J.T. Xing

A power–flow analysis based on continuum dynamics

Based on the governing equations of continuum mechanics, a power–flow analysis is presented. In developing the mathematical model, the concept of energy–flow density vector is introduced, which uniquely defines the energy transmission between one part of a material body/system and another. This approach allows the energy–flow line, the energy–flow potential and the equipotential surface to be defined. From this model, the local equation of energy–flow balance, the equation of energy exchange between two or many subsystems and the time–average equations are derived to describe the characteristics of energy flow and energy exchange within the continuum. To demonstrate the applicability of the proposed mathematical model, the energy–flow relation between two simple oscillators is discussed and the concept generalized to sequential and non–sequential multiple systems. Such multiple systems are examined and for non–sequential systems, which are analogous to statically indeterminate structural systems, an approach is developed for the solution of their power flow and energy exchange. It is further shown that the governing equation of energy flow is a first–order partial differential equation which does not directly correspond to the equation describing the flow of thermal energy in a heat–conduction problem.

A general linear mathematical model of power flow analysis and control for integrated structure-control systems

Abstract Generalized integrated structure–control dynamical systems consisting of any number of active/passive controllers and three-dimensional rigid/flexible substructures are investigated. The developed mathematical model assessing the behaviour of these complex systems includes description of general boundary conditions, the interaction mechanisms between structures, power flows and control characteristics. Three active control strategies are examined. That is, multiple channel absolute/relative velocity feedback controllers, their hybrid combination and an existing passive control system to which the former control systems are attached in order to improve overall control efficiency. From the viewpoint of continuum mechanics, an analytical solution of this generalized structure–control system has been developed allowing predictions of the dynamic responses at any point on or in substructures of the coupled system. Absolute or relative dynamic response or receptance, transmissibility, mobility, transfer functions have been derived to evaluate complex dynamic interaction mechanisms through various transmission paths. The instantaneous and time-averaged power flow of energy input, transmission and dissipation or absorption within and between the source substructure, control subsystems and controlled substructure are presented. The general theory developed provides an integrated framework to solve various vibration isolation and control problems and provides a basis to develop a general algorithm that may allow the user to build arbitrarily complex linear control models using simple commands and inputs. The proposed approach is applied to a practical example to illustrate and validate the mathematical model as well as to assess control effectiveness and to provide important guidelines to assist vibration control designers.

Mixed finite element substructure—subdomain methods for the dynamical analysis of coupled fluid-solid interaction problems

Based on a variational method adopting the dynamic pressure in the fluid and the acceleration in the solid as arguments in the functional, several substructure-subdomain methods are developed to describe the dynamical behaviour of fluid-structure systems excited externally. Formulations are presented of a displacement consistency model and a hybrid displacement model for the solid structure, a pressure equilibrium model for the fluid and a mixed substructure-subdomain model for the fluid-structure interacting system. These substructure—subdomain methods make the mixed finite-element approaches developed to analyse fluid-solid interactions more effective and efficient especially in calculating solutions to large complex engineering problems. This is achieved through the suitable selection of mode vectors to reduce the number of degrees of freedom accepted in the finite-element method to a manageable size without reducing significantly the accuracy of solution; by synthesis of the equations modelling the dynamic interactions and by developing techniques to eliminate mathematical difficulties occurring in the matrix formulations. By these means, consistent and unifying theoretical models are developed to describe the dynamical behaviour of the solid, fluid and their interactions which are in forms adaptable for solution on a personal computer. This is demonstrated by analysing a wide selection of fluid-structure (e.g. dam-water system excited by earthquake or explosion) and air-structure (e.g. structural-borne noise in a fuselage) interacting systems using purposely written computer software.

A mixed finite element method for the dynamic analysis of coupled fluid-solid interaction problems

On the basis of a variational principle, a mixed finite element approach is developed to describe the linear dynamics of coupled fluid–structure interactions. The variables of acceleration in the elastic solid and pressure in the fluid are adopted as the arguments of the variational principle. These are chosen since they directly relate to many practical fluid–structure interaction dynamic problems involving free surface disturbances, e. g. a dam-water system, a fuel cell in an aircraft, etc. Matrix equations describing the motions are presented and four methods of solution discussed, each simplifying and approximating the matrix equations for easier application to solve various types of engineering problems. This is demonstrated by analysing a selection of fluid–structure interaction problems of practical interest. The examples illustrate the general principle and application of the described functional approach without need to resort to more complex dynamic problems which can be analysed in a similar manner.

A power flow mode theory based on a system's damping distribution and power flow design approaches

A power flow mode theory is developed to describe the natural power flow behaviour of a dynamic system based on its inherent damping distribution. The systems characteristic-damping matrix is constructed and it is shown that the eigenvalues and eigenvectors of this matrix identify natural power flow characteristics. These eigenvectors, or power flow mode vectors, are chosen as a set of base-vectors spanning the power flow space and completely describe the power flow in the system. The generalized coordinate of the velocity vector decomposed in this space defines the power flow response vector. A time-averaged power flow expression and theorems relating to its estimation are presented. Based on this theory, power flow design approaches are proposed to identify energy flow patterns satisfying vibration control requirements. The mode control factor defines the measure of the correlation between a power flow mode and a natural vibration mode of the system. Power flow design theorems are presented providing guidelines to construct damping distributions maximizing power dissipation or to suppress/retain a particular vibration mode and/or a motion. The developed damping-based power flow mode theory is compared with a mobility-based power flow model. It is shown that the proposed power flow model provides insight into the power flow dissipation mechanisms in dynamic systems. Examples are presented to demonstrate the applicability of the power flow mode theory and the power flow design approach. These examples demonstrate the generality of the theory, including non-symmetric damping matrices, and illustrate power flow design applications through modifications of the systems damping distribution using passive and/or active control components.

Hybrid active and passive control of vibratory power flow in flexible isolation system

A hybrid active and passive vibration control strategy is developed to reduce the total power flows from machines, subject to multiple excitations, to supporting flexible structures. The dynamic interactions between machines, controllers, and receiving structures are studied. A force feedback control process governed by a proportional control law is adopted to produce active control forces to cancel the transmitted forces in the mounts. Computational simulations of a simple and a multiple dimensional hybrid vibration isolation system are performed to study the force transmissibility and the total power flows from vibration sources through active and passive isolators to the supporting structures. The investigation focuses on the effects of a hybrid control approach to the reduction of power flow transmissions and the influence of the dynamic characteristics of the control on power flow spectra. The hybrid control mechanism is synthesised from the power flow analysis. Conclusions and control strategies, well supported by numerical simulations, are deduced providing very useful guidelines for hybrid vibration isolation design.

A study of power flow in a coupled plate–cylindrical shell system

A substructure approach is formulated to investigate the power flow characteristics of a plate–cylindrical shell system subject to both conservative and dissipative coupling conditions. The system is divided into a shell substructure and a plate substructure. The theoretical receptance function of each substructure with a free–free interface condition is formulated by modal analysis to describe the dynamical behaviour of each substructure. The displacement components induced by external forces and the interface coupling forces are deduced, permitting determination of the coupling forces and power flow through the interface between the two substructures. On the basis of the dynamic information of the two substructures and through a synthesis analysis using the geometrical compatibility and force balance conditions on the coupling interfaces, the dynamic characteristics of power flow excited and transmitted within the system are calculated. A power flow density vector and the corresponding energy flow line are defined for this coupled system. The numerical example demonstrates the applicability of the proposed method and illustrates the power flow characteristics associated with the complex coupled plate–cylindrical shell system.

A mixed finite-element finite-difference method for nonlinear fluid-structure interaction dynamics. I. Fluid-rigid structure interaction

A mixed finite–element finite–difference numerical method is developed to calculate nonlinear fluid–solid interaction problems. In this study, the structure is assumed to be rigid with large motion and the fluid flow is governed by nonlinear, viscous or non–viscous, field equations with nonlinear boundary conditions applied to the free surface and fluid–solid interaction interfaces. A moving coordinate system fixed at a point in the structure is used to describe the fluid flow, and for numerical analysis purposes, an arbitrary Lagrangian–Eulerian mesh system is constructed relative to this moving system. This provides a convenient method of overcoming the difficulties of matching fluid meshes with large solid motion. Nonlinear numerical equations describing nonlinear fluid–solid interaction dynamics are derived through a numerical discretization scheme of study. A coupling iteration process is used to solve these numerical equations. A selection of numerical examples illustrates the developed mathematical model and through numerical simulations it is shown that the proposed approach is practical and useful.

Variational principles of nonlinear dynamical fluid–solid interaction systems

Based on the fundamental equations of continuum mechanics, the concept of Hamiltons principle and the adoption of Eulerian and Lagrangian descriptions of fluid and solid, respectively,variational principles admitting variable boundary conditions are developed to model mathematically the nonlinear dynamical behaviour of the responses and interactions between fluid and solid. The nonlinearity of the fluid is introduced through nonlinear field equations and nonlinear boundary conditions on the free surface and fluid–solid interaction interface. The structure is treated as a nonlinear elastic body. This model assumes the fluid inviscid, incompressible or compressible and the fluid motion irrotational or rotational but isentropic along the flow path of each fluid particle. The stationary conditions of the variational principles include the governing equations of nonlinear elastic dynamics, fluid dynamics and those relating to the fluid–structure interaction interface as well as the imposed boundary conditions. A family of variational principles are obtained depending on the assumptions introduced into the mathematical model (i.e. fluid incompressible, motion irrotational, etc.) and these provide a foundation to construct numerical schemes of study to assess the dynamical behaviour of nonlinear fluid–solid interaction systems. Two simple illustrative examples are presented demonstrating the applicability of the proposed theoretical approach.

Natural vibration of a flexible beam-water coupled system with a concentrated mass attached at the free end of the beam

The dynamical behaviour of a flexible beam-water interaction system having a concentrated mass (with accompanying moment of inertia) at the free end of the beam is examined. In the water domain, the coupled system is subject to an undisturbed boundary condition at infinity and a zero surface wave or linear surface disturbance condition on the free surface. The governing equations describing the behaviour of the system are analysed using the separation of variables method and their solutions presented. The eigenvalue equation of the natural vibration of the beam-water system is derived and exact solutions for each combination of boundary conditions are obtained. Calculations show that, for the undisturbed condition at infinity in the water domain, the natural frequencies of the coupled beam-concentrated mass dynamical system are lower than those of the beam alone. With constant ratio of concentrated mass to mass of the beam, as the beam changes from thick to thin, the concentrated mass and moment of inertia become less influential on the natural vibration behaviour of the coupled system.