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)COS A)DEL aerodynamic forces aerodynamic stability derivatives aeronautical aircraft angular velocity ax(i ax¹ ax² ax³ B)SIN(B base vectors bivector body axes Christoffel symbols coefficients coordinate transformation equations COS(A COS(A)SIN(A COS(A)SIN(A)SIN(B COS(B cos² covariant and contravariant curvilinear coordinate denoted derivatives with respect differential DM₁ dt dt dx¹ dx² dx³ Earth-fixed equa equations of motion əx(i Əx¹ Əx² Əx³ əxi əyi əyi əyi forces and moments formulation function I:1 THRU inertia inputs MACSYMA mathematical metric tensors mixed tensor models msec obtained P₁ physical components polar coordinate system rate of change Ricci tensor ROW 3 COLUMN scalar SIN(A SIN(B sin² substitution from equation summation convention tensor components tensor of rank thrust vector tion transformation law U₁ U₂ x¹ sin x2 x¹)² аха ду дуг дуз дуі дут дха ахв
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Stran 146 - 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 167 - 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 95 - U^(x) be a motion vector component in this system of axes. Then the stability derivatives with respect to motion components, as measured in the y system of axes, are related to the corresponding derivatives in the x system of axes by the following equation: (2.7.1) It should be noted that force, moment, and motion components obey the same transformation law as the system coordinates, that is Hence (2.7.2) (2.7.3) Therefore, from equations (2.5.2) and (2.7.3) lt (2.7.4) Substitution from (2.7.2) and...
Stran 7 - ... covariance because if general coordinate transformations are contemplated, the transformation law for the components of a contravariant vector denoted by superscripts differs from that for a covariant vector denoted by subscripts. It must be emphasized, however, that the covariance or contravariance of tensor components is not an intrinsic property of the entity under consideration. The distinction is due to the way in which the entity is related to its environment, or the coordinate system,...
Stran 174 - More importantly, if the functions g,y(x) are such that the system of equations (1.13.12) has no solution, then no admissible transformation of coordinates exists, which reduces equation (3.2.1) to the Pythagorean form. In this case, the manifold is nonEuclidean and the use of coordinate transformation equations as inputs will fail. by The manual derivation of the Christoffel symbols of the first kind for a cylindrical polar coordinate system would proceed as follows: Referring to equation (3.2.6),...
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.
Stran 182 - C(3,3,3) = 0 3.3 THE VELOCITY VECTOR Three methods of obtaining the metric tensors have been indicated: one of these uses the method of vector calculus; another uses the known differential coefficients from the coordinate transformation equations; and the method described in the preceding section uses the coefficients of the fundamental quadratic form. Since the coordinate transformation method is more adaptable to digital logic than the vector method, it can be used for all Euclidian applications....
Stran 224 - This is 4.2 (4.1.16) 4.2 THE PHYSICAL FORM OF THE NAVIER-STOKES EQUATIONS Because equation (4.1.16) is a tensor equation, all the velocity and force components occurring in this equation are tensor components. This form of the equation is well suited to theoretical studies. However, in practical applications, it is the physical components that are of interest (ref. 1 ). The tensor components of a vector, or first-order tensor, are related to its physical components as follows (ref. 2): (4.2.1) 3x(')...
Stran 332 - A statement of the fact that the /th component of the linear velocity vector is a function of time, requires the use of the DEPENDENCIES function. The use of this function permits the system to differentiate the components C/.- with respect to time.
Stran 219 - ... Celestial Mechanics. Second ed. The Macmillan Company, 1959. APPLICATIONS TO FLUID MECHANICS 4. 1 Formulation of the Navier-Stokes Equations and the Continuity Equation 4.2 The Physical Form of the Navier-Stokes Equations 4.3 Formulation of the Christoffel Symbols of the Second Kind 4.4 References 4.1 4.1 FORMULATION OF THE NAVIER-STOKES EQUATIONS AND THE CONTINUITY EQUATION The Navier-Stokes equations form the basis of the whole science of fluid mechanics (ref. 1). The technique described in...