Showing papers in "Smart Materials and Structures in 1993"
TL;DR: The linearity between the volume resistivity change and the compressive stress was good for mortar containing carbon fibers together with either methylcellulose or latex as dispersants as discussed by the authors.
Abstract: Electrically conducting concrete, as provided by the addition of a short carbon fibers (0.2-0.4 vol.%) to concrete, can function as smart structure material that allows non-destructive electrical probing for the monitoring of flaws. The electrical signal is related to an increase in the concrete's volume resistivity during crack generation or propagation and a decrease in the resistivity during crack closure. The linearity between the volume resistivity change and the compressive stress was good for mortar containing carbon fibers together with either methylcellulose or latex as dispersants. However, the linearity was poor for mortar containing carbon fibers together with both methylcellulose and silica fume, as this mortar required a minimum compressive stress for crack closure, whereas the other two mortars did not.
349 citations
TL;DR: In this paper, the bending beam method was used to detect and make use of volume changes in conjugated polymers such as polyaniline (PANi) in response to some external stimuli.
Abstract: Electrically conducting conjugated polymers such as polyaniline (PANi) usually show gel properties with volume changes in the range 1-100% in response to some external stimuli. The bending beam method provides an effective and sensitive way to detect and make use of these volume changes. Bipolymer laminate strips (2.5*0.2 cm2), made of a PANi layer (10 mu m thick) and a substrate polyimide layer (45 mu m thick) bound together, were prepared. These strips bend, corresponding to volume changes in the PANi layer, during electrochemical redox of the PANi in aqueous solutions. The extension (
117 citations
TL;DR: In this paper, a finite-element model is developed for the active buckling control of laminated composite plates using piezoelectric materials, which can be surface bonded or embedded and can be either continuous or segmented.
Abstract: A finite-element model is developed for the active buckling control of laminated composite plates using piezoelectric materials. The finite-element model is based on the first-order shear deformation plate theory in conjunction with linear piezoelectric theory. The piezoelectric sensors and actuators can be surface bonded or embedded and can be either continuous or segmented. The dynamic buckling behavior of the laminated plate subjected to a linearly increasing uniaxial compressive load is investigated. The sensor output is used to determine the input to the actuator using a proportional control algorithm. The forces induced by the piezoelectric actuators under the applied voltage fields enhance the critical buckling load. Finite-element solutions are presented for composite plates with clamped and simply supported boundary conditions and the effectiveness of piezoelectric materials in enhancing the buckling loads is demonstrated.
99 citations
TL;DR: In this paper, an analytical model is developed for structural control optimization in which both noncollocated sensor/actuator placement and feedback control gain are considered as independent design variables, and two different control algorithms are applied to derive the control input, and the optimal set of solutions is determined by the modified method of feasible direction.
Abstract: An analytical model is developed for structural control optimization in which both noncollocated sensor/actuator placement and feedback control gain are considered as independent design variables. Two different control algorithms-active damping control and optimal output feedback control-are applied to derive the control input, and the optimal set of solutions is determined by the modified method of feasible direction. Analytical results show that noncollocated sensor/actuator location can lead to instability of the closed-loop system; however, the optimal sensor/actuator location, together with the optimal feedback control gain, can stabilize the structural control systems. In addition, the performance of both active damping control and LQ control can be maximized.
65 citations
TL;DR: In this article, short carbon fibers were found to exhibit volume electrical resistivity of 103-105Omega cm and contact resistivity between the cured concrete and stainless steel at zero contact pressure.
Abstract: Concrete containing 0.2-0.4 vol.% short carbon fibers was found to exhibit volume electrical resistivity of 103-105Omega cm and contact resistivity (between the cured concrete and stainless steel) of 103-106Omega cm at zero contact pressure. Increasing the contact pressure from 0 up to 0.05 MPa was sufficient to lower the contact resistivity to a minimum value. Increasing the fiber content to >0.4 vol.% did not decrease the contact resistivity, but decreased the volume resistivity. The values of the volume and contact resistivities depended on the non-fiber additives (i.e., latex, methylcellulose and silica fume) needed for fiber dispersion. Using latex gave a higher volume resistivity (1 × 105Omega cm) and a lower contact resistivity (5 × 103Omega cm2) than methylcellulose and silica fume; the high volume resistivity was due to the large proportion of latex used; the low contact resistivity was due to the lack of adherent on the surface of fibers protruding from the concrete containing latex.
58 citations
TL;DR: In this paper, the authors used nonlinear constitutive relations describing the fundamental material behavior, along with supporting to test results, to establish a foundation for an engineering description of electrostrictive materials for the design of integrated actuator and structure systems.
Abstract: Lead magnesium niobate (Pb(Mg1/3Nb2/3)O3 or PMN) exhibits many attractive properties required by actuators for precision submicron control. At room temperature hysteresis is negligible, thermal expansion is less than 1 microstrain degrees C-1, and the longitudinal strain sensitivity is 375 microstrain at 600 V mm-1. There has been recent interest in using PMN actuators in applications near 0 degrees C, which is near the Curie temperature of the material. The purpose of this paper is to use the nonlinear constitutive relations describing the fundamental material behavior, along with supporting to test results, to establish a foundation for an engineering description of electrostrictive materials for the design of integrated actuator and structure systems. This paper presents an experimental characterization of PMN strain response, hysteresis and dielectric permittivity for temperatures between -50 degrees C and 100 degrees C. Measurements were made of the transverse strain of a thin PMN plate at bias fields of 0 and 400 V mm-1 and for frequencies between 1 and 1000 Hz. Field dependent effects are discussed in terms of the nonlinear electrostrictive equations derived from a parametric elastic Gibbs free energy function. The temperature dependence of the permittivity and induced piezoelectric coefficient are modelled by a modified Curie-Weiss law.
44 citations
TL;DR: In this article, an analytical model is developed for structure vibration suppression with collocated piezoelectric sensor/actuator in optimal placement and optimal feedback gain, and two different control algorithms-active damping control and output feedback control-are applied to obtain the control law.
Abstract: An analytical model is developed for structure vibration suppression with collocated piezoelectric sensor/actuator in optimal placement and optimal feedback gain. Two different control algorithms-active damping control and output feedback control-are applied to obtain the control law, and the optimal set of sensor/actuator location and feedback gain is determined by the modified method of feasible direction. Analytical results show that a collocated piezoelectric actuator/sensor can lead to instability of the closed loop system. In active damping control, the optimal sensor/actuator location is not at the root of a cantilever beam as commonly believed. However, for LQ formulation of output feedback control, the optimal location is at the beam root with the feedback gain determined by the Ricatti equation. The analytical predictions are verified by experiments.
38 citations
TL;DR: It is shown that a network trained on data from finite-element simulation of the structure can successfully locate faults in the framework itself.
Abstract: A neural network is trained to report the position of a fault in a framework structure. It is shown that a network trained on data from finite-element simulation of the structure can successfully locate faults in the framework itself.
37 citations
TL;DR: In this paper, the development of an active aeroservoelastic missile fin using directionally attached piezoelectric (DAP) actuator elements is detailed, and a torque plate was constructed from 0.2032 mm thick DAP elements bonded to a 0.127 mm thick AISI 1010 steel substrate.
Abstract: The development of an active aeroservoelastic missile fin using directionally attached piezoelectric (DAP) actuator elements is detailed. Several different types of actuator elements are examined, including piezoelectric polymers, piezoelectric fiber composites, and conventionally attached piezoelectric (CAP) and DAP elements. These actuator elements are bonded to the substrate of a torque plate. The root of the torque plate is attached to a fuselage hard point or folding pivot. The tip of the plate is bonded to an aerodynamic shell which undergoes a pitch change as the plate twists. The design procedures used on the plate are discussed. A comparison of the various actuator element shows that DAP elements provide the highest deflections with the highest torsional stiffness. A torque plate was constructed from 0.2032 mm thick DAP elements bonded to a 0.127 mm thick AISI 1010 steel substrate. The torque plate produced static twist deflections in excess of +/- 3 deg. An aerodynamic shell with a modified NACA 0012 profile was added to the torque plate. This fin was tested in a wind tunnel at speeds up to 50 ms/sec. The static deflection of the fin was predicted to within 6 percent of the experimental data.
33 citations
TL;DR: The use of fiber optic sensors in the civil structures arena is increasing, particularly in the instrumentation of high-performance structures as mentioned in this paper, where the authors feel that multi-media sensors (i.e., fiber optic sensor configurations in which multiple parameters of interest can be measured simultaneously with a single fiber) represent a potentially better method of sensor multiplexing.
Abstract: The use of fiber optic sensors in the civil structures arena is increasing, particularly in the instrumentation of high-performance structures. However, the physical configurations of many such sensor platforms necessitates the use of fiber optic sensor multiplexing. Typically fiber optic sensor multiplexing has involved the grouping of similar or identical sensors along single transmit/receive fiber optic leads coupled with some form of time-, phase-, frequency- or polarization-based multiplexing. Based on work on embedding sensors into multiple large civil structures, the authors feel that multi-media sensors (i.e. fiber optic sensor configurations in which multiple parameters of interest can be measured simultaneously with a single fiber) represent a potentially better method of sensor multiplexing. Results based on sensor multiplexing and interrogation of pressure sensors and vibration sensors used in the health monitoring of a modest-sized hydroelectric dam are described.
32 citations
TL;DR: In this paper, a 3D formulation of flexible rods with embedded line actuators is presented, where both the rod and the line actuator are assumed to be initially curved and relatively positioned in an arbitrary way.
Abstract: The 3D formulation of flexible rods with embedded line actuators is presented in this paper. Both the rod and the line actuator are assumed to be initially curved and relatively positioned in an arbitrary way. The deformed configurations of the rod and actuator are connected by assuming that the principal planes of the rod remain plane and inextensible in the deformed configuration. The resultant forces and moments in the rod and actuator are found by solving the equilibrium equations for both the rod and actuator and applying continuity of tractions on the rod-actuator interface. The formulation is reduced to 2D and applied to the case of a shape memory alloy (SMA) fiber embedded in a cylindrical rod. The deformed shapes of the rod under repeated thermal actuation and the resulting shape memory loss due to the development of residual stresses are evaluated. Finally, the inverse problem of finding the required actuation force and initial curvature, to acquire a predetermined deformed shape, is solved in closed form for the 2D case.
TL;DR: In this article, a multiobjective optimization technique is used to formulate the problem of actuator location and continuous optimization problem involving control and structure interaction, which leads to a combined problem which includes the discrete actuators location problem and the continuous optimization, and the results obtained indicate that improved structural and control performance can be obtained with only a few optimally placed actuators.
Abstract: Intelligent structures are structures which can actively react to an unpredictable environmental disturbance in a controlled manner. Piezoelectric materials are an excellent choice for the development of sensors and actuators for these structures due to their special properties. It is important to locate these discrete actuators optimally on the structure in order to achieve the most efficient implementation of their special properties. It is also necessary to design the structure to be controlled for optimum performance. This leads to a combined problem which includes the discrete actuator location problem and the continuous optimization problem involving control and structure interaction. A multiobjective optimization technique is used to formulate the problem. Optimum piezoelectric actuator thicknesses for the static case are determined for the active piezoelectric elements development in this work. An optimization procedure is then presented which includes actuator locations, vibration reduction, power consumption, minimization of dissipated energy and maximization of the natural frequency as design objectives. The procedure is demonstrated through a cantilever beam problem. Results obtained indicate that improved structural and control performance can be obtained with only a few optimally placed actuators.
TL;DR: In this paper, a testbed consisting of a flexible frame slewed by a DC motor is modeled for active vibration suppression, where two active members are used as self-sensing actuators in feedback control loops.
Abstract: A testbed consisting of a flexible frame slewed by a DC motor is modeled for active vibration suppression. This presents a challenging control problem since the primary action of slewing induces both bending and torsional vibrations in the structure. Inserted into the frame are two active members that can be used as self-sensing actuators in feedback control loops. First, a model for the slewing frame is developed using Lagrange's equations and finite element approximations. The interactions between the structure and the slewing actuator are then derived from the equations for a DC motor. Similar expressions are obtained for the forces applied to the structure by the active elements. A detailed model of the self-sensing actuator is provided, which includes the terms due to actuator and sensor dynamics. A theoretical study is then conducted to obtain control laws that simultaneously slew the frame and suppress the residual vibrations. Simulation results indicate that the DC motor is effective in slewing the frame and suppressing the bending motion but not the torsional motion. Hence, the torsional vibrations are suppressed using the active members in collocated feedback loops.
TL;DR: In this article, a liquid-crystalline state of the lubricant in the friction zone of synovia was discovered, which was used for the development of synthetic lubricants.
Abstract: This investigation was devoted to the study of a natural phenomenon-low friction in living joints-and development of synthetic lubricants. The experiments conducted allowed us to establish that the known property of synovia to ensure the high antifrictional ability of joint cartilages is provided by the realization of a liquid-crystalline state of the lubricant in the friction zone that was unknown before. An idea was advanced that molecules of liquid-crystalline cholesterol compounds found in synovia are arranged with their longer axis aligned with prevailing microgroove locomotions on cartilage surfaces to make a liquid-crystalline nematic phase, thus reducing energy dissipation during relative motion of contacting cartilages and leading to the medicative effect revealed by experimentation.
TL;DR: In this article, the use of a limited number of actuator/sensor pairs to alter the shape or behavior of a truss structure to other desired states is explored analytically and experimentally.
Abstract: The use of a limited number of actuator/sensor pairs to alter the shape or behavior of a truss structure to other desired states is explored analytically and experimentally. Feasibility of the concept is established by numerical simulation. As a demonstration, a sequence of validation tests is performed and the correlation with analysis is discussed. An existing full scale, space-erectable high precision truss structure is used for this purpose. For the most part, the test results agreed well with the analysis. However, micron-level nonlinearities were discovered in the truss behavior. The significance of these nonlinearities in precision structures and their impact on the basic premise of adaptivity is discussed.
TL;DR: In this paper, a study of piezoelectric films as sensors for detecting fluctuating area-averaged strain has been conducted, and both theoretical and experimental results on the characterization of piezo properties are presented for KYNAR Piezo Films.
Abstract: A study of piezoelectric films as sensors for detecting fluctuating area-averaged strain has been conducted. Both theoretical and experimental results on the characterization of piezoelectricity are presented for KYNAR Piezo Films. The theoretical formula relating the in-plane area-averaged strain to the electric output of a piezo film through a charge amplifier was derived, and the relevant constants were determined experimentally. The present results show that if the films are prepared and adhered properly, they can be used satisfactorily for fluctuating strain measurement. The present work has also investigated the possibility of using PVDF films as sensors for monitoring crack growth on a metallic specimen.
TL;DR: In this paper, a finite element formulation is developed to analyze laminated plates with arbitrarily placed piezoceramic patches, and the shape of the patches has a direct effect on the stress field within the patch and in the surrounding material.
Abstract: A finite element formulation is developed to analyze laminated plates with arbitrarily placed piezoceramic patches. However, only isotropic layered plates are analyzed as illustrative examples because of the primary emphasis on the effect of piezoceramic patch shapes. The technique is applied to obtain static response and stress fields due to the application of electric field to the piezoceramic patches. Different shapes of piezoceramic patches are inserted in a square aluminium plate. The actuation stress fields due to the piezoceramic patches are discussed in detail. It is observed that the shape of the patches has a direct effect on the stress field within the patch and in the surrounding material. The choice of the shape of the patches depends on the desired stress field, however, sharp corners should be avoided. The circular patch is subjected to the in-plane hydrostatic stress field. The stresses are homogeneous in the elliptical patch. The stresses are higher near the edges of the square and rectangular patches.
TL;DR: In this paper, fiber-optic balanced double-polarization Michelson interferometers with fringe-counting readout using passive-quadrature demodulation are employed for remote sensing of the surface strain of plates made from carbon-fiber composites, which are screwed to the main wing spar of a Cessna C207A.
Abstract: Fiber-optic balanced double-polarization Michelson interferometers with fringe-counting read-out using passive-quadrature demodulation are employed for remote sensing of the surface strain of plates made from carbon-fiber composites, which are screwed to the main wing spar of a Cessna C207A. Two series of flight tests have been performed with three different sensors. During parabolic flight maneuvers, quasistatic strain variations of up to 500 mu in are measured along with superimposed engine- and aerodynamically induced vibrations with strain amplitudes of 5 mu in at (sub-) harmonics of the engine speed (40 Hz), and compared with the read-out of a conventional resistive strain gauge.
TL;DR: A modified active acoustic coating system using an adaptive digital controller has been developed for absorbing the sound incident on an underwater object, featuring a digital delay-line feedforward scheme.
Abstract: A modified active acoustic coating system using an adaptive digital controller has been developed for absorbing the sound incident on an underwater object. The redesigned control system, featuring a digital delay-line feedforward scheme, is predicted theoretically and verified experimentally to have a broader control bandwidth than the analog phase-shifter system which was developed previously. Another significant improvement to the system is the use of a multichannel delay-line network to achieve on-board monitoring of sound reflection from the coating, so that the system performance can be self-evaluated and the optimized by adjusting its control parameters. The adaptive adjustment function is realized through software on a personal computer interfaced with the digital delay-line network. This adaptive digital controller design enables the active absorbing coating to achieve a high echo reduction over a wide frequency band.
TL;DR: In this paper, an electrorheological (ER) fluid is a suspension of semiconducting particles in a nonconducting carrier fluid, which can change its rheological properties upon the application of an external electric field.
Abstract: An electrorheological (ER) fluid is a suspension of semiconducting particles in a non-conducting carrier fluid. The novel feature of this fluid is its ability to change its rheological properties upon the application of an external electric field. To investigate the properties of an activated ER fluid an experimental oscillatory electroviscometer was commissioned and a fluid tested. Conclusions drawn from the results were that the activated ER fluid changed from a viscous system, to one with a yield stress, elasticity and enhanced viscosity. These rheological changes were catalogued for a variety of electric field strengths, oscillatory frequencies and amplitudes.
TL;DR: In this paper, a new optical system for measuring the slope, thickness and wave velocity of a liquid film flow is presented, and the equations derived are employed in simulations involving film flow with low and high-amplitude waves.
Abstract: A new optical system for measuring the slope, thickness and wave velocity of a liquid film flow is presented. The theory of the system is discussed and the equations derived are employed in simulations involving film flow with low- and high-amplitude waves. The results, which show that it is possible to determine the three parameters for a low-amplitude wave, while only the wave velocity may be found for a high-amplitude wave, may have potential applications in multi-phase flow systems.
TL;DR: In this article, the authors examined various fiber sensor multiplexing techniques, e.g., frequency-, time-, and coherence-multiplexing, in an attempt to ascertain the method best suited for interrogation of multiple sensors scattered throughout a modern civil structure.
Abstract: Various fiber sensor multiplexing techniques, e.g., frequency-, time-, and coherence-multiplexing, have been examined in an attempt to ascertain the method best suited for interrogation of multiple sensors scattered throughout a modern civil structure. Based on embedded fiber sensor results conducted at a multistory reinforced concrete structure, where more than eighty single-mode and multimode fiber optic sensors have been embedded into the structure, it has been determined that in many instances a radio telemetry method of interrogating the sensors is optimal. Many real-world factors influence the overall nature of the use of multiplexed fiber sensors in civil structures. In instances where intensity-modulating fiber sensors have been multiplexed onto a single transmit/receive fiber, radio telemeterized command and data acquisition from the fiber sensor 'network' may be achieved. The development of the interrogation of the multiplexed fiber optic sensors is described.
TL;DR: The moony transducers as discussed by the authors are a type of modified Flextensional transducers, which consists of a thin piezoceramic disk, two thick metal end plates, and shallow cavities between the disk and the plates.
Abstract: The moony transducer is a type of modified flextensional transducer, which consists of a thin piezoceramic disk, two thick metal end plates, and shallow cavities between the disk and the plates. The shallow cavity is designed to redistribute the thickness direction stress field to the radial and tangential directions. Flextensional transducers are good candidates for low frequency, large displacement actuators, high sensitivity hydrophones and low frequency underwater sound sources. Finite element analysis is used to investigate the actuator and sensor function of these transducers when subjected to hydrostatic loading. Modal analysis of such a transducer is performed and the admittance spectrum and the voltage sensitivity under hydrostatic pressure are studied as a function of frequency and cavity shape and the performance is compared with the ordinary piston type, Langevin transducer.
TL;DR: In this article, an investigation of the properties of photoelastic sensing materials via the stress-induced effect using an interferometric technique is presented, with a view to using such a material as a stress/pressure sensing device.
Abstract: An investigation of the properties of photoelastic sensing materials via the stress-induced effect using an interferometric technique is presented. Sensing material properties such as stress relaxation, elastic limit, and its temperature dependency are investigated with a view to using such a material as a stress/ pressure sensing device.
TL;DR: In this paper, a robust control algorithm was proposed for active damping enhancement of an aluminium cantilever beam and a three-mass system with three natural frequencies below 3 Hz.
Abstract: Smart structures is now a vigorous area of study. One type of smart structure employs shape memory alloys, such as Nitinol, in the form of thin wire actuators to provide enhanced damping. To the authors knowledge, the only control algorithm that has been implemented heretofore has been a simple on-off control law. In this study, other control algorithms are employed, ranging from classical to a robust controller based on the LQG/LTR algorithm, and their relative effectiveness is compared. The robust control algorithm gave the best active damping enhancement. Two mechanical systems are studied. The first is an aluminium cantilever beam for which the free-vibration damping of the fundamental mode was enhanced. The second system studied is a three-mass system with its three natural frequencies below 3 Hz.
TL;DR: In this article, a single-mode polarization-maintaining optical fiber sensor is proposed to monitor the complete strain state of the host material provided that the sensor direction and polarization axes are properly orientated.
Abstract: Summarizes the normal strain coupling from an isotropic material into an embedded optical fiber which serves as a sensor. The sensor is based on single-mode polarization-maintaining fibers and is sensitive to all three normal strains. It is shown analytically that six such sensors can monitor the complete strain state of the host material provided that the sensor direction and polarization axes are properly orientated.
TL;DR: Discusses the capabilities of a mathematical tool useful in the simulation of physical and diagnostic reasoning, particularly with application to smart structures logic, and a few sample problems in structural analysis and control are formulated and solved arithmetically in terms of interval-valued variables.
Abstract: Discusses the capabilities of a mathematical tool useful in the simulation of physical and diagnostic reasoning, particularly with application to smart structures logic. More precisely, a few sample problems in structural analysis and control are formulated and solved arithmetically in terms of interval-valued variables, i.e. in terms of variables that can take any numerical value within a given set. Interval variables can also be used to represent uncertainties of a nonprobabilistic nature and enable the formulation of semi-qualitative physical models consistent with typical reasoning paradigms, where the ability to discern between different behavioural states and causal paths is an important feature.
TL;DR: In this article, the authors explored the ramifications of optical isotropy in structurally embedded optical fiber sensors, in particular the tensor transformations between the principal strain coordinate system defined by the mechanical boundary conditions and the coordinate system defining by the guiding axis of an optical fiber sensor are used to explain the influence of shear strains on the optical response of embedded sensors.
Abstract: The author's work explores the ramifications of optical isotropy in structurally embedded optical fiber sensors. In particular, the tensor transformations between the principal strain coordinate system defined by the mechanical boundary conditions and the coordinate system defined by the guiding axis of an optical fiber sensor are used to explain the influence of shear strains on the optical response of embedded sensors. Both isotropic silica fibers and anisotropic sapphire fibers are considered. Analogies to common mechanical constitutive models are provided to aide in interpreting the results.
TL;DR: In this article, a maximum entropy approach is used to derive linear and nonlinear responses to simultaneously applied electric, magnetic, stress, and temperature fields, which are compared with classical linear-response theory.
Abstract: Linear- and nonlinear-response theories are developed using a maximum-entropy approach. The approach is compared with classical linear-response theory. Expressions for linear and nonlinear responses to simultaneously applied electric, magnetic, stress and temperature fields are derived. The statistical-mechanical theories of dynamic and thermally driven systems are used to obtain generalized equations of evolution for the driven quantities. In a linear approximation, the Kubo expression is obtained. These equations are valid far from equilibrium. The time evolution of the electric polarization vector can be separated into a relaxation term and an external source term. Expressions for time-dependent entropy are developed and analysed. In the very special case of the relaxation approximation, commonly used in the Boltzmann equation, the equation reduced to Debye's equation. Linear constitutive relations are given for electro-acoustic interactions at low driving fields.
TL;DR: In this article, a systematic approach is presented to formulate the structured uncertainties due to variations in the values of natural frequencies and the modal damping ratios, i.e. the parameters of the A matrix of the structure's state-space model, as well as the unstructured uncertainty resulting from the fact that only a limited number of modes are included in the model of the smart structure.
Abstract: A systematic approach is presented to formulate the structured uncertainties due to variations in the values of natural frequencies and the modal damping ratios, i.e. the parameters of the A matrix of the structure's state-space model, as well as the unstructured uncertainty resulting from the fact that only a limited number of modes are included in the model of the smart structure. A computer program has been developed that takes the percentage of the variation on the parameters, as well as the desired number of modes to be included in the model and provides the user with the appropriate formulation of the uncertainty.