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Number of results: 11
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Abstract

The analytical approach is used for checking the stability of laterally unrestrained bisymmetric beams. The stability equations for simply supported beams are solved approximately using the Bubnov–Galerkin method [4]. The lateral buckling moment depends on bending distribution and on the load height effect. Each of applied concentrated and distributed loads, may have arbitrary direction and optional coordinate for the applied force along the cross section’s height. Derived equations allow for simple, yet fast control of lateral buckling moment estimated by FEM [15].

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Authors and Affiliations

R. Bijak
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Abstract

Double-beam model is considered in many investigations both theoretical and typically engineering ones. One can find different studies concerning analysis of such structures behaviour, especially in the cases where the system is subjected to dynamic excitations. This kind of model is successfully considered as a reliable representation of railway track. Inclusion of nonlinear physical and geometrical properties of rail track components has been justified by various computational studies and theoretical analyses. In order to properly describe behaviour of real structures their nonlinear properties cannot be omitted. Therefore a necessity to search appropriate analytical nonlinear models is recognized and highlighted in published literature. This paper presents essential extension of previously carried out double-beam system analysis. Two nonlinear factors are taken into account and parametrical analysis of the semi-analytical solution is undertaken with special emphasis on different range of parameters describing nonlinear stiffness of foundation and layer between beams. This study is extended by preliminary discussion regarding the dynamic effects produced by a series of loads moving along the upper beam. A new solution for the case of several forces acting on the upper beam with different frequencies of their variations in time is presented and briefly discussed.
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Authors and Affiliations

Piotr Koziol
1
ORCID: ORCID
Rafał Pilecki
2
ORCID: ORCID

  1. PhD, DSc, Assoc. Prof., Cracow University of Technology, Faculty of Civil Engineering, ul. Warszawska 24, 31-155 Kraków, Poland
  2. MSc, Eng., former student of Cracow University of Technology
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Abstract

The paper presents two analytical solutions namely for Fanning friction factor and for Nusselt number of fully developed laminar fluid flow in straight mini channels with rectangular cross-section. This type of channels is common in mini- and microchannel heat exchangers. Analytical formulae, both for velocity and temperature profiles, were obtained in the explicit form of two terms. The first term is an asymptotic solution of laminar flow between parallel plates. The second one is a rapidly convergent series. This series becomes zero as the cross-section aspect ratio goes to infinity. This clear mathematical form is also inherited by the formulae for friction factor and Nusselt number. As the boundary conditions for velocity and temperature profiles no-slip and peripherally constant temperature with axially constant heat flux were assumed (H1 type). The velocity profile is assumed to be independent of the temperature profile. The assumption of constant temperature at the channel’s perimeter is related to the asymptotic case of channel’s wall thermal resistance: infinite in the axial direction and zero in the peripheral one. It represents typical conditions in a minichannel heat exchanger made of metal.
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Authors and Affiliations

Jarosław Mikielewicz
Witold Rybiński
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Abstract

The aim of the present work is to verify a numerical implementation of a binary fluid, heat conduction dominated solidification model with a novel semi-analytical solution to the heat diffusion equation. The semi-analytical solution put forward by Chakaraborty and Dutta (2002) is extended by taking into account variable in the mushy region solid/liquid mixture heat conduction coefficient. Subsequently, the range in which the extended semi-analytical solution can be used to verify numerical solutions is investigated and determined. It has been found that linearization introduced to analytically integrate the heat diffusion equation impairs its ability to predict solidus and liquidus line positions whenever the magnitude of latent heat of fusion exceeds a certain value.
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Authors and Affiliations

Tomasz Wacławczyk
Michael Schäfer
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Abstract

In this paper, the logarithmic mean temperature difference method is used to determine the heat power of a tube-in-tube exchanger. Analytical solutions of the heat balance equations for the exchanger are presented. The considerations are illustrated by an example solution of the problem. In particular, the heat power of the tube-in-tube heat exchanger is determined taking into account the variants of work in the co-current and counter-current mode. Apart from the analytical solutions, appropriate numerical calculations in Matlab environment have been carried out.
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Bibliography

[1] Andrzejczyk R., Muszynski T.: Thermal and economic investigation of straight and U-bend double tube heat exchanger with coiled wire turbulator. Arch. Thermodyn. 40(2019), 2, 17–33.
[2] Bury T., Składzien J., Widziewicz K.: Experimental and numerical analyses of finned cross flow heat exchangers efficiency under non-uniform gas inlet flow conditions. Arch. Thermodyn. 31(2010), 4, 133–144.
[3] Hobler T.: Heat Transfer and Exchangers. Warszawa 1971 (in Polish).
[4] Kuppan T.: Heat Exchanger Design Handbook (2nd Edn.). CRC Press Taylor & Francis Group, Boca Raton 2013.
[5] Nitsche M., Gbadamosi R.O.: Heat Exchanger Design Guide. Elsevier, New York 2016.
[6] Pakowski Z., Adamski R.: Fundamentals of MATLAB in Process Engineering. Lodz Univ. Technol. Press, Łódz 2014 (in Polish).
[7] Roetzel W., Luo X.: Thermal analysis of heat exchanger networks. Arch. Thermodyn. 26(2005), 1, 5–16.
[8] Shah R.K., Sekulic D.P.: Fundamentals of Heat Exchanger Ddesign. Wiley, Hoboken 2003.
[9] Smith E.M.: Thermal Design of Heat Exchangers. A Numerical Approach: Direct- Sizing and Step-Wise Rating. Wiley, Chichester 1997.
[10] Taler D.: Numerical Modeling and Experimental Testing of Heat Exchangers. Springer, Berlin 2018.
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Authors and Affiliations

Kazimierz Rup
1

  1. Cracow University of Technology, al. Jana Pawła II 37, Cracow, Poland
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Abstract

In this paper, a semi-analytical solution for free vibration differential equations of curved girders is proposed based on their mathematical properties and vibration characteristics. The solutions of in-plane vibration differential equations are classified into two cases: one only considers variable separation of non-longitudinal vibration, while the other is a synthesis method addressing both longitudinal and non-longitudinal vibrationusing Rayleigh’s modal assumption and variable separation method. A similar approach is employed for the out-of-plane vibration, but further mathematical operations are conducted to incorporate the coupling effect of bending and twisting. In this case study, the natural frequencies of a curved girder under different boundary conditions are obtained using the two proposed methods, respectively. The results are compared with those from the finite element analysis (FEA) and results show good convergence.

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Authors and Affiliations

Y. Song
X. Chai
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Abstract

Sensitive MEMS-based thermal flow sensors are the best choice for monitoring the patient’s respiration prompt diagnosis of breath disturbances. In this paper, open space micro-calorimetric flow sensors are investigated as precise monitoring tools. The differential energy balance equation, including convection and conduction terms, is derived for thermal analysis of the considered sensor. The temperature-dependent thermal conductivity of the thin silicon-oxide membrane layer is considered in the energy balance equation. The derived thermal non-linear differential equation is solved using a well-known analytical method, and a finite-element numerical solution is used for the confirmation. Results show that the presented analytical model offers a precise tool for evaluating these sensors. The effects of flow and thin membrane film parameters on thermo-resistive micro-calorimetric flow sensors’ performance and sensitivity are evaluated. The optimization has been performed at different flow velocities using a genetic algorithm method to determine the optimum configuration of the considered flow sensor. The geometrical parameters are selected as a decision variable in the optimization procedure. In the final step, using optimization results and curve-fitting, the expressions for the optimum decision variables have been derived. The sensor’s optimum configuration is achieved analytically based on flow velocity with the analytical terms for optimum decision variables.
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Authors and Affiliations

Mojtaba Babaelahi
1
Somayyeh Sadri
2

  1. Department of Mechanical Engineering, University of Qom, Qom, Iran
  2. Thermal Cycle and Heat Exchangers Department, Niroo Research Institute, Tehran, Iran
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Abstract

Vibration in rotating machinery leads to a series of undesired effects, e.g. noise, reduced service life or even machine failure. Even though there are many sources of vibrations in a rotating machine, the most common one is mass unbalance. Therefore, a detailed knowledge of the system behavior due to mass unbalance is crucial in the design phase of a rotor-bearing system. The modelling of the rotor and mass unbalance as a lumped system is a widely used approach to calculate the whirling motion of a rotor-bearing system. A more accurate representation of the real system can be found by a continuous model, especially if the mass unbalance is not constant and arbitrarily oriented in space. Therefore, a quasi-analytical method called Numerical Assembly Technique is extended in this paper, which allows for an efficient and accurate simulation of the unbalance response of a rotor-bearing system. The rotor shaft is modelled by the Rayleigh beam theory including rotatory inertia and gyroscopic effects. Rigid discs can be mounted onto the rotor and the bearings are modeled by linear translational/rotational springs/dampers, including cross-coupling effects. The effect of a constant axial force or torque on the system response is also examined in the simulation.
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Bibliography

  1.  J.W. Lund and F.K. Orcutt, “Calculations and Experiments on the Unbalance Response of a Flexible Rotor,” J. Eng. Ind., vol. 89, no. 4, pp. 785–796, 1967.
  2.  A. Vollan and L. Komzsik, Computational Techniques of Rotor Dynamics with the Finite Element Method. Boca Raton: CRC Press, 2012.
  3.  J.S. Rao, Rotor Dynamics. New Delhi: New Age International, 1996.
  4.  A.-C. Lee and Y.-P. Shih, “The Analysis of Linear Rotor-Bearing Systems: A General Transfer Matrix Method,” J. Vib. Acoust., vol. 115, no. 4, pp. 490–497, 1993.
  5.  T. Yang and C. Lin, “Estimation of Distributed Unbalance of Rotors,” J. Eng. Gas Turbines Power, vol. 124, no. 4, pp. 976‒983, 2002.
  6.  J.-S. Wu and H.-M. Chou, “A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses,” J. Sound Vib., vol. 81, no. 3, pp.  1–10, 1999.
  7.  J.-S. Wu, F.-T. Lin, and H.-J. Shaw, “Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks,” J. Appl. Mech., vol. 220, no.  3, pp. 451–468, 2014.
  8.  M. Klanner and K. Ellermann, “Steady-state linear harmonic vibrations of multiple-stepped Euler-Bernoulli beams under arbitrarily distributed loads carrying any number of concentrated elements,” Appl. Comput. Mech., vol. 14, no. 1, pp. 31–50, 2020.
  9.  M. Klanner, M.S. Prem, and K. Ellermann, “Steady-state harmonic vibrations of a linear rotor-bearing system with a discontinuous shaft and arbitrary distributed mass unbalance,” in Proceedings of ISMA2020 International Conference on Noise and Vibration Engineering and USD2020 International Conference on Uncertainty in Structural Dynamics, Leuven, Belgium, Sep. 2020, pp. 1257–1272.
  10.  H. Ziegler, “Knickung gerader Stäbe unter Torsion,” J. Appl. Math. Phys. (ZAMP), vol. 3, pp. 96–119, 1952.
  11.  V.V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability. New York: Pergamon Press, 1963.
  12.  H. Ziegler, Principles of Structural Stability. Basel: Springer Basel AG, 1977.
  13.  L. Debnath and D. Bhatta, Integral Transforms and Their Applications. CRC Press, 2015.
  14.  D. Mitrinović and J.D. Kečkić, The Cauchy Method of Residues. D. Reidel Publishing, 1984.
  15.  S.I. Hayek, Advanced Mathematical Methods in Science and Engineering. CRC Press, 2010.
  16.  B. Adcock, D. Huybrechs, and J. Martín-Vaquero, “On the Numerical Stability of Fourier Extensions,” Found. Comput. Math., vol. 14, no. 4, pp. 638–687, 2014.
  17.  R. Matthysen and D. Huybrechs, “Fast Algorithms for the Computation of Fourier Extensions of Arbitrary Length,” SIAM J. Sci. Comput., vol. 38, no. 2, pp. A899–A922, 2016.
  18.  A.-C. Lee, Y. Kang, and L. Shin-Li, “A Modified Transfer Matrix Method for Linear Rotor-Bearing Systems,” J. Appl. Mech., vol. 58, no. 3, pp. 776–783, 1991.
  19.  M.I. Friswell, J.E. T. Penny, S.D. Garvey, and A.W. Lees, Dynamics of Rotating Machines. New York: Cambridge University Press, 2010.
  20.  A. De Felice and S. Sorrentino, “On the dynamic behaviour of rotating shaftsunder combined axial and torsional loads,” Meccanica, vol. 54, no. 7, pp. 1029–1055, 2019.
  21.  R.L. Eshleman and R.A. Eubanks, “On the Critical Speeds of a Continuous Rotor,” J. Manuf. Sci. Eng., vol. 91, no. 4, pp. 1180‒1188, 1969.
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Authors and Affiliations

Michael Klanner
1
ORCID: ORCID
Marcel S. Prem
1
Katrin Ellermann
1

  1. Graz University of Technology, Institute of Mechanics, Kopernikusgasse 24/IV, 8010 Graz, Austria
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Abstract

Based on wave mechanics theory, the dynamic response characteristics of cantilever flexible wall in two-dimensional site are analyzed. The partial derivative of the vibration equation of soil layer is obtained, and the general solution of the volume strain is obtained by the separation of variables method. The obtained solution is substituted back to the soil layer vibration equation to obtain the displacement vibration general solution. Combined with the soil-wall boundary condition and the orthogonality of the trigonometric function, the definite solution of the vibration equation is obtained. The correctness of the solution is verified by comparing the obtained solution with the existing simplified solution and the solution of rigid retaining wall, and the applicable conditions of each simplified solution are pointed out. Through parameter analysis, it is shown that when the excitation frequency is low, the earth pressure on the wall is greatly affected by the soil near the wall. When the excitation frequency is high, the influence of the far-field soil on the earth pressure of the wall gradually increases. The relative stiffness of the wall, the excitation frequency and the soil layer damping factor have a significant effect on the dynamic response of the flexible retaining wall.
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Authors and Affiliations

Xiuzhu Yang
1
ORCID: ORCID
Xinyuan Liu
1
ORCID: ORCID
Shuang Zhao
1
ORCID: ORCID
Jun Yu
1
ORCID: ORCID

  1. Central South University, School of Civil Engineering, Changsha, 410075, China
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Abstract

In investigations constituting Part I of this paper, the effect of approximations in the flexural-torsional buckling analysis of beam-columns was studied. The starting point was the formulation of displacement field relationships built straightforward in the deflected configuration. It was shown that the second-order rotation matrix obtained with keeping the trigonometric functions of the mean twist rotation was sufficiently accurate for the flexural-torsional stability analysis. Furthermore, Part I was devoted to the formulation of a general energy equation for FTB being expressed in terms of prebuckling stress resultants and in-plane deflections through the factor k 1. The energy equation developed there was presented in several variants dependent upon simplified assumptions one may adopt for the buckling analysis, i.e. the classical form of linear eigenproblem analysis (LEA), the form of quadratic eigenproblem analysis (QEA) and refined (non-classical) forms of nonlinear eigenproblem analysis (NEA), all of them used for solving the flexural-torsional buckling problems of elastic beamcolumns. The accuracy of obtained analytical solutions based on different approximations in the elastic flexural–torsional stability analysis of thin-walled beam-columns is examined and discussed in reference to those of earlier studies. The comparison is made for closed form solutions obtained in a companion paper, with a scatter of results evaluated for k 1 = 1 in the solutions of LEA and QEA, as well as for all the options corresponding to NEA. The most reliable analytical solution is recommended for further investigations. The solutions for selected asymmetric loading cases of the left support moment and the half-length uniformly distributed span load of a slender unrestrained beam-column are discussed in detail in Part II. Moreover, the paper constituting Part II investigates how the buckling criterion obtained for the beam-column laterally and torsionally unrestrained between the end sections might be applied for the member with discrete restraints. The recommended analytical solutions are verified with use of numerical finite element method results, considering beam-columns with a mid-section restraint. A variant of the analytical form of solutions recommended in these investigations may be used in practical application in the Eurocode’s General Method of modern design procedures for steelwork.
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Authors and Affiliations

Marian Giżejowski
1
ORCID: ORCID
Anna Barszcz
1
ORCID: ORCID
Paweł Wiedro
1
ORCID: ORCID

  1. Warsaw University of Technology, Faculty of Civil Engineering, Al. Armii Ludowej 16, 00-637 Warsaw, Poland
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Abstract

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (knowfrom the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant M�� and the buckling minor axis stress resultant Mz. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II.
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Authors and Affiliations

Marian Giżejowski
1
ORCID: ORCID
Anna Barszcz
1
ORCID: ORCID
Paweł Wiedro
1
ORCID: ORCID

  1. Warsaw University of Technology, Faculty of Civil Engineering, Al. Armii Ludowej 16, 00-637 Warsaw, Poland

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