[1]
The velocity and angular velocity are as follows.
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[2]
According to the relative description, flight kinematics could be expressed by rigid movement in the body coordinate frame and elasticity movement relative to the body coordinate frame. Rigid movement expressed by the vector and . Easticity movement expressed by the vector and. This paper only consider linear elasticity deformations. By mode coordinate method, and are as follows.
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[3]
Define generalized coordinate vector and quasi-coordinates vector of the vehicle as follows.
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[4]
From above, we can easily achieve as follows.
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[5]
The matrixes and are coordinate transform of inertia reference frame and body coordinate frame. Flight Dynamics Modeling Based on the method of analytical mechanics, kinetic energy and flexibility potential energy of the vehicle were firstly developed, and then concise flight dynamics equations were achieved by Lagrange's equations under quasi-coordinates. Vehicle Kinetic Energy. The kinetic energy of vehicle is as follows.
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[6]
is kinetic energy density, is integral district, andare one and two order inertia tensors respectively which are as follows.
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[7]
So, the kinetic energy of vehicle is as follows.
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[8]
Generalized mass matrixes are as follows.
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[9]
Vehicle Flexibility Potential Energy. Elasticity deformations of vehicle airframe include torsion deformations around , bend and cut deformations, , around and, we can easily get as follows.
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[10]
and are partial derivative relative to respectively. flexibility potential energy are as follows.
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[11]
Generalized elasticity stiffness matrixes are as follows.
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[12]
Lagrange's equations based on quasi-coordinates. From 3. 1 we know that kinetic energy of vehicle can be easily expressed as the function of generalized velocity . In order to get concise flight dynamics equations, we deduce as follows.
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[13]
Lagrange's equations.
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[14]
Above equations multiply, so get the following equations.
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[15]
By the expressions of 3. 1 and 3. 2, flight dynamics equations using Lagrange's equations based on quasi-coordinates can be expressed as follows.
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[16]
Results and analysis The vehicle parallel，rotation and elasticity deformations movements exist inertia couplings. Many inertia couplings can be eliminated by choosing appropriate body coordinate frame. For example, if one order inertia tensor, momentum and angular momentum are all equal to zero as follows.
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[17]
From above, flight dynamics of flexible flight vehicles are same as the ones of rigid flight vehicles and parallel movements，rotation movements and Elasticity deformations movements of the vehicle are as follows respectively.
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[18]
Using the model of hypersonic flight vehicles and relative parameters which were achieved under the balance state by Schmidt, the flight dynamics simulation model of the vehicle were established. The results of pith angular velocity, pith angle, velocity and attack angle are as follows: Fig 2. Pith angular velocity response of AHFV Fig 3. Pith angular velocity response of AHFV Fig4. Velocity response of AHFV Fig5. Attack angle response of AHFV Conclusions Based on the method of analytical mechanics, kinetic energy and flexibility potential energy of the vehicle were firstly developed, and then concise flight dynamics equations were achieved by Lagrange's equations under quasi-coordinates. The model can fully include coupling characteristics of flight dynamics. If viscidity friction was considered, Rayleigh dissipation function should be included in the Lagrange's equations. But the flight dynamics equations don't changed substantially, so the modeling method brought forward in this paper is not only efficient but also essential for the research of dynamics, stability, control, couplings and so on. Acknowledgements The financial support of the School of Astronautics and Northwestern Polytechnical University are gratefully acknowledged. Referencesd.
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