@article{mollik_kennedy_ul shougat_li_fitzgerald_echols_kirk_silverberg_perkins_2022, title={Discrete element method simulator for joint dynamics: a case study using a red-tailed hawk's hallux digit}, volume={6}, ISSN={["1573-272X"]}, DOI={10.1007/s11044-022-09828-x}, journal={MULTIBODY SYSTEM DYNAMICS}, author={Mollik, Tushar and Kennedy, Scott and Ul Shougat, Md Raf E. and Li, Xiaofu and Fitzgerald, Timothy and Echols, Scott and Kirk, Nick and Silverberg, Larry and Perkins, Edmon}, year={2022}, month={Jun} }
 @article{coppola_silverberg_2021, title={Isotropic packing algorithm for particle simulations}, volume={122}, ISSN={["1097-0207"]}, DOI={10.1002/nme.6742}, abstractNote={This article introduces the R2 Fill method, which is an algorithm for filling a region with particles isotropically. The method's isotropy is tested geometrically and then mechanically with the Discrete Element Method (DEM). It applies to planar and three-dimensional problems and is simpler to implement and has competitive metrics compared to other Advancing Front packing algorithms. The DEM formulation used in this paper provides a way to model solid materials and eliminate residual stress created in any particle packing algorithm. The R2 Fill method was tested over ranges of particle sizes. At large size ranges the method generated a higher coordinate number and denser particle packing, with 79.718% fill at the maximum tested size range. The geometric isotropy was improved at higher size ranges yielding less than a 1% isotropic error. The method also had fewer particles than the similar and more complex algorithm by Yongjun Li. Lastly we tested mechanical isotropy by using our DEM formulation to measure the stress–strain response under a tensile load at different orientations. The R2 Fill method produced mechanical isotropy within 2% error for the elastic moduli, shear moduli, and Poisson ratio.}, number={17}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Coppola, Stephen and Silverberg, Larry M.}, year={2021}, month={Sep}, pages={4698–4708} }
 @article{powers_silverberg_gopalarathnam_2020, title={Artificial Lumbered Flight for Autonomous Soaring}, volume={43}, ISSN={["1533-3884"]}, DOI={10.2514/1.G004397}, abstractNote={Soaring strategies are redefining the flight capabilities of small-class fixed-wing unmanned aerial vehicles. This paper presents an autonomous soaring strategy that exploits updraft energy indepen...}, number={3}, journal={JOURNAL OF GUIDANCE CONTROL AND DYNAMICS}, author={Powers, Thomas C. and Silverberg, Larry M. and Gopalarathnam, Ashok}, year={2020}, month={Mar}, pages={553–566} }
 @article{silverberg_eischen_2020, title={On a new field theory formulation and a space-time adjustment that predict the same precession of Mercury and the same bending of light as general relativity}, volume={33}, ISSN={["2371-2236"]}, DOI={10.4006/0836-1398-33.4.489}, abstractNote={This article introduces a new field theory formulation. The new field theory formulation recognizes vector continuity as a general principle and begins with a field that satisfies vector continuity equations. Next, independent of the new formulation, this article introduces a new space-time adjustment. Then, we solve the one-body gravitational problem by applying the space-time adjustment to the new field theory formulation. With the space-time adjustment, the new formulation predicts precisely the same precession of Mercury and the same bending of light as general relativity. The reader will find the validating calculations to be simple. The equations of motion that govern the orbital equations are in terms of Cartesian coordinates and time. An undergraduate college student, with direction, can perform the validations.}, number={4}, journal={PHYSICS ESSAYS}, author={Silverberg, Larry M. and Eischen, Jeffrey W.}, year={2020}, month={Dec}, pages={489–512} }
 @article{zhu_pritchard_silverberg_2018, title={A New System Development Framework Driven by a Model-Based Testing Approach Bridged by Information Flow}, volume={12}, ISSN={["1937-9234"]}, DOI={10.1109/JSYST.2016.2631142}, abstractNote={As complexity increases, model-based approaches have become standard in the systems engineering community. However, engineers from different domains approach the system/component development at different levels of abstraction. Information gaps are created because of their different perspectives. In addition, although systems have become more complex, requiring more attention to testing, system methodologies have not kept up. A new framework is needed to bridge the information gaps between the different domains. To take full advantage of both model-based approaches and testing, we propose a new framework, in the form of a new design structure matrix, for complex system development (CSD) that more effectively utilizes information flow from model-based testing (MBT). Toward the goal of being broadly applicable, the framework was designed to be fundamental, flexible, and scalable. A case study of a powertrain design demonstrates how the proposed framework bridges gaps in early development stages. The analysis of both simulation and field testing results has shown that utilization of feedback from MBT refines requirements by closing information gaps earlier. All the information loops in the framework match the information loops in the case study. Furthermore, the framework, because of CSD's iterative nature, has potential in improving information reuse.}, number={3}, journal={IEEE SYSTEMS JOURNAL}, author={Zhu, Di and Pritchard, Ewan G. D. and Silverberg, Larry M.}, year={2018}, month={Sep}, pages={2917–2924} }
 @article{silverberg_tran_laue_2018, title={Gender comparison in consistency in the basketball free throw by an event-driven approach}, volume={21}, ISSN={["1460-2687"]}, DOI={10.1007/s12283-018-0276-z}, number={4}, journal={SPORTS ENGINEERING}, author={Silverberg, Larry M. and Tran, Chau M. and Laue, Christopher}, year={2018}, month={Dec}, pages={333–340} }
 @article{tschida_silverberg_2013, title={Cellular growth algorithms for shape design of truss structures}, volume={116}, ISSN={["1879-2243"]}, DOI={10.1016/j.compstruc.2012.10.006}, abstractNote={This paper introduces a novel approach to truss structure shape design, drawing inspiration from biological processes, that treats each joint or member of a truss as a cell and the entire truss as a multicellular organism. The approach mimics plant tropisms in which, through cellular growth, cells change their size and shape in response to scripted gene commands. Examples illustrate the effectiveness of the approach; it is shown to produce arching and double-arching of pinned-pinned trusses, bulging of cantilever trusses and a plant-like tropic response.}, journal={COMPUTERS & STRUCTURES}, author={Tschida, Colin E. and Silverberg, Larry M.}, year={2013}, month={Jan}, pages={1–6} }
 @article{silverberg_tran_adams_2011, title={Optimal Targets for the Bank Shot in Men's Basketball}, volume={7}, ISSN={["1559-0410"]}, DOI={10.2202/1559-0410.1299}, abstractNote={The purpose of this study was to gain an understanding of the bank shot and ultimately determine the optimal target points on the backboard for the bank shot in men’s basketball. The study used over one million three-dimensional simulations of basketball trajectories. Four launch variables were studied: launch height, launch speed, launch angle, and aim angle. The shooter’s statistical characteristics were prescribed to yield a 70 percent free throw when launching the ball seven feet above the ground with 3 Hz of back spin. We found that the shooter can select a bank shot over a direct shot with as much as a 20 percent advantage. The distribution over the court of preferences of the bank shot over the direct shot was determined. It was also shown that there is an aim line on the backboard independent of the shooter’s location on the court. We also found that at 3.326 inches behind the backboard, there exists a vertical axis that aids in finding the optimal target point on the backboard. The optimal target point is the crossing of the vertical axis and the aim line that is in the shooter’s line of sight.}, number={1}, journal={JOURNAL OF QUANTITATIVE ANALYSIS IN SPORTS}, author={Silverberg, Larry M. and Tran, Chau M. and Adams, Taylor M.}, year={2011}, month={Jan} }
 @article{edwards_silverberg_2010, title={Autonomous Soaring: The Montague Cross-Country Challenge}, volume={47}, ISSN={["1533-3868"]}, DOI={10.2514/1.c000287}, abstractNote={A novel method was developed for locating and allowing gliders to stay in thermals (convective updrafts). The method was applied to a 5 kg, glider, called ALOFT (autonomous locator of thermals), that was entered in the 2008 Montague Cross-Country Challenge held on 13-15 June 2008 in Montague, California. In this competition, remote-controlled gliders in the 5 kg class competed on the basis of speed and distance. ALOFT was the first known autonomously soaring aircraft to enter a soaring competition and its entry provided a valuable comparison between the effectiveness of manual soaring and autonomous soaring. ALOFT placed third in the competition in overall points, outperforming manually-flown aircraft in its ability to center and use updrafts, especially at higher altitudes and in the presence of wind, to fly more optimal airspeeds, and to fly directly between turn points. The results confirm that autonomous soaring is a bona fide engineering subdiscipline, which is expected to be of interest to engineers, who might find that this has some utility in the aviation industry.}, number={5}, journal={JOURNAL OF AIRCRAFT}, author={Edwards, Daniel J. and Silverberg, Larry M.}, year={2010}, pages={1763–1769} }
 @article{levedahl_silverberg_2009, title={Control of Underwater Vehicles in Full Unsteady Flow}, volume={34}, ISSN={["1558-1691"]}, DOI={10.1109/JOE.2009.2027798}, abstractNote={In this paper, a general formulation of the problem of control of underwater vehicles in full unsteady flow is presented. First, a reduced-order model of the coupled fluid vehicle (CFV) system is developed. The inability to observe fluid motion motivates a fluid compensation control (FCC) approach that compensates for the hydrodynamic loads synthesized from surface measurements. The FCC consists of a tracker, a regulator, and a fluid compensator. A condition is provided that guarantees vehicle stability. The tradeoff between regulation and fluid compensation is also examined. A numerical example of an elliptically shaped vehicle illustrates the results.}, number={4}, journal={IEEE JOURNAL OF OCEANIC ENGINEERING}, author={Levedahl, Blaine A. and Silverberg, L.}, year={2009}, month={Oct}, pages={656–668} }
 @article{morton_silverberg_2009, title={Fourier series of half-range functions by smooth extension}, volume={33}, ISSN={["1872-8480"]}, DOI={10.1016/j.apm.2007.12.009}, abstractNote={This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L ] of the interval [− L, L ] in one-dimensional problems and over the sub-domain [0, L x ] × [0, L y ] of the domain [− L x , L x ] × [− L y , L y ] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.}, number={2}, journal={APPLIED MATHEMATICAL MODELLING}, author={Morton, Jeremy and Silverberg, Larry}, year={2009}, month={Feb}, pages={812–821} }
 @article{silverberg_levedahl_2008, title={Characterizing Hydrodynamic Loads in Full Unsteady Flow}, volume={46}, ISSN={["0001-1452"]}, DOI={10.2514/1.36626}, number={12}, journal={AIAA JOURNAL}, author={Silverberg, Larry and Levedahl, Blaine}, year={2008}, month={Dec}, pages={3159–3163} }
 @article{tran_silverberg_2008, title={Optimal release conditions for the free throw in men's basketball}, volume={26}, ISSN={["1466-447X"]}, DOI={10.1080/02640410802004948}, abstractNote={Abstract The purpose of this study was to determine the optimum release conditions for the free throw in men's basketball. The study used hundreds of thousands of three-dimensional simulations of basketball trajectories. Five release variables were studied: release height, release speed, launch angle, side angle, and back spin. The free throw shooter was assumed to shoot at 70% and to release the ball 2.134 m (7 ft) above the ground. We found that the shooter should place up to 3 Hz of back spin on the ball, should aim the ball towards the back of the ring, and should launch the ball at 52° to the horizontal. We also found that it is desirable to release the ball as high above the ground as possible, as long as this does not adversely affect the player's launch consistency.}, number={11}, journal={JOURNAL OF SPORTS SCIENCES}, author={Tran, Chau M. and Silverberg, Larry M.}, year={2008}, pages={1147–1155} }
 @article{silverberg_levedahl_2005, title={Autonomous coordination of aircraft formations using direct and nearest-neighbor approaches}, volume={42}, ISSN={["1533-3868"]}, DOI={10.2514/1.6868}, abstractNote={Two approaches are developed for autonomous coordination of aircraft formations. The development of the approaches relies on past work in the areas of distributed control (modal, robust, optimal, and decentralized). The formation coordination problem is divided into a tracking problem (changing the formation) and a regulation problem (maintaining the formation). How to separate the spatial parts of the tracking problem from the temporal parts is demonstrated. With respect to the regulation problem, it is shown that the goal of the regulation problem is to dampen uniformly the motion of the aircraft. It is pointed out that for fuel-optimality the closed-loop damping factors of the aircraft need to be less than π/2. Two types of decentralized coordination are examined: direct coordination (using inertial measurements) and nearest-neighbor coordination (using relative measurements). A perturbation analysis is developed for the efficient calculation of control gains that minimize power and uniformly dampen motion. A numerical example illustrates robust formation changes from nine-aircraft (3 × × 3) grids to V-type formations. Why the performance of direct coordination is generally better than the performance of nearestneighbor coordination and why implementing direct coordination is simpler than implementing nearest-neighbor coordination are explained. However, nearest-neighbor coordination can be used in collision avoidance, and so it must still be considered as a viable option.}, number={2}, journal={JOURNAL OF AIRCRAFT}, author={Silverberg, L and Levedahl, BA}, year={2005}, pages={469–477} }
 @book{basic experiments in pid control for engineers_2005, ISBN={053697571X}, publisher={Boston : Pearson/Custom Publishing}, year={2005} }
 @article{silverberg_duval_2005, title={Transpermanent magnetics using alternating uniform linear stacks}, volume={127}, ISSN={["1050-0472"]}, DOI={10.1115/1.1897743}, abstractNote={Abstract In this paper we apply recent developments in transpermanent magnetics to the problem of ultra-low-power valve control. Whereas the traditional approach to ultra-low-power valve control is based on latching mechanisms that turn off valves during inactive periods, in this paper we describe an approach that eliminates the need for latching mechanisms. Instead of latching mechanisms, the principles of transpermanent magnetics are employed to switch the states of permanent magnets; the use of permanent magnets instead of electromagnets eliminates power loads during inactive periods, thereby reducing power consumption to ultralow levels. The permanent magnets in a transpermanent magnet valve are configured in a stack. The relationships between the strength and number of permanent magnets in the stack and the stroke and resolution of the valve are developed. In this paper we show that the alternating uniform linear stack is well suited for digital process valves having a small number of states. Then in the paper we report on the design and testing of a laboratory prototype valve that uses an alternating uniform linear stack. The prototype valve had five states yielding a range of flow rates between 0 and 1.58m∕s with a resolution of 0.3m∕s. In this paper we find that transpermanent valves represent a promising valve technology for digital process valves.}, number={4}, journal={JOURNAL OF MECHANICAL DESIGN}, author={Silverberg, L and Duval, L}, year={2005}, month={Jul}, pages={679–687} }
 @article{kiefer_silverberg_gonzalez_2004, title={A case study of prototyping methods and design for manufacture: electrostatic window blinds}, volume={15}, ISSN={["0954-4828"]}, DOI={10.1080/0954482021000050820}, abstractNote={This paper presents a case study of the design development of a new product, electrostatically actuated window blinds. The text presents a description of each stage in the prototype development followed by a discussion of different prototyping methods and design for manufacture and assembly issues. Results from electromechanical and thermal analyses of the product itself are also included. This case study determined that, for this product development, the most efficient, cost-effective prototyping solution was simple plastic injection moulding. It also concluded that following good design for manufacture and assembly practices during prototype construction will save time and money when transitioning to a production design.}, number={1}, journal={JOURNAL OF ENGINEERING DESIGN}, author={Kiefer, S and Silverberg, L and Gonzalez, M}, year={2004}, month={Feb}, pages={91–106} }
 @article{silverberg_farmer_2004, title={Transpermanent magnetic actuation for spacecraft pointing, shape control, and deployment}, volume={41}, ISSN={["1533-6794"]}, DOI={10.2514/1.1350}, abstractNote={Transpermanent magnetic actuators are systems consisting of one or more permanent magnets, some of whose magnetic strengths can be switched onboard by surrounding pulse-coils. Transpermanent magnetic actuators are shown to be particularly wellsuited for spacecraft pointing, shape control, and deployment applications. In many spacecraft pointing, shape control, and deployment applications, it is desirable to hold displacements or forces between two points to within specified requirements (the regulation problem) and periodically to change or remove these requirements (the tracking problem). Furthermore, the interest generally lies in satisfying the dynamic performance requirements while expending minimal power, while meeting tight tolerances (particularly in optical applications), and while experiencing little wear and fatigue. The transpermanent magnetic actuator is shown to expend no power during regulation, and the transpermanent magnetic actuator is shown to be able to change periodically or remove the strength of its own magnets, thereby enabling both fine-tune adjustments and large-scale adjustments during tracking. The fine-tune adjustments are necessary in thermally varying space environments, and the large-scale adjustments are necessary in deployment problems in which pivot points experience large-angle rotations. A transpermanent magnetic actuator concept is demonstrated.}, number={6}, journal={JOURNAL OF SPACECRAFT AND ROCKETS}, author={Silverberg, L and Farmer, D}, year={2004}, pages={1051–1055} }
 @article{silverberg_duval_2003, title={Analysis of trans-permanent magnetic systems}, volume={125}, ISSN={["0022-0434"]}, DOI={10.1115/1.1543553}, abstractNote={This paper develops a method of switching the states of permanent magnets through an on-board magnetization process. The resulting dynamic systems are referred to as trans-permanent magnetic systems. This paper first describes the governing equations needed for the analysis of trans-permanent magnetic systems. Then, the paper demonstrates their feasibility in an experiment. In doing so, this paper presents a method that has the potential of leading to new ultra-low-power designs for electromechanical devices.}, number={1}, journal={JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME}, author={Silverberg, L and Duval, L}, year={2003}, month={Mar}, pages={143–146} }
 @article{silverberg_tran_adcock_2003, title={Numerical analysis of the basketball shot}, volume={125}, ISSN={["1528-9028"]}, DOI={10.1115/1.1636193}, abstractNote={This paper numerically analyzes the dynamics of the basketball shot. The focus of the paper is on the development of a general formulation for the dynamics of the shot beginning when the ball leaves the shooter’s hand and ending when the shot is made or missed. The numerical analysis developed in this paper can be used to conduct a parametric study of the dynamics of the basketball shot, which in turn, can be used to improve individual shooting and team strategy. The individual skill level of the shooter enters the formulation through the statistical accuracy of the release. The paper then shows how to determine the shooter’s probability of making a given shot.}, number={4}, journal={JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME}, author={Silverberg, L and Tran, C and Adcock, K}, year={2003}, month={Dec}, pages={531–540} }
 @article{kiefer_silverberg_gonzalez_2001, title={Electrostatically actuated window blinds}, volume={50}, DOI={10.1016/s0304-3886(00)00038-3}, abstractNote={An electrostatic method for actuating window blinds was recently developed by the authors in work that is documented elsewhere. The method was developed in an experimental effort without the help of predictive tools. This paper examines that problem in more detail and formulates an electrostatic model for these systems. The complexity of the charge distribution over the slats and the glass demands a relatively large model (hundreds of thousands of degrees of freedom). This paper formulates the large-order model and then shows how to reduce the order of the model significantly. The charge distribution is represented as a linear combination of assumed modes. The charge distribution is assumed to be periodic in the vertical direction, which is tantamount to neglecting end effects. Experimental results verify the accuracy of the predictions in the presence of the order-reduction assumptions. The electrostatic model predicts the slat angles to within the accuracy of the measurement system. The conductive and convective heat transfer across the air space are also analyzed. The flow patterns explain the heat transfer (thermal efficiency) of windows with slats enclosed.}, number={4}, journal={Journal of Electrostatics}, author={Kiefer, S. F. and Silverberg, Larry M. and Gonzalez, M. L.}, year={2001}, pages={229–248} }
 @book{silverberg_thrower_2001, title={Marks' mechanics problem-solving companion}, ISBN={0071362789}, publisher={New York: McGraw-Hill}, author={Silverberg, L. and Thrower, J. P.}, year={2001} }
 @article{stanley_silverberg_1999, title={Paraboloidally shaped antenna possessing pointing and beam-width control}, volume={36}, ISSN={["0022-4650"]}, DOI={10.2514/2.3487}, abstractNote={The surface shape of a paraboloidal ree ector was recone gured using electrostatic forces of repulsion. The experiment consisted of a recone gurable electrostatically shaped ree ector and a measurement system. A set of test cases demonstrated pointing and beam-width control of the radiating surface. The surface ree ector angles of rotation µx and µy were plotted vs eight test cases. The resulting sine functions show the near-linear relationship between voltage and pointing, indicating that expected rotations are in agreement with measurements and that a simple interpretation of Coulomb’ s law dominates the behavior of the system. Nomenclature a = vector of undetermined coefe cients C = coefe cient matrix Ex = error function for rotation about the x axis Ey = error function for rotation about the y axis i; j;k = counting indices p = number of sample points in one quadrant Rx;Ry = rotation matrices xn; yn;zn = coordinates in a new reference frame z = vertical position of ith point in the (j;k)th quadrant z = new vertical position of ith point in the (j;k)th quadrant z = vector of measured displacements z (x; y) = function to be curve e tted µx = angle of rotation about the x axis µy = angle of rotation about the y axis}, number={5}, journal={JOURNAL OF SPACECRAFT AND ROCKETS}, author={Stanley, RJ and Silverberg, LM}, year={1999}, pages={736–744} }
 @article{silverberg_washington_1999, title={Weighted-residual discretization for uniform damping and uniform stiffening of structural systems}, volume={22}, ISSN={["0731-5090"]}, DOI={10.2514/2.7638}, number={4}, journal={JOURNAL OF GUIDANCE CONTROL AND DYNAMICS}, author={Silverberg, L and Washington, G}, year={1999}, pages={614–618} }
 @article{silverberg_1998, title={Comment on "conjecture about orthogonal functions" - Reply}, volume={21}, ISSN={["0731-5090"]}, DOI={10.2514/2.4223}, abstractNote={I N “ConjectureAbout OrthogonalFunctions,”1 I presenteda conjecture “...to provoke insight that could lead to a proof of the conjecture at a later date.” That insight came from Stephen Hendry in his comment about that paper.Hendry showed that the conjecture follows froma theoremgivenin the1939 textbookentitledOrthogonal Polynomialsby Szego. The theorem is called the Gauss–Jacobi quadraturetheorem,althoughcontributionsto the theoremwere also made by Christoffel, Mehler, and Heine.3i8 The theorem is stated as follows. If x1 < x2 < ¢ ¢ ¢ < xn denote the zeros of the polynomial pn.x/, there exist real numbers31; 32; : : : ; 3n such that Z b}, number={1}, journal={JOURNAL OF GUIDANCE CONTROL AND DYNAMICS}, author={Silverberg, L}, year={1998}, pages={189–190} }
 @article{wang_eischen_silverberg_1998, title={On control and optimization of elastic multilink mechanisms}, volume={67}, ISSN={["0045-7949"]}, DOI={10.1016/s0045-7949(98)00060-1}, abstractNote={Abstract This paper presents a simple method for control of nonlinear elastic multilink mechanisms. The associated control law consists of open and closed loop components. The open loop component produces the desired overall rigid body motion, while the closed loop component suppresses the elastic motion relative to rigid body motion. Both the control law and the structural dynamics are referred to an inertial coordinate system. As a consequence, the control forces and control moments are easily simulated in a dynamic finite element analysis program. A series of numerical simulations illustrate the generality of this new method.}, number={6}, journal={COMPUTERS & STRUCTURES}, author={Wang, HL and Eischen, JW and Silverberg, LM}, year={1998}, month={Jun}, pages={483–502} }
 @article{silverberg_1997, title={Conjecture about orthogonal functions}, volume={20}, ISSN={["0731-5090"]}, DOI={10.2514/2.4021}, number={1}, journal={JOURNAL OF GUIDANCE CONTROL AND DYNAMICS}, author={Silverberg, L}, year={1997}, pages={198–202} }
 @article{washington_silverberg_1997, title={Uniform damping and stiffness control of structures with distributed actuators}, volume={119}, ISSN={["0022-0434"]}, DOI={10.1115/1.2801294}, abstractNote={The research in this paper introduces an algorithm for uniformly damping and stiffening structures using distributed actuators. Inherent in this algorithm is the ability to eliminate structural biases, which can cause steady state error. The method involves the utilization of modal techniques to calculate the proportional, integral, and derivative gains in a feedback control law. The gains are calculated by first converting the system into the modal domain, second by employing the principles of uniform damping and stiffening to the system in the modal domain, and finally by converting that system back into the physical domain. The beauty of this technique lies in the fact that even though the principles of modal control are used in gain setting, no modal information is needed in the fined control law. The theory is validated by simulations and is illustrated in an example. Since a modal approach and distributed actuation is utilized, the benefit to active control and adaptive structures is discussed.}, number={3}, journal={JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME}, author={Washington, GN and Silverberg, L}, year={1997}, month={Sep}, pages={561–565} }
 @misc{silverberg_1994, title={Electrostatically shaped membranes}, volume={5307082}, publisher={Washington, DC: U.S. Patent and Trademark Office}, author={Silverberg, L. M.}, year={1994} }