Mathematical Modeling of Diverse PhenomenaNational Aeronautics and Space Administration, Scientific and Technical Information Brach, 1979 - 394 strani Tensor calculus is applied to the formulation of mathematical models of diverse phenomena. Aeronautics, fluid dynamics, and cosmology are among the areas of application. The feasibility of combining tensor methods and computer capability to formulate problems is demonstrated. The techniques described are an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer. |
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A)sin² aerodynamic forces aerodynamic stability derivatives aircraft angular velocity ax(i ax¹ ax² ax³ B)COS B)SIN(B base vectors bivector body axes Christoffel symbols coefficients contravariant coordinate transformation equations corresponding COS(A COS(A)SIN(A COS(B cos² covariant covariant and contravariant covariant derivative curvilinear coordinate denoted derivatives with respect differential dt dt dx¹ dx² dx³ Earth-fixed equa equations of motion Əx¹ Əx² Əx³ əxi əyi əyi əyi forces and moments formulation I:1 THRU inputs MACSYMA mathematical metric tensors mixed tensor models Navier-Stokes equations obtained P₁ physical components polar coordinate system rate of change Ricci tensor ROW 1 COLUMN scalar SIN(A SIN(B sin² substitution from equation tensor components tensor of rank thrust vector tion transformation law U₂ velocity components x¹ sin x² x¹)² аха ахв ду ду³ дув дуг дуз дуі дут дха
Priljubljeni odlomki
Stran 142 - Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Stran 163 - The tensor components of a vector are not, in general, the same as the physical components. Instead, they are components that obey transformation laws corresponding to their variance. The transformation laws for covariant and contravariant vectors are given by equations (1.6.5) and (1.6. 3), respectively. It may be noted that when the base vectors define an orthogonal Cartesian reference frame, the tensor components are the same as the physical components. As a consequence of the geometrical simplification...
Stran 323 - COS2(A)SIN(B) 2' 3 r3' 2 330 DP p COS(A)SIN(A)COS2(B) - DP D SIN2(A)COS(B) COS2(A)COS(B) - D. D COS(A)SIN(A))DEL(P, F'* 6.14 FORCES PRODUCED BY LINEAR ACCELERATION PERTURBATIONS The procedure used in the preceding two sections may, with equal facility, be used to formulate the aerodynamic forces produced by linear acceleration perturbations. However, in this case the required forces are obtained by multiplying the transformed aerodynamic stability derivatives, with respect to acceleration components,...
Stran 77 - For this system, the x-axis is in the same horizontal plane as the relative wind at all times. In addition to the wind axes and the wind-tunnel stability axes, there are other systems of axes fixed in the body and moving with the body. These are referred to as body axes. In aerospace applications, a body axis system has the x-axis fixed along the longitudinal center line of the body, the y-axis normal to the plane of symmetry, and the z-axis in the plane of symmetry. It should also be noted that...
Stran 21 - Vector Components To facilitate the computer processing of vectors and dyadics, all such entities should be expressed in terms of their tensor components and a corresponding set of base vectors, rather than in terms of their physical components and a set of unit base vectors. When referred to a general curvilinear coordinate system, a vector A...
Stran 18 - 4> • a where 4> is the inertia tensor or, as it is sometimes called, the inertia dyadic, that is, As in the case of the stress tensor, it is seen that the inertia tensor assumes the form of a dyadic, or a two linear form in the vector f. This justifies its classification as a...
Stran 214 - In these equations the summation convention is again assumed. That is to say, if in any term an index occurs twice, the term is to be summed with respect to that index for all admissible values of the index. In relativistic mechanics, equation (5.1.1) is replaced by the following trajectory equation, which is the equation of a geodesic (ref.
Stran 309 - The formulation of models of aeronautical systems for simulation and other purposes involves at least 12 equations: 3 force equations; 3 moment equations; 3 Euler angle equations, or 9 direction cosine equations to determine the spatial orientation of the body; and 3 equations to determine the location of the body in inertial space. In view of this complexity, it is important to mechanize as much of the formulation as possible.
Stran 214 - To complete the system, two more equations are added. One of these is the equation of state that relates the pressure and the density. It may be written as follows: p=p(p) (4.1.2) The other equation expresses the principle of conservation of mass and assumes the form (ref . 1 ) Furthermore, if the process is not isothermal it is necessary to make use of the energy equation, which draws up a balance between mechanical and thermal energy and furnishes a differential equation for the temperature distribution....
Stran 25 - This is seen to satisfy the mathematical definition of a covariant vector given in equation (1.4.6). 1.5 BASE VECTORS The transformation laws and, hence, the covariant and contravariant character of the base vectors and their reciprocals may be obtained as follows: Let the differential of a position vector be denoted by dr. Then if...