Journal articles: 'Lagrange's method'

1

Srinivasan, Gopala Krishna. "A Note on Lagrange's Method of Variation of Parameters." Missouri Journal of Mathematical Sciences 19, no. 1 (February 2007): 11–14. http://dx.doi.org/10.35834/mjms/1316092232.

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Peng, Hui Lian. "Modeling and Simulating of Multi-Rigid-Body System with Friction-Affected Translational Joints." Applied Mechanics and Materials 88-89 (August 2011): 250–56. http://dx.doi.org/10.4028/www.scientific.net/amm.88-89.250.

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In this paper, a modeling and simulating method is presented for the planar rigid multi-body system with friction-affected translational joints. The dynamical equations of the non-smooth system are formulated by the first kind of Lagrange's equations in Cartesian coordinate system. The normal forces are expressed by the Lagrange multipliers, and the frictional forces are characterized by Coulomb's friction law. The state transition problem of stick-slip and normal forces of bilateral constraints is solved by an improved trial-and-error method. Finally, the planar rigid multi-body system with two translational joints is considered as a illustrative application example.

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TOMITA, Masakazu. "Analysis of forward extrusion by grid method (Application of Lagrange's method of undetermined multipliers)." Transactions of the Japan Society of Mechanical Engineers Series A 52, no. 477 (1986): 1435–39. http://dx.doi.org/10.1299/kikaia.52.1435.

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Giacaglia, G. E. O., and W. Q. Lamas. "On Euler-Lagrange's Equations: A New Approach." TEMA (São Carlos) 21, no. 2 (July 22, 2020): 359. http://dx.doi.org/10.5540/tema.2020.021.02.359.

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A new formalism is proposed to study the dynamics of mechanical systems composed of N connected rigid bodies, by introducing the concept of $6N$-dimensional composed vectors. The approach is based on previous works by the authors where a complete formalism was developed by means of differential geometry, linear algebra, and dynamical systems usual concepts. This new formalism is a method for the description of mechanical systems as a whole and not as each separate part. Euler-Lagrange's Equations are easily obtained by means of this formalism.

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TOMITA, Masakazu. "Analysis of forward extrusion by the grid method. Application of Lagrange's method of undetermined multipliers." JSME international journal 30, no. 260 (1987): 242–47. http://dx.doi.org/10.1299/jsme1987.30.242.

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Prasad Mondal, Sankar, Dileep Kumar Vishwakarma, and Apu Kumar Saha. "Solution of second order linear fuzzy difference equation by Lagrange's multiplier method." Journal of Soft Computing and Applications 2016, no. 1 (2016): 11–27. http://dx.doi.org/10.5899/2016/jsca-00063.

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Ibănescu, Radu, and Cătălin Ungureanu. "Lagrange's Equations versus Bond Graph Modeling Methodology by an Example of a Mechanical System." Applied Mechanics and Materials 809-810 (November 2015): 914–19. http://dx.doi.org/10.4028/www.scientific.net/amm.809-810.914.

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The bond graph modeling method was discovered by Henry Painter in 1959 and has quickly become a wide spread method all over the modeling engineering world. The method is based on the analysis of power circulation in systems and has some indisputable advantages over other modeling methods, based in principle on mathematical aspects. The paper proposes a comparison between the bond graph method and Lagrange's equations method, by applying both methods to model a mechanical system. The bond graph model is a graphical model. There are three possibilities to exploit the bond-graph diagram. The first one consists in deducing a system of differential or algebraic-differential equations from the diagram. The second one consists in obtaining the block diagram model from the bond graph diagram, without additionally writing any equations, followed by the block diagram implementation in the appropriate software, which permits to perform simulations at once. The third one consists in implementing the bond graph diagram directly in the appropriate software, where simulations can immediately run. The advantages and the disadvantages of the methods are emphasized, but the decision about the most appropriate method is up to the modeler.

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MANOFF, S. "EINSTEIN'S THEORY OF GRAVITATION AS A LAGRANGIAN THEORY FOR TENSOR FIELDS." International Journal of Modern Physics A 13, no. 12 (May 10, 1998): 1941–67. http://dx.doi.org/10.1142/s0217751x98000846.

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Einstein's theory of gravitation (ETG) is considered as a Lagrangian theory of tensor fields over (pseudo) Riemannian spaces without torsion (Vn spaces, n=4) by means of the method of Lagrangians with covariant derivarives (MLCD). In a trivial manner Euler–Lagrange's equations as Einstein's equations are obtained. The corresponding energy–momentum tensors (EMT's) are found for the standard for the ETG Lagrangian invariant on the basis of the covariant Noether identities. The symmetric energy–momentum tensor of Hilbert appears as an element irrelevant to the whole scheme of the considered Lagrangian thoery of tensor fields over Vn spaces despite of the fact that it has some elements of the structure of the variational EMT of Euler–Lagrange. The notion of the active gravitational rest mast density is related to the variational EMT of Euler–Lagrange and on this basis to a certain extent to the EMT of Hilbert.

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Hah, Chong-Ku. "Simulation of Scooped Swing in High Bar Using Lagrange's Method : A Case Study." Journal of the Korea Contents Association 7, no. 4 (April 28, 2007): 234–40. http://dx.doi.org/10.5392/jkca.2007.7.4.234.

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Shagal, G., and S. A. Meguid. "Effect of payloads contact upon the coupled dynamic response of two cooperating robots." Robotica 14, no. 6 (November 1996): 659–65. http://dx.doi.org/10.1017/s026357470001852x.

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The coupled dynamic response of two cooperating robots handling two flexible payloads is treated using a new algorithm. In this algorithm, the dynamic equations describing the system are obtained using Lagrange's method for the rigid robot links and the finite element method for the flexible payloads. The contact between the flexible payloads is modelled using the penalty function method and a contact search algorithm is employed to identify the contact region.

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OHISHI, Masahiro, Fumio OHTOMO, Masaaki YABE, Mitsuru KANOKOGI, Takaaki SAITO, Yasuaki SUZUKI, and Chikao NAGASAWA. "Speed Enhancement Technique for Pulsed Laser Rangefinders Based on Lagrange's Theorem Using an Undersampling Method." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E94-A, no. 8 (2011): 1738–46. http://dx.doi.org/10.1587/transfun.e94.a.1738.

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Himayetul, Islam Syed Md. "An Alternative Method for Solving Lagrange's First-Order Partial Differential Equation with Linear Function Coefficients." Journal of Partial Differential Equations 28, no. 3 (June 2015): 208–24. http://dx.doi.org/10.4208/jpde.v28.n3.2.

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13

Ge, Z.-M., C.-W. Jen, and F.-N. Ku. "Dynamic analysis of a two-gyro Anschütz compass." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 211, no. 3 (March 1, 1997): 239–49. http://dx.doi.org/10.1243/0954406971521818.

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In this paper, the dynamics of the two-gyro Anschütz compass in two cases, i.e. the earth spin velocity and zero vehicle velocity case, as well as the earth spin velocity and non-zero vehicle case, are studied. The detailed exact equations of motion of this compass are obtained by Lagrange's equations. The system is studied by the linear approximation method, and these equations are solved as eigenvalue problems. The stabilities of these motions are also discussed. The analytical stabilities for two cases from the linear approximation method are checked by the numerical solutions.

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Szabó, Tamás, and László Rónai. "Generalized displacements and momenta formulations of an electromechanical plunger." Journal of Computational and Applied Mechanics 15, no. 2 (2020): 173–84. http://dx.doi.org/10.32973/jcam.2020.010.

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This paper deals with four different derivations of the governing equations of a solenoid plunger with lumped-parameter. Energy-based modeling is employed with extended Hamilton's principle with independent generalized coordinates and generalized momenta in order to be applicable to composite Lagrange's equations. In the electromechanical models, displacements and charges are regarded to be generalized coordinates, mechanical momenta and flux linkages are the generalized momenta. The derived systems of differential equations are solved numerically with the Runge-Kutta method.

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Prakash, S., J. Vanualailai, and T. Soma. "Obtaining approximate region of asymptotic stability by computer algebra: A case study." South Pacific Journal of Natural and Applied Sciences 20, no. 1 (2002): 56. http://dx.doi.org/10.1071/sp02011.

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One of the classical problems in nonlinear control system analysis and design is to find a region of asymptotic stability by the Direct Method of Lyapunov. This paper tentatively shows, via a numercial example, that this problem can be easily solved using Quantifier Elimination (QE). In particular, if the governing equations are described by differential equations containing only polynomials, then the problem can be conveniently solved by a computer algebra software packages such as Qepcad or Redlog. In our case study, we use a simple Lyapunov function and Qepcad to estimate the stability region, and the results are verified by an optimization method based on Lagrange's method.

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Eftekhari, Mojtaba, Mahdi Eftekhari, and Hossein Karimpour. "Neuro-fuzzy adaptive control of a revolute stewart platform carrying payloads of unknown inertia." Robotica 33, no. 9 (May 22, 2014): 2001–24. http://dx.doi.org/10.1017/s0263574714001222.

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SUMMARYIn this research, a Stewart parallel platform with rotary actuators is simulated and a prototype is tested under different operative conditions. The purpose is to make the robot robust against inertia variations considering the fact that different payloads of unknown size may be transported. Due to the complexity issued by expressing the equations of motion with independent variables, the governing equations are derived by Lagrange's method using Lagrange multipliers for imposing the kinematic constraints imposed on this parallel robot. Eliminating Lagrange multipliers by projecting the equations onto the orthogonal complement of the space of constraints, the equations of motion are transformed to a reduced form suitable for the purpose of controller design. The control approach considered here is based on a neuro-fuzzy interference method. As a first step, each revolute arm link are individually trained under different loadings and diverse maneuvers. It is purposed that once employed together, the links will have learned how to collaborate with each others for performing a common task. Training data are divided to several clusters by using a subtractive clustering algorithm. For every cluster, a fuzzy rule is derived so that the output follows the desired trajectory. In the last stage, these rules are employed by utilizing back propagation algorithms and the effectiveness of the neuro-fuzzy system becomes approved by performing multiple maneuvers and its robustness is checked under various inertia loads. The controller has ultimately been implemented on a prototype of the Stewart mechanism in order to analyze the reliability and feasibility of the method.

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Sanh, Do, and Do Dang Khoa. "A new method for determining the reactions of mechanical constraints." Vietnam Journal of Mechanics 28, no. 1 (April 17, 2006): 35–42. http://dx.doi.org/10.15625/0866-7136/28/1/5477.

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In the present paper it is introduced the method for determining the reactions of mechanical constraints (holonomic and nonholonomic constraints).As is known, for studying dynamical characters of a mechanical system it is necessary to determine the constraint reactions acting on the system. Up to now, the reactions are calculated through Lagrange's multipliers. By such a way the reactions are determined only indirectly. In the [3, 4], two methods of determining directly the reactions are discussed. However, for applying these methods, it is necessary to compute the inverse matrix of the matrix of inertia. This thing in general is not convenient, specially when the matrix of inertial is of large size and dense.In the present paper it is represented the method for determining the constraint reactions, by which it is possible to avoid inertia the computation of the inverse matrix of the matrix of inertia is avoided. For this in the paper it is used the middle variables by which we obtain a closed set of algebraic equations for directly determining reactions.

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Balaji, K., B. S. Nataraju, and H. N. Sureshakumar. "Matrix approach to deployment dynamics of an n-panel solar array." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 217, no. 1 (March 1, 2003): 15–27. http://dx.doi.org/10.1243/146441903763049414.

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The equations of motion of a deploying solar array are derived using Lagrange's method and solved numerically. With increase in the number of panels, the mathematical modelling becomes complicated, involving the derivation of lengthy equations which can be error prone. A matrix approach has been adopted for automatic derivation of lengthy equations of motion. This facilitates the accommodation of n panel formulation by increasing the number of rows and columns. The complexity in derivation of the air drag and damper are discussed here.

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19

Li, M. "Nonlinear Vibration of a Multirotor System Connected by a Flexible Coupling Subjected to the Holonomic Constraint of Dynamic Angular Misalignment." Mathematical Problems in Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/243758.

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This paper proposes a mathematical model of the multirotor system with a flexible coupling on spring supports on Lagrange's approach, which has taken into account the effects of dynamic angular misalignment and mass unbalance. Then its nonlinear dynamic behaviors of the system are discussed based on the method of multiple scales and numerical technique, respectively. The results show that the responses of the system in lateral directions contain a similar component to that of the mass unbalanced system on both the vibrating frequency and amplitude and involve the typical nonlinear components such as the ones from some combined frequencies; the results also reveal that the numerical agreements on the above-mentioned methods are perfect for the transient responses.

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Abd El-Bar, S. E., and F. A. Abd El-Salam. "Analytical and Semi-Analytical Treatment of the Satellite Motion in a Resisting Medium." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/489849.

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The orbital dynamics of an artificial satellite in the Earth's atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange's planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.

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Rudenko, Yu V. "APPLICATION OF THE AVERAGE METHOD FOR ANALYSIS OF DC VOLTAGE CONVERTERS WITH INTERRUPTED CURRENTS IN INDUCTIVE ELEMENTS." Praci elektrodinamiki Nacionalanoi akademii nauk Ukraini Institutu 2020, no. 57 (December 2, 2020): 55–64. http://dx.doi.org/10.15407/publishing2020.57.055.

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The fundamentals of the state-space averaging method using Lagrange's theorems for calculating dc semiconductor converters, which operate in modes of intermittent currents in inductive elements, are considered. To implement the assumption introduced in the study, it is proposed to replace the capacitor in the output circuits of the converter equivalent circuit with a source of constant emf. This makes it possible to obtain a simplified mathematical model of the converter for calculating the integral characteristics, which corresponds to a jointly defined system of algebraic equations with averaged variables. Using the developed models, the parameters of the buck, boost, and inverting converters are calculated, the analytical and graphic dependencies of the limiting values of these parameters are determined, which ensure the specified operating mode. The obtained dependencies make it possible to compare the properties of converters and select their effective parameters. References 10, figures 1, table 1.

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Tomczuk, Bronisław, and Andrzej Waindok. "A coupled field-circuit model of a 5-phase permanent magnet tubular linear motor." Archives of Electrical Engineering 60, no. 1 (March 1, 2011): 5–14. http://dx.doi.org/10.2478/v10171-011-0001-z.

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A coupled field-circuit model of a 5-phase permanent magnet tubular linear motor A method for modeling of the dynamics characteristics for a 5-phase permanent magnet tubular linear motor (PMTLM) is presented. Its electromagnetic nonlinear field analysis with finite element method (FEM) has been coupled with the circuit model. The calculation model includes the equations for electrical circuits and mechanical quantities as well. They have been obtained using Lagrange's method. The calculated and measured waves of the mover position have been compared for several values of the excitation current. This comparison yields a good agreement. Presented calculation model is very useful in designing and optimization of the PMTLM and in the calculation of the parameters for the control algorithms intended for such a type of actuators.

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Ly, Sidy, Fulgence Mansal, Diaraf Seck, and Moussa Balde. "A Location Problem of Obstacles in Population Dynamics." Journal of Mathematics Research 8, no. 4 (August 1, 2016): 211. http://dx.doi.org/10.5539/jmr.v8n4p211.

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The aim of this paper is to determine the optimal locations where Fish Aggregating Devices (F.A.D) or artificial traps must be placed in a given place of the sea and to preverse resources. Our work focuses on two parts: the first one is the study of static optimization problem with a functional taking into account the distance between the sites or F.A.D and the second one is devoted to solving an optimization problem with constraints expressed in classical model of fishery: Lagrange's method and Pontryagin's maximum principle the main mathematical tools to get characterization results of the location of artificial traps.

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Tokhi, M. O., and A. K. M. Azad. "Modelling of a single-link flexible manipulator system: theoretical and practical investigations." Robotica 14, no. 1 (January 1996): 91–102. http://dx.doi.org/10.1017/s026357470001897x.

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SummaryThis paper presents theoretical and experimental investigations into modelling a single-link flexible manipulator system. An analytical model of the manipulator, characterised by an infinite number of modes, is developed using the Lagrange's equation and modal expansion method. This is used to develop equivalent time-domain and frequency-domain working models of the system in state-space and transfer function forms respectively. The model parameters are then estimated experimentally using system's measured input/output data. The model thus obtained is validated through experimentation and results including the effect of payload on system characteristics presented and discussed.

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Xu, Wei Min, Yun Lin Li, Jian Xin Chu, and Xian Wen Zhou. "Modeling of an Underactuated Nonlinear Double-Container Overhead Crane." Advanced Materials Research 479-481 (February 2012): 680–87. http://dx.doi.org/10.4028/www.scientific.net/amr.479-481.680.

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Compared with a traditional single-container overhead crane, the double-container overhead crane is a type of new efficient container stevedoring machinery, however, it is actually a complicated nonlinear system. In this brief, both the kinematic and dynamic characteristics concerning the double-container crane are investigated, and then an accurate dynamic model of the underactuated double-container overhead crane is established by utilizing Euler-Lagrange's method. Not only is the variation of the rope and payloads, but also some uncertainties such as mechanical frictions and air-resistance in the environment are taken into account in modeling. Simulation results show the validity of the established double-container overhead crane model.

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XIN, Liang, Zixue DU, Junchao ZHOU, Zhen YANG, and Zhouzhou XU. "Study on Dynamic Response of Straddle-Type Monorail Vehicle with Single-Axle Bogie Under Curve Condition." Mechanics 27, no. 2 (April 15, 2021): 122–29. http://dx.doi.org/10.5755/j02.mech.25319.

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This paper is concerned with the dynamic response of straddle-monorail with single-axle bogie under curve condition. A 15 degrees-of-freedom(DOF) dynamic model is established for straddle-type monorail vehicle with single-axle bogie, which consists driving wheels, steering wheels and stabilizing wheels. The motion equations of the straddle-type monorail vehicle are derived using the Lagrange's equation, and the wheel-rail contact model and the curving track beam model are created. Compared with the test results, the accuracy of the method is verified. Finally, the influence of curve radius, curve superelevation rate, number of passengers and stiffness of driving wheels on dynamic response is discussed.

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Cardona Foix, Salvador, Lluïsa Jordi Nebot, and Joan Puig-Ortiz. "Reduced inertial parameters in system of one degree of freedom obtained by Eksergian's method." Mechanical Sciences 8, no. 1 (March 30, 2017): 91–100. http://dx.doi.org/10.5194/ms-8-91-2017.

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Abstract. The mechanisms of one degree of freedom can be dynamically analysed by setting out a single differential equation of motion which variable is the generalized coordinate selected as independent. In front of the use of a set of generalized dependent coordinates to describe the system, the method exposed in this work has the advantage of working with a single variable but leads to complex analytical expressions for the coefficients of the differential equation, even in simple mechanisms. The theoretical approach, in this paper, is developed from Eksergian's method and Lagrange's equations. The equation of motion is written by means of a set of parameters – reduced parameters – that characterize the dynamic behaviour of the system. These parameters are function of the independent coordinate chosen and its derivative and can be obtained numerically by direct calculus or by means of a kinetostatic analysis, as is proposed. Two cases of study of the method are presented. The first example shows the study of pedalling a stationary bicycle used in a rehabilitation process. The second one shows the analysis of a single dwell bar mechanism which is driven by an electric motor.

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Le Chau, Ngoc, Nhat Linh Ho, Ngoc Thoai Tran, and Thanh-Phong Dao. "Analytical Model and Computing Optimization of a Compliant Gripper for the Assembly System of Mini Direct-Current Motor." International Journal of Ambient Computing and Intelligence 12, no. 1 (January 2021): 1–28. http://dx.doi.org/10.4018/ijaci.2021010101.

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This study proposes a combination of kinematics-based design method, statistic method, and TLBO algorithm to solve computing optimization for a compliant gripper. It is employed in an assembly system of mini direct current motor. First, the kinematic models of the gripper are developed. The static model is paid attention. Next, the dynamic model is established based on Lagrange's principle. Then, a multi-criteria optimization for the gripper is conducted by Taguchi method integrated with TLBO algorithm. Finally, the FEA and experiments are implemented to verify the optimal results and evaluate the performances of the compliant gripper. The results indicated that theoretical models are in good accord with the results from simulations and experiments. Additionally, the performance of the present method is superior to PSO algorithm. The results revealed that the compliant gripper allows a displacement up to 3000 µm and the amplification ratio of 12 times. The compliant gripper is a potential application for the mini direct-current motor assembly system.

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Yan, Y. L., and B. Z. Yue. "Study on Rigid-Liquid Coupling Dynamics of the Spacecraft With Arbitrary Axisymmetrical Tanks." Journal of Mechanics 34, no. 6 (May 10, 2018): 817–28. http://dx.doi.org/10.1017/jmech.2018.15.

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AbstractThis article focus on the rigid-liquid coupling dynamics of the spacecraft with arbitrary axisymmetrical tanks with curved walls and top. The carrier velocity potential function was obtained according to the motion equations of a representative point in the tank of spacecraft, and the relative velocity potential function can be expressed as Gauss hypergeometric series. The Hamilton's variational principle was applied to derive the governing equations of liquid sloshing based on the profile of hydrostatic shape of free liquid surface. The dynamic equations of modal coordinates were established through the Galerkin method. The state equations of coupled motion of main rigid platform of the spacecraft were deduced by using the Lagrange's equations in term of general quasi-coordinates. Thruster firing was actuated to the coupled system to analyze the rigid-liquid coupled dynamic behaviors. Computer numerical simulations was carried out to confirm the validity of the method developed in this paper.

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Sanh, Do, and Do Dang Khoa. "The method of determining internal forces at any cross section of links in mechanisms." Vietnam Journal of Mechanics 26, no. 2 (July 1, 2004): 110–21. http://dx.doi.org/10.15625/0866-7136/26/2/5695.

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In the paper it is introduced a method of determining internal forces at any cross section of the links of mechanisms. As known, so far it is used the method of D'Alembert, which consists of two steps, the determination of the acceleration states of links and the establishment of the equilibrium equations for the set of forces including the forces of inertia and the internal forces at the cross section. A. I. Lurie proposed a method of analytical mechanics for this problem. Its concept is to make a new system called the released one by cutting the link at a cross section under consideration and adding some coordinates. Only one condition putting restriction on the released system is the additional coordinates must equal zero. Under this restriction the new created system is coincided to the original one. This restriction is equivalent to put the mechanical constraints, whose reaction forces are the components of internal forces at the cross section under consideration. It is necessary emphasize that the Lurie's method is convenient only for opened loops, but is not applied for closed ones. Moreover, the Lagrange's multiplier equations applied by A. I. Lurie are unsuitable. In this paper it is presented the generalized Lurie's method, which is applied for the opened and closed loops by using the Principle of Compatibility.

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Mhaske, Ashok Sahebrao, and Kirankumar Laxmanrao Bondar. "Fuzzy Transportation Problem by Using Triangular, Pentagonal and Heptagonal Fuzzy Numbers With Lagrange's Polynomial to Approximate Fuzzy Cost for Nonagon and Hendecagon." International Journal of Fuzzy System Applications 9, no. 1 (January 2020): 112–29. http://dx.doi.org/10.4018/ijfsa.2020010105.

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The transportation problem is a main branch of operational research and its main objective is to transport a single uniform good which are initially stored at several origins to different destinations in such a way that the total transportation cost is minimum. In real life applications, available supply and forecast demand, are often fuzzy because some information is incomplete or unavailable. In this article, the authors have converted the crisp transportation problem into the fuzzy transportation problem by using various types of fuzzy numbers such as triangular, pentagonal, and heptagonal fuzzy numbers. This article compares the minimum fuzzy transportation cost obtained from the different method and in the last section, the authors introduce the Lagrange's polynomial to determine the approximate fuzzy transportation cost for the nanogon (n = 9) and hendecagon (n = 11) fuzzy numbers.

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Cheng, Guangfeng, Chuh Mei, and Raymond Y. Y. Lee. "Large Amplitude Vibration of a Cantilever Beam with Tip Mass under Random Base Excitation." Advances in Structural Engineering 4, no. 4 (October 2002): 203–10. http://dx.doi.org/10.1260/136943301320896679.

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Nonlinear large amplitude random vibration of cantilever beam with lumped mass and rotary inertia under zero mean, stationary, Gaussian random base excitation is studied, using the inextensional beam theory. Single-mode approximation is employed to discretize the Lagrange's equation. The resulting nonlinear governing modal equation of motion is solved with application of the stochastic linearization method. Two examples, a cantilever beam with/without tip mass, are analyzed as application of the developed methodology. Effects of mass and rotary inertia variation on system response are investigated in detail. Results showed that increasing rotary inertia could reduce the random response of the beam structure and the random response of the structure is quite sensitive to the tip mass variation. The nonlinearities of the inextensional beam vibration result in a spring hardening system.

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Forysiewicz, Monika, Leon Kukielka, and Katarzyna Gotowala. "Finite Element Simulation of Physical Phenomena in Real Conditions of a Single Grain Cutting Process." Materials Science Forum 862 (August 2016): 288–97. http://dx.doi.org/10.4028/www.scientific.net/msf.862.288.

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The cutting process was presented as a real object as well as its physical and mathematical modelling. For the description of the non-linear phenomena, at the typical increment ratio, the updated Lagrange's description was used. Adequate deformation and stress increments measurements were used, e.g. Green-Lagrange's deformation tensor increment and the increment of the Piola-Kirchhoff's second symmetrical tensor. Nonlinearity of the material was described by means of the increment model taking into consideration the deformation and deformation rate records. The workpiece is treated as a body in which there may be elastic deformation (in the range of reversible deformation) and visco and plastic (in terms of irreversible deformation), with nonlinear hardening. For the construction of the material model Huber-Mises-Hencky's non-linear plasticity condition was used, associated principle of flow as well as mixed hardening (isotopic-kinematic). The condition of the material after pre-machining processes was also taken into account by means of implementation of initial conditions of: displacement, strain and stress. Yield stress of the body was described by a Cowper-Symonds' model allows for linear-isotropic, kinematic or mixed plastic strain hardening and the effect of the intensity of plastic strain velocity. The applications in ANSYS program and results of numerical calculations were presented. A method of generating a three-dimensional abrasive grain with a geometry close to actual were describes. The influence of the process parameters on the states of strains and stresses and on the quality of the product was presented. Numerical calculations of cutting process with single abrasive grain were made and investigated the deformation and stress occurring in the workpiece. The experimental test stand of single abrasive grain cutting process, the test plan and the verifications of results of numerical simulations were describes. The results were statistically developed and that’s give the models in the regression function form.

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Kalanta, S. "MIXED AND COMPATIBLE FINITE ELEMENTS IN THE ANALYSIS PROBLEM OF ELASTOPLASTIC STRUCTURES." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 2, no. 5 (March 31, 1996): 29–43. http://dx.doi.org/10.3846/13921525.1996.10531546.

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A problem of ideal elastoplastic structures stress-strain field determination is considered The dual general and discrete mathematical models of analysis problem are made on the basis of the extremal energy principles and finite element method. In these models the possible discontinuity of displacements and the dissipation of energy in the place of those discontinuities, also the different external effects (load, initial strains and support settlements) are estimated. At first, on the basis of the mixed functional and mixed finite elements the discrete expressions of fundamental relationships (geometric equations, yield conditions) and the discrete mathematical model of mixed formulation of the problem are made. This mathematical model corresponds to the minimum total energy principle for a kinematically admissible displacements. The dual static formulation of the problem is obtained by Lagrange's multipliers method; this corresponds to the minimum complementary energy principle. The kinematic formulation of the problem is obtained in the case of linear yield conditions. These mathematical models permit to determine the lower values of the stress and displacements of structures. It has shown that the approximation of geometric equations and yield conditions by Bubnov-Galiorkin's collocation method gives the more accurate results.

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WANG, JINJIE, GUOHUA CAO, YANDONG WANG, and RONGHUA WU. "A NOVEL DRIVING STRATEGY FOR DYNAMIC SIMULATION OF HOISTING ROPE WITH TIME-VARYING LENGTH." International Journal of Modeling, Simulation, and Scientific Computing 04, no. 03 (August 19, 2013): 1350009. http://dx.doi.org/10.1142/s1793962313500098.

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A simulation method for investigating the vibration behavior of hoisting rope with time-varying length is improved. By previously creating markers in the MSC.ADAMS software package, the parametric model of the rope wound along helix is established based on the concentrated-mass theory with multi-degree of freedom (multi-DOF). A novel driving strategy, cooperating fixed joints with angle sensors under the control of driving script, is proposed to substitute conventional contact force. Researching on the hoisting rope in the sinking winch mechanism, an equivalent discretization model is obtained with complicated boundary conditions considered. The differential equations of motion of the hoisting system are formulated employing Lagrange's equation and numerically solved using Runge–Kutta method. The simulation indicates that the horizontal swing is decreased in principle and the simulation with 800 discrete ropes is not performed more than 61 min. Therefore, this feasible strategy could not only guarantee the accuracy but also promote simulation efficiency and stability. The motion curves exported from ADAMS simulation coincide with one in numerical simulation, which validates both the numerical model and the driving strategy.

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36

Wang, Y. R., and T. H. Chen. "The Vibration Reduction Analysis of a Rotating Mechanism Deck System." Journal of Mechanics 24, no. 3 (September 2008): 253–66. http://dx.doi.org/10.1017/s1727719100002318.

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AbstractIn this paper, an optimized position of a mass-spring-damper vibration absorber is proposed for a rotating mechanism device (such as optical disk drive or rotary-wing and deck coupled system). A nonlinear 3-D theoretical model for a deck is established by Lagrange's equation. A 2-bladed rotor and deck (foundation) coupled aeroelastic system with vibration reduction device is presented and studied as well. The analytical solution is obtained by the Multiple-scales method for the case of no vibration absorber. The numerical results in time and frequency domain and with/no absorber are acquired. This research provides a theoretical background for the preliminary vibration reduction design for industries. It is found that the existing disk drives vibration can be reduced by simply adding the absorber at the end corner isolator of the deck, but without changing the main configurations. This will not only save costs but also increase testing efficiency, achieving the most cost-effective vibration reduction result.

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37

Chung, J. "Effect of gravity and angular velocity on an automatic ball balancer." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 219, no. 1 (January 1, 2005): 43–51. http://dx.doi.org/10.1243/095440605x8333.

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The effects of gravity and an angular velocity profile on the performance of an automatic ball balancer (ABB) are studied in this paper. In order to investigate these effects, a physical model of a Jeffcott rotor with an ABB is adopted in this study, in which gravity as well as the angular acceleration is considered. With the polar coordinates, the non-linear equations of motion are derived by using Lagrange's equation. These equations include gravity, the angular acceleration, and the angular jerk. Based on the equations derived, time responses are computed by using the generalized α method. The effects of gravity on the balancing performance are analysed. For various angular velocity profiles, the ABB performance is also evaluated. The analysis of results shows that the balancing of the rotor with an ABB can be achieved regardless of gravity. It is also shown that a smooth velocity profile results in less vibration compared with a non-smooth velocity profile.

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38

AKOUR, SALIH N., and JAMAL F. NAYFEH. "NONLINEAR DYNAMICS OF POLAR-ORTHOTROPIC CIRCULAR PLATES." International Journal of Structural Stability and Dynamics 06, no. 02 (June 2006): 253–68. http://dx.doi.org/10.1142/s0219455406001940.

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The dynamics of nonlinear polar orthotropic circular plates with simply supported boundary condition are investigated. Kirchhoff strain displacement relations for thin plates plus next higher-order nonlinear terms (von Karman type geometric nonlinearity) are considered. Lagrangian density function and Hamilton's principle are utilized to derive Lagrange's equations, from which the equations of motion and associated boundary conditions are derived. Analytical solution is obtained by the perturbation techniques and numerical solution by the Runge–Kutta method. Phase diagrams, discrete Fast Fourier Transform (FFT diagrams) and time history responses are presented for studying the forced vibration behavior. The sub-harmonic and primary resonances are studied as well as the effect of adding damping foil layers. The quadratic term in the governing equation plays a softening role on the overall behavior of the plate due to its relatively large coefficient. The increase of damping tends to smooth out the unstable region (i.e. jump phenomenon) in the system.

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39

Lv, Binglin, Wanyou Li, and Huajiang Ouyang. "Moving Force-Induced Vibration of a Rotating Beam with Elastic Boundary Conditions." International Journal of Structural Stability and Dynamics 15, no. 01 (January 2015): 1450035. http://dx.doi.org/10.1142/s0219455414500357.

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In this paper, an analytical technique, the so-called Fourier Spectral method (FSM), is extended to the vibration analysis of a rotating Rayleigh beam considering the gyroscopic effect. The model presented can have arbitrary boundary conditions specified in terms of elastic constraints in the translations and rotations or even in terms of attached lumped masses and inertias. Each displacement function is universally expressed as a linear combination of a standard Fourier cosine series and several supplementary functions introduced to ensure and accelerate the convergence of the series expansion. Lagrange's equation is established for all the unknown Fourier coefficients viewed as a set of independent generalized coordinates. A numerical model is constructed for the rotating beam. First, a numerical example considering simply supported boundary conditions at both ends is calculated and the results are compared with those of a published paper to show the accuracy and convergence of the proposed model. Then, the method is applied to one real work piece structure with elastically supported boundary conditions updated from the modal experiment results including both the frequencies and mode shapes using the method of least squares. Several numerical examples of the updated model are studied to show the effects of some parameters on the dynamic characteristics of the work piece subjected to moving loads at different constant velocities.

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40

Lu, Jian-Wei, Xiao-Ming Sun, Alexander F. Vakakis, and Lawrence A. Bergman. "Influence of backlash in gear reducer on dynamic of single-link manipulator arm." Robotica 33, no. 08 (April 29, 2014): 1671–85. http://dx.doi.org/10.1017/s0263574714000915.

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SUMMARYThe dynamic modeling of a flexible single-link manipulator arm with consideration of backlash in the planetary gear reducer at the joint is presented, and the influence of backlash on the dynamic response of the system is evaluated. A 2K-H planetary gear reducer with backlash was employed as an example to discuss the dynamic modeling of the sub-model of the planetary gear reducer, and the sub-model of the planetary gear reducer was established based on the lumped mass method. The flexible manipulator was regarded as an Euler--Bernoulli beam, and the dynamic model of the flexible manipulator arm with backlash in the planetary gear reducer was determined from Lagrange's equations. Based on the this model, the influence of the backlash in the planetary gear reducer and excitation frequency on the dynamic response of the system were evaluated through simulation, and the results showed that the dynamic response of the system is sensitive to the backlash and the excitation frequency simultaneously, which provides a theoretical foundation for improvement of dynamic modeling and control of the flexible manipulator arm.

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41

Dey, Sudip, and Amit Karmakar. "Finite-element impact response of debonded composite turbine blades." International Journal of Computational Materials Science and Engineering 03, no. 01 (March 2014): 1450001. http://dx.doi.org/10.1142/s2047684114500018.

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This paper investigates on the transient behavior of debonded composite pretwisted rotating shallow conical shells which could be idealized as turbine blades subjected to low velocity normal impact using finite-element method. Lagrange's equation of motion is used to derive the dynamic equilibrium equation and the moderate rotational speeds are considered neglecting the Coriolis effect. An eight-noded isoparametric plate bending element is employed in the finite element formulation incorporating rotary inertia and effects of transverse shear deformation based on Mindlin's theory. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the impact parameters. The time-dependent equations are solved by using Newmark's time integration scheme. Parametric studies are performed to investigate the effects of triggering parameters like angle of twist, rotational speed, laminate configuration and location of debonding considering low velocity normal impact at the center of eight-layered graphite-epoxy composite cantilevered conical shells with bending stiff [Formula: see text], torsion stiff ([45°/-45°/-45°/45°]s) and cross-ply ([0°/90°/0°/90°]s) laminate configurations.

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Kalanta, S. "DUAL MATHEMATICAL MODELS OF ELASTOPLASTIC STRUCTURES ANALYSIS PROBLEM ESTIMATING DISCONTINUITY OF DISPLACEMENTS." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 1, no. 3 (September 30, 1995): 5–29. http://dx.doi.org/10.3846/13921525.1995.10531517.

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A problem of ideal elastoplastic structures stress-strain field determination is considered in the article. The general dual mathematical models (static and kinematic formulation) of analysis problem is derived on the basis of the extremal energy principles and theory of duality. The different external effects are estimated, namely: load, initial strains, prestressing and support settlements. At first, on the basis of the complementary energy principle the mathematical model of static formulation of the problem is made. The kinematic formulation of the problem is obtained on the basis of Lagrange's multipliers method; this corresponds to the minimum total energy principle for a kinematically admissible displacements. In these mathematical models the possible discontinuity of displacements and the dissipation of energy in the place of those discontinuities are estimated what was not done in the previous publications. The discrete expressions of fundamental relationships (equilibrium and geometric equations, yield conditions) and a dual pair of discrete equilibrium mathematical models are obtained on the basis of general static formulation of the problem using equilibrium finite elements. They permit to determine the upper meanings of the stress and displacements of structures. In the article it has been shown, that the approximation of yield conditions by Bubnov-Galiorkin's collocation method gives the more accurate results.

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Yao, X., X. Guo, Y. Feng, and C. Yu. "Research on the uprising dynamics of a double-carriage overhead crane system." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225, no. 1 (January 1, 2011): 112–22. http://dx.doi.org/10.1177/09544062jmes2131.

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The dynamic characteristics, in the uprising phase, of an overhead crane carrying two carriages and a cylindrical payload are studied in this article. The crane system investigated mainly consists of a twin beam, two carriages, a rigid payload, and two wire ropes. A new analytical model, in which the beam, carriages, payload, and wire ropes are, respectively, considered as uniform Euler—Bernoulli beam, lumped masses, rigid body, and springs, is presented to describe the uprising dynamics of the specific crane system. The most distinguished characteristic of the model is the dynamic coupling deriving from the presence of a flexible beam, carriages, and the cylindrical payload. The kinetic and potential energies, during the pre-tensing and lifting phases, of the components in the system are presented in particular. Then, utilizing the Rayleigh—Ritz method, one can obtain the differential equations of the dynamic system substituting the energy into Lagrange's equation. The differential equations are numerically solved by the fourth-order Runge—Kutta method, and then some useful results representing the dynamic features of the system are obtained according to the calculation. The validity of the analytical model is demonstrated by an equivalent FE model created by ANSYS and ADAMS. Comparison of the results, obtained by two distinct approaches, indicates good agreements, which can be the validity evidence of the analytical model.

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44

Kim, W., and J. Chung. "Performance of automatic ball balancers on optical disc drives." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216, no. 11 (November 1, 2002): 1071–80. http://dx.doi.org/10.1243/095440602761609443.

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The balancing performance of an automatic ball balancer fitted on the spindle motor of optical disc drives, for instance CD-ROM or DVD drives, is analysed. It is well known that the operating speed of an optical disc drive with an automatic ball balancer should be higher than the translational natural frequency of the system to be balanced. However, the influence of the rotational natural frequency on the performance of automatic ball balancers has not yet been investigated. In this study, considering not only the translational but also the rotational motion of the system, the non-linear equations of motion are derived using Lagrange's equation. Applying a perturbation method to the non-linear equations, the linearized equations. Applying a perturbation method to the non-linear equations, the linearized equations in the neighbourhood of a balanced equilibrium position are obtained. These equations have time-dependent periodic coefficients, which require application of the Floquet theory for stability analysis. From the stability analysis the effects of the stiffness and damping of rubber suspensions on the automatic ball balancer's performance are evaluated for the variation of operating speed. To verify the results, time responses are also computed from the non-linear equations. It is finally found that the natural frequency of not only the translational but also the rotational motion has an influence on the balancing performance of automatic ball balancers.

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Dey, Sudip, and Amit Karmakar. "Effect of Location of Delamination on Free Vibration of Cross-Ply Conical Shells." Shock and Vibration 19, no. 4 (2012): 679–92. http://dx.doi.org/10.1155/2012/726986.

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Location of delamination is a triggering parameter for structural instability of laminated composites. In this paper, a finite element method is employed to determine the effects of location of delamination on free vibration characteristics of graphite-epoxy cross-ply composite pre-twisted shallow conical shells. The generalized dynamic equilibrium equation is derived from Lagrange's equation of motion neglecting Coriolis effect for moderate rotational speeds. The formulation is exercised by using an eight noded isoparametric plate bending element based on Mindlin's theory. Multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The standard eigen value problem is solved by applying the QR iteration algorithm. Finite element codes are developed to obtain the numerical results concerning the effects of location of delamination, twist angle and rotational speed on the natural frequencies of cross-ply composite shallow conical shells. The mode shapes are also depicted for a typical laminate configuration. Numerical results obtained from parametric studies of both symmetric and anti-symmetric cross-ply laminates are the first known non-dimensional natural frequencies for the type of analyses carried out here.

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Wang, Yi-Ren, and Ko-En Hung. "Damping Effect of Pendulum Tuned Mass Damper on Vibration of Two-Dimensional Rigid Body." International Journal of Structural Stability and Dynamics 15, no. 02 (February 3, 2015): 1450041. http://dx.doi.org/10.1142/s0219455414500412.

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This study investigated the effect of a pendulum tuned mass damper (PTMD) on the vibration of a slender two-dimensional (2D) rigid body with 1:2 internal resonance. Focus is placed on the damping effect of various parameters of the PTMD on preventing the internal resonance of the system. The instruments used include fixed points plots, time response and Poincaré maps, which were compared for confirmation of accuracy. The Lagrange's equation is employed to derive the equations of motion for the system. The method of multiple scales (MOMS) is applied to analyzing this nonlinear vibration model. The internal resonance conditions of the rigid body in vibration are obtained by the eigen-analysis. Moreover, a 3D internal resonance contour plot (3D-IRCP) aided by various amplitude analysis tables is proposed for identification of the parameter combinations of the PTMD for preventing internal resonance. This approach enables the designers to evaluate the effectiveness of various parameter combinations of the PTMD prior to the design process. The present study indicates that without changing the main configuration, the vibration amplitudes in the main body can be greatly reduced under certain parameter combinations of the PTMD.

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Nhan, TRAN Huu, and Nguyễn Văn Hoàng. "Turning dynamics analysis of a heavy articulated vehicle by the vehicle planar dynamic model with two steering input signals." Science & Technology Development Journal - Engineering and Technology 4, no. 2 (April 17, 2021): first. http://dx.doi.org/10.32508/stdjet.v4i2.767.

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The vehicle planar single track dynamic model with two input steering angle parameters is derived by using Lagrange's method with the basis of equations for calculating the tire's force components. Dynamic analysis of a heavy articulated vehicle in case of turing is carried out by the vehicle planar dynamic model, in which two input steering angles are taken into account. Simulation with the selected velocity value to make sure that the stability according to the friction conditions at all axles of the vehicle is satisfied. Turning spacing, lateral forces at each axle of the vehicle are determined and analyzed for all three different cases of steering angles, respectively with steering angle of the semi-trailer is in the same direction, in the opposite direction and is locked or not steered in comparision with the steering angle of the tractor. The obtained results show that the derived model could employ to determine the planar kinematic and dynamic parameters, and analyze the dynamic safety features of an articulated vehicle, too. In addition, the derived mathematical model could also employ to develop a computational model that controls the planar articulated vehicle dynamics.

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48

Rudenko, Yu V. "MODELING OF PROCESSES ON THE BASIS OF STATE SPACE AVERAGING IN BOOST CONVERTER WITH MAGNETICALLY COUPLED ELEMENTS." Praci Institutu elektrodinamiki Nacionalanoi akademii nauk Ukraini 2021, no. 58 (May 19, 2021): 44–54. http://dx.doi.org/10.15407/publishing2021.58.044.

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The features of the state-space average method using Lagrange's theorems for calculation of processes in the boost converter with magnetically coupled elements (with autotransformer connection of inductors) in continuous currents mode are considered. A mathematical model of the converter based on relative variables has been developed, which makes it possible to calculate processes in a wide range of parameters taking into account the degree of magnetic coupling between inductive elements, including the definition of constant and pulsating components. It is shown that to obtain a complete system of the algebraic equations as a mathematical model of converters, which provides its single solution, it is advisable to additionally use the ratio according to energy balance in reactive elements in a steady state at the operation intervals of the converter during commutation period. The boundary dependences of boost converter parameters with magnetically coupled elements between the modes of discontinuous and continuous currents are calculated. The relative characteristics of the converter have been studied, which demonstrate the influence of imperfection of the magnetic coupling in the regulation zone of the relative duration value of switching interval. Ref. 21, fig. 6.

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Gu, Dawei, Juqian Zhang, Bangchun Wen, Xueliang Zhang, and Yunshan Liu. "Theoretical and experimental investigation on controlled synchronization of four co-rotating coupled exciters driven by induction motors in a vibrating system." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 13 (February 18, 2019): 4556–76. http://dx.doi.org/10.1177/0954406219830437.

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This paper aims at theoretically and experimentally investigating the controlled synchronization of four co-rotating coupled exciters in a vibrating system driven by induction motors. Using the Lagrange's equations, the motion equations of the vibrating system are derived. Combining the dynamic model of an induction motor with the dynamic model of a vibrating system, an electromechanical coupling model is developed. By virtue of the average method of modified small parameters and the Routh–Hurwitz principle, the self-synchronization criterion for four exciters and the stability criterion of synchronous states are obtained. Based on the numerical results, the stable inphase motion of four exciters fails to be implemented by means of self-synchronization, and as a result, the desired motion type of the vibrating system cannot be realized. Hence, the controlled synchronization is introduced into the vibrating system. Owing to the coupling characteristics of the vibrating system, the control challenge can be turned into a synchronization control problem between four exciters driven by induction motors. To perform the synchronized motion of zero phase differences between four exciters, sliding mode control algorithm and field-oriented control method on four induction motors are applied to develop the controlled synchronization scheme by adopting the master–slave control strategy. The stability of the closed loop system is proved by Lyapunov theorem. Experiments on a corresponding controlled synchronization bedstand are performed to examine the effectiveness of the developed controllers, including a comparison with self-synchronization method. Additionally, experimental results show the robustness of the proposed control scheme against the influence of parameter perturbations and external disturbances. The controlled synchronization method provides a novel approach to the development of vibrating machines.

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Sohn, Jin-Seung, Jin Woo Lee, Eun-Hyoung Cho, No-Cheol Park, and Young-Pil Park. "Dynamic Analysis of a Pendulum Dynamic Automatic Balancer." Shock and Vibration 14, no. 2 (2007): 151–67. http://dx.doi.org/10.1155/2007/452357.

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The automatic dynamic balancer is a device to reduce the vibration from unbalanced mass of rotors. Instead of considering prevailing ball automatic dynamic balancer, pendulum automatic dynamic balancer is analyzed. For the analysis of dynamic stability and behavior, the nonlinear equations of motion for a system are derived with respect to polar coordinates by the Lagrange's equations. The perturbation method is applied to investigate the dynamic behavior of the system around the equilibrium position. Based on the linearized equations, the dynamic stability of the system around the equilibrium positions is investigated by the eigenvalue analysis.The stability analysis provides the design requirements for the pendulum automatic dynamic balancer to achieve a balancing of the system. The efficiency of ball automatic dynamic balancer, for reducing the total vibration of the system, is better than one of pendulum automatic balancer, if the rotating speed is above critical speed. However, pendulum automatic dynamic balancer can achieve balancing even if the rotating speed is below critical speed.The time response analysis demonstrates the stability analysis from computing the radial displacement of the rotating system and the positions of pendulums. Furthermore, in order to confirm the theoretical analysis, various experiments are made on pendulum automatic dynamic balancer.

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Journal articles on the topic 'Lagrange's method'

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