cg is at c/2);
You should realize, that I_CG≠I_EA;
Since the moment center and the c.g. do not coincide a static mass moment that couples the plunge and pitch motion should be introduced. This means that it is crucial to draw a correct kinetic diagram that shows the dynamic forces and moments;
Do not forget to draw a correct free body diagram as well.
Note 1: when deriving the equations of motion first use a general symbol for the lift L and the aerodynamic moment M_ac, in part C2 the expressions for these loads should be substituted.
Note 2: leave the analytical expressions for the mass moment of inertia and the static mass moment in the mass matrix.

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