IRIS Università degli Studi di Bolognahttps://cris.unibo.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Thu, 27 Jan 2022 19:47:39 GMT2022-01-27T19:47:39Z10321Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forceshttp://hdl.handle.net/11585/80407Titolo: Elliptic integral solutions of spatial elastica of a thin straight rod bent under concentrated terminal forces
Abstract: In this article, we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod’s centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11585/804072006-01-01T00:00:00ZEOQ under exogenous and periodic demandhttp://hdl.handle.net/11585/75770Titolo: EOQ under exogenous and periodic demand
Abstract: In this paper we give a sufficient condition for the existence of the economic batch to a Wilson-type inventory model loaded by a fully exogenous continuous demand function of time. After some cases solvable in closed form, the computational problem is introduced of inverting the reordering time versus the ordered quantity as necessary step to obtain the cost function to be minimized. Such a mixed (theoretical/numerical) approach is applied to a demand consisting of three different behaviors: growth, decrease and prolonged zero. Such a wave-form is assumed to iterate itself periodically and the relevant seasonal demand is expanded in a Fourier series of time. Performing the integration and reverting the reordering time, the cost function is computed and its minimizing EOQ detected. Finally an example shows that the above conditions guarantee the existence but not uniqueness to solution.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11585/757702009-01-01T00:00:00ZMathematical properties of EOQ models with special cost structurehttp://hdl.handle.net/11585/133755Titolo: Mathematical properties of EOQ models with special cost structure
Abstract: An existence-uniqueness theorem is proved about a minimum cost order for a class of inventory models whose holding costs grow according to a stock level power law. The outcomes of a previous work of two of the authors are then extended to different environments: i.e. when the holding costs change during time generalizing the Weiss model, or with invariable holding costs but adopting a backordering strategy. Application cases are provided assuming several functional behaviors of demand versus the stock level
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11585/1337552013-01-01T00:00:00ZThe meridian curve of a wetting drop: a boundary value problem and its elliptic integrals solutionhttp://hdl.handle.net/11585/389303Titolo: The meridian curve of a wetting drop: a boundary value problem and its elliptic integrals solution
Abstract: In this article we study the shape of free surfaces of a static fluid under gravity. We consider the meridian curve of a heavy liquid drop standing on a horizontal base: the main assumption concerns the liquid wetting capability, namely its contact angle well below pi/2. The nonlinear differential boundary problem is solved through the shooting method. Our treatment is self-consistent as holding all demonstrations of existence, uniqueness, and computability. We conclude providing the eigenvalues set to the radius and the meridian curve of the drop through elliptic integrals: such a new exact solution—see (3.9) and (3.10) —is enriching the literature on capillarity.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11585/3893032014-01-01T00:00:00ZExact solution to a first-fifth power nonlinear unforced oscillatorhttp://hdl.handle.net/11585/96121Titolo: Exact solution to a first-fifth power nonlinear unforced oscillator
Abstract: A one-dimensional, undamped, anharmonic oscillator whose restoring force is an odd polynomial function of displacement, is solved exactly through the elliptic functions and inverted, providing oscillation amplitude through time, and then revealing a new fully integrable nonlinear system. The closed form relationship linking the period $mathbb{T}$ to the initial motion amplitude textit{a} can then play as a benchmark to the approximate values of literature
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/11585/961212010-01-01T00:00:00ZA thin heavy flagpole bent under a transverse wind: his elastica by hypergeometric functionshttp://hdl.handle.net/11585/112740Titolo: A thin heavy flagpole bent under a transverse wind: his elastica by hypergeometric functions
Abstract: In this article we are dealing with a uniform cylindrical pole fixed in a
vertical direction at its lowest point, and carried to such a height that
the vertical position becomes unstable and flexure begins. Simultaneously
the wind is assumed to blow so that being active as a transverse load, and the relevant elastica is investigated. The consequent nonlinear
differential equation cannot be treated analytically: but its
linearized variant, small deflections, can. We have been led to a Airy type equation to be integrated in closed form through the first kind Bessel
and the 1F2 generalized hypergeometric functions.
The problem has then completely solved: so we make a useful thing for engineering purposes, or better, for testing the relevant numerical algorithms.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1127402012-01-01T00:00:00ZFIRST ORDER DIFFERENTIAL EQUATIONS: EARLY HISTORY FOLLOWING JACOPO RICCATI AND GABRIELE MANFREDIhttp://hdl.handle.net/11585/752730Titolo: FIRST ORDER DIFFERENTIAL EQUATIONS: EARLY HISTORY FOLLOWING JACOPO RICCATI AND GABRIELE MANFREDI
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11585/7527302019-01-01T00:00:00ZJohann Bernoulli's first lecture from the first integral calculus textbook ever written: an annotated translationhttp://hdl.handle.net/11585/781533.1Titolo: Johann Bernoulli's first lecture from the first integral calculus textbook ever written: an annotated translation
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11585/781533.12019-01-01T00:00:00ZPi and the hypergeometric functions of complex argumenthttp://hdl.handle.net/11585/105990Titolo: Pi and the hypergeometric functions of complex argument
Abstract: In this article we derive some new identities concerning pi; algebraic
radicals and some special occurrences of the Gauss hypergeometric function 2F1 in the analytic continuation. All of them have been
derived by tackling some elliptic or hyperelliptic known integral, and
looking for another representation of it by means of hypergeometric
functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing an algebraic radical as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11585/1059902011-01-01T00:00:00ZExact Solutions of Nonlinear Equation of Rod Deflections Involving the Lauricella Hypergeometric Functionshttp://hdl.handle.net/11585/105991.1Titolo: Exact Solutions of Nonlinear Equation of Rod Deflections Involving the Lauricella Hypergeometric Functions
Abstract: The stress induced in a loaded beam will not exceed some threshold, but also its maximum deflection, as for all the elastic systems, will be controlled. Nevertheless, the linear beam theory fails to describe the large deflections; highly flexible linear elements, namely, rods, typically found in aerospace or oil applications, may experience large displacements—but small strains, for not leaving the field of linear elasticity—so that geometric nonlinearities become significant. In this article, we provide analytical solutions to large deflections problem of a straight, cantilevered rod under different coplanar loadings. Our researches are led by means of the elliptic integrals, but the main achievement concerns the Lauricella FD3 hypergeometric functions use for solving elasticity problems. Each of our analytic solutions has been individually validated by comparison with other tools, so that it can be used in turn as a benchmark, that is, for testing other methods based on the finite elements approximation.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11585/105991.12011-01-01T00:00:00Z