Integrated measurement system using accelerometers and gyros as peripheral sensors to estimate the motional state of an elastic beam

Integrated navigation devices for vehicle guidance are the most common example of an integrated motion measurement system combining the signals from an inertial measurement unit (IMU consisting of three accelerometers and three gyros) and a GPS receiver with a single antenna. For this, the vehicle is traditionally assumed to be a single rigid body with six motional degrees of freedom to be determined. During periods of low vehicle dynamics the common integrated navigation systems show, however, stability problems. Nevertheless, the stability of the system can be guaranteed by distributing sensors over the vehicle structure. In this case the rigid body assumption has to be expanded to take the distributed sensors and the flexibility of the structure into account. Integrated systems in general are fusing different measuring signals by combining their benefits and blinding out their disadvantages. For instance, gyros and accelerometers are used to obtain reliable signals with a good time resolution. On the other hand, aiding sensors like radar units and strain gauges are known to be long-term accurate. Furthermore, the kernel of such integrated systems consists of an extended Kalman filter that estimates the motion state of the structure. Besides the sensor signals, the basis for the filter is an additional kinematical model of the structure which has to be developed individually. The example of the motion of an elastic beam being considered here is meant to be an approach to obtain motional measurements of a wing of a large airplane during flight. By means of a modal approach, a kinematical model of the beam was developed. This paper will compare integrated systems utilising accelerometers as peripheral sensors with systems using gyros and systems with a combination of both peripheral sensor types. Based on simulation the paper shows this approach, different sensor configurations, and estimated motion results of an elastic beam. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrated Motional Measurement for a Flexible Structure

Integrated navigation systems based on gyros and accelerometers are well established devices for vehicle guidance. The system design is traditionally based on the assumption that the vehicle is a rigid body. However, generalizing such integrated systems to flexible structures is possible. The example of the motion of a simple beam being considered here is meant to be a first approach to obtain sophisticated motional measurements of a wing of a large airplane during flight. The principle of integrated navigation systems consists of combining different measuring methods by using their specific advantages. Gyros and accelerometers are used to obtain reliable signals within a short period of time. On the other hand, aiding sensors like radar units and strain gauges are used because of their long-term accuracy. The kernel of the integrated system consists, however, of an extended Kalman filter that estimates the motion state of the structure. Besides the sensor signals, the basis for the filter is an additional kinematical model of the structure. By means of a model reduction, a kinematical model of the beam was developed. Based on simulation the paper presents this approach, the appropriate sensor set, and first estimated motion results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

A reduced beam and joint concept modeling approach to optimize global vehicle body dynamics

Ideally, NVH simulations become available already in the concept phase of vehicle development. The initial computer-aided design (CAD) can then be improved (by already including countermeasures), and the feasibility to balance NVH with other performance attributes is increased. In this early design stage, when exact geometrical information is not or scarcely available, conventional virtual prototyping techniques based on detailed CAD and FE models are not directly applicable. A state-of-the-art overview of concept NVH simulation methods in vehicle industry is given.This paper then presents a “Reduced Beam and Joint Modeling” approach to analyze and optimize the global bending and torsion modes of a vehicle body. Concept modifications in the body beam-like sections and in the joints are analyzed using the body reduced modal model. This small-sized model can be used to quickly and accurately optimize the low-frequency vehicle performance. The modifications are considered with respect to the existing (predecessor) model. Equivalent beam properties are estimated from the body FE model; modifications in the beam-like sections are then implemented with beam elements from a standard FE library. The joint modifications are considered through static superelements: stiffness formulations between the end points of the joint connected to the beam layout. The validity of the approach is first demonstrated on simple example models. An industrial vehicle BIW application case is subsequently presented. A beam and joint layout is created, and used for a fast sensitivity analysis to identify suitable modifications to improve the global modes. Next, two application cases are presented. First, a fast optimization analysis is performed to optimize the global body modes. Subsequently, suitable physical modifications are identified and applied to the full FE model; it is shown that the effect of these physical modifications is accurately predicted with the fast sensitivity analysis.     

Dynamical Modeling and Flatness Based Control of a Belt Drive System

Belt driven systems are part of many industrial applications, like computerized numerical control (CNC) machines in particular cutting machines and 3D-printers. In this paper the dynamical modeling and a flatness based controller design for belt driven systems are proposed. Due to the special kinematics, the stiffness of the belt is nonlinear, leading to nonlinear equations of motion. By neglecting some minor dynamical effects, the resulting system simplifies to a differentially flat one. This allows to calculate nominal feed-forward control torques by using the flat output of the system. To stabilize the error dynamics, an additional PD control law is introduced. The proposed method is compared with a controller, where elastic deflections for the feed forward part are neglected and elastic deformations are compensated by modifying the desired trajectories in a model-based manner. The tracking performance of both methods is evaluated in certain simulations and experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Analytical study on dynamic responses of a curved beam subjected to three-directional moving loads

Considering the warping resistance, inertia force and moving three-directional loads, a more comprehensive set of governing equations for vertical, torsional, radial and axial motions of the curved beam are derived. The analytical solutions for vertical, torsional, radial and axial responses of the curved beam subjected to three-directional moving loads are obtained, using the Galerkin method to discretize the partial differential equations and the modal superposition method to decouple the ordinary differential equations. The analytical results are compared with the numerical integration and a published work to verify the validity of the proposed solutions. Effects of Galerkin truncation terms and damping ratio on solution convergence are also discussed. Considering first-mode and higher-mode truncation respectively, the conditions of resonance and cancellation are analyzed for vertical, torsional, radial and axial motions of the curved beam. Taking a curved bridge under passage of a vehicle as an example, the influences of system parameters, such as vehicle speed, braking acceleration, bridge curve radius, bridge span and bridge deck elastic modulus, on bridge midpoint vibration are explored. The proposed approach and results may be beneficial to enhance understanding the three-directional vehicle-induced dynamic responses of curved bridges. It is shown that when the axial motion, or the multiple moving loads are involved, the first-order truncation are not accurate enough and one should use higher-mode truncation to study the responses of curved beams. In addition, it is necessary to consider damping in the vibration study of curved beams.     

Degenerate billiards

In an ordinary billiard system, trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than 1, we say that the billiard is degenerate. We study those trajectories of degenerate billiards that have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as small-mass limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems that shadow the trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit.     

On the representation of switched systems with inputs by perturbed control systems

This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.     

Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow

New sandwich panels and tubes have widely applications in nanotechnology such as transportation, naval, aerospace industries, micro and nanoelectromechanical systems and fluid storage. For example, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In the current study, stability and free vibration analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow embedded in an orthotropic foundation under electro-thermo-mechanical loadings are presented. In fact, this work can be provided a valuable background for more research and further experimental investigation. It is assumed that the micro tubes are made of flexible material and smart piezoelectric composites reinforced by carbon nanotubes as core and face sheets, respectively. Energy method and Hamilton's principle are applied to derive the governing equations of motions based on Euler–Bernoulli beam model and using modified strain gradient theory. Moreover, generalized differential quadrature method is used to discretize and solve the governing equations of motions. Numerical results are investigated to predict the influences of length-to-radius, thickness of face sheets-to-thickness of core ratio, temperature changes, orthotropic elastic medium, Knudsen number, and carbon nanotubes volume fraction on the dimensionless natural frequencies and critical flow velocity of sandwich double-bonded piezoelectric micro composite tubes. The results of this article show that increasing the thickness ratio, volume fraction carbon nanotubes and orthotropic elastic constants lead to enhance the dimensionless natural frequency and stability of system, while decrease these parameters with increasing the temperature and length-to-radius ratio.     

Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fish's body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.     

An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty

The traditional Krasnoselskii’s fixed point theorem in Banach spaces does not reproduce the rich and varied forms of operator equations in abstract spaces which are not linear structure. Consequently, its applications to integral equations and differential equations have met many obstacles. The present alternative Krasnoselskii’s fixed point theorem in generalized semilinear Banach spaces overcomes this deficiency and opens up for profitable investigation such as differential systems with uncertainty. An application to the existence of solutions of nonlocal problems for fuzzy implicit fractional differential systems under Caputo generalized Hukuhara differentiability with demonstrated example is given to validate the effectiveness of our theoretical results.     

On the Application of Accurate Distance Algorithms to Multibody Simulations

Distance algorithms are most frequently used in robotics to determine the distance between two obstacles in the environment of a robot or between a sensor point and an object. We extend the multibody simulation package MOBILE for an application of accurate algorithms for distance computation between objects represented by convex or non-convex polyhedra. These objects are represented by their vertices and oriented facets. As an application example, a multibody system is discussed where a sensor point moves close to a non-convex obstacle. The computed results show that the algorithms developed are suitable for accurate real-time multibody simulations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Infinite-Dimensional Optimal Filtering Problem with Mobile and Stationary Sensor Networks

In this article, we introduce a framework to address filtering and smoothing with mobile sensor networks for distributed parameter systems. The main problem is formulated as the minimization of a functional involving the trace of the solution of a Riccati integral equation with constraints given by the trajectory of the sensor network. We prove existence and develop approximation of the solution to the Riccati equation in certain trace-class spaces. We also consider the corresponding optimization problem. Finally, we employ a Galerkin approximation scheme and implement a descent algorithm to compute optimal trajectories of the sensor network. Numerical examples are given for both stationary and moving sensor networks.     

A spatial shear deformable beam finite element based on the absolute nodal coordinate formulation

In the present paper a three-dimensional beam finite element undergoing large deformations is proposed. Since the definition of the proposed finite element is based on the absolute nodal coordinate formulation (ANCF), no rotational coordinates occur in the formulation. In the current approach, the orientation of the cross section is parameterized by means of slope vectors. Since those are no unit vectors, the cross-section can deform, similar to existing thick beam and shell elements. The nodal displacements and the directional derivatives of the displacements are chosen as nodal coordinates, but in contrast to standard ANCF elements, the proposed formulation is based on the two transversal slope vectors per node only. Different approaches for the virtual work of elastic forces are presented: a continuum mechanics based formulation, as well as a structural mechanics based formulation, which is in accordance with classical nonlinear beam finite elements. Since different interpolation functions as in standard ANCF elements are used, a much better convergence rate (up to order four) can be obtained. Therefore, the present element has high potential for application in geometrically nonlinear problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design of a coordinated adaptive backstepping tracking control for nonlinear uncertain active suspension system

This paper presents a novel adaptive backstepping tracking control for nonlinear uncertain active suspension system, which can achieve the coordinated control over the sprung-mass acceleration and suspension dynamic displacement for nonlinear uncertain active suspension system based on a developing model-reference system. First, according to adaptive backstepping control principle, this model-reference system is designed with purpose of providing the ideal reference trajectories for the sprung-mass displacement and vertical velocity, respectively. Then, the design of a coordinated adaptive backstepping tracking controller is conducted to make the control plant accurately track the prescribed performances of the model-reference system by virtue of the backstepping technique and Lyapunov stability theory, in which a virtual controller with online parameter regulation rules is designed and implemented to guarantee the stability of vehicle body. Finally, a numerical example is provided to verify the effectiveness of our designed adaptive backstepping tracking controller under various operating scenarios.     

Three-dimensional vehicle-ballasted track-subgrade interaction: Model construction and numerical analysis

In this work a three-dimensional vehicle-ballasted track-subgrade interaction model is developed, where the vehicle is modeled as a multi-rigid-body system, the track-subgrade interaction is modelled by finite element method (FEM) with the rail modelled as beam elements, the sleeper and the subgrade layers modelled as solid elements, and the vehicle and the track-subgrade system are coupled by unified matrix formulations and solved simultaneously and time-dependently. As the modelling advancement, the versatility for FEM construction of track-subgrade systems has been promoted, where the elemental scale can be arbitrarily chosen without consideration of the node-to-node matching principle as the conventional methods; besides the vehicle and the track-subgrade systems have been coupled effectively by wheel-rail contact models. The applicability of various wheel-rail contact models has been discussed, and to reveal the effectiveness of this model in solving engineering problems such as the soil elasticity unevenness and the contact break of “hanging sleepers”, numerical examples have also been presented with referencable conclusions.     

线性主动结构及模态(Ⅱ)——离散系统及梁

在“线性主动结构及模态(Ⅰ)”中给出了主动结构的基本概念及模态的若干属性的基础上,对主动离散系统及主动梁的属性做了进一步的讨论,包括稳定性和正交性,并用具体实例对模态做了解释．将伴随结构概念在梁结构中推广,具体讨论了两种配置的主动梁,它们分别代表离散传感和作动及分布传感和作动的配置,并给出了用主动梁振型和伴随主动梁振型表示的正交性条件．用实例给出了同位和非同位主动刚度梁的特征值随反馈大小的变化．