A classical approach to structural control with piezoelectrics

Analytical dynamics of mechanisms—a computer oriented overview

Designers of lightweight flexible space structures are faced with the significant challenge of achieving position and shape control while avoiding structural vibration. A potentially attractive approach to the vibration and shape control problems lies in the use of piezoelectric films as actuators and sensors. Piezoelectric materials deform when exposed to an electric field, or conversely, when deformed they produce an electric field. This, property considered along with the fact that they are lightweight, inexpensive and exhibit a very wide dynamic response bandwidth, make piezoelectric films attractive as actuators and sensors for certain applications. In this paper, some fundamental relationships for beams incorporating piezoelectric film actuators and sensors are examined. The differential equation of motion for a beam with piezoelectric film bonded to both sides is used to develop Laplace domain transfer function models of the system. These transfer functions are exact Laplace domain representations of the system equations of motion. The transfer functions are cast into a closed rational form using Maclaunn series expansions representing a specific number of modes. In this form, the transfer functions lend themselves to classical control analysis. It is shown that the transfer function relating a voltage applied to a full coverage actuating layer, to the voltage induced in a full coverage sensing layer on the opposite beam face, behaves like a classic colocated system with alternating poles and zeros and accordingly the system is easy to stabilize with low order compensation. In contrast to this result, it is shown that in spite of the effective colocation of actuator and sensor in the case of the transfer function from actuating voltage to tip position, the desirable alternating pole/zero pattern is not exhibited due to incompatibility of actuating and sensing signals. This result is verified experimentally.

The static stability of an asymmetric sandwich beam with viscoelastic core on viscoelastic supports at the ends and subjected to an axial pulsating load and a steady, one-dimensional temperature gradient is investigated by computational method. The equations of motion and associated boundary conditions are obtained using the Hamilton's energy principle. Then, these equations of motion and the associated boundary conditions are non- dimensionalised. A set of Hill's equations is obtained from the non-dimensional equations of motion by the application of the general Galerkin method. The static buckling loads are obtained from the Hill's equations. The effects of shear parameter, geometric parameters, core loss factors, and thermal gradient on the non-dimensional static buckling loads zones have been investigated.

This paper presents design and implementation of digital parametric equalizer. The design method starts with analog filter prototypes consisting of peak filters, low-pass shelving filter, and high-pass shelving filter in the form of transfer function in the s-domain. The bilinear Pascal matrix is used for converting the transfer function in s-domain using bilinear s-z transformation to obtain all three types of digital filter as a transfer function in z-domain. By cascading of these digital filters result in obtaining a digital parametric equalizer. The simulation results are shown and hardware implementation on STM32F769 for real-time testing is also demonstrated to ensure the successfully of the proposed design and implementation.

A (mathematical) model of the RF system is essential for the design and analysis of LLRF systems. The model here describes the (dynamical) relationship between the outputs and inputs of the RF system in the form of transfer functions. The transfer functions provide insights into the behavior and limitations of the RF system. For example, we may predict the RF system outputs in the presence of RF drive, beam loading and external distrubances (e.g., electromagnetic noise, thermal drifts or mechanical vibrations) using the transfer functions. Another useful information that the model can provide is the stability limits for the loop phase, loop gain and loop delay, which helps to determine these operational parameters of the RF controller. We will derive the RF system model through its physical principles, which can, therefore, also be used to optimize the operational parameters and to identify the characteristics of the RF system.

The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series that transform the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The resulting math model is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton’s principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition–based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models.

Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation

Compartmental models have been widely applied to biological systems where material is transferred between compartments. The simple assumption of linear transfer functions results in a mathematically tractable system, but can underestimate the variation. Processes with some particular forms of nonlinear transfer functions are analysed, and numerical results suggest that concave transfer functions increase variability, whereas convex transfer functions decrease variability.

Introduction.Ensuring the safety of country food industry in terms of the duration of storage and the quality of products is impossible without sterilizing products in autoclaves. The effectiveness of the sterilization processes depends on the degree of their automation. In the last twenty years, the improvement of automatic and automated control systems was primarily based on the development of technical means for automation without theoretical justification of decision-making. The proposed work is aimed at identifying the links between the parameters and connections of the sterilization process and the choice of structural and parametric features of the control system. Materials and Methods. A qualitative analysis is carried out based on the modern theory of automatic control for an approximative model of the thermal process of steam heating in an autoclave, taking into account the laws of heat transfer and the sufficiency of using a twodimensional model depending upon the structural and functional features of the model, which have regard to the parameters and relationships of the process, namely, the Kalman’s controllability properties of the model in the time domain in the state-space representation (the transition from the transfer function with zeros in the numerator to the normal differential system differential equations is also described). There were also analized the stability properties of the model in the frequency domain by means of transfer functions and structural transformations and the relationship of parameters in the form of inequalities with the subsequent choice of proportional-integral-differential configuration components for a real autoclave using the matrix of expert estimates. Results. It is shown that to make a qualitatively study of the issues of controllability and stability of the approximative model of the thermal process of water heating by steam in an autoclave, depending on the process parameters, it is necessary to represent the model the time domain (in the state-space representation) and in the frequency domain (in the form of transfer functions). The analysis of the controllability of the process is based on three approaches: the first (formalized) approach is based on the representation of the model in the form of a normal system of ordinary differential equations in the Cauchy form with the development of a method of decreasing the order of the higher derivatives of coordinates and introducing additional control signals taking into account the control derivatives; the second (unformalized) is based on the exclusion of management derivatives through structural transformation; the third (direct) approach uses the first-order heat balance and heat conduction equations derived from physical considerations. Under the conditions of Kalman’s controllability, dependencies between the parameters of the process and the degree of its controllability have been obtained.The analysis of the stability of the process is based on studying the poles of the transfer functions in the frequency domain and the characteristic roots of the equations of state in the time domain. On the basis of structural transformations, a closed canister heating loop with water with inertia, depending on the autoclave charging parameters, is isolated. Transient processes in this circuit take an amplifying, aperiodic or integral character, which affects the nature of the transient processes of the control system as a whole. The formalized choice of the components of the proportional-integral-differential regulation law is carried out depending on the frequency of application of the degree of loading and the need for the components of the proportional-integral-differential regulator using the matrix of expert estimates. Conclusions. The results of the research will serve as the material for the development of a real model of the autoclaving process, taking into account the static and dynamic characteristics of measuring, conversion and actuating elements, investigating the influence and compensation of inertia and nonlinearities of real elements, followed by the development of an automated system for controlling the sterilization process in autoclaves. The results of the work can be used to study general and applied problems of optimal control in both food and other industries, for example, in the production of building materials and the production of rubber products.

This paper deals with irrational transfer functions having monotonic nondecreasing step responses. Firstly, some results on the monotonicity of step responses in irrational transfer functions describing fractional- or distributed-order systems are presented. Then, some conditions guaranteeing the existence of monotonic nondecreasing step responses in more general forms of irrational transfer functions are found. Various examples are brought to show the usefulness of the obtained results in time response analysis of fractional/distributed-order systems. The achievements of the paper can be used in the design of control systems having monotonic step responses.

About the Author. Preface . Glossary . 1 Introduction . 1.1 Historical Perspective. 1.2 The Case for Simulation. 1.3 The Changing Role of Simulation. 1.4 The Organization of a Flight Simulator. 1.5 The Concept of Real-time Simulation. 1.6 Pilot Cues. 1.7 Training versus Simulation. 1.8 Examples of Simulation. References. 2 Principles of Modelling . 2.1 Modelling Concepts. 2.2 Newtonian Mechanics. 2.3 Axes Systems. 2.4 Differential Equations. 2.5 Numerical Integration. 2.6 Real-time Computing. 2.7 Data Acquisition. 2.8 Flight Data. 2.9 Interpolation. 2.10 Distributed Systems. 2.11 A Real-time Protocol. 2.12 Problems in Modelling. References. 3 Aircraft Dynamics . 3.1 Principles of Flight Modelling. 3.2 The Atmosphere. 3.3 Forces. 3.4 Moments. 3.5 Axes Systems. 3.6 Quaternions. 3.7 Equations of Motion. 3.8 Propulsion. 3.9 The Landing Gear. 3.10 The Equations Collected. 3.11 The Equations Revisited - Long Range Navigation. References. 4 Simulation of Flight Control Systems . 4.1 The Laplace Transform. 4.2 Simulation of Transfer Functions. 4.3 PID Control Systems. 4.4 Trimming. 4.5 Aircraft Flight Control Systems. 4.6 The Turn Coordinator and the Yaw Damper. 4.7 The Auto-throttle. 4.8 Vertical Speed Management. 4.9 Altitude Hold. 4.10 Heading Hold. 4.11 Localizer Tracking. 4.12 Auto-land Systems. 4.13 Flight Management Systems. References. 5 Aircraft Displays . 5.1 Principles of Display Systems. 5.2 Line Drawing. 5.3 Character Generation. 5.4 D Graphics Operations. 5.5 Textures. 5.6 OpenGL (R) . 5.7 Simulation of Aircraft Instruments. 5.8 Simulation of EFIS Displays. 5.9 Head-up Displays. References. 6 Simulation of Aircraft Navigation Systems . 6.1 Principles of Navigation. 6.2 Navigation Computations. 6.3 Map Projections. 6.4 Primary Flight Information. 6.5 Automatic Direction Finding (ADF). 6.6 VHF Omnidirectional Range (VOR). 6.7 Distance Measuring Equipment (DME). 6.8 Instrument Landing Systems (ILS). 6.9 The Flight Director. 6.10 Inertial Navigation Systems. 6.11 Global Positioning Systems. References. Further Reading. 7 Model Validation . 7.1 Simulator Qualification and Approval. 7.2 Model Validation Methods. 7.3 Latency. 7.4 Performance Analysis. 7.5 Longitudinal Dynamics. 7.6 Lateral Dynamics. 7.7 Model Validation in Perspective. References. 8 Visual Systems . 8.1 Background. 8.2 The Visual System Pipeline. 8.3 3D Graphics Operations. 8.4 Real-time Image Generation. 8.5 Visual Database Management. 8.6 Projection Systems. 8.7 Problems in Visual Systems. References. 9 The Instructor Station . 9.1 Education, Training and Instruction. 9.2 Part-task Training and Computer-based Training. 9.3 The Role of the Instructor. 9.4 Designing the User Interface. 9.5 Real-time Interaction. 9.6 Map Displays. 9.7 Flight Data Recording. 9.8 Scripting. References. 10 Motion Systems . 10.1 Motion or No Motion? 10.2 Physiological Aspects of Motion. 10.3 Actuator Configurations. 10.4 Equations of Motion. 10.5 Implementation of a Motion System. 10.6 Hydraulic Actuation. 10.7 Modelling Hydraulic Actuators. 10.8 Limitations of Motion Systems. 10.9 Future Motion Systems. References. Index.

This paper presents a detailed study on the impact source identification of a plate structure using time-reversal (T-R) method. Prior to impact monitoring, the plate is calibrated (or characterized) by transfer functions at discrete locations on the plate surface. Both impact location and impact loading time-history are identified using T-R technique and associated signal processing algorithms. Numerical verification for finite-size isotropic plates under low velocity impacts is performed to demonstrate the versatility of T-R method for impact source identification. The tradeoff between accuracy of the impact location detection and calibration spacing is studied in detail. In particular, the effect of plate thickness on calibration spacing has been examined. A number of parameters selected for determining the impact location, approximated transfer functions and steps taken for reconstructing the impact loading time-history are also examined. Two types of noise with various intensities contaminated in strain response and/or transfer functions are investigated for demonstrating the stability and reliability of the T-R method. The results show that T-R method is robust against noise in impact location detection and force reconstruction in circumventing the inherent ill-conditioned inverse problem. Only transfer functions are needed to be calibrated and four sensors are requested in T-R method for impact identification.

Identification of the equivalent linear dynamics and controller design for an unmanned underwater vehicle

When digital computer-based parameter estimation techniques are used to identify linear, dynamic systems, the results are usually obtained in the form of a high order Z-domain transfer function. For many practical applications, however, a model transfer function of second or third order is sufficient to represent the prominent features of the system dynamics. A computationally simple transfer function reduction method, based on the Matsuburu time delay theorem, is presented herein. This method is compared with a recently reported Padé approximation approach in both the time and frequency domains.

This paper presents the analytical and experimental results on optimal placement of piezoceramic actuators for shape control of beam structures. The objective is to determine the optimum piezoceramic actuator locations and voltages to minimize the error between the desired shape and the achieved shape. The analytical model for predicting beam deformation due to a piezoelectric actuator is based on the Euler-Bernoulli model. The cost function has fifth-order polynomials in the actuator locations and second-order polynomials in actuator voltages. This difference resulted in difficulty in simultaneous optimization of actuator locations and voltages. Using embedded Nader and Mead simplex algorithms to separately optimize actuator locations and voltages was found to produce reliable results, converging to the same optimum solution for a variety of initial conditions. Experimental results show that the analytical model provides a reasonable prediction of actuator performance at low input voltage, but does not account for the nonlinear behavior of the piezoceramic and effects of hysteresis.