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1
Piché, R., and A. Ellman. "Numerical Integration of Fluid Power Circuit Models Using Two-Stage Semi-Implicit Runge-Kutta Methods." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 208, no. 3 (May 1994): 167–75. http://dx.doi.org/10.1243/pime_proc_1994_208_114_02.
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Fluid power circuits that contain fluid volumes of different orders of magnitude are difficult to simulate because the system of ordinary differential equations is numerically stiff. Even algorithms specially designed for stiff systems require excessively small time steps to avoid numerical oscillation in simulations of some circuits. In this paper the accuracy and numerical stability of several two-stage semi-implicit Runge-Kutta methods that have been proposed in circuit simulation literature are analysed and compared. It is shown that, for integration of very stiff circuits, the best method in this class is an L-stable method. A simple numerical example is used to verify the theoretical results. The example includes a novel way of modelling orifice flow that is especially suitable for numerical simulations.
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Dear, J., Z. Shi, and J. Lin. "An efficient numerical integration system for stiff unified constitutive equations for metal forming applications." IOP Conference Series: Materials Science and Engineering 1270, no. 1 (December 1, 2022): 012008. http://dx.doi.org/10.1088/1757-899x/1270/1/012008.
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Unified constitutive equations have been developed in recent years to predict viscoplastic flow and microstructural evolution of metal alloys for metal forming applications. These equations can be implemented into commercial FE code, such as ABAQUS and PAMSTAMP, to predict mechanical and physical properties of materials in a wide range of metal forming processes. These equations are normally stiff and need significant computer CPU time to solve. In this research, a series of numerical analyses are performed to investigate the difficulties within MATLAB of solving these stiff unified constitutive equations. A metric is introduced to allow evaluation of the numerical stiffness to assess the most appropriate numerical integration method. This metric is based on the ratio of maximum to minimum eigenvalue. This metric allows for an appropriate numerical method to be chosen giving more effective modelling of deformation and plasticity processes. Based on the theoretical work described above, a user-friendly system, based on MATLAB, is then developed for numerically integrating these types of stiff constitutive equations. This is particularly useful for metal forming engineers and researchers who need an effective computational tool to determine constitutive properties well based on numerical integration theories.
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Asnor, Mohd Yatim, and Ibrahim. "Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae." Symmetry 11, no. 10 (October 14, 2019): 1289. http://dx.doi.org/10.3390/sym11101289.
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Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.
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Braileanu, G. "Matrix operators for numerically stable representation of stiff linear dynamic systems." IEEE Transactions on Automatic Control 35, no. 8 (1990): 974–80. http://dx.doi.org/10.1109/9.58516.
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Solovarova, Liubov S., and Ta D. Phuong. "On the numerical solution of second-order stiff linear differential-algebraic equations." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 24, no. 2 (June 30, 2022): 151–61. http://dx.doi.org/10.15507/2079-6900.24.202202.151-161.
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This article addresses systems of linear ordinary differential equations with an identically degenerate matrix in the main part. Such formulations of problems in literature are usually called differential-algebraic equations. In this work, attention is paid to the problems of the second order. Basing on the theory of matrix pencils and polynomials, sufficient conditions for existence and uniqueness of the equations’ solution are given. To solve them numerically, authors investigate a multistep method and its version based on a reformulated notation of the original problem. This representation makes it possible to construct methods whose coefficient matrices can be calculated at previous points. This approach has delivered good results in numerical solution of first-order differential-algebraic equations that contain stiff and rapidly oscillating components and have singular matrix pencil. The stability of proposed numerical algorithm is investigated for the well-known test equation. It is shown that this difference scheme has the first order of convergence. Numerical calculations of the model problem are presented.
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Gao, Pan, Zhihui Liu, Ji Zeng, Yiting Zhan, and Fei Wang. "A Random Forest Model for the Prediction of Spudcan Penetration Resistance in Stiff-Over-Soft Clays." Polish Maritime Research 27, no. 4 (December 1, 2020): 130–38. http://dx.doi.org/10.2478/pomr-2020-0073.
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Abstract Punch-through is a major threat to the jack-up unit, especially at well sites with layered stiff-over-soft clays. A model is proposed to predict the spudcan penetration resistance in stiff-over-soft clays, based on the random forest (RF) method. The RF model was trained and tested with numerical simulation results obtained through the Finite Element model, implemented with the Coupled Eulerian Lagrangian (CEL) approach. With the proposed CEL model, the effects of the stiff layer thickness, undrained shear strength ratio, and the undrained shear strength of the soft layer on the bearing characteristics, as well as the soil failure mechanism, were numerically studied. A simplified resistance profile model of penetration in stiff-over-soft clays is proposed, divided into three sections by the peak point and the transition point. The importance of soil parameters to the penetration resistance was analysed. Then, the trained RF model was tested against the test set, showing a good prediction of the numerical cases. Finally, the trained RF was validated against centrifuge tests. The RF model successfully captured the punch-through potential, and was verified using data recorded in the field, showing advantages over the SNAME guideline. It is supposed that the trained RF model should give a good prediction of the spudcan penetration resistance profile, especially if trained with more field data.
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Grenestedt, Joachim L., and Mikael Danielsson. "Elastic-Plastic Wrinkling of Sandwich Panels With Layered Cores." Journal of Applied Mechanics 72, no. 2 (March 1, 2005): 276–81. http://dx.doi.org/10.1115/1.1828063.
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Elastic-plastic wrinkling of compression loaded sandwich panels made with layered cores was studied analytically and experimentally. A core with a stiff layer near the sandwich skins can improve various properties, including wrinkling and impact strengths, with only a minor weight penalty. The 2D plane stress and plane strain bifurcation problems were solved analytically, save for a determinantal equation which was solved numerically. Experiments were performed on aluminum skin/foam core sandwich panels with different combinations of stiff and soft core materials. Good correlation between experiments and theory was obtained.
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Chen, Shanqin. "Krylov SSP Integrating Factor Runge–Kutta WENO Methods." Mathematics 9, no. 13 (June 24, 2021): 1483. http://dx.doi.org/10.3390/math9131483.
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Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.
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Albi, Giacomo, Young-Pil Choi, and Axel-Stefan Häck. "Pressureless Euler alignment system with control." Mathematical Models and Methods in Applied Sciences 28, no. 09 (August 2018): 1635–64. http://dx.doi.org/10.1142/s0218202518400018.
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We study a non-local hydrodynamic system with control. First, we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that lead to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally, we use a finite volume scheme, coupled with an implicit–explicit time integrator to solve numerically the stiff scale of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.
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Tudor, M. "A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models." Geoscientific Model Development 6, no. 4 (July 5, 2013): 901–13. http://dx.doi.org/10.5194/gmd-6-901-2013.
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Abstract. Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection schemes allow for long time steps. These longer time steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme, but small oscillations remain in operational weather forecasts using ARPÉGE and ALADIN models. In this paper, a simple test is designed to reveal if the formulation of particular a physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. However, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of the NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test identified the set of equations in the stratiform precipitation scheme (a diagnostic Kessler-type scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.
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Gao, He, Zhongming Gu, Shanjun Liang, Tuo Liu, Jie Zhu, and Zhongqing Su. "Enhancing ultrasound transmission and focusing through a stiff plate with inversely optimized auxiliary meta-lens." Applied Physics Letters 120, no. 11 (March 14, 2022): 111701. http://dx.doi.org/10.1063/5.0085462.
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Effective sound energy transmission and beam manipulation through stiff and dense materials such as metal remain daunting tasks. It is in part attributable to the vast impedance mismatch between those materials and ambient media. Adding openings may facilitate to better bridge energy over, yet ineffective in many applications and may also damage the structural integrity. Here, we present an auxiliary ultrasound focusing meta-lens for stiff and dense materials. It offers significantly enhanced ultrasound transmission and focusing through a stiff metal plate yet without enforcing any through holes or openings. The simple, one-sided only meta-structures are designed and optimized by an inverse strategy based on the genetic algorithm. We numerically and experimentally demonstrate the much enhanced ultrasound transmission when the meta-lens is added to a flat brass plate, along with the capability to offer simultaneous ultrasound focusing. This design methodology can be easily extended to deal with more complex shaped target in a straightforward manner, offering a practical solution to the efficient tunneling of ultrasound energy through stiff and dense materials. With simple grating structures, the meta-lens can be easily fabricated, showing great application prospects in medical imaging and disease treatment.
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WNEK, MACIEJ, and JOSEF K. MOŚCICKI. "Numerical Study of the Phase Equilibria and Molecular Dynamics of Stiff-Chain Polymers in Solution." International Journal of Modern Physics C 02, no. 01 (March 1991): 541–45. http://dx.doi.org/10.1142/s0129183191000858.
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The zero-shear viscosity and dielectric relaxation are studied numerically for several model solutions of polydisperse rods. The phase equilibrium calculation is based on the Warner-Flory lattice theory [1]. In the biphasic range of the solutions two different models of phase mixing are considered.
13
Zhu, Liyong. "Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations." Advances in Applied Mathematics and Mechanics 9, no. 1 (October 11, 2016): 157–72. http://dx.doi.org/10.4208/aamm.2015.m1045.
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AbstractIn this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.
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Stewart, David E. "A numerical method for friction problems with multiple contacts." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 37, no. 3 (January 1996): 288–308. http://dx.doi.org/10.1017/s0334270000010675.
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AbstractFriction problems involving “dry” or “static” friction can be difficult to solve numerically due to the existence of discontinuities in the differential equations appearing in the right-hand side. Conventional methods only give first-order accuracy at best; some methods based on stiff solvers can obtain high order accuracy. The previous method of the author [16] is extended to deal with friction problems involving multiple contact surfaces.
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Chen, Zhou, Weiqiu Chen, and Jizhou Song. "Buckling of a stiff thin film on an elastic graded compliant substrate." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2208 (December 2017): 20170410. http://dx.doi.org/10.1098/rspa.2017.0410.
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The buckling of a stiff film on a compliant substrate has attracted much attention due to its wide applications such as thin-film metrology, surface patterning and stretchable electronics. An analytical model is established for the buckling of a stiff thin film on a semi-infinite elastic graded compliant substrate subjected to in-plane compression. The critical compressive strain and buckling wavelength for the sinusoidal mode are obtained analytically for the case with the substrate modulus decaying exponentially. The rigorous finite element analysis (FEA) is performed to validate the analytical model and investigate the postbuckling behaviour of the system. The critical buckling strain for the period-doubling mode is obtained numerically. The influences of various material parameters on the results are investigated. These results are helpful to provide physical insights on the buckling of elastic graded substrate-supported thin film.
16
Tudor, M. "A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models." Geoscientific Model Development Discussions 5, no. 4 (December 11, 2012): 4233–68. http://dx.doi.org/10.5194/gmdd-5-4233-2012.
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Abstract. Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection scheme in the model dynamics allows for long time-steps. These longer time-steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time-step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme. But small oscillations remain in an operational weather forecasts using ARPÉGE and ALADIN models. It is needed to test of the complete model formulation, as implemented in the operational forecast. In this paper, a simple test is designed. The test reveals if the formulation of particular physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. But, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of a NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test has identified the stratiform precipitation scheme (a diagnostic Kessler type scheme) as a stiff problem, particularly the term that describes the evaporation of snow.
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Shornikov, Yury V., and Konstantin A. Timofeev. "Analysis of numerical methods for integrating high-dimensional stiff systems in SimInTech." Analysis and data processing systems, no. 4 (December 27, 2022): 63–74. http://dx.doi.org/10.17212/2782-2001-2022-4-63-74.
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When modeling complex dynamic processes, it becomes necessary to numerically solve the Cauchy problem for systems of ordinary differential equations (ODEs). The efficiency of the applied numerical methods depends on the degree of stiffness and dimension of the problem [1–3]. Depending on the task class, different methods behave differently.This article provides a comparative analysis of explicit adaptive and diagonal-implicit integration methods implemented in the SimInTech software [4]. The SimInTech software package is designed to simulate complex dynamic processes in systems of various classes. The system supports the ability to develop models in the form of block diagrams, as well as describe systems of differential equations using the built-in programming language and simulate event-driven systems and finite automata.It is shown that the most effective in solving problems of the considered class are the diagonal-implicit Runge-Kutta type integration methods – DIRK2 and DIRK4 from the SimInTech package. The DIRK3 method is inferior due to the large number of calculations of the Jacobian matrix. The preferred method is rather The DIRK2 method is preferable in this case, because it has a greater number of time steps with almost the same performance and does not increase the integration step so much with relatively low accuracy settings of the integration method. Of the explicit methods of the Runge-Kutta type with an adaptive numerical scheme, the “Adaptive-5” method is the most effective for solving problems of this class. For problems of this class, we can recommend the use of explicit integration methods “Adaptive-5”, “Adaptive-1” with a small system dimension.The traditional implicit Gear and Euler integration methods also effectively solve this problem, provided that the algorithm for calculating the Jacobian matrix is effectively implemented.
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Sayed-Ahmed, M. E., and A. S. El-Yazal. "Laminar fully developed flow and heat transfer of RobertsonStiff fluids in a rectangular duct." Canadian Journal of Physics 83, no. 2 (February 1, 2005): 165–82. http://dx.doi.org/10.1139/p04-063.
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The laminar fully developed flow and heat transfer through a rectangular duct of a viscous incompressible RobertsonStiff fluid is investigated. RobertsonStiff fluids are time independent non-Newtonian materials possessing a yield value. The governing momentum and energy equations are solved numerically using finite-difference approximations. We consider two cases of thermal boundary conditions: H1 the "circumferentially constant wall temperature and axially constant wall heat flux" and H2 the "circumferentially and axially constant wall heat flux". The velocity, temperature, the average friction factor and Nusselt numbers for the two cases are computed for various values of the physical parameters. The present results have been compared with the known solutions for Newtonian and power-law fluids and are found to be in good agreement.PACS Nos.: 47.50.+d, 47.15.x
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Ren, Y., and C. F. Beards. "Identification of ’Effective’ Linear Joints Using Coupling and Joint Identification Techniques." Journal of Vibration and Acoustics 120, no. 2 (April 1, 1998): 331–38. http://dx.doi.org/10.1115/1.2893835.
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Almost all real-life structures are assembled from components connected by various types of joints. Unlike many other parts, the dynamic properties of a joint are difficult to model analytically. An alternative approach for establishing a theoretical model of a joint is to extract the model parameters from experimental data using joint identification techniques. The accuracy of the identification is significantly affected by the properties of the joints themselves. If a joint is stiff, its properties are often difficult to identify accurately. This is because the responses at both ends of the joint are linearly-dependent. To make things worse, the existence of a stiff joint can also affect the accuracy of identification of other effective joints (the term “effective joints” in this paper refers to those joints which otherwise can be identified accurately). This problem is tackled by coupling these stiff joints using a generalized coupling technique, and then the properties of the remaining joints are identified using a joint identification technique. The accuracy of the joint identification can usually be improved by using this approach. Both numerically simulated and experimental results are presented.
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Gao, Feng, Wang Guo, Peijian Chen, Chengzheng Cai, and Guangjian Peng. "Numerical Analysis on the Wrinkling Instability of a Stiff Film Adhering to an Elastic Substrate with a Graded Coating." International Journal of Applied Mechanics 11, no. 02 (March 2019): 1950015. http://dx.doi.org/10.1142/s1758825119500157.
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The wrinkling instability of a stiff film adhering to a pre-strained inhomogeneous bi-layer substrate consisting of a homogeneous substrate and a graded coating is investigated in the present paper. The critical strain, wavelength and amplitude of the film/inhomogeneous substrate system are calculated numerically and analyzed comprehensively. Compared with the numerical result, a theoretical model is introduced to approximately predict the wrinkling responses of the system. The influence of various geometric and material parameters on the wrinkling behavior is mainly focused. The wrinkling responses are found to be highly related to the graded laws and the thickness of the inhomogeneous coating as well as the Poisson’s ratio. What is more, a proper choice of graded properties of a substrate can improve the wrinkling response of a film/substrate system. The present finding should be very meaningful to guide the design of various stretchable and flexible electronics.
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Slepukhin, Valentin M., Maximilian J. Grill, Qingda Hu, Elliot L. Botvinick, Wolfgang A. Wall, and Alex J. Levine. "Topological defects produce kinks in biopolymer filament bundles." Proceedings of the National Academy of Sciences 118, no. 15 (April 5, 2021): e2024362118. http://dx.doi.org/10.1073/pnas.2024362118.
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Bundles of stiff filaments are ubiquitous in the living world, found both in the cytoskeleton and in the extracellular medium. These bundles are typically held together by smaller cross-linking molecules. We demonstrate, analytically, numerically, and experimentally, that such bundles can be kinked, that is, have localized regions of high curvature that are long-lived metastable states. We propose three possible mechanisms of kink stabilization: a difference in trapped length of the filament segments between two cross-links, a dislocation where the endpoint of a filament occurs within the bundle, and the braiding of the filaments in the bundle. At a high concentration of cross-links, the last two effects lead to the topologically protected kinked states. Finally, we explore, numerically and analytically, the transition of the metastable kinked state to the stable straight bundle.
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Witkowski, Krzysztof, Grzegorz Kudra, Grzegorz Wasilewski, and Jan Awrejcewicz. "Modelling and experimental validation of 1-degree-of-freedom impacting oscillator." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 233, no. 4 (October 4, 2018): 418–30. http://dx.doi.org/10.1177/0959651818803165.
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In this article, a mechanical 1-degree-of-freedom oscillator with harmonic forcing and impacts was analysed both numerically and experimentally. A special attention was paid to the mathematical modelling and realistic numerical simulations of a real system. The developed experimental rig consists of a cart mounted on a guide, connected with springs to the support, and equipped with one-sided stiff limiter of motion. The cart was excited by an unbalanced disc mounted on the shaft of a stepper motor. Modelling was focused on the mathematical description of the impact process, where soft obstacle with Hertzian stiffness was assumed and different forms of non-linear damping were tested, including original modifications of the already used models. Model parameters are identified based on two experimental solutions corresponding to free vibrations – with and without impacts. Then, the model was validated by means of experimental and numerical analysis of bifurcation dynamics of the forced system.
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Faraci, David, Claudia Comi, and Jean-Jacques Marigo. "Band Gaps in Metamaterial Plates: Asymptotic Homogenization and Bloch-Floquet Approaches." Journal of Elasticity 148, no. 1 (January 2022): 55–79. http://dx.doi.org/10.1007/s10659-022-09879-3.
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AbstractIn this work, we study the transversal vibration of thin periodic elastic plates through asymptotic homogenization. In particular, we consider soft inclusions and rigid inclusions with soft coatings embedded in a stiff matrix. The method provides a general expression for the dynamic surface density of the plate, which we compute analytically for circular inclusions or numerically for two-way ribbed plates. Through asymptotic homogenization, we find that band gaps related to in-plane propagating transversal waves occur for frequency intervals in which the effective surface density is negative. The same result is obtained via an asymptotic analysis of the Bloch-Floquet problem on a unit cell, showing the equivalence of the two approaches. Finally, we validate the method by comparing in several examples the predicted band gaps with those obtained from numerical Bloch-Floquet analyses on the real unit cell.
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Ferri, A. A., and B. S. Heck. "Vibration Analysis of Dry Friction Damped Turbine Blades Using Singular Perturbation Theory." Journal of Vibration and Acoustics 120, no. 2 (April 1, 1998): 588–95. http://dx.doi.org/10.1115/1.2893868.
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The dynamic behavior of a turbine blade with an attached dry friction damper is investigated. Due to the presence of both high and low frequency resonances, the system is numerically stiff and time consuming to simulate. Three reduced-order models that remove the numerical stiffness problem are developed using singular perturbation theory. Two of these models are developed for the case of small damper mass, and the third model is developed for the case of high damper stiffness. It is shown that these models are consistent with two physically-motivated approximate models that have appeared in the literature. The accuracy of each approximate model is examined for a range of system and excitation parameters through comparison with the original full-order model. It is seen that inclusion of damper mass has a significant effect on the blade system response.
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WAI, CHUA KAH, LOY KAK CHOON, and RUWAIDIAH IDRIS. "ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS." Universiti Malaysia Terengganu Journal of Undergraduate Research 3, no. 1 (January 31, 2021): 25–36. http://dx.doi.org/10.46754/umtjur.v3i1.189.
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Ordinary Differential Equations (ODEs) are usually used in numerous fields especially in solving the modelling problem. Numerical methods are one of the vital mathematical tools to solve the ODEs that appear in various modelling problems by determining the approximation solution close to the in exact solution if it exists. Runge-Kutta methods (RK) are the numerical methods used to integrate the ODEs by applying multistage methods at the midpoint of an interval which can efficiently produce a more accurate result or small magnitude of error. We proposed Runge-Kutta methods (RK) to solve the 1st_ order nonlinear stiff ODEs. The RK methods used in this research are known as the RK-2, RK-4, and RK-5 methods. We proved the existence and uniqueness of the ODEs before we solved it numerically. We also proved the absolute-stability of the RK methods to determine the overall stability of these methods. We found two suitable test cases which are the standard test problem and manufactured solution. We proved that by combining the adaptive step size with RK methods can result in more efficient computation. We implemented the 2nd_, 4th_ and 5th_ order of RK methods with step size adaptively algorithm to solve the test problem and manufactured solution via Octave programming language. The resulting numerical error and the stability of each method can be studied. We compared our results using several error plots versus the Central Processing Unit (CPU) time required to compute a given nonlinear 1st_ order stiff ODE problem. In a conclusion, RK methods which combine with the adaptive step size can result in more efficient computation and accuracy compare with the fixed step size RK methods.
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Liu, Yuchuan, W. Wayne Chen, Dong Zhu, Shuangbiao Liu, and Q. Jane Wang. "An Elastohydrodynamic Lubrication Model for Coated Surfaces in Point Contacts." Journal of Tribology 129, no. 3 (February 2, 2007): 509–16. http://dx.doi.org/10.1115/1.2736433.
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An elastohydrodynamic lubrication (EHL) model for coated surfaces in point contacts has been developed by combining the elastic deformation formulation for the coated surfaces with an EHL model. Inverse fast Fourier transform (IFFT) is employed first to obtain the influence coefficients (ICs) from the frequency response function (FRF). The subsequent calculation of elastic deformation is performed using the efficient algorithm of discrete convolution and fast Fourier transform (DC-FFT). The coating EHL model is verified by the comparison to available numerical results. The effects of coating on lubrication under various loads, speeds, rheological models, and pressure-viscosity behaviors are numerically investigated. Similar to the observations from dry contact, stiffer coatings in EHL tend to reduce the nominal contact radius but increase the maximum contact pressure, and vice versa for more compliant coatings. However, as coating thickness increases, the influence of coatings on film thickness, including the central and the minimum film thicknesses, does not follow a monotonic variation, and therefore, cannot be predicted by any simple film thickness equation. The reason for that is the pressure viscosity effect which tends to counterbalance the effect of coating. The average friction coefficient in lubricant film increases in stiff coating cases but decreases for compliant coating cases. Furthermore, two possible approaches to improving the minimum film thickness thus reducing friction and wear in mixed lubrication are indicated: a thin stiff coating for conventional EHL and a thick compliant coating for soft EHL.
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Micunovic, Milan, and Ljudmila Kudrjavceva. "On inelasticity of damaged quasi-rate-independent orthotropic materials." Theoretical and Applied Mechanics, no. 00 (2022): 1. http://dx.doi.org/10.2298/tam211007001m.
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The paper deals with a body having a random 3D-distribution of two-phase inclusions: spheroidal mutually parallel voids as well as differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth-order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by modified Vakulenko?s approach to endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is given. The results of the theory are applied to the damage-elasto-viscoplastic strain of reactor stainless steel AISI 316H.
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Mićunović, Milan, and Ljudmila Kudrjavceva. "On inelasticity of damaged quasi rate independent anisotropic materials." Mathematics and Mechanics of Solids 24, no. 3 (February 26, 2018): 778–95. http://dx.doi.org/10.1177/1081286518755228.
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This paper deals with a body that has a random 3D-distribution of two phase inclusions: spheroidal mutually parallel voids, and differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by a modified Vakulenko approach to the endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is considered. The results of the theory are applied to the damage-elasto-viscoplastic strain of a reactor stainless steel AISI 316H.
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Torabi, Seyed Mousa, Abolfazl Tari, and Sedaghat Shahmorad. "Two-step collocation methods for two-dimensional Volterra integral equations of the second kind." Journal of Applied Analysis 25, no. 1 (June 1, 2019): 1–11. http://dx.doi.org/10.1515/jaa-2019-0001.
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Abstract In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.
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Alsaidi, Bashir, Woong Yeol Joe, and Muhammad Akbar. "Computational Analysis of 3D Lattice Structures for Skin in Real-Scale Camber Morphing Aircraft." Aerospace 6, no. 7 (July 7, 2019): 79. http://dx.doi.org/10.3390/aerospace6070079.
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Conventional or fixed wings require a certain thickness of skin material selection that guarantees structurally reliable strength under expected aerodynamic loadings. However, skin structures of morphing wings need to be flexible as well as stiff enough to deal with multi-axial structural stresses from changed geometry and the coupled aerodynamic loadings. Many works in the design of skin structures for morphing wings take the approach either of only geometric compliance or a simplified model that does not fully represent 3D real-scale wing models. Thus, the main theme of this study is (1) to numerically identify the multi-axial stress, strain, and deformation of skin in a camber morphing wing aircraft under both structure and aerodynamic loadings, and then (2) to show the effectiveness of a direct approach that uses 3D lattice structures for skin. Various lattice structures and their direct 3D wing models have been numerically analyzed for advanced skin design.
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Jin Sian, Chai, Yeak Su Hoe, and Ali H. M. Murid. "Some Numerical Methods and Comparisons for Solving Mathematical Model of Surface Decontamination by Disinfectant Solution." MATEMATIKA 34, no. 2 (December 2, 2018): 271–91. http://dx.doi.org/10.11113/matematika.v34.n2.1055.
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A mathematical model is considered to determine the effectiveness of disinfectant solution for surface decontamination. The decontamination process involved the diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing effect. The mathematical model is a reaction-diffusion type. Finite difference method and method of lines with fourth-order Runge-Kutta method are utilized to solve the model numerically. To obtain stable solutions, von Neumann stability analysis is employed to evaluate the stability of finite difference method. For stiff problem, Dormand-Prince method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB programming is selected for the computation of numerical solutions. From the results obtained, fourth-order Runge-Kutta method has a larger stability region and better accuracy of solutions compared to finite difference method when solving the disinfectant solution model. Moreover, a numerical simulation is carried out to investigate the effect of different thickness of disinfectant solution on bacteria reduction. Results show that thick disinfectant solution is able to reduce the dimensionless bacteria concentration more effectively
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Lee, S. L., T. S. Chen, and B. F. Armaly. "Mixed Convection Along Vertical Cylinders and Needles With Uniform Surface Heat Flux." Journal of Heat Transfer 109, no. 3 (August 1, 1987): 711–16. http://dx.doi.org/10.1115/1.3248147.
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Mixed convection along vertical cylinders and needles with uniform surface heat flux is investigated for the entire mixed convection regime. A single modified buoyancy parameter χ and a single curvature parameter Λ are employed in the analysis such that a smooth transition from pure forced convection (χ = 1) to pure free convection (χ = 0) can be accomplished. For large values of the curvature parameter and/or Prandtl number, the governing transformed equations become stiff. Thus, a numerically stable finite-difference method is employed in the numerical solution in conjunction with the cubic spline interpolation scheme to overcome the difficulties that arise from the stiffness of the equations. Local Nusselt numbers are presented for 0.1 ≤ Pr ≤ 100 that cover 0 ≤ χ ≤ 1 (∞ ≥ Ωχ ≥ 0) and 0 ≤ Λ ≤ 50. For needles (Λ ≥ 5), the local Nusselt numbers Nuχ/(Reχ1/2 + Grχ*1/5) are found to be nearly independent of the buoyancy parameter χ. Correlation equations for the local Nusselt numbers are also presented.
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Naserkhaki, Sadegh, and Hassan Pourmohammad. "SSI AND SSSI EFFECTS IN SEISMIC ANALYSIS OF TWIN BUILDINGS: DISCRETE MODEL CONCEPT." Journal of Civil Engineering and Management 18, no. 6 (November 20, 2012): 890–98. http://dx.doi.org/10.3846/13923730.2012.734850.
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This paper presents a numerical study of soil-structure interaction (SSI) and structure-soil-structure interaction (SSSI) effects on response of twin buildings during earthquake excitations. The buildings are modeled as shear buildings and the soil is simulated by a discrete model representing a visco-elastic half-space subjected to earthquake acceleration. Equation of motion of twin buildings with different conditions, fixed based (FB), SSI and SSSI, are developed via an analytical procedure and solved numerically. Buildings responses are evaluated for aforementioned three conditions considering various soil types and compared together. One must say that soil causes change in distribution of responses throughout the buildings while ignoring soil interaction may lead to detrimental effects on buildings. Anyway, interaction between twin buildings with SSSI condition slightly mitigates soil unfavorable effects compare to one building with SSI condition. In addition, it is found that influence of soil is very significant for soft to stiff soils whereas negligible for hard soils.
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Braun, K. T., N. Bakas, G. Markou, and S. W. Jacobsz. "Advanced numerical modelling of the nonlinear mechanical behaviour of a laterally loaded pile embedded in stiff unsaturated clay." Journal of the South African Institution of Civil Engineering 65, no. 2 (June 27, 2023): 28–38. http://dx.doi.org/10.17159/2309-8775/2023/v65n2a4.
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Capturing and understanding the ultimate limit state behaviour of reinforced concrete piles embedded in soil requires the use of advanced tools or the performance of expensive tests. An experiment was performed where reinforced concrete piles embedded in a stiff unsaturated clay profile were load-tested on-site. However, even though in-situ experiments can provide engineers with valuable insight, their cost and time limitations come with restrictions, especially when dealing with a parametric investigation on the soil's material properties, the size of the piles, or the piles' material properties. The objective of this research work was to numerically model the nonlinear mechanical behaviour of laterally loaded full-scale piles through detailed 3D modelling, and perform an in-depth parametric investigation to provide answers to unknown factors that the actual physical experiment could not answer. Furthermore, this work serves as a pilot project that will be used to pave the way in developing multiple soil-structure interaction models that will be used to generate a dataset that helps the creation of predictive models through machine learning algorithms. For the needs of this research work, the reinforced concrete piles were discretised with 8-noded isoparametric hexahedral elements that accounted for cracking through the smeared crack approach. Steel reinforcement bars and stirrups were simulated as embedded rebar elements, while the soil domain was also discretised through 8-noded hexahedral elements. Most of the required material properties assumed during the nonlinear analyses were defined according to relevant laboratory experiments. According to the numerical investigation, it was found that the proposed numerical model has the ability to reproduce the experimental results with high accuracy, while providing in-depth insight on the failure mechanisms for both the soil and reinforced concrete domains.
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GAEBLER, HARRY J., and HERMANN J. EBERL. "A simple model of biofilm growth in a porous medium that accounts for detachment and attachment of suspended biomass and their contribution to substrate degradation." European Journal of Applied Mathematics 29, no. 6 (April 12, 2018): 1110–40. http://dx.doi.org/10.1017/s0956792518000189.
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We derive a macroscopic model for biofilm formation in a porous medium reactor to investigate the role of suspended bacteria on reactor performance. The starting point is the mesoscopic one-dimensional Wanner–Gujer biofilm model. The following processes are included: hydrodynamics and transport of substrate in the reactor, biofilm and suspended bacteria growth in the pore space, attachment of suspended cells to the biofilm, and detachment of biofilm cells. The mesoscopic equations are up-scaled from the biofilm scale to the reactor scale, yielding a stiff system of balance laws, which we study numerically. We find that suspended bacteria and attachment can have a significant effect on biofilm reactor performance.
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Gao, He, and Hao Tang. "Numerical Simulation of Gaseous Detonation Performance and Wall Reflection Effect of Acetylene-Rich Fuel." Energies 15, no. 14 (July 7, 2022): 4985. http://dx.doi.org/10.3390/en15144985.
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The efficient utilization of energy in rich fuel detonation processes and the effective control method of soot are important topics in combustion research. In this paper, we numerically study the detonation wave behavior of acetylene–air systems in rich fuel condition by using a reduced reaction mechanism. The non-stiff terms of the governing equations are solved explicitly using the gas kinetic scheme, and the stiff terms are solved implicitly. Our results show that the acetylene pyrolysis is the dominant reaction process. The oxidation reaction is exploited to initiate the reaction induction process, providing the required energy to overcome the potential energy barrier. The secondary detonation structure is due to the stable interaction of the transverse waves and the combined action of the vinyl reaction system, thus effectively improve the energy release rate and providing a powerful solution for the fuel-rich high-energy release of advanced heat engines. The area of the unreacted pocket increases with the acetylene concentration, resulting in an irregular wave-front and detonation cell. The reflected shock wave impacting on the wall induces the secondary reaction of the detonation products. The concentration of polycyclic aromatic hydrocarbons decreases significantly and regenerates near the wall. Our approach provides an effective tool for controlling detonation soot and the preparation of carbon particles.
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Potapenko, Tatiana, Joseph Burchell, Sandra Eriksson, and Irina Temiz. "Wave Energy Converter’s Slack and Stiff Connection: Study of Absorbed Power in Irregular Waves." Energies 14, no. 23 (November 24, 2021): 7892. http://dx.doi.org/10.3390/en14237892.
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Two different concepts of wave energy converter coupled to the novel C-GEN linear generator have been studied numerically, including the evaluation of different buoy sizes. The first concept has a slack connection between the buoy and the generator on the seabed. Another concept is based on a stiff connection between the buoy and the generator placed on an offshore platform. Three different approaches to calculate the damping force have been utilized within this study: the optimal damping coefficient, R-load, and RC-load. R-load is a model for the load applied to a grid-connected generator with passive rectification. RC-load is a model for a phase angle compensation applied to a system with active rectification. The radiation forces originating from the oscillatory motion of the buoy have been approximated using the transfer function in the frequency domain and the vector fitting algorithm. A comparison of the approximation methods is presented, and their accuracy has been evaluated. The advantage of the vector fitting method has been shown, especially for higher approximation orders which fit the transfer function with high accuracy. The study’s final results are shown in terms of the absorbed power for the sea states of March 2018 at Wave Hub, UK.
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TRIMBORN, TIMO, KARL-JOSEF KOCH, and THOMAS M. STEGER. "MULTIDIMENSIONAL TRANSITIONAL DYNAMICS: A SIMPLE NUMERICAL PROCEDURE." Macroeconomic Dynamics 12, no. 3 (June 2008): 301–19. http://dx.doi.org/10.1017/s1365100507070034.
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We propose the relaxation algorithm as a simple and powerful method for determining the transition process in growth models numerically. This method has a number of important advantages: (1) It can easily deal with a wide range of dynamic systems including stiff differential equations and systems giving rise to a continuum of stationary equilibria. (2) The application of the procedure is fairly user-friendly. The only input required consists of the dynamic system. (3) The variant of the relaxation algorithm we propose exploits in a natural manner the infinite time horizon, which usually underlies optimal control problems in economics. As an illustrative application, we compute the transition process of the models of Jones [Jones, C.I. (1995) R&D-based models of economic growth. Journal of Political Economy 103 (3), 759–784] and Lucas [Lucas, R.E., Jr. (1988) On the mechanics of economic development. Journal of Monetary Economics 22, 3–42].
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Caputo, Francesco, Giuseppe Lamanna, and Alessandro Soprano. "Energy Absorption Capabilities of a Square Tube System." Key Engineering Materials 488-489 (September 2011): 561–64. http://dx.doi.org/10.4028/www.scientific.net/kem.488-489.561.
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In the present paper the authors refer about a series of experimental tests, where an aluminium alloy square tube, filled with an aluminium foam, was crushed by a longitudinal load at a speed of 10 m/s. The test apparatus consisted of a sled installed on a very stiff frame moving on appropriate guides, as the specimen was set on a home-made fixture. Two arrangements of square tubes were considered as specimens: a “standard” one and an “optimized” one. Both crushing behaviours and energy absorption capabilities were analyzed experimentally and numerically simulated by means of the explicit FE code LS-DYNA®; the complete numerical model consisted of the striker, the assemblage of square tubes and the base. A high-speed video recording system was used to capture the images from the physical test. The results from the numerical analyses were compared to those obtained from the experiments: those results showed that the force–deflection response had been overestimated by the numerical model. The authors attempted to justify this inconsistency by considering the influence of the strain rate parameters of the considered Cowper-Symonds analytical model on the results. It was shown that the “optimized” energy absorber exhibited a more desirable force–deflection response than the standard one due to some easy design changes, which involved the insertion of aluminium foam dampers.
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Assaad, M. C., B. Kimble, Y. M. Huang, R. Burgan, G. C. Fralick, J. D. Wrbanek, and J. M. Gonzalez. "Thin-Film Heat Flux Sensor for Measuring the Film Coefficient of Rubber Components of a Rolling Tire." Tire Science and Technology 36, no. 4 (December 1, 2008): 275–89. http://dx.doi.org/10.2346/1.2999702.
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Abstract Heat flux is one of a number of parameters, together with pressure, temperature, flow, etc., of interest to tire engineers, engine designers and fluid dynamicists. The ability to measure heat flux magnitude and direction was incorporated into a resistance bridge design fabricated using thin-film techniques to allow fast response. The result is a sensor that does not need the large area and stiff packaging required for the thermopile design, nor does it have a low output, but has nearly as fast response. The development of this sensor offers a new laboratory procedure to establish heat transfer coefficients for different regions of a tire. Testing generated heat transfer coefficients that were within the range reported in the literature, and the numerically predicted temperatures from this data agree well with the experimentally generated values.
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Do, Nguyen B., Aldo A. Ferri, and Olivier A. Bauchau. "Efficient Simulation of a Dynamic System with LuGre Friction." Journal of Computational and Nonlinear Dynamics 2, no. 4 (March 18, 2007): 281–89. http://dx.doi.org/10.1115/1.2754304.
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Friction is a difficult phenomenon to model and simulate. One promising friction model is the LuGre model, which captures key frictional behavior from experiments and from other friction models. While displaying many modeling advantages, the LuGre model of friction can result in numerically stiff system dynamics. In particular, the LuGre friction model exhibits very slow dynamics during periods of sticking and very fast dynamics during periods of slip. This paper investigates the best simulation strategies for application to dynamic systems with LuGre friction. Several simulation strategies are applied including the explicit Runge–Kutta, implicit Trapezoidal, and implicit Radau-IIA schemes. It was found that both the Runge–Kutta and Radau-IIA methods performed well in simulating the system. The Runge–Kutta method had better accuracy, but the Radau-IIA method required less integration steps.
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Bui, Duc Van, Manh Van Nguyen, Trong Dang Nguyen, and Truong Nho Vu. "A numerically investigate of the improvement of load carrying capacity of square footings utilizing micropiles." Journal of Mining and Earth Sciences 63, no. 5 (October 31, 2022): 106–17. http://dx.doi.org/10.46326/jmes.2022.63(5).10.
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This paper is aimed to address an actual case study on the use of micropile technology for improving the bearing capacity of an old building. The numerical simulation results show that the load-carrying capacity of square footing utilizing micropiles is notably increased. The improvement of the bearing capacity of the foundations depends on the strengthening methods, such as inclination angle ((), length (L), and distance of micropile from the edge of footings (S). Specifically, with the same length value of pile used, the bearing capacity reaches the largest magnitude at the S/B ratios of (0.5(0.75). The use of inclined piles yields a larger magnitude of bearing capacity than the vertical ones, these obtained results are contributed to the contribution of the “confining effects” of soil mass underneaths the footing as subjected to vertical loads. Additionally, if the soil mass below the footing has a high bearing capacity (firm to stiff clayey soils, medium to dense sandy soils…) , the design value of L/B ratio in the strengthening method should be in range of (2.0÷3.0), chosing beyond that optimal range is uneconomical since the improvement of bearing capacity is insignificant. In other words, the relationship between stress bulb in soil under the footing and the length of micropile should be taken into consideration to achieve a higher economic efficiency of the strengthening method.
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COSTA, R. R. R., M. C. V. LIMA, G. M. S. ALVA, and E. S. MAGALHÃES. "Optimization of the bending stiffness of beam-to-column and column-to-foundation connections in precast concrete structures." Revista IBRACON de Estruturas e Materiais 10, no. 5 (September 2017): 985–97. http://dx.doi.org/10.1590/s1983-41952017000500003.
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Abstract This work involved the structural optimization of precast concrete rigid frames with semi-rigid beam-to-column connections. To this end, several frames were simulated numerically using the Finite Element Method. Beams and columns were modeled using bar elements and their connections were modeled using spring elements, with variable bending stiffness. The objective function was based on the search of the least stiff connection able to ensure the global stability of the building. Lastly, a connection model with optimal stiffness was adopted to design the frame. Semi-rigid beam-to-column connections with a constraint factors of 0.33 sufficed to ensure the maximum allowable horizontal displacement and bending moment of the connection, with a global stability parameter of 1.12. This confirms that even connections with low constraints generate significant gains from the structural standpoint, without affecting construction and assembly-related aspects.
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Kutuk, M. Akif, and Ibrahim Gov. "Application of Topology Optimization to the Tibial Osteotomy Fixation Plates." Applied Bionics and Biomechanics 10, no. 2-3 (2013): 125–33. http://dx.doi.org/10.1155/2013/762374.
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Application of topology optimization to fixation plates is the main consideration of this paper. The interbody fusion plates are required to give mechanical support to tibia with minimally invasive surgical procedure. Topology optimization is used to obtain fixation plates with possible minimum material usage. Topology optimization is applied to three types of plates which are used in upper tibial osteotomy. Initial design of the plates are first numerically modelled and then investigated for stresses under possible highest load values. The results of the analysis indicated that the plates are very stiff even under high loads. Application of topology optimization to plates yielded minimized weight and material usage while keeping the plates still adequate for possible high load values. It was also revealed that up to 50% of mass could be saved by an optimal implant design.
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Kheiri, Mojtaba. "Dynamics and Stability of a Flexible, Slender Cylinder Flexibly Restrained at One End and Free at the Other and Subjected to Axial Flow." Archive of Mechanical Engineering 63, no. 3 (September 1, 2016): 379–96. http://dx.doi.org/10.1515/meceng-2016-0022.
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Abstract In this paper, Lagrange’s equations along with the Ritz method are used to obtain the equation of motion for a flexible, slender cylinder subjected to axial flow. The cylinder is supported only by a translational and a rotational spring at the upstream end, and at the free end, it is terminated by a tapering end-piece. The equation of motion is solved numerically for a system in which the translational spring is infinitely stiff, thus acting as a pin, while the stiffness of the rotational spring is generally non-zero. The dynamics of such a system with the rotational spring of an average stiffness is described briefly. Moreover, the effects of the length of the cylinder and the shape of the end-piece on the critical flow velocities and the modal shapes of the unstable modes are investigated.
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O’Shea, Michael J. "A climbers pendulum fall." Physics Education 58, no. 1 (December 9, 2022): 015025. http://dx.doi.org/10.1088/1361-6552/aca312.
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Abstract We analyse a lead climber pendulum fall, a fall that can occur after the climber traverses horizontally away from their highest anchor. The model is based on a spring pendulum with a nonlinear component and incorporates air resistance. The resulting set of equations can be solved numerically using an Excel spreadsheet that is easily implemented by a student. The maximum tension and speed during the fall were calculated for a real dynamic climbing rope. The inclusion of rope elasticity can, under the right conditions, lead to a speed at the bottom of the swing that is less than that found for a stiff rope. The path of the climber is not closed as they swing back and forth, and we analyse radial oscillations to gain further insight. For multiple swings, the air resistance is important for removing energy from the system.
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Tlemissov, A., Zh Tlemissova, K. Boshkayev, A. Urazalina, and H. Quevedo. "ANALYSIS OF THE EQUATIONS OF STATE FOR NEUTRON STARS." PHYSICO-MATHEMATICAL SERIES 5, no. 333 (October 15, 2020): 43–52. http://dx.doi.org/10.32014/2020.2518-1726.81.
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In this work we consider various equations of state of neutron star matter, which include from the point of neutron drops formation to supra nuclear densities. Particular attention is paid to the nucleon – nucleon interaction since, in addition to the kinetic energies of the particles, the interactions among nucleons play a key role. Moreover, we investigate the properties of super-dense matter with diverse sets of particles such as electrons, protons, and the contribution of various particles-carriers of interaction. In order to achieve these goals, different potentials were considered, which are in a good agreement with experimental data. Furthermore, we find the energy of the system by using a variety of multi-particle methods, including the interaction of nucleons. Thanks to this information, thermodynamic parameters such as pressure, energy density and the speed of sound in the star are calculated. We compared similar equations of state of matter so that we could demonstrate the difference from each other. The Tolman-Oppenheimer-Volkoff system of equations has been solved numerically to construct mass-central density, radius-central density and mass-radius relations using different equations of state. In conclusion, the latest observational constraints on the equation of state are taken into account and we show that the observational data require that the equation of state be stiff, despite the fact that all stiff equations of state violate the principle of causality at high central densities, unlike soft ones.
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Rusinek, Rafal, Joanna Rekas, Katarzyna Wojtowicz, and Robert Zablotni. "Polyharmonic Vibrations of Human Middle Ear Implanted by Means of Nonlinear Coupler." Materials 14, no. 18 (September 7, 2021): 5121. http://dx.doi.org/10.3390/ma14185121.
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This paper presents a possibility of quasi-periodic and chaotic vibrations in the human middle ear stimulated by an implant, which is fixed to the incus by means of a nonlinear coupler. The coupler represents a classical element made of titanium and shape memory alloy. A five-degrees-of-freedom model of lumped masses is used to represent the implanted middle ear for both normal and pathological ears. The model is engaged to numerically find the influence of the nonlinear coupler on stapes and implant dynamics. As a result, regions of parameters regarding the quasi-periodic, polyharmonic and irregular motion are identified as new contributions in ear bio-mechanics. The nonlinear coupler causes irregular motion, which is undesired for the middle ear. However, the use of the stiff coupler also ensures regular vibrations of the stapes for higher frequencies. As a consequence, the utility of the nonlinear coupler is proven.
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Pelc, J. "Static three-dimensional modelling of pneumatic tyres using the technique of element overlaying." Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 216, no. 9 (September 1, 2002): 709–16. http://dx.doi.org/10.1243/09544070260340808.
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The aim of the paper is to propose and verify an effective method for the realistic modelling of pneumatic tyres. A static, three-dimensional model of a pneumatic tyre, based on the finite element (FE) program MARC, is presented. The cord-rubber composite was modelled by overlaying the elements characterizing the cord and those representing incompressible rubber. A two-dimensional model was used to simulate the tyre mounting and inflation process. The model was then developed into a three-dimensional one. The displacements, radial stiffness and delamination stresses caused by the vertical load were determined. The shape of the tyre footprint and the pressure distribution in this zone were also predicted. Good correlation between measured and computed stiffness characteristics was observed. The proposed technique of element overlaying yields a tyre model that is numerically more stable than that using only orthotropic elements (very stiff cord merged with compliant rubber).
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Li, Changquing, and Christopher D. Rahn. "Design of Continuous Backbone, Cable-Driven Robots." Journal of Mechanical Design 124, no. 2 (May 16, 2002): 265–71. http://dx.doi.org/10.1115/1.1447546.
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Continuous backbone robots driven by cables have many potential applications in dexterous manipulation for manufacturing and space environments. Design of these robots requires specification of a stiff yet bendable backbone, selection of cable support heights and spacings, and development of a cable drive system. The robot arm divides into sections that are subdivided into segments bounded by cable supports. Cable pairs attach to the end of each section and provide two axis bending. Thus, with many sections, the arm can be bent into complex shapes to allow redundant positioning of the end effector payload. The kinematics of the entire arm are determined from the segment kinematics. This paper derives and numerically solves the nonlinear kinematics for a single segment of a continuous backbone robot. Optimal spacing of the cable supports maximizes displacement, load capacity, and simplicity of the robot kinematics. An experimental system verifies the theoretically predicted performance.