a b s t r a c tA reformulation of the Discrete Energy-Averaged model for the calculation of 3D hysteretic magnetization and magnetostriction of iron-gallium (Galfenol) alloys is presented in this paper. An analytical solution procedure based on an eigenvalue decomposition is developed. This procedure avoids the singularities present in the existing approximate solution by offering multiple local minimum energy directions for each easy crystallographic direction. This improved robustness is crucial for use in black-box finite element codes. Analytical simplifications of the 3D model to 2D and 1D applications are also presented. In particular, the 1D model requires calculation for only one easy direction, while all six easy directions must be considered for general applications. Compared to the approximate solution procedure, it is shown that the resulting robustness comes at no expense for 1D applications, but requires almost twice the computational cost for 3D applications. To find model parameters, we employ the average of the hysteretic data, rather than anhysteretic curves, which would require additional measurements. An efficient optimization routine is developed that retains the dimensionality of the prior art. The routine decouples the parameters into exclusive sets, some of which are found directly through a fast preprocessing step to improve accuracy and computational efficiency. The effectiveness of the model is verified by comparison with existing measurement data.
Motivated by the problem of synthesizing a pattern of flexures that provide a desired constrained motion, this paper presents a new screw theory that deals with “line screws” and “line screw systems.” A line screw is a screw with a zero pitch. The set of all line screws within a screw system is called a line variety. A general screw system of rank m is a line screw system if the rank of its line variety equals m. This paper answers two questions: (1) how to calculate the rank of a line variety for a given screw system and (2) how to algorithmically find a set of linearly independent lines from a given screw system. It has been previously found that a wire or beam flexure is considered a line screw, or more specifically a pure force wrench. By following the reciprocity and definitions of line screws, we have derived the necessary and sufficient conditions of line screw systems. When applied to flexure synthesis, we show that not all motion patterns can be realized with wire flexures connected in parallel. A computational algorithm based on this line screw theory is developed to find a set of admissible line screws or force wrenches for a given motion space. Two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.
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