The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.
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22Analysis of cracked bars with rectangular cross-section and isotropic coating layer under torsion
Analysis of cracked bars with rectangular cross-section and isotropic coating layer under torsion
11Semianalytic Stress-Function Variational Approach for the Interfacial Stresses in Bonded Joints
High interfacial stresses near the edges of bonded joints are responsible for their debonding failure. This paper reports a new semianalytic stress-function variational approach to the interfacial stresses of a bonded joint, which is made of a straight tension bar covered with a reinforcing patch and subjected to mechanical loads and/or uniform change of temperature. The process introduces two interfacial shear and normal stress functions, which are correlated via the approximately same radius of curvature of the slender adherends. All the stress components in the joint are expressed in terms of the interfacial stress functions based on the classic Euler-Bernoulli beam theory and equilibrium equations of elasticity. Deformation compatibility of the joint is satisfied by minimizing the complementary strain energy, which leads to a fourth-order ordinary differential equation (ODE) of the interfacial shear stress function. The interfacial shear and normal stresses are determined explicitly and compar...
Read more15Thermal postbuckling analysis of variable angle tow composite cylindrical panels
The advanced Automated Fiber Placement (AFP) technologies allow the fiber trajectories to be curvilinearly steered over the panel platform, which brings out a novel type of cylindrical panel structure, termed as Variable Angle Tow (VAT) composite cylindrical panel. So far, the thermal postbuckling problem of VAT composite cylindrical panels under temperature loadings remain unexplored. The aim of this article is to cover this lack by developing a methodology based on a generalized Rayleigh–Ritz formulation in the framework of the principle of thermoelastic complementary energy. The proposed panel model is based on Donnell’s shallow shell theory and accounts for von Kármán’s geometrical nonlinearity. With the aid of the Lagrange multiplier method and the introduction of Airy’s stress function, a mixed variational formula that governs the nonlinear thermoelastic problem of VAT composite cylindrical panels is derived from the principle of thermoelastic potential energy. The Rayleigh-Ritz formulation combined with the mixed variational formula is then applied to derive the simultaneous nonlinear thermoelastic equations, and the Crisfield’s arc-length method, which can cope well with snap-through or snap-back problems, is employed to trace the equilibrium path that characterizes the thermal postbuckling features. Two different modeling strategies, namely, semi-inverse method and Lagrangian multiplier method, are adopted to formulate the nonlinear thermoelastic problem of VAT composite cylindrical panels undergoing temperature change. The application of Lagrangian multiplier method overcomes some limitations inherent in semi-inverse method in terms of constructing Rayleigh-Ritz formulation, and thus provides generality to model general in-plane boundary constraint. The accuracy and robustness of the proposed Rayleigh-Ritz model are validated against finite element solutions and previously published results. Effects of geometric imperfection, curvature radius, aspect ratio and fiber orientation angle on thermal postbuckling responses of VAT cylindrical plates are discussed by several examples. The study presented here may be helpful for the preliminary design of VAT curved panels in thermal environments.
Read more81Maximum principle for the general optimal control problem with phase and regular mixed constraints
We prove the maximum principle for the problem outlined in the title with the aid (or more precisely in the class) of so-called sliding variations. INTRODUCTION The by-now classical maximum principle for the simplest nonlinear optimal control problem was formulated by Pontryagin and proved by Boltyanskii over 30 years ago (see [1, 2, 3] and also [8]). Since then, the maximum principle has been derived for various optimal control problems in an enormous number of studies. This is well-known and we do not give a full list of references here; we only mention [4-7, 9-24, 27-30]. In our view, the deepest and most thorough studies are those of Milyutin and Dubovitskii [9-14]. These authors advance very far in their development of the maximum . principle for problems with ordinary differential equations - first to problems with phase constraints [9] and then to the general problem with phase and arbitrary (in general, nonregular) mixed constraints [10-14]. These authors completely resolve the issue of deriving the necessary first-order conditions for these problems. Despite these achievements, we are still witnessing the publication of studies that derive the maximum principle for various problems which are particular cases of the general Dubovitskii-Milyutin problem. The authors of some of these studies possibly do not realize that they are rederiving well-established results.** This is partly due to the fact that, on the one hand, optimal control results are primarily interesting for applied engineers, while the references [9-14] are far from being accessible to every engineer, because they use a fairly complex mathematical apparatus which is not familiar to engineers; on the other hand, optimal control theory as such so far has failed to attract a sufficiently wide followign of mathematicians. "Optimizers" still feel that if a problem originates in engineering practice, then its solution (and preferably the process of its analysis) should remain in the domain of concepts which are familiar in the engineering environment (compare this situation with the totally different state of things in the theory of partial differential equations or in probability theory). This, in our view, accounts for repeated attempts to simplify the proof of the maximum principle. However, even if some studies achieve a certain simplification of the proof, this is the result of extreme simplification of the problem, and not an achievement of the particular method of analysis. The positive impact of these attempts is that they highlight the complexity of the problem. They have shown that the problem cannot be solved by elementary techniques, without invoking new methods (specifically, methods of functional analysis). We are thus forced to admit that the complexity of the work of Dubovitskii-Milyutin is not contrived: it is inherent to the complexity of the underlying problem. Yet there is a fairly wide class of problems for which the maximum principle can be derived relatively simply. These are the problems with pure phase constraints and so-called "regular" mixed constraints. They cover most of the problems considered in the literature. The maximum principle for these problems was derived by Milyutin and Dubovitskii back in the 1960s, immediately after their derivation of the maximum principle for problems with phase constraints [9]. However, it has never been published because, in the authors' opinion, the addition of regular mixed constraints to phase constraints did not constitute a significant advance.
Read more65Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid
Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid
3Elastic-plastic torsion of circular ring sectors
The solution is presented to the problem of uniform torsion of circular ring sectors with circular cross section under the assumption of perfect plasticity. The elastic-plastic problem is solved by a semi-inverse method. When the entire ring is plastic a discontinuity of stress appears which may be regarded as the limiting case of an elastic core; such discontinuities have recently been discussed in the literature [1]. The solution for the discontinuous fully plastic stress distribution is exact, that for the elastic-plastic case approximate in the sense that it is found exactly for cross sections differing slightly from the circular. This difference is negligible for ratios of ring radius R R to cross section radius ρ \rho occurring in practical applications to helical springs of small pitch.
Read moreANALYSIS OF POSSIBLE CAUSES AND MECHANISMS OF DESTRUCTION OF BUILDING STRUCTURES
In order to better understand the wave properties of the Timoshenko equation, the derivation of the refined equation from the equations of the plane elasticity problem for a long band is carried out. The simple iterations method is used for the derivation. It includes known methods: the semi-inverse method of Saint-Venant and Picard operator. In accordance with the semi-inverse method, a part of the unknowns is defined, which are interpreted as the values of the initial (zero) approximation. Proceeding from them, a sequential computation is carried out using a sequence of the four Picard operators in such a way that the outputs of the one operator are the inputs for the next. Calculating in this way all the unknowns in the zeroth approximation by the direct integration over the transverse coordinate, the values of the initial approximation are calculated in the first approximation. These quantities are small of the second order with respect to the dimensionless thickness. Expressions for the unknowns are obtained as power functions of the transverse coordinate and as a function of the deriva-tives along the longitudinal coordinate. By the Banach fixed point theorem, the computation process is asymptot-ically convergent one. After this, boundary conditions on the long sides are satisfied by means of the derivatives of the arbitrariness, depending only on the longitudinal coordinate. This gives us the ordinary differential equations for the determination of these arbitrary functions. In turn, the integration constants of the last equations can be found from the conditions on the short sides of the strip. The ordinary differential equations are split into equations for slowly varying and quickly varying quantities. The slowly changing values give the classical solu-tion of the beam oscillations. The quickly varying solutions give the perturbed solutions describing high-frequency oscillations and singularly perturbed wave solutions for time-concentrated effects. Some of these soluions are absent in the Timoshenko equation. It is assumed that the selected shear waves provoke in the buildings subjected to the rapid impacts (shock by airplane, explosions, and seismic movements of the base) the interruptions of interlayers between the floors and subsequent progressive collapse.
Read more11Solution of Thermal Stress Problems in Tube Sheets by the Boundary Point Least Squares Method
This paper describes the application of the boundary point least squares approach to the plane stress analysis of tube sheets with either mechanical or thermal loads. The paper includes a derivation of appropriate stress functions, a discussion of the point matching and boundary point least squares methods, and a description of the application of the method to the analysis of different hole configurations in tube sheets. It concludes with numerical results obtained from the analysis of the thermal stresses near the divider lane of a tube sheet from a two-pass heat exchanger.
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1Calculation of the green’s function of boundary value problems for linear ordinary differential equations
The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found
Read more4Mathematical Analysis of the Process Forces Effect on Collet Chuck Holders
Chuck holders are widely used for jobs with high precision. A chuck holder consists of a nut with a tapered surface and a thin-slotted clamping sleeve typically made of hardened steel and named a collet. Chuck holders are, essentially, wedge mechanisms. In this paper, we investigated the reactions and strains due to the forces during the chip removal process in the contact elements or jaws of the collet by means of mathematical analysis. Deflections in the jaws of the collet arise with a high influence from the precision of the workpieces. The cutting or process forces cause an axial force, a radial force, a torsional moment, and a bending moment on the chuck collet, and, consequently, displacements and inclinations of the clamping system are caused. Therefore, the proposed analytical models are based on elasticity and contact theories. The mathematical model for determining the deflections of the clamping system force was developed and implemented using MATLAB. The results showed that the variation in the clamping force during rotation in a collet chuck holder mainly depends on the stiffness of the collet chuck holder and the stiffness of the workpiece. The results indicated that the collet should be vulcanized to minimize the deformations that affect the final product. The deflections of a collet chuck holder due to process forces depend strongly on the clearances, wedge angle, and stiffness of the collet.
Read more2On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations
There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.
Read more1Effects of functionally graded coating layer on the torsional behavior of circular cross-section bars with multiple cracks and cavities
Effects of functionally graded coating layer on the torsional behavior of circular cross-section bars with multiple cracks and cavities
Read more23Ordinary Differential Equations with Applications to Mechanics
PREFACE. INTRODUCTION. Generalities. Ordinary differential equations. Supplementary conditions associated to ODEs. The Cauchy (initial) problem.The two-point problem. 1: LINEAR ODEs OF FIRST AND SECOND ORDER. 1.1 Linear first order ODEs. 1.1.1 Equations of the form . 1.1.2 The linear homogeneous equation. 1.1.3 The general case. 1.1.4 The method of variation of parameters (Lagrange's method). 1.1.5 Differential polynomials. 1.2 Linear second order ODEs. 1.2.1 Homogeneous equations. 1.2.2 Non-homogeneous equations. Lagrange's method. 1.2.3 ODEs with constant coefficients. 1.2.4 Order reduction. 1.2.5 The Cauchy problem. Analytical methods to obtain the solution. 1.2.6 Two-point problems (Picard). 1.2.7 Sturm-Liouville problems. 1.2.8 Linear ODEs of special form. 1.3. Applications 2: LINEAR ODEs OF HIGHER ORDER (n >2). 2.1 The general study of linear ODEs of order . 2.1.1 Generalities. 2.1.2 Linear homogeneous ODEs. 2.1.3 The general solution of the non-homogeneous ODE. 2.1.4 Order reduction. 2.2 Linear ODEs with constant coefficients. 2.2.1 The general solution of the homogeneous equation. 2.2.2 The non-homogeneous ODE. 2.2.3 Euler type ODEs. 2.3 Fundamental solution. Green function. 2.3.1 The fundamental solution. 2.3.2 The Green function. 2.3.3 The non-homogeneous problem. 2.3.4 The homogeneous two-point problem. Eigenvalues. 2.4 Applications. 3: LINEAR ODSs OF FIRST ORDER. 3.1 The general study of linear first order ODSs. 3.1.1 Generalities. 3.1.2 The general solution of the homogeneous ODS. 3.1.3 The general solution of the non-homogeneous ODS. 3.1.4 Order reduction of homogeneous ODSs. 3.1.5 Boundary value problems for ODSs. 3.2 ODSs with constant coefficients. 3.2.1 The general solution of the homogeneous ODS. 3.2.2 Solutions in matrix form for linear ODSs with constant coefficients. 3.3 Applications. 4: NON-LINEAR ODEs OF FIRST AND SECOND ORDER. 4.1 First order non-linear ODEs. 4.1.1 Forms of first order ODEs and oftheir solutions. 4.1.2 Geometric interpretation. The theorem of existence and uniqueness. 4.1.3 Analytic methods for solving first order non-linear ODEs. 4.1.4. First order ODEs integrable by quadratures. 4.2 Non-linear second order ODEs. 4.2.1 Cauchy problems. 4.2.2 Two-point problems. 4.2.3 Order reduction of second order ODEs. 4.2.4 The Bernoulli-Euler equation. 4.2.5 Elliptic integrals. 4.3 Applications. 5: NON-LINEAR ODSs OF FIRST ORDER. 5.1 Generalities. 5.1.1 The general form of a first order ODS. 5.1.2 The existence and uniqueness theorem for the solution of the Cauchy problem. 5.1.3 The particle dynamics. 5.2 First integrals of an ODS. 5.2.1 Generalities. 5.2.2 The theorem of conservation of the kinetic energy. 5.2.3 The symmetric form of an ODS. Integral combinations. 5.2.4 Jacobi's multiplier. The method of the last multiplier. 5.3 Analytical methods of solving the Cauchy problem for non-linear ODSs. 5.3.1 The method of successive approximations (Picard-Lindeloff). 5.3.2 The method of the Taylor series expansion. 5.3.3 The linear equivalence method (LEM). 5.4 Applications. 6: VARIATIONAL CALCULUS. 6.1 Necessary condition of extremum for functionals of integral type. 6.1.1 Generalities. 6.1.2 Functionals of the form..... 6.1.3 Functionals of the form..... 6.1.4 Functionals of integral type, depending on n functions. 6.2 Conditional extrema. 6.2.1 Isoperimetric problems. 6.2.2 Lagrange's problem. 6.3 Applications. 7: STABILITY. 7.1 Lyapunov Stability. 7.1.1 Generalities. 7.1.2 Lyapunov's theorem of stability. 7.2 The stability of the solutions of dynamical systems. 7.2.1 Autonomous dynamical systems. 7.2.2 Long term behaviour of the solutions. 7.3 Applications. INDEX. REFERENCES.
Read moreResearch on method of vibration analysis of fan bridge truss in operation
To research the effect of fans of direct air cooling system on the fan bridge truss when the fans are in operation, 1∶1 full-scale fan test model is designed. By using strain resolution method, the oscillation forces in operation are measured, including bending moment, torsion moment and centrifugal force. Accordingly, the formula is provided. A thermal power plant of direct air cooling system being used is selected. Theoretically, analyze the vibration of fan bridge truss by the formula of oscillation forces obtained from the test. Then measure the vibration of fan bridge truss resulted from fans in operation on the spot. Compared the theoretical analysis with the measured data, there is small difference. So the reasons for that are analyzed. And more research work shall be done for perfection of the fan bridge truss calculation theory.
Read moreNotice of Retraction: Torsion of uniform bars using the improved Hybrid Boundary Node Method
An improved Hybrid Boundary Node Method (Hybrid BNM) is developed for solving the torsion problem of uniform bars. The governing equation of the torsion problem is Poisson's equation. Based on the dual reciprocity method (DRM), the solution of problem is divided into complementary and particular solutions. The complementary solution is solved by the Hybrid BNM, and the DRM is employed to solve the particular one using radial basis functions (RBF). Hybrid BNM is a boundary type meshless method, which based on moving least squares (MLS) approximation. The method has many advantages, such as simple postprocess and high accuracy. However, shape functions for the classical MLS approximation lack the delta function property. Thus in this method, the boundary condition cannot be enforced easily and directly, and its computational cost is high for the inevitable transformation strategy of boundary condition. In the method we proposed, a regularized weight function is adopted, which leads to the MLS shape functions fulfilling the interpolation condition exactly, which enables a direct application of essential boundary conditions without additional numerical effort. Numerical results for the torsion of uniform bar are presented to demonstrate the efficiency and accuracy of the present method.
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