Set Of Constraint Equations Research Articles

Quenching Vibration on a Symmetric Laminated Composite Plate Excited by Multiple Harmonics Using Inerter-based Dynamic Vibration Absorbers

Abstract In this paper a simple and efficient method is developed to quench the steady state vibration of a symmetric laminated composite rectangular plate subjected to multiple harmonics with distinct excitation frequencies. The introduction of the inerter into the traditional dynamic vibration absorber enlarges the solution space for both the feasible attachment locations and the absorber parameters. For a given set of attachment locations, when the traditional vibration absorbers yield physically unrealizable absorber parameters, i.e., negative absorber parameters, an inerter-based vibration absorber can be used such that the absorber parameters become strictly positive. Using the assumed modes method, the governing equations of the laminated composite plate carrying the inerter-based dynamic vibration absorbers are first formulated. A set of constraint equations is established by enforcing nodes at user-specified locations on the plate. Harmonic excitations consisting of a single or multiple distinct excitations are considered. In addition, two attachment scenarios are analyzed: when the attachment and node locations coincide, and when they are distinct. Numerical experiments show that the absorbers parameters can be readily determined for both cases, and validate the proposed method of enforcing the desired node locations when the plate is subjected to either a single or multiple harmonic excitations.

Quenching Vibration on a Harmonically Excited Symmetric Laminated Composite Plate

Abstract In this paper, a simple and efficient method is developed to quench the steady-state vibration of a harmonically excited, damped and symmetric laminated composite rectangular plate. This is achieved by enforcing points of zero displacement, or nodes, at some specified locations on the laminated composite plate using properly tuned damped oscillators. Using the assumed-modes method, the governing equations of the laminated composite plate carrying the damped oscillators are first formulated. A set of constraint equations is established by enforcing nodes at user-specified locations on the plate. Two attachment scenarios are considered: when the attachment and node locations coincide and when they are distinct. Numerical experiments show that for both cases, the damped oscillator parameters can be readily determined and the desired node locations can be successfully imposed. More importantly, enforcing nodes can suppress vibration in the vicinity of the node locations, thereby keeping that region of the laminated composite plate nearly stationary.

A time-optimal wellbore trajectory design for slide drilling systems

For decades, the slide drilling system (SDS) has played a critical role in the area of directional drilling. Drilling in the sliding mode is much more time-consuming than drilling in the rotating mode for a SDS. Therefore, it generally takes more time to drill a curved wellbore segment than a straight segment of the same length. Based on this fact, a novel wellbore trajectory design for SDSs is proposed in this paper to reduce the total drilling time and improve economy. The basic idea of our strategy is to replace a long curved wellbore section, e.g., the build-up and drop-off sections, with a series of short circular-arc and straight-line segments in the design stage. Assuming that rotating and sliding modes are only used for straight-line and circular-arc segments, respectively, the total drilling time is reduced by increasing the average rate of penetration. Therefore, the problem lies in how to replace a long circular arc with multiple short circular arcs and straight lines to minimize the total drilling time, which is the aim of this research. In this paper, a piecewise trial strategy is proposed, where the optimized wellbore trajectory is generated by adding circular-arc and straight-line segments with equal length one by one according to the “line first” principle. This strategy converts the original optimization problem, which involves both combinatorial and parameter optimization of unknown dimension, to a much easier one, where there are only three design variables and one set of constraint equations. Due to the multimodal characteristics of the objective function, genetic algorithm (GA) is adopted to obtain the optimal solution. A great reduction of the drilling time is obtained by applying our method to two typical cases, which shows a clear potential for future field applications.

Stability and Accuracy Analysis of Real-Time Hybrid Simulation (RTHS) with Incomplete Boundary Conditions and Actuator Delay

The main purpose of this paper is to examine the effects of incomplete boundary conditions and actuator delay on the dynamic responses of seismically excited civil structures. A set of constraint equations representing the reserved interface degrees-of-freedom (DOFs) and the delay are introduced to form a mechanical model of real-time hybrid simulation (RTHS) (referred to as RTHS-I&A) for a multi-degree-of-freedom (MDOF) system based on dynamic substructure method (DSM). Then, the RTHS-I&A system is modeled by a discrete closed-loop transfer function based on discrete control theory, using a selected integration algorithm, and the stability of the system is investigated by examining the poles of the function. Three typical cases with different structural properties are utilized to investigate the effects of incomplete boundary conditions and actuator delay. The results show that both incomplete boundary conditions and actuator delay greatly affect the dynamic responses of structures, and the combination of the two factors will amplify their influence especially on the nodes at the interface. The numerical model of RTHS-I&A proposed in this paper is quite useful for evaluating the responses of structures with different interface conditions and loading schemes that are preliminarily designed before a physical testing is conducted, and provides guidance for future relevant researches.

Dynamic trajectory planning of reconfigurable suspended cable robot

This paper focuses on the dynamic trajectory planning of reconfigurable suspended cable robot. By using the approach of the dynamic equilibrium of suspended cable robot, a set of constraint equations are derived. These equations represent the cable tensions in terms of the cable directions, moving-platform payload and the instantaneous position of the moving-platform. While fulfilling the dynamic equilibrium, the position of cable contact points are reconfigured thanks to the pulley mechanism. Through the pulley reconfiguration, the robot may adapt its tension to be always positive while executing the task for a given trajectory. Two robot structures will be compared for various parametric trajectories. A prototype has been fabricated for both robot structures to demonstrate the tension distributions along the trajectory.

Transversal vibration analysis of the upper span of nonlinear closed-loop track systems

In order to analyze the effect of track pre-tension on the transversal vibration of the upper track, a multibody dynamics model of a tracked vehicle needs to be established. In view of the complex structure of tracked vehicles, the computational efficiency of conventional methods cannot meet engineering design requirements. In this paper, the Riccati Transfer Matrix Method for nonlinear closed-loop multibody systems is proposed and applied to the modeling and simulation of tracked vehicle systems. It avoids the overall dynamic equations of the system and achieves efficient computation with a smaller matrix scale. By selecting accelerations and forces as elements of the state vectors instead of coordinates and forces, linearization is avoided. The track chain is cut in a revolute joint. Its effect on the closed-loop system is represented by a set of constraint equations and internal constraint forces. An improved Riccati transformation for closed-loop systems is proposed to connect the internal force and acceleration parts in the state vector with the system constraint internal forces. The track pre-tension can be adjusted by moving the position of the idler relative to the chassis. The root mean square of vertical acceleration of mid-span with respect to the chassis is selected to evaluate the transversal vibration of the upper track. Simulation of a tracked vehicle driving on different road conditions is carried out with different idler displacements. Finally, a reasonable adjustment strategy for the track tensioner is proposed corresponding to different running conditions.

Quenching Vibration by Imposing Nodes on a Plate Subjected to Multiple Harmonics With Distinct Excitation Frequencies

Abstract In this paper, a simple and efficient method to enforce nodes, or points of zero vibration, on an arbitrarily supported rectangular plate subjected to multiple harmonics with distinct excitation frequencies is developed. This vibration suppression is achieved by attaching a chain of properly tuned oscillators, configured in series, to the host plate. The governing equations of the combined system are first obtained using the assumed-modes method. Enforcing the node conditions for the given excitation frequencies, a set of constraint equations are formulated, from which the sprung mass parameters can be determined. Two special cases are considered: when the attachment and node locations coincide (or collocated) and when they do not (or non-collocated). For the former case, determining the required oscillator parameters requires one to solve an inverse eigenvalue problem, and for the latter case, it requires one to use a generic optimization algorithm. Procedures to tune the parameters of the oscillator chains are outlined in detail, and numerical experiments validate the proposed method of enforcing nodes to mitigate excess vibration when the plate is subjected to multiple harmonic excitations.

Reconfiguration Analysis of a 3-DOF Parallel Mechanism

This paper deals with the reconfiguration analysis of a 3-DOF (degrees-of-freedom) parallel manipulator (PM) which belongs to the cylindrical parallel mechanisms family. The PM is composed of a base and a moving platform shaped as equilateral triangles connected by three serial kinematic chains (legs). Two legs are composed of two universal (U) joints connected by a prismatic (P) joint. The third leg is composed of a revolute (R) joint connected to the base, a prismatic joint and universal joint in sequence. A set of constraint equations of the 1-RPU−2-UPU PM is derived and solved in terms of the Euler parameter quaternion (a.k.a. Euler-Rodrigues quaternion) representing the orientation of the moving platform and of the Cartesian coordinates of the reference point on the moving platform. It is found that the PM may undergo either the 3-DOF PPR or the 3-DOF planar operation mode only when the base and the moving platform are identical. The transition configuration between the operation modes is also identified.

Symmetries and physically realisable ensembles for open quantum systems

A D-dimensional Markovian open quantum system will undergo stochastic evolution which preserves pure states, if one monitors without loss of information the bath to which it is coupled. If a finite ensemble of pure states satisfies a particular set of constraint equations then it is possible to perform the monitoring in such a way that the (discontinuous) trajectory of the conditioned system state is, at all long times, restricted to those pure states. Finding these physically realisable ensembles (PREs) is typically very difficult, even numerically, when the system dimension is larger than 2. In this paper, we develop symmetry-based techniques that potentially greatly reduce the difficulty of finding a subset of all possible PREs. The two dynamical symmetries considered are an invariant subspace and a Wigner symmetry. An analysis of previously known PREs using the developed techniques provides us with new insights and lays the foundation for future studies of higher dimensional systems.

Structural Synthesis of Baranov Trusses With Up to 13 Links

The structural synthesis of Baranov trusses is still an open problem, although Baranov trusses are widely used in the design and analysis of mechanisms and robots. This paper proposes a systematic method for the structural synthesis of Baranov trusses. First, the definition review and the graph-form representations of Baranov trusses are proposed. Second, seven propositions on structural characteristics of Baranov trusses are concluded. Then, based on the set of constraint equations and a rigid subchain detection algorithm, a systematic method is presented to synthesize the complete set of Baranov trusses with a specified number of links. Finally, the synthesis results of contracted graphs (including valid and rigid contracted graphs) and topological graphs of Baranov trusses with up to 13 links are provided, and the synthesis methods and results between ours and the ones in the existing literature are compared and discussed in detail.

Operator product expansion for conformal defects

We study the operator product expansion (OPE) for scalar conformal defects of any codimension in CFT. The OPE for defects is decomposed into “defect OPE blocks”, the irreducible representations of the conformal group, each of which packages the contribution from a primary operator and its descendants. We use the shadow formalism to deduce an integral representation of the defect OPE blocks. They are shown to obey a set of constraint equations that can be regarded as equations of motion for a scalar field propagating on the moduli space of the defects. By employing the Radon transform between the AdS space and the moduli space, we obtain a formula of constructing an AdS scalar field from the defect OPE block for a conformal defect of any codimension in a scalar representation of the conformal group, which turns out to be the Euclidean version of the HKLL formula. We also introduce a duality between conformal defects of different codimensions and prove the equivalence between the defect OPE block for codimension-two defects and the OPE block for a pair of local operators.

Imposing points of zero displacement and zero slopes on a plate subjected to steady-state harmonic excitation

In this paper, a simple and effective method to enforce fixed nodes, or points of zero displacement and zero slope, on an arbitrarily supported rectangular plate subjected to steady-state harmonic excitations is developed. This is achieved by attaching properly tuned translational and rotational oscillators at specified locations. The governing equations of the combined system are first derived using the assumed-modes method. By enforcing the conditions of zero displacements and zero slopes simultaneously, a set of constraint equations are formulated, from which the oscillator parameters can be determined. When the attachment locations coincide with the desired fixed node locations, it is always possible to select the oscillator parameters such that one or multiple fixed nodes are induced at any locations on the plate for any excitation frequency. When the attachment and the desired node locations are not collocated, it is only possible to induce nodes at certain locations on the plate. When the fixed node locations are judiciously chosen, a selected region of the plate can be made to remain nearly stationary. Thus, the proposed method provides a simple and yet effective means to passively control excessive vibrations.

Identification and Comparison for Continuous Motion Characteristics of Three Two-Degree-of-Freedom Pointing Mechanisms

Two-degree-of-freedom (2DOF) pointing mechanisms, including the gimbal structure, the 1-RR&2-RRR spherical parallel mechanism (SPM), and the Omni-Wrist III, are increasingly applied in tracking devices, mechanical transmission, and artificial joint. Though they share the same number of degree-of-freedom at any given configuration, they will exhibit and transfer different motion characteristics, such as rotation and rolling, when moving continuously. Thanks to the concept of operation mode, these three mechanisms' distinct continuous motion characteristics can be identified and further compared through Euler parameter quaternions, Euler angles, algebraic geometry, and axodes so that the appropriate mechanism for tracking or transmission can be selected. At first, elementary operation modes are numerated based on the number of zero components in a quaternion. In order to acquire all possible operation modes, a set of constraint equations relating to each mechanism are formulated, and an algebraic geometry method is adopted to solve the constraint equations that are much too complicated. For rotation, namely, 1DOF (one-degree-of-freedom) operation mode, its continuous rotation axes are investigated. As to rolling, namely, 2DOF operation mode, allowing for the fact that the difference in 2DOF operation mode of the three mechanisms is not intuitive, axode characteristics of the three mechanisms are investigated and compared. It is found that from the above process of identification and comparison on rotation and rolling, the three mechanisms' distinctive motion characteristics can be effectively obtained.

Model-Based Analysis of Friction-Induced Subsynchronous Whirl for a Rotor Contacting With Clearance Bearings Under Axial Load

For rotors supported by active magnetic bearings (AMBs), clearance bearings are commonly used to provide backup support under loss of AMB functionality. Test data from real machines shows that vibration during touchdown on backup bearings may involve steady forward whirling of the rotor with a subsynchronous frequency. This excitation is believed to be due to friction forces transmitted between the rotor and a bearing end-face under axial load. This paper proposes a new analytical approach to model and predict such friction-driven forward whirl behaviors. A set of constraint equations are derived that relate a circular whirl motion of arbitrary orbital speed to the frequency response functions for the rotor-housing structure. This model is coupled with an evaluation of Coulomb friction associated with slip between the rotor and the supporting end-face of a thrust bearing. The resulting equations can be used to compute a set of possible whirl motions via a root-finding procedure. A case study is undertaken for a 140 kW energy storage flywheel. Model-based predictions are compared with measured data from spin-down tests and show a good level of agreement. The study confirms the role of friction-related forces in driving forward-whirl response behaviors. It also highlights the key role of housing and machine support characteristics in response behavior. This influence is shown to be complex and not open to simple physical interpretation. Therefore, the proposed analytical method is seen as a useful tool to investigate this influence while avoiding the need for time consuming numerical simulations.

Automatic generation of the complete set of planar kinematic chains with up to six independent loops and up to 19 links

The synthesis of all feasible topological (kinematic) structures of kinematic chains still has broad research spaces although various methods have been developed in the past fifty years. In this paper, a systematic method is proposed to automatically synthesize the complete set of planar non-fractionated kinematic chains with up to six independent loops. First, the contracted graphs of kinematic chains are synthesized and classified by their planar and non-planar characteristics. Then, the set of constraint equations is derived from the contracted graphs and used to synthesize topological graphs of kinematic chains. And an improved algorithm is developed to obtain the simplified characteristic codes of topological graphs to increase the efficiency of isomorphism identification. Finally, a systematic synthesis method is proposed and the complete set of planar non-fractionated kinematic chains with up to six independent loops and up to 19 links is achieved for the first time. In addition, a comparative analysis between our synthesis results and the ones in the literature is conducted, illustrating the validity and efficiency of the method.

Reconfiguration analysis of a 4-RUU parallel manipulator

This paper deals with the characterization of the operation modes of the 4-RUU parallel manipulator with an algebraic approach, namely the Study's kinematic mapping of the Euclidean group SE(3). As the 4-RUU parallel manipulator is an over-constrained manipulator, it can be decomposed into two 2-RUU parallel manipulators. The manipulators are described by a set of constraint equations and the primary decomposition is computed. By combining the results of primary decomposition from two 2-RUU parallel manipulators, it reveals that the 4-RUU parallel manipulator has two Schönflies modes (4-dof) and one lower dimension operation mode (2-dof). The singularity conditions are obtained by deriving the determinant of the Jacobian matrix of the constraint equations with respect to the Study parameters in each operation mode. It is shown that there exist singular configurations in which the manipulators may switch from one operation mode to another operation mode. All the singular configurations are mapped onto the joint space, i.e., the actuated joint angles, and are geometrically interpreted. Eventually, the 4-RUU parallel manipulator may switch from the 1st Schönflies mode to the 2nd Schönflies mode, or vice versa, via the 2-dof third mode that contains self-motions.

An efficient formulation based on the Lagrangian method for contact–impact analysis of flexible multi-body system

In this paper, an efficient formulation based on the Lagrangian method is presented to investigate the contact–impact problems of flexible multi-body systems. Generally, the penalty method and the Hertz contact law are the most commonly used methods in engineering applications. However, these methods are highly dependent on various non-physical parameters, which have great effects on the simulation results. Moreover, a tremendous number of degrees of freedom in the contact–impact problems will influence the numerical efficiency significantly. With the consideration of these two problems, a formulation combining the component mode synthesis method and the Lagrangian method is presented to investigate the contact–impact problems in flexible multi-body system numerically. Meanwhile, the finite element meshing laws of the contact bodies will be studied preliminarily. A numerical example with experimental verification will certify the reliability of the presented formulation in contact–impact analysis. Furthermore, a series of numerical investigations explain how great the influence of the finite element meshing has on the simulation results. Finally the limitations of the element size in different regions are summarized to satisfy both the accuracy and efficiency. In this paper, an efficient formulation based on the Lagrangian method is presented to investigate the contact–impact problems of flexible multi-body systems. A transformation from finite element coordinates p to component mode synthesis generalized coordinates y can be performed. When a contact pair is activated, a set of constraint equations should be added to the system

The estimation of a unique solution for steady-state diffuse optical tomography by applying mechanical pressure

Abstract The accuracy of diffuse optical tomography (DOT) highly depends on two important factors: first, the knowledge of the tissue optical heterogeneities for accurate modeling of light propagation, and second, the uniqueness of reconstructed values of optical properties. Previous studies illustrated that the inverse problem associated with steady-state DOT does not have unique solutions. In this study, we propose a simple method that can be applied to improve this challenging problem of steady-state DOT. In this method, we study the propagation of photons through compressed breast phantoms. The applied mechanical pressure can change the values of optical properties and this pressure dependence of optical properties as a set of constraint equations can be used to improve the inverse problem. The applied pressure can help us to restrict the distribution of possible values of depth and radius of defect inside breast phantom reconstructed by inverse problem.

Flexible and Operable Heat Exchanger Networks

This paper presents the application of the proposed model-based methodology in solving integrated process design and control (IPDC ) of heat exchanger networks ( HENs ). Many methods for HENs synthesis have been developed over the past decades, which aim to provide HENs designs that yield a reasonable trade-off between capital and operating costs. However, in most of HENs synthesis activities, the sole consideration in solution derivation is about design cost. Process operational issue especially controllability is frequently not a concern in the process design. As a result, the controllability of a designed HEN may be questionable. Industrial practice has made it clear that process controllability should be considered during process synthesis. The HENs design can be further improved to ensure that the design is more cost efficient and controllable. This can be achieved by developing a new model-based integrated process design and control methodology, which includes cost optimality and controllability aspects at the early HEN design stage. The IPDC for HEN problem is decomposed into four hierarchical sequential stages: (i) target selection, (ii) HEN design analysis, (iii) controllability analysis, and (iv) optimal selection and verification. The set of constraint equations in the IPDC problem for HEN design is decomposed into four sub-problems which correspond to four hierarchical stages. The capability of the proposed methodology in solving IPDC of HEN problem was tested using biomethanol production plant. The results show that the proposed methodology was able to find the best solution which satisfied design, control and economic criteria in easy, efficient and systematic manner.

Min-max criterion to the optimal design of vibration absorber in a system with Coulomb friction and viscous damping

In this paper, an optimal design of damped dynamic vibration absorber using minimax method for a primary system with Coulomb friction and viscous damping is presented. Meanwhile, to reduce sensitivity of the primary system response to the variations of the forcing frequency, a set of constraint equations that include the distribution characteristic of the resonant frequencies of the primary system is established for the optimal design. Effectiveness of the proposed design is illustrated through numerical simulations. The formulations and results in this paper extend previous results given by Den Hartog and Pennestri.