Equations Of Motion For System Research Articles

Adaptive Control of a Cable-Actuated Parallel Manipulator Mounted on a Platform with Differential Wheels under Payload Uncertainty

Cable-actuated parallel Manipulators (CPMs) are widely employed for object handling applications. In order to displace the carried object along the ground to an unlimited distance, the CPMs can be mounted on wheeled mobile robots (WMRs). The derivation of the dynamic equations of motion for this integrated system is presented using Lagrange method. Since in load carrying task, the inertia of the moving load is the main source of uncertainty, an adaptive control approach is considered for the CPM, whereas the WMR uses a feedback linearized sliding mode approach. In order to maintain the end-effector of the CPM in within its relative workspace in the WMR frame, the convergence rate in the two controllers should be similar. Decentralization of the control law can be accomplished if the inertia of the CPM motors are negligible compared with the other inertias of the system. This assumption is shown to be applicable if an introduced index is small enough to have noticeable effect on the tracking error.

Vibration mode decomposition response analysis of large floating roof tank isolation considering swing effect

To solve the seismic response problem of a vertical floating roof tank with base isolation, the floating roof is assumed to experience homogeneous rigid circular plate vibration, where the wave height of the vibration is linearly distributed along the radius, starting from the theory of fluid velocity potential; the potential function of the liquid movement and the corresponding theoretical expression of the base shear, overturning the moment, are then established. According to the equivalent principle of the shear and moment, a simplified mechanical model of a base isolation tank with a swinging effect is established, along with a motion equation of a vertical storage tank isolation system that considers the swinging effect based on the energy principle. At the same time, taking a 150,000 m3 large-scale storage tank as an example, a numerical analysis of the dampening effect was conducted using a vibration mode decomposition response spectrum method, and a comparative analysis with a simplified mechanical model with no swinging effect was applied.

A new method for establishing motion equations in nonholonomic systems

In this paper, we present a new method of establishing the fundamental motion equations of nonholonomic systems without using any variational principles. According to the invariance of motion equations under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, the double index tensor analysis method is proposed cooperating with the product rules of matrix. Then we let first integrals equivalent to the corresponding nonholonomic constraints imposed upon the system to explain the self consistency of nonholonomic systems, it is shown that the self consistency is the core of nonholonomic mechanics and can verify which model is appropriate for describing nonholonomic systems. Therefore, the form of nonholonomic constraint force is determined only by three qualities: the invariance of the motion equations under the constraint of first integrals, the invariance of nonholonomic constraint force under the generalized coordinates transformation and the nonholonomic constraint function groups transformation, to establish the fundamental motion equation of nonholonomic systems. Using the double index tensor analysis method, the three qualities and the Moore-Penrose generalized inverse of matrix theory, the equation of motion of nonholonomic systems are derived. Some classical equations in nonholonomic systems, such as the Routh equation, the Nielsen equation and the Chaplygin equation are also derived to prove our method. The method is totally based on the self consistency of nonholonomic systems and the covariation of motion equations, these two characters are naturally derived from mathematical and mechanical requirements of nonholonomic systems. Not only does not use any transcendental variational principles, such as D’Alembert-Lagrange principle, Gauss principle or Jourdian principle, but also illustrate that the validity of the three principles are attributed to the two characters. Further more, this method even provides a reasonable explanation for Chetaev conditions in nonholonomic systems, and shows that the vakonomic mechanics derived by Hamilton’s principle is not an appropriate model for nonholonomic systems.

A rigorous proof for the equivalence of the projective Newton–Euler equations and the Lagrange equations of second kind for spatial rigid multibody systems

It is well known that the projective Newton–Euler equation and the Lagrange equation of second kind lead to the same result when deriving the dynamical equations of motion for holonomic rigid multibody systems. It can be shown that both approaches follow from the principles of d’Alembert or Jourdain. However, as to the author’s knowledge, no direct rigorous proof for the equivalence of these approaches is given in literature so far when it comes to spatial systems of rigid bodies. In this paper, we present a novel proof that directly addresses the projective Newton–Euler equation and the Lagrange equation of second kind without the detour via variational principles. The proof is mainly based on vector and matrix manipulations and elementary concepts of differential geometry. Although the mathematical framework is thereby kept simple, the argumentation is considerably more complex compared to the case of planar systems of rigid bodies or spatial systems of particles. An illustrative example is presented.

Numerical study of flows in axial compressors of aircraft gas-turbine engines

Design and adjustment of compressors at modern gas-turbine engines are based on the wide use of numerical analysis methods of a various level of complexity. Such approaches make it possible to analyze the alignment of joint operation of stages and to perform the required correction of geometrical parameters. The methods for calculating the 1D and 2D flow in compressors are distinguished by high flexibility, which allows the utilization of considerable experience in designing and experimental research. These methods are consequently in demand at all stages of the engine life cycle: when creating, adjusting, at operation.We present methods for the calculation of parameters and structure of current, as well as the summary characteristics of axial stages and multistage compressors. To solve a system of motion equations, we employed a matrix method, which makes it possible to apply small computation grids in a flow-through part of the compressor. The method is designed to numerically simulate the sub-, trans-, and supersonic flows in the flow-through part of axial compressor stages and multistage axial compressors at aircraft engines. The methods are implemented in the form of software complexes.The article reports certain results related to the verification of these complexes. We use data from experimental studies into various multi-stage compressors and high-head fan stages. We show a satisfactory agreement between calculated and experimental data over a wide range of modes of consumption and rotation frequency.The developed software package was used to improve the geometrical parameters for an axial multi-stage compressor, aimed at increasing the air flow rate through the compressor and enhancing the reserve of its gas-dynamic stability.Employing small computational grids made it possible to undertake a series of studies, previously available only for the calculation methods of spatial flow. We considered different variants for the execution of the bushing surface at a high-head fan stage. When analyzing the structure of flow in a stage, we show the change in the axial component of velocity in a blade-to-blade channel of the impeller along its axis.Development and application of the new methods of calculation will improve the quality of design of axial compressors and enhance competitiveness of the Ukrainian aircraft gas turbine engines.

Action Function Formulation for Conservative Systems with Second-Order Lagrangian

The Euler Lagrange equation is studied to obtain the equations of motion for conservative systems with second order Lagrangian. The solutions of these equations are substituted in the given Lagrangian. The action function is then derived by calculating the time integral of the Lagrangian. To explain the application of our formalism two examples are discussed.

Variational order for forced Lagrangian systems

We are able to derive the equations of motion for forced mechanical systems in a purely variational setting, both in the context of Lagrangian or Hamiltonian mechanics, by duplicating the variables of the system as introduced by Galley (2013 Phys. Rev. Lett. 110 174301) and Galley et al (2014 arXiv:1412.3082). We show that this construction is useful to design high-order integrators for forced Lagrangian systems and, more importantly, we give a characterization of the order of a method applied to a forced system using the corresponding variational order of the duplicated one.

System modeling and instability control of wind turbine blade based on hydraulic pitch system and radial basic function neural network proportional–integral–derivative controller

System modeling and aeroservoelastic control for divergent instability of stall-induced composite wind turbine blade modeled as thin-walled symmetric layup beam analysis have been investigated based on hydraulic pitch system and radial basic function neural network control. The blade is modeled as single-cell thin-walled beam structure with the circumferentially asymmetric stiffness design, exhibiting flap/lead-lag bending coupling deformation. The stall flutter and aeroservoelastic control of composite blade are investigated based on dynamic stall Beddoes–Leishman aerodynamic model and radial basic function neural network proportional–integral–derivative controller, with pitch actuator performed by hydraulic system. The system motion equations consist of the aeroelastic equations and the six-order pitch equation. The nonlinear aeroelastic responses, including both flap/lead-lag responses and pitch angle responses under different parameters, are solved by Galerkin method and the nonlinear time integration scheme with nonlinear residual analysis. To verify the effectiveness of the control scheme and realize visualized display of large thin-walled blade in the laboratory, experimental platform based on hardware-in-the-loop simulation is built to realize real-time control and virtual simulation. The platform structure consists of PLC hardware, monitoring interface in configuration software, and simulation environment that is connected by the OPC server with PLC system. The platform lays the foundation for vibrational behavior research on visualization of large wind turbine blade under divergent stall situation and verifies the real-time feasibility of the control algorithm proposed.

A novel computer-oriented dynamical approach with efficient formulations for multibody systems including ignorable coordinates

• Minimum number of equations for systems possessing ignorable coordinates. • Derivation of motion equations for unconstrained or constrained mechanical systems. • Use of generalized speeds as motion variables and employing the matrix notation. • Better satisfaction of the motion constraints compared to conventional methods. • Low computational errors during numerical integration. Based on Lagrangian mechanics, we present a novel and computationally efficient set of equations of motion in the matrix notation, for unconstrained or constrained mechanical systems including ignorable coordinates. The equations are applicable to multibody systems including holonomic or nonholonomic constraints. It is shown that by appropriate selection of generalized speeds as a new set of motion variables, the constraint reaction forces can be automatically eliminated from the set of developed reduced dynamical equations in a straightforward manner, resulting in a minimal set of dynamic equations. We present simulation results on one constrained and one unconstrained system to demonstrate the advantages of the proposed dynamic modeling approach for these different classes of dynamical systems. According to the results, the proposed dynamic equations not only reduce the number of equations and computational effort, but also better satisfy the invariance of constraints and conserved quantities of systems in comparison to the other conventional methods.

Flexible multibody dynamic modeling of a floating wind turbine

This paper presents a modeling approach for the nonlinear flexible dynamics simulation of a floating offshore wind turbine (FOWT). The system consists of a floating platform, the moorings, the wind turbine tower, nacelle and the rotor. The floating platform is modeled as a six-degrees-of-freedom (6-DOF) rigid body subject to buoyancy, hydrodynamic and mooring loads. The wind turbine tower is modeled as a three-dimensional (3D) damped tapered Euler–Bernoulli beam undergoing coupled general rigid body and elastic motions. The beam bending-bending and twist motions are considered. The nacelle is modeled as a rigid body attached to the tower tip. The rotor is also modeled as rigid body spinning around its axis and subject to aerodynamic load. The generator torque control is integrated into the model to capture the rotor spin dynamics. The equations of coupled rigid body and elastic motions are derived using Lagrange’s Equation in terms of the generalized flexible coordinates and the platform quasi coordinates. The external loads are formulated and incorporated into the system equations of motion. The dynamic model is extensively validated against the most popular FOWT simulation tools with an excellent agreement. Finally, the influence of the tower flexibility on the FOWT dynamic behavior is investigated.

Parametric Instability and Localization of Vibrations in Three-Blade Wind Turbines

Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees-of-freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.

Novel piezoelectric-based ocean wave energy harvesting from offshore buoys

Ocean wave energy is one of the huge energy sources, which easily wasted around us. Because of low frequency of the ocean waves, less attention has been paid on vibration-based energy harvesting from this energy source. In this study, a novel beam-column piezoelectric-based energy harvesting system is studied, which can be optimally used as an Ocean Wave Energy Harvester (OWEH). In doing so, the electromechanical equations of motion for the energy harvesting system are accurately derived. The results, which are obtained using the governing equations, are validated by experimental results. Then application of the presented energy harvesting system, which is installed in offshore buoys, is studied. It is shown that harvested energy is enough to provide needed energy of the electrical devices in the buoy. It is shown that the energy harvesting system in the buoy, which is subjected to large wave height and low frequency, is more efficient. Finally, application of the self-tuning buoy, which works based on the ocean wave frequency, is studied. The presented novel system opens new field of research that helps to use the ocean wave energy in a proper way.

Vibration, Buckling and Aeroelastic Analyses of Functionally Graded Multilayer Graphene-Nanoplatelets-Reinforced Composite Plates Embedded in Piezoelectric Layers

This paper deals with the vibration characteristics and nonlinear aeroelastic response of the functionally graded (FG) multilayer composite plate reinforced with graphene nanoplatelets (GPLs) subjected to in-plane excitations and applied voltage. The different GPL nanofillers distribution patterns across the thickness are discussed, in which the effective Young’s modulus is determined by modified Halpin–Tsai model. Based on high-order shear deformation theory, the motion equations of the FG plate system considering the von Kármán geometric nonlinearity are derived using the Hamilton’s principle. The Galerkin method is applied to discretize the partial differential governing equations into the ordinary differential nonlinear system. The effects of many influential parameters, i.e., GPLs weight fraction, distribution pattern, geometry size, applied voltage and the number of layers, on the vibration and aeroelastic behaviors are presented in detail. Numerical results show that a small amount of GPLs reinforcement can have a significant enhancement effect on the performance of the composite plate structure. Moreover, the in-plane force and aerodynamic pressure play an opposite effect on the dynamic stability, and the jumping phenomena, quasi-periodic motion can be observed with the compressive force increased further.

Three-dimensional vortex-induced vibrations of supported pipes conveying fluid based on wake oscillator models

The present study develops a new three-dimensional nonlinear model for investigating vortex-induced vibrations (VIV) of flexible pipes conveying internal fluid flow. The unsteady hydrodynamic forces associated with the wake dynamics are modeled by two distributed van der Pol wake oscillators. In particular, the nonlinear partial differential equations of motion of the pipe and the wake are derived, taking into account the coupling between the structure and the fluid. The nonlinear equations of motion for the coupled system are then discretized by means of the Galerkin technique, resulting in a high-dimensional reduced-order model of the system. It is shown that the natural frequencies for in-plane and out-of-plane motions of the pipe may be different at high internal flow velocities beyond the threshold of buckling instability. The orientation angle of the postbuckling configuration is time-varying due to the disturbance of hydrodynamic forces, thus yielding sometimes unexpected results. For a buckled pipe with relatively low cross-flow velocity, interestingly, examining the nonlinear dynamics of the pipe indicates that the combined effects of the cross-flow-induced resonance of the in-plane first mode and the internal-flow-induced buckling on the IL and CF oscillation amplitudes may be significant. For higher cross-flow velocities, however, the effect of internal fluid flow on the nonlinear VIV responses of the pipe is not pronounced.

A New Analytical Model for Estimating the Natural Frequency of Pile-Supported Bridges

The common design for earthquake loading assumes that bridges are fixed at their bases. In reality, interaction between soil and structure decreases the overall stiffness of the structural system. This is not always desirable and may elevate the fundamental period of the system. The current study set to assess soil-structure interaction (SSI) impacts on dynamic performance of pile-supported bridge piers. The bridge foundation is assumed as a distributed mass pile, and a new analytical model is proposed to predict seismic analysis of pile-supported bridges. Applying the dynamic equations of motion for system and using virtual work principles, SSI effects were estimated. Based on the analytical model, a new equation is proposed for calculating the natural frequency of pile-supported bridges. Since the proposed model is very similar to real soil-pile-pier systems, this equation can be used to estimate the fundamental period of the single-pier bridges.

Improved HHT integration method for index-2 over-determined equations of motion in multibody system dynamics

Equations of motion for multibody systems with holonomic constraints are index-3 differential-algebraic equations (DAEs). Velocity constraint equations can be obtained by differentiating the position constraint equations with respect to time. When the velocity constraint equations are combined with index-3 equations of motion, the equations of motion in the form of index-2 over-determined DAEs can be obtained. HHT-SI2 method, which is extended from HHT (Hilbert-Hughs-Taylor) method in the community of structural dynamics and used for the numerical integration of equations of motion in the form of index-2 over-determined DAEs, is improved. Firstly, only the correction term of the HHT-SI2 method is improved, and two unknown vectors are introduced as before. The condition number of Jacobian matrix in Newton-Raphson method is decreased during the numerical solution of nonlinear equations from discretization, and then the limitation of step-size for the numerical integration method is reduced. Secondly, the form of the correction term in HHT-SI2 method is completely changed, and only one unknown vector is introduced. The quantity of equation in nonlinear equations form discretization is decreased, and the integration speed of the method is increased obviously. Then the validity of the two improved methods is verified by numerical experiments, and the improved methods are compared with HHT-SI2 method in the view of integration speed and step-size limitation. Numerical experiments also show that the two improved methods are second-order accuracy and numerical damping can be controlled. In the end, the two improved methods are also analyzed and compared with other existed similar methods in the conclusion section. The advantages of the new methods are pointed out.

Analysis of motion of the system «electric motor – mechanism of сlass IV with dwell driven links» during acceleration

In this article electromechanical system motor – mechanism IV class with dwell driven is considered. The problem of finding the motion equations of system using the second type Lagrange equations with indefinite multipliers is solved. In work the method given in A.Myklebust is used, therewere obtained the equation of speed ratio. Numerical values of the lengths of mechanism links and physical parameters of the electric motor are given. The graphs of the links of mechanism IV class and the motor shaft in MatLab program are obtained. Analysis of the graphs showed that the steady speed of all the links is achieved in 1.5 seconds. The driving link or the motor speed is equal to 10 rad/s and achieved by 0,4 seconds, followed by speed fluctuations, which make up about 30-35%. High initial current is valid during the time equal to 1.5 seconds. DOI: http://dx.doi.org/10.5755/j01.mech.23.6.16261

On some aspects of the geometry of non integrable distributions and applications

We consider a regular distribution \begin{document}$\mathcal{D}$\end{document} in a Riemannian manifold \begin{document}$(M, g)$\end{document} . The Levi-Civita connection on \begin{document}$(M, g)$\end{document} together with the orthogonal projection allow to endow the space of sections of \begin{document}$\mathcal{D}$\end{document} with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of \begin{document}$\mathcal{D}$\end{document} , one directly with the connection in \begin{document}$(M, g)$\end{document} and the other one with this intrinsic connection. Their difference is the second fundamental form of \begin{document}$\mathcal{D}$\end{document} and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.

Improvement of field method and its application to integrating motion equation in Riemann-Cartan space

Like the Hamilton-Jacobi method, the Vujanović field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanović field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanović field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanović field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m (m n) first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution (the complete solution is a special case of m=n) of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanović field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanović field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subjected to an active force, its motion equation in its Riemann-Cartan configuration space can be obtained by a first order nonlinear nonholonomic mapping. Since the motion equations in Riemann-Cartan configuration space contain quasi-speeds, they are often considered to be difficult to solve directly. In this paper we give a briefing of how to construct the motion equations of a first order linear nonholonomic constraint system in its Riemann-Cartan configuration space, and how to obtain the first integrals of the motion equations in the Riemann-Cartan configuration space by the improved field method. This is an effective method to study some nonholonomic nonconservative motions.

Semi-analytical research on acoustic-structure coupling calculation of partially submerged cylindrical shell

Vibroacoustic analysis of a partially submerged cylindrical shell-liquid coupling system is a typical acoustic-structure interaction problem in an acoustic half-space. Generally, the vibration and acoustic solutions of this problem almost depend on numerical method in previous studies. However, whether from the aspect of verifying the numerical methods or from the aspect of revealing the vibroacoustic mechanism of the acoustic-structure interaction system, the development of an analytical or semi-analytical method is indispensable. In this study, a semi-analytical method is proposed to address the vibroacoustic response of a horizontal cylindrical shell partially immersed in water The acoustic coordinate system is established on the free surface, and the sine series function is introduced to express the sound pressure to meet the pressure release boundary condition on the free surface automatically. Then based on the two-dimensional Flgge shell theory, the motion equation of the shell-liquid coupling system is established with using the center of the shell as the origin of the coordinate system. By using the Galerkin method, the velocity continuous condition on the fluid-structure interface is established. Then the relation matrix between the acoustic pressure amplitude and the shell displacement amplitude is derived after converting the sound pressure from different coordinate systems into that in the same coordinate system by using the geometry relationship of two coordinate systems. Then the vibration and sound radiation of this coupling system are predicted by solving the coupled matrix equations, and the coupled free vibration can also be addressed by solving the characteristic equation after setting the determinant of the coefficient matrix to be zero. To verify the accuracy and reliability of the present method, the coupled natural frequencies, the vibration response and the distribution of the sound pressure obtained from the present method are compared with those obtained from the finite element method, showing that they are in good agreement with each other. Then the shapes of the first four order modes for the coupled system are presented and compared with those of the system submerged in unbounded fluid field, and the couple between different modes, and the couple between the symmetric and antisymmetric modes are observed due to the effect of the free surface. The characteristics of the root mean square velocity of the shell with different immersed depths are discussed, which reveals that the peaks of response curves shift to the lower frequencies with increasing immersed depth. The characteristics of far field sound pressure directivity are presented and explained by the image method in detail. This study provides a novel method to analytically predict the vibroacoustic response of an elastic structure partially coupling with the fluid field when bounded in a sound half space.