In this paper, a new method for analyzing rigid body motion from measured data is presented. The approach is numerically stable, explicitly accounts for the errors inherent in measured data and those introduced by floating point arithmetic, automatically accommodates any number of rigid body particles, and is computationally efficient. The sole restriction on the data is that it represent 3 noncollinear particles of a rigid body.

The location and orientation of a rigid body in space can be defined in terms of three noncollinear points in the body. As the rigid body is moved through space, the motion may be described by a series of rotations and translations. The sequence of displacements may be conveniently represented in matrix form by a series of displacement matrices that describe the motion of the body between successive positions. If the rotations and translations (and hence the displacement matrix) are known then succeeding positions of a rigid body may be easily calculated in terms of the initial position. Conversely, if successive positions of three points in the rigid body are known, it is possible to calculate the parameters of the corresponding rotation and translation. In this paper, a new solution is presented which provides explicit formulas for the rotation and translation of a rigid body in terms of the initial and final positions of three points fixed in the rigid body. The rotation matrix is determined directly whereupon appropriate rotation angles and other information can subsequently be calculated if desired.

In Part I of this paper, a combination of Airy stress functions and direct displacement functions was utilized to obtain the plane elasticity solution for the stresses and displacements in a multilayer laminated orthotropic strip subjected to a temperature gradient that is arbitrarily symmetric in the longitudinal direction. The solution is exact for the specific boundary conditions associated with satisfaction of zero slope of transverse displacement at the strip ends and shear traction free edges at the strip ends. In Part II of this paper, numerical results are presented for several examples and compared to detailed finite element analyses for approximate zero edge slope and free edge boundary conditions. The results indicate the shear and peel stress concentrations and axial stress distributions for the boundary condition are in excellent agreement with the finite element analysis results for the zero edge slope case and correlate with the free edge boundary condition over a broad range of temperature gradient cases. This correlation with the free edge numerical analysis indicates the theoretical approach is feasible as a conceptual design tool for a reasonable range of free edge engineering problems.

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In Part I of this paper, a combination of Airy stress functions and direct displacement functions is utilized to obtain the plane elasticity solution for the stresses and displacements in a multilayer laminated anisotropic strip subjected to a temperature gradient that is arbitrarily symmetric in the longitudinal direction. The method of analysis utilized departs from previous works in that an eigenfunction solution is developed assuming a length coordinate expansion of the stresses and displacements, with an exponential variation in the thickness coordinate direction. This avoids the shortcoming of the nonorthogonal Fadle-Papkovitsch eigenfunctions in that an orthogonal series repesentation suitable for modeling of nonuniform as well as uniform temperature distributions is obtained. The resulting eigenfunctions satisfy the necessary equations of equilibrium, conditions of displacement compatibility, requirements on interlaminar stress and displacement continuity, traction free surface conditions, and shear traction free edge conditions. As interfacial conditions and strain compatibility are satisfied exactly, the approach overcomes the inter-facial approximations and average sense compatibility invoked in previous complementary virtual work approaches. The solution does not exactly satisfy the free edge normal traction requirement since only resultant force is enforced to zero; however, convergence for enforced zero transverse slope at the strip ends can be established, as the eigenfunctionsare orthogonal. Thus the solution is exact for these edge conditions. In Part II of this paper, numerical results are presented for several examples and compared to those obtained from our own MSC/NASTRAN finite element analyses. The results indicate the shear and peel stress concentrations and axial stress distributions are in excellent agreement with the finite element analyses for the zero edge slope boundary condition. Also, good correlation was determined with finite element analysis results for the free edge boundary condition over the range of problems considered. This correlation with the finite element numerical results verifies the solution and indicates application of the solution as an approximation to free edge engineering problems is reasonable for abroad range of practical cases.