Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order

A new singularity-free analytical formula has been developed for the gravity field of arbitrary 3D polyhedral mass bodies with horizontally and vertically varying density contrast using third-order polynomial functions. First, the observation sites are moved to the origin of the coordinate system. Then, the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies. Furthermore, singularity-free closed-form solutions are derived for these line integrals over the edges. Thus, the observation sites can be located inside, on, or outside the 3D distributions. A synthetic prismatic mass body is adopted to verify the accuracy and singularity-free property of our newly developed analytical expressions. Excellent agreements are obtained between our solutions and other published closed-form solutions with relative errors in the order of [Formula: see text] to [Formula: see text]. In addition, an octahedral model and a near-Earth asteroid model are used to verify the accuracy of the presented method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach.

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Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

Adaptive Structures and Materials Systems ◽

10.1115/imece2002-33996 ◽

2002 ◽

Author(s):

William S. Oates ◽

Christopher S. Lynch

Keyword(s):

Closed Form ◽

Piezoelectric Materials ◽

Stress Singularity ◽

Closed Form Solution ◽

Form Solution ◽

Governing Equations ◽

Closed Form Solutions ◽

Order Polynomial ◽

Work Done ◽

Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

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Analytical and Graphical Solutions to the Forward Position Problem of Two Robots Manipulating a Spatial Linkage Payload

Journal of Mechanical Design ◽

10.1115/1.2826355 ◽

1997 ◽

Vol 119 (3) ◽

pp. 349-358

Author(s):

G. R. Pennock ◽

K. G. Mattson

Keyword(s):

Closed Form ◽

Orthogonal Transformation ◽

Solution Technique ◽

Closed Form Solutions ◽

Transformation Matrices ◽

Kinematic Geometry ◽

Order Polynomial ◽

Position Problem ◽

Four Bar Linkage ◽

Insight Into

This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a spatial four-bar linkage payload. To simplify the kinematic analysis, the Bennett linkage, which is a special geometry spatial four-bar, will be regarded as the payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown joint displacements in the wrist subassembly of each robot. An important result is that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. Repeated roots of the polynomials are shown to correspond to the stationary configurations of the system. The paper emphasizes that an understanding of the kinematic geometry of the system is essential to verify the number of possible solutions to the forward position problem. Graphical methods are also presented to provide insight into the assembly and stationary configurations. A numerical example of the two robots manipulating the Bennett linkage is included to demonstrate the importance of the polynomial and closed-form solutions.

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Relaxation and creep in twist and flexure

Multidiscipline Modeling in Materials and Structures ◽

10.1108/mmms-11-2013-0067 ◽

2014 ◽

Vol 10 (3) ◽

pp. 304-327 ◽

Cited By ~ 4

Author(s):

V. Kobelev

Keyword(s):

Strain Rate ◽

Closed Form ◽

Stress Function ◽

Constitutive Models ◽

Analytic Description ◽

Secondary Creep ◽

Common Procedure ◽

Content Type ◽

Closed Form Solutions ◽

Analytical Expressions

Purpose – The purpose of this paper is to derive the exact analytical expressions for torsion and bending creep of rods with the Norton-Bailey, Garofalo and Naumenko-Altenbach-Gorash constitutive models. These simple constitutive models, for example, the time- and strain-hardening constitutive equations, were based on adaptations for time-varying stress of equally simple models for the secondary creep stage from constant load/stress uniaxial tests where minimum creep rate is constant. The analytical solution is studied for Norton-Bailey and Garofalo laws in uniaxial states of stress. Design/methodology/approach – The creep component of strain rate is defined by material-specific creep law. In this paper the authors adopt, following the common procedure Betten, an isotropic stress function. The paper derives the expressions for strain rate for uniaxial and shear stress states for the definite representations of stress function. First, in this paper the authors investigate the creep for the total deformation that remains constant in time. Findings – The exact analytical expressions giving the torque and bending moment as a function of the time were derived. Research limitations/implications – The material isotropy and homogeneity preimposed. The secondary creep phase is considered. Practical implications – The results of creep simulation are applied to practically important problem of engineering, namely for simulation of creep and relaxation of helical and disk springs. Originality/value – The new, closed form solutions with commonly accepted creep models allow a deeper understanding of such a constitutive model's effect on stress and deformation and the implications for high temperature design. The application of the original solutions allows accurate analytic description of creep and relaxation of practically important problems in mechanical engineering. Following the procedure the paper establishes closed form solutions for creep and relaxation in helical, leaf and disk springs.

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The eigenbuckling analysis of hexagonal lattices: closed-form solutions

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0244 ◽

2021 ◽

Vol 477 (2251) ◽

pp. 20210244

Author(s):

S. Adhikari

Keyword(s):

Closed Form ◽

Elastic Properties ◽

Axial Force ◽

Hexagonal Lattice ◽

Force Parameter ◽

Closed Form Solutions ◽

Analysis And Design ◽

Lattice Materials ◽

Design Calculations ◽

Analytical Expressions

Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

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Closed-Form Solutions to the Kinematics of a Parallel Locomotion Mechanism With Actuated Spoke Wheels

Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2011-48557 ◽

2011 ◽

Author(s):

Ping Ren ◽

Dennis Hong

Keyword(s):

Motion Planning ◽

Numerical Simulations ◽

Closed Form ◽

Real Time ◽

Convex Surface ◽

Closed Form Solutions ◽

Time Motion ◽

Discrete Contact ◽

Parallel Configuration ◽

Analytical Expressions

A parallel locomotion mechanism can be defined as “a mechanism with parallel configuration that has discrete contact with respect to the ground which renders a platform the ability to move”. The actuated spoke wheel robot IMPASS (Intelligent Mobility Platform with Active Spoke System) presented in this paper serves as an example of such locomotion mechanisms. The current prototype of IMPASS has two actuated spoke wheels and one passive tail with its lower portion designed as convex surface. The robot is considered as a mechanism with variable topologies (MVTs) because of its metamorphic configuration. Closed-form solutions to the kinematics of the variable topologies are developed and verified with numerical simulations. The analytical expressions to these solutions allow themselves to be used directly in the real-time motion planning and monitoring of the robot.

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The Forward Position Problem of Two Puma-Type Robots Manipulating a Bennett Linkage Payload

Volume 2A: 24th Biennial Mechanisms Conference ◽

10.1115/96-detc/mech-1146 ◽

1996 ◽

Author(s):

Gordon R. Pennock ◽

Keith G. Mattson

Keyword(s):

Closed Form ◽

Orthogonal Transformation ◽

Solution Technique ◽

Closed Form Solutions ◽

Transformation Matrices ◽

Kinematic Geometry ◽

Order Polynomial ◽

Position Solution ◽

Position Problem ◽

Insight Into

Abstract This paper presents a solution to the forward position problem of two PUMA-type robots manipulating a Bennett linkage payload. The orientation of a specified payload link is described by a sixth-order polynomial and a specified angular joint displacement in the wrist subassembly of one of the robots is described by a second-order polynomial. A solution technique, based on orthogonal transformation matrices with dual number elements, is used to obtain closed-form solutions for the remaining unknown angular joint displacements in the wrist subassembly of each robot. The paper shows that, for a given set of robot input angles, twenty-four assembly configurations of the robot-payload system are possible. The polynomials provide insight into these configurations, and also reveal stationary configurations of the system. The paper emphasizes that insight into the kinematic geometry of the system is essential in developing the forward position solution. Graphical methods are presented which provide insight into the geometry, and a check of the analytical approach. For illustrative purposes, a numerical example of the two robots manipulating a Bennett linkage is included in this paper to demonstrate the importance of the polynomials and the closed-form solutions.

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