Vibration reduction for vision systems on board unmanned aerial vehicles using a neuro-fuzzy controller

In this paper, an intelligent control approach based on neuro-fuzzy systems performance is presented, with the objective of counteracting the vibrations that affect the low-cost vision platform onboard an unmanned aerial system of rotating nature. A scaled dynamical model of a helicopter is used to simulate vibrations on its fuselage. The impact of these vibrations on the low-cost vision system will be assessed and an intelligent control approach will be derived in order to reduce its detrimental influence. Different trials that consider a neuro-fuzzy approach as a fundamental part of an intelligent semi-active control strategy have been carried out. Satisfactory results have been achieved compared to those obtained by means of vibration reduction passive techniques.

Calibration of accelerometers for the measurement of microvibrations

The accelerometers used for the measurement of microvibrations or microgravity applications, such as active control of space structures, attitude control, scientific payloads, or even on-Earth testing of structures at very low-excitation levels, require a dedicated calibration procedure that includes the gravitational effects. Otherwise, on-Earth calibrations can be inaccurate due to the collateral projection of the local gravity onto the sensitive axis. An on-Earth calibration technique for the 107102s amplitude range and 0-100-Hz frequency range is described. Special attention has been given to the modeling of gravitational effects on the response of the calibration device and the accelerometer itself. The sensitivity and resolution tests performed on piezoelectric accelerometers showthe accuracy andthe potential of thistechnique. Typical scale factorun certainty, which hasbeen carefully analyzed, is of the order of 2% at acceleration levels of 10sg.

A comparative study of SISO and MIMO control strategies for floor vibration damping

Civil engineering structures such as floor systems with open-plan layout or lightweight footbridges are susceptible to excessive level of vibrations caused by human loading. Active vibration control (AVC) via inertial mass actuators has been shown to be a viable technique to mitigate vibrations, allowing structures to satisfy vibration serviceability limits. Most of the AVC applications involve the use of SISO (single input single-output) strategies based on collocated control. However, in the case of floor structures, in which mostof the vibration modes are locally spatially distributed, SISO or multi-SISO strategies are quite inefficient. In this paper, a MIMO (multi-inputs multi-outputs) control in decentralised and centralised configuration is designed. The design process simultaneously finds the placement of multiple actuators and sensors and the output feedback gains. Additionally, actuator dynamics, actuator nonlinearities and frequency and time weightings are considered into the design process. Results with SISO and decentralised and centralised MIMO control (for a given number of actuators and sensors) are compared, showing the advantages of MIMO control for floor vibration control.

Control de vibraciones en pasarelas peatonales

En los últimos años se ha construido un gran número de pasarelas peatonales como respuesta a la demanda de nuevas vías de paso en las ciudades. Estas estructuras tienen requisitos constructivos menos exigentes en comparación con otros tipos de puentes, lo cual ha facilitado el desarrollo de diseños con nuevos esquemas resistentes, complicadas geometrías y el empleo de nuevos materiales. En general estas estructuras son esbeltas, ligeras y poco amortiguadas, lo que en ocasiones ha generado problemas de vi-braciones al paso de peatones una vez puestas en servicio. Las normativas actuales son cada vez más sensibles a esta problemática, recomendando diseños cuyas frecuencias naturales deben estar alejadas de los rangos de frecuencia de paso típicos de los peatones y fijando límites de confort en forma de valores máximos de aceleración permitidos, asegurándose así un correcto comportamiento de la estructura. En el presente artículo se analiza esta problemática desde un punto de vista práctico. Para ello se muestran los puntos clave de las normativas y guías de diseño de pasarelas que se pueden encontrar actualmente en la bibliografía, se presentan las técnicas que habitualmente se emplean en el análisis dinámico experimental de estas estructuras, y se comentan las soluciones a las que generalmente se recurre para mejorar su comportamiento dinámico. Por último, se muestran los trabajos llevados a cabo por el Centro Tecnológico CARTIF en colaboración con las Universidades de Valladolid y Castilla-La Mancha en la pasarela peatonal del Museo de la Ciencia de Valladolid. Estos trabajos incluyen: (1) el estudio dinámico de los tres vanos metálicos de dicha pasarela, (2) el diseño e implementación de un amortiguador de masa sintonizado en el vano más sensible a las vibraciones, (3) la implementación de un amortiguador de masa activo utilizando un excitador electrodinámico, y (4) el desarrollo de pruebas para la verificación del estado de servicio de la pasarela. In the last years, a wide number of footbridges have been built as demand response of more direct pathways in cities. These structures have lower building requirements as compared with standard bridges. This circumstance has facilitated the development of new structural design with complex geometries and innovative materials. As a result, these structures may be slender, light and low damped, leading to vibration problems once in service. The current codes take into account this problem, and recommend designs with natural frequencies away from the typical pedestrian pacing rates and fix comfort limits to guarantee the serviceability of the structure.This paper studies this problem from a practical point of view. Thus, the key points of codes and footbridges guidelines are showed, the typical experimental dynamic analysis techniques are presented, and the usual solutions adopted to improve the dynamic performance of these structures are discussed. Finally, the works carried out on the Valladolid Science Museum Footbridge by Centro Tecnológico CARTIF in collaboration with the Universities of Valladolid and Castilla-La Mancha are showed. These works include: (1) the dynamic study of the three steel spans of the footbridge, (2) the design and implementation of a tuned mass damper in the liveliest span, (3) the implementation of an active mass damper using an electrodynamic shaker, and (4) the development of field tests to assess the serviceability of such span.

Adaptive Control for ZVS Three Phase Full Active Bridge Converter with ARCN

This paper presents an adaptive control for the auxiliary circuit, called ARCN (Auxiliary Resonant Commutating Network), used to achieve ZVS in full active bridge converters under a wide load range. Depending on the load conditions, the proposed control adapts the timing of the ARCN to minimize the losses. The principle of operation and implementation considerations are presented for a three phase full active bridge converter, proposing different methods to implement the control according to the specifications. The experimental results shown verify the proposed methodology.

Soporte de redes activas para un servicio integrado de multipunto fiable con control de congestión

Las limitaciones de las tecnologías de red actuales, identificadas en la Agencia de Proyectos de Investigación Avanzados para la Defensa (DARPA) durante 1995, han originado recientemente una propuesta de modelo de red denominado Redes Activas. En este modelo, los nodos proporcionan un entorno de ejecución sobre el que se ejecuta el código asociado a cada paquete. El objetivo es disponer de una tecnología de red que permita que nuevos servicios de red sean desarrollados e instalados rápidamente sin modificar los nodos de la red. Un servicio de red que se puede beneficiar de esta tecnología es la transmisión de datos en multipunto con diferentes grados fiabilidad. Las propuestas actuales de servicios de multipunto fiable proporcionan una solución específica para cada clase de aplicaciones, y los protocolos existentes extremo a extremo sufren de limitaciones técnicas relacionadas con una fiabilidad limitada, y con la ausencia de mecanismos de control de congestión efectivos. Esta tesis realiza propuestas originales conducentes a solucionar parte de las limitaciones actuales en el ámbito de Redes Activas y multipunto fiable con control de congestión. En primer lugar, se especificará un servicio genérico de multipunto fiable que, basándose en los requisitos de una serie de aplicaciones consideradas relevantes, proporcione diferentes clases de sesiones y grados de fiabilidad. Partiendo de la definición del servicio genérico especificado, se diseñará un protocolo de comunicaciones sobre la tecnología de Redes Activas que proporcione dicho servicio. El protocolo diseñado estará dotado de un mecanismo de control de congestión para que la fuente ajuste dinámicamente el tráfico inyectado a las condiciones de carga de la red. En esta tesis se pretende también profundizar en el estudio y análisis de la tecnología de Redes Activas, experimentando con dicha tecnología para proporcionar una realimentación a sus diseñadores. Dicha experimentación se realizará en tres ámbitos: el de los servicios y protocolos que puede soportar, el del modelo y arquitectura de las Redes Activas y el de las plataformas de ejecución disponibles. Como aportación adicional de este trabajo, se validarán los objetivos anteriores mediante una implementación piloto de las entidades de protocolo y de su interfaz de servicio sobre uno de los entornos de ejecución disponibles. Abstract The limitations of current networking technologies identified in the Defense Advance Research Projects Agency (DARPA) along 1995 have led to a recent proposal of a new network model called Active Networks. In this model, the nodes provide an execution environment over which the code used to process each packet is executed. The objective is a network technology that allows the fast design and deployment of new network services without requiring the modification of the network nodes. One network service that could benefit from this technology is the transmission of multicast data with different types of loss tolerance. The current proposals for reliable multicast services provide specific solutions for each application class, and existing end-to-end protocols suffer from technical drawbacks related to limited reliability and lack of an effective congestion control mechanism. This thesis contains original proposals that aim to solve part of the current drawbacks in the scope of Active Networks and reliable multicast with congestion control. Firstly, a generic reliable multicast network service will be specified. This service will be designed from the requirements of a relevant set of applications, and will provide different session classes and different types of reliability. Then, a network protocol based on Active Network technology will be designed such that it provides the specified network service. This protocol will incorporate a congestion control mechanism capable of performing an automatic adjustment of the traffic injected by the source to the available network capacity. This thesis will also contribute to a deeper study and analysis of Active Network technology, by experimenting with the technology in order to provide feedback to its designers. This experimentation will be done attending to three different scopes: support of Active Network for services and protocols, Active Network model and architecture, and currently available Active Network execution environments. As an additional contribution of this work, the previous objectives will be validated through a prototype implementation of the protocol entities and the service interface based on one of the current execution environments.

Combining series elastic actuation and magneto-rheological damping for the control of agile locomotion

All-terrain robot locomotion is an active topic of research. Search and rescue maneuvers and exploratory missions could benefit from robots with the abilities of real animals. However, technological barriers exist to ultimately achieving the actuation system, which is able to meet the exigent requirements of these robots. This paper describes the locomotioncontrol of a leg prototype, designed and developed to make a quadruped walk dynamically while exhibiting compliant interaction with the environment. The actuation system of the leg is based on the hybrid use of series elasticity and magneto-rheological dampers, which provide variable compliance for natural-looking motion and improved interaction with the ground. The locomotioncontrol architecture has been proposed to exploit natural leg dynamics in order to improve energy efficiency. Results show that the controller achieves a significant reduction in energy consumption during the leg swing phase thanks to the exploitation of inherent leg dynamics. Added to this, experiments with the real leg prototype show that the combined use of series elasticity and magneto-rheologicaldamping at the knee provide a 20 % reduction in the energy wasted in braking the knee during its extension in the leg stance phase.

Neural network controller for active demand side management with PV energy in the residential sector

In this paper, we describe the development of a control system for Demand-Side Management in the residential sector with Distributed Generation. The electrical system under study incorporates local PV energy generation, an electricity storage system, connection to the grid and a home automation system. The distributed control system is composed of two modules: a scheduler and a coordinator, both implemented with neural networks. The control system enhances the local energy performance, scheduling the tasks demanded by the user and maximizing the use of local generation.

Transductive neurofuzzy-based torque control of a milling process: results of a case study

This paper presents the design and implementation of an intelligent control system based on local neurofuzzy models of the milling process relayed through an Ehternet-based application. Its purpose is to control the spindle torque of a milling process by using an internal model control paradigm to modify the feed rate in real time. The stabilization of cutting cutting torque is especially necessary in milling processes such as high-spedd roughing of steel moulds and dies tha present minor geometric uncertainties. Thus, maintenance of the curring torque increaes the material removal rate and reduces the risk of damage due to excessive spindle vibration, a very sensitive and expensive component in all high-speed milling machines. Torque control is therefore an interesting challenge from an industrial point of view.

Control motion approach of a lower limb orthosis to reduce energy consumption

By analysing the dynamic principles of the human gait, an economic gait‐control analysis is performed, and passive elements are included to increase the energy efficiency in the motion control of active orthoses. Traditional orthoses use position patterns from the clinical gait analyses (CGAs) of healthy people, which are then de‐normalized and adjusted to each user. These orthoses maintain a very rigid gait, and their energy cosT is very high, reducing the autonomy of the user. First, to take advantage of the inherent dynamics of the legs, a state machine pattern with different gains in eachstate is applied to reduce the actuator energy consumption. Next, different passive elements, such as springs and brakes in the joints, are analysed to further reduce energy consumption. After an off‐line parameter optimization and a heuristic improvement with genetic algorithms, a reduction in energy consumption of 16.8% is obtained by applying a state machine control pattern, and a reduction of 18.9% is obtained by using passive elements. Finally, by combining both strategies, a more natural gait is obtained, and energy consumption is reduced by 24.6%compared with a pure CGA pattern.

Relative Dynamics and Control of an Ion Beam Shepherd Satellite

The ion beam shepherd (IBS) is a recently proposed concept for modifying the orbit and/or attitude of a generic orbiting body in a contactless manner, which makes it a candidate technology for active space debris removal. In this paper we deal with the problem of controlling the relative position of a shepherd satellite coorbiting at small separation distance with a target debris. After deriving the orbit relative motion equations including the effect of the ion beam perturbation we study the system stability and propose different control strategies.

Dual Active Bridge based Battery Charger for Plug-in Hybrid Electric Vehicle with Charging Current Containing Low Frequency Ripple

High power density is strongly preferable for the on-board battery charger of Plug-in Hybrid Electric Vehicle (PHEV). Wide band gap devices, such as Gallium Nitride HEMTs are being explored to push to higher switching frequency and reduce passive component size. In this case, the bulk DC link capacitor of AC-DC Power Factor Correction (PFC) stage, which is usually necessary to store ripple power of two times the line frequency in a DC current charging system, becomes a major barrier on power density. If low frequency ripple is allowed in the battery, the DC link capacitance can be significantly reduced. This paper focuses on the operation of a battery charging system, which is comprised of one Full Bridge (FB) AC-DC stage and one Dual Active Bridge (DAB) DC-DC stage, with charging current containing low frequency ripple at two times line frequency, designated as sinusoidal charging. DAB operation under sinusoidal charging is investigated. Two types of control schemes are proposed and implemented in an experimental prototype. It is proved that closed loop current control is the better. Full system test including both FB AC-DC stage and DAB DC-DC stage verified the concept of sinusoidal charging, which may lead to potentially very high power density battery charger for PHEV.

Contribution to active multi-beam reconfigurable antennas for L and S bands

La presente tesis doctoral con título "Contribution to Active Multi-Beam Reconfigurable Antennas for L and S Bands" ha sido desarrollada por el investigador ingeniero de telecomunicación estudiante de doctorado Javier García-Gasco Trujillo en el Grupo de Radiación del Departamento de Señales, Sistemas y Radiocomunicaciones de la ETSI de Telecomunicación de la Universidad Politécnica de Madrid bajo la dirección de los doctores Manuel Sierra Pérez y José Manuel Fernández González. Durante décadas, el desarrollo de antenas de apuntamiento electrónico ha estado limitado al área militar. Su alto coste y su gran complejidad eran los mayores obstáculos que frenaban la introducción de esta tecnología en aplicaciones comerciales de gran escala. La reciente aparición de componentes de estado sólido prácticos, fiables, y de bajo coste ha roto la barrera del coste y ha reducido la complejidad, haciendo que las antenas reconfigurables de apuntamiento electrónico sean una opción viable en un futuro cercano. De esta manera, las antenas phased array podrían llegar a ser la joya de la corona que permitan alcanzar los futuros retos presentes en los sistemas de comunicaciones tanto civiles como militares. Así pues, ahora es el momento de investigar en el desarrollo de antenas de apuntamiento electrónico de bajo coste, donde los nuevos componentes de estado sólido comerciales forman el núcleo duro de la arquitectura. De esta forma, el estudio e implementación de estos arrays de antenas activas de apuntamiento electrónico capaces de controlar la fase y amplitud de las distintas señales implicadas es uno de los grandes retos de nuestro tiempo. Esta tesis se enfrenta a este desafío, proponiendo novedosas redes de apuntamiento electrónico e innovadores módulos de transmisión/recepción (T/R) utilizando componentes de estado sólido de bajo coste, que podrán integrar asequibles antenas activas reconfigurables multihaz en bandas L y S. En la primera parte de la tesis se realiza una descripción del estado del arte de las antenas phased array, incluyendo su base teórica y sus ventajas competitivas. Debido a que las contribuciones obtenidas en la presente tesis han sido realizadas dentro de distintos proyectos de investigación, donde se han manejada antenas de simple/doble polarización circular y simple/doble banda de trabajo, se describen detenidamente los dos proyectos más relevantes de la investigación: el radar de basura espacial de la Agencia Espacial Europea (ESA), Space Situational Awareness (SSA); y la estación base de seguimiento y control de satélites de órbita baja, GEOdesic Dome Array (GEODA). Sin lugar a dudas, los dispositivos desfasadores son uno de los componentes clave en el diseño de antenas phased arrays. Recientemente se ha observado una gran variación en el precio final de estos dispositivos, llegando en ocasiones a límites inasequibles. Así pues, se han propuesto distintas técnicas de conformación de haz alternativas a la utilización de componentes desfasadores comerciales: el desfasador de líneas conmutadas, la red de haz conmutado, y una novedosa red desfasadora divisora/combinadora de potencia. Para mostrar un uso práctico de las mismas, se ha propuesto el uso de las tres alternativas para el caso práctico del subarray de cinco elementos de la celda GEODA-SARAS. Tras dicho estudio se obtiene que la novedosa red desfasadora divisora/combinadora de potencia propuesta es la que mejor relación comportamiento/coste presenta. Para verificar su correcto funcionamiento se construye y mide los dos bloques principales de los que está compuesta la red total, comprobando que en efecto la red responde según lo esperado. La estructura más simple que permite realizar un barrido plano es el array triangular de tres elementos. Se ha realizado el diseño de una nueva red multihaz que es capaz de proporcionar tres haces ortogonales en un ángulo de elevación _0 y un haz adicional en la dirección broadside utilizando el mencionado array triangular de tres elementos como antena. En primer lugar se realizar una breve introducción al estado del arte de las redes clásicas multihaz. Así mismo se comentan innovadores diseños de redes multihaz sin pérdidas. El estudio da paso a las redes disipativas, de tal forma que se analiza su base matemática y se muestran distintas aplicaciones en arrays triangulares de tres elementos. Finalmente, la novedosa red básica propuesta se presenta, mostrando simulaciones y medidas de la misma para el caso prácticoo de GEODA. También se ha diseñado, construido y medido una red compuesta por dos redes básicas complementarias capaz de proporcionar seis haces cuasi-ortogonales en una dirección _0 con dos haces superpuestos en broadside. La red propuesta queda totalmente validada con la fabricación y medida de estos con prototipos. Las cadenas de RF de los módulos T/R de la nueva antena GEODA-SARAS no son algo trivial. Con el fin de mostrar el desarrollo de una cadena compleja con una gran densidad de componentes de estado sólido, se presenta una descripción detallada de los distintos componentes que integran las cadenas de RF tanto en transmisión como en recepción de la nueva antena GEODA-SARAS. Tras presentar las especificaciones de la antena GEODA-SARA y su diagrama de bloques esquemático se describen los dos bloques principales de las cadenas de RF: la celda de cinco elementos, y el módulo de conversión de panel. De la misma manera también se presentará el módulo de calibración integrado dentro de los dos bloques principales. Para comprobar que el funcionamiento esperado de la placa es el adecuado, se realizará un análisis que tratará entre otros datos: la potencia máxima en la entrada del transmisor (comprobando la saturación de la cadena), señal de recepción mínima y máxima (verificando el rango de sensibilidad requerido), y el factor G/T (cumpliendo la especificación necesaria). Así mismo se mostrará un breve estudio del efecto de la cuantificación de la fase en el conformado de haz de RF. Los estudios muestran que la composición de las cadenas de RF permite el cumplimiento de las especificaciones necesarias. Finalmente la tesis muestra las conclusiones globales del trabajo realizado y las líneas futuras a seguir para continuar con esta línea de investigación. ABSTRACT This PhD thesis named "Contribution to Active Multi-Beam Reconfigurable Antennas for L and S Bands", has been written by the Electrical Engineer MSc. researcher Javier García-Gasco Trujillo in the Grupo de Radiación of the Departamento de Señales, Sistemas y Radiocomunicaciones from the ETSI de Telecomunicación of the Universidad Politécnica de Madrid. For decades, the implementation of electronically steerable phased array antennas was confined to the military area. Their high cost and complexity were the major obstacles to introduce this technology in large scale commercial applications. The recent emergence of new practical, low-cost, and highly reliable solid state devices; breaks the barrier of cost and reduces the complexity, making active phased arrays a viable future option. Thus, phased array antennas could be the crown jewel that allow to meet the future challenges in military and civilian communication systems. Now is time to deploy low-cost phased array antennas, where newly commercial components form the core of the architecture. Therefore, the study and implementation of these novel low-cost and highly efficient solid state phased array blocks capable of controlling signal phase/amplitude accurately is one of the great challenges of our time. This thesis faces this challenge, proposing innovative electronic beam steering networks and transmitter/ receiver (T/R) modules using affordable solid state components, which could integrate fair reconfigurable phased array antennas working in L and S bands. In the first part of the thesis, a description of the state of art of phased array antennas, including their fundamentals and their competitive advantages, is presented. Since thesis contributions have been carried out for different research projects, where antennas with single/double circular polarization and single/double working frequency bands have been examined, frameworks of the two more important projects are detailed: the Space Situational Awareness (SSA) programme from the European Space Agency (ESA), and the GEOdesic Dome Array (GEODA) project from ISDEFE-INSA and the ESA. Undoubtedly, phase shifter devices are one of the key components of phased array antennas. Recent years have witnessed wide fluctuations in commercial phase shifter prices, which sometimes led to unaffordable limit. Several RF steering technique alternatives to the commercial phase shifters are proposed, summarized, and compared: the switched line phase shifter, the switched-beam network, and the novel phase shifter power splitter/combiner network. In order to show a practical use of the three different techniques, the five element GEODA-SARAS subarray is proposed as a real case of study. Finally, a practical study of a newly phase shifter power splitter/combiner network for a subarray of five radiating elements with triangular distribution is shown. Measurements of the two different phase shifter power splitter/combiner prototypes integrating the whole network are also depicted, demonstrating their proper performance. A triangular cell of three radiating elements is the simplest way to obtain a planar scanner. A new multibeam network configuration that provides three orthogonal beams in a desired _0 elevation angle and an extra one in the broadside steering direction for a triangular array of three radiating elements is introduced. Firstly, a short introduction to the state of art of classical multi-beam networks is presented. Lossless network analysis, including original lossless network designs, are also commented. General dissipative network theory as well as applications for array antennas of three radiating elements are depicted. The proposed final basic multi-beam network are simulated, built and measured to the GEODA cell practical case. A combined network that provides six orthogonal beams in a desired _0 elevation angle and a double seventh one in the broadside direction by using two complementary proposed basic networks will be shown. Measurements of the whole system will be also depicted, verifying the expected behavior. GEODA-SARAS T/R module RF chains are not a trivial design. A thorough description of all the components compounding GEODA-SARAS T/R module RF chains is presented. After presenting the general specifications of the GEODA-SARAS antenna and its block diagrams; two main blocks of the RF chains, the five element cell and the panel conversion module, are depicted and analyzed. Calibration module integrated within the two main blocks are also depicted. Signal flow throw the system analyzing critical situations such as maximum transmitted power (testing the chain unsaturation), minimum and maximum receiving signal (verifying sensitivity range), maximum receiver interference signals (assuring a proper reception), and G/T factor (fulfilling the technical specification) are evaluated. Phase quantization error effects are also listed. Finally, the manuscript contains the conclusions drawn of the present research and the future work.

Health monitoring of web plastifying dampers subjected to cyclic loading through vibration tests

This article investigates experimentally the application of health monitoring techniques to assess the damage on a particular kind of hysteretic (metallic) damper called web plastifying dampers, which are subjected to cyclic loading. In general terms, hysteretic dampers are increasingly used as passive control systems in advanced earthquake-resistant structures. Nonparametric statistical processing of the signals obtained from simple vibration tests of the web plastifying damper is used here to propose an area index damage. This area index damage is compared with an alternative energy-based index of damage proposed in past research that is based on the decomposition of the load?displacement curve experienced by the damper. Index of damage has been proven to accurately predict the level of damage and the proximity to failure of web plastifying damper, but obtaining the load?displacement curve for its direct calculation requires the use of costly instrumentation. For this reason, the aim of this study is to estimate index of damage indirectly from simple vibration tests, calling for much simpler and cheaper instrumentation, through an auxiliary index called area index damage. Web plastifying damper is a particular type of hysteretic damper that uses the out-of-plane plastic deformation of the web of I-section steel segments as a source of energy dissipation. Four I-section steel segments with similar geometry were subjected to the same pattern of cyclic loading, and the damage was evaluated with the index of damage and area index damage indexes at several stages of the loading process. A good correlation was found between area index damage and index of damage. Based on this correlation, simple formulae are proposed to estimate index of damage from the area index damage.