This paper presents an algorithm to find natural frequencies and mode shapes of a uniform Reddy beam. In this method, spatial state equations are developed and the spatial state transition matrix is computed, which is independent of the boundary conditions of the beam. Then, natural frequencies and mode shape equations are easily derived for any boundary conditions in terms of elements of the state transition matrix evaluated at the right end of the beam. As examples, these equations are derived for pinned-pinned and clamped-clamped beams. Numerical difficulties in implementing this method are recognized, and a reduced-order spatial state space model is developed by analyzing the nature of roots of the sixth-order characteristic equation. Numerical results are presented for rectangular beams and compared to those for Timoshenko beams.

Although the electromagnetic transient (EMT) framework can capture subsynchronous oscillations (SSOs), it faces scalability issues for large-scale systems. Thus motivated, we propose a generalized dynamic phasor (DP) framework to analyze SSOs in multi-machine systems with inverter-based resources (IBRs) and large loads such as artificial intelligence data centers (AI DCs) under balanced and unbalanced conditions. The grid-following (GFL) and grid-forming (GFM) IBRs are modeled in their respective dq-frame DPs. In contrast, the detailed model of multi-mass turbine driven synchronous generators (SGs) along with dynamic transmission network models and loads are represented in pnz-frame DPs. The linearizability and time-invariance of the framework enable us to perform eigen decomposition, which is a powerful tool for root-cause analysis of SSO modes and the design of damping controllers. In addition, the DP modeling approach facilitates faster simulation of large-scale systems. The generalized framework is validated with EMTDC/PSCAD simulations using the IEEE first benchmark model for subsynchronous resonance and the modified IEEE 4-machine system. Several use cases are presented on the modified IEEE 68-bus system with two GFL IBRs to show the applicability of the framework. First, time- and frequency-domain analyses of the IBR-induced SSO mode are presented. Then, two solutions are proposed to damp the poorly damped SSO mode: (a) a decentralized controller is designed using particle swarm optimization, and (b) the control of one GFL IBR is replaced by GFM control. Finally, the impact of AI DC load on primary frequency response of the system and the multi-mass turbines of the SGs are studied.

This paper deals with the stability and forced response of a bladed disk under fluctuating rotor speed which leads to time varying blades’ stiffnesses. Inevitable small variations in blades’ stiffnesses due to manufacturing tolerances result in time-varying mistuning. Using the method of slowly varying parameters, stability conditions are obtained analytically and verified by numerical solution of governing differential equations. The stabilizing effects of random and alternate mistuning are investigated in the presence of time varying blades’ stiffnesses. For stable bladed disks with time-varying random mistuning and with constant random mistuning, statistical distributions of the peak maximum amplitude are generated via Monte Carlo simulations at 0.1% and 1% damping ratios.

This paper develops a method to compute the frequency response of a nonuniform Euler-Bernoulli beam without modal decomposition or finite elements discretization. This method is applicable to any types of linear damping, nonuniformities, discontinuities or constraints. The method utilizes the concept of spatial state transition matrix, which is independent of boundary conditions of the beam. The steady state response is obtained by the solution of spatial state equations, which has two components: contributions from initial states, and convolution integral between spatial state transition matrix and amplitudes of sinusoidal external forcing functions. This approach yields the frequency response function of an infinite-dimensional Euler-Bernoulli beam from any applied input to any measured output. The method is illustrated via two numerical examples. The first example is a black hole attached to a fixed-free uniform beam with a viscous damper at the tip, which contains nonuniform geometry with a discontinuity, collocated input force and output transverse displacement. An algorithm is presented for the generation of a virtual black hole. The second example is the stepped piezoelectric cantilever beam with multiple discontinuities and non-collocated piezoelectric voltage input and transverse displacement output of the beam.

Phthalates, a sort of plasticizer, are widely utilized in various consumer products and pose significant environmental and health risks due to their persistence and potential toxicity. This review explores the occurrence, sources, environmental impact, and remediation strategies for phthalates. Various remediation techniques have been investigated to address phthalate contamination. Among these, photocatalysis, an advanced oxidation process (AOP), has emerged as a viable approach due to its ability to mineralize organic contaminants into innocuous byproducts. The review discusses the recent advancements in photocatalytic processes, the underlying mechanisms, and degradation pathways for phthalate removal. The mechanism of photocatalytic degradation includes the generation of reactive oxygen species like hydroxyl radicals (OH•) and superoxide radicals (O2-•) and their role in breaking down phthalate molecules. It also highlights recent advancements in photocatalytic materials, such as metal-doped semiconductors and composite materials, which enhance the removal efficiency. The review concludes by emphasizing the need for integrated approaches to achieve effective and sustainable phthalate remediation. Future research should focus on developing efficient and cost-effective photocatalytic materials, optimizing reactor design, and scaling up photocatalytic processes for practical applications. The review also highlights the challenges and limitations of photocatalytic processes, including low quantum efficiency, catalyst deactivation, and mass transfer limitations. Potential areas of study are put forward to address these challenges and further advance the application of photocatalysis for phthalate removal. This review intends to help the development of efficient photocatalytic technologies for the remediation of phthalate-contaminated water by providing a complete overview of the present state of the art.

Abstract A simple algorithm is presented to find near-minimum-time control inputs for a controllable continuous-time linear time-invariant system with any number of states and inputs. The well-known digital deadbeat control algorithm is modified to satisfy input constraints, and the design issue is the selection of sampling period to minimize the time required to reach the desired final state. During each sampling period, the input required to place all the poles at the origin of z-plane (deadbeat control) via state feedback is computed. If the infinity norm of the deadbeat input exceeds the input constraint, the input is found by placing the poles as close to the origin as possible. A sufficient condition for the system stability is developed. Numerical examples illustrate that this algorithm can lead to true time-optimal control or near time-optimal control.

Optimal sliding Gaussian control of a quadrotor aerial vehicle with a flexible structure has been demonstrated in six-degree-of-freedom flight simulation to be more robust to structural excitation than a standard proportional-derivative control scheme. A general derivation of the optimal sliding Gaussian controller, including an optimal gain selection algorithm and observer design, is presented. Additionally, a brief development of the equations of motion for a quadrotor with a flexible structure is provided as background for the nonlinear six-degree-of-freedom flight simulation results. Robustness properties of the optimal sliding Gaussian controller are evaluated using singular value analysis of linearized models of the closed-loop quadrotor with a flexible structure. Flight simulation and singular value analysis results illustrate the robustness improvements of optimal sliding Gaussian control over the standard proportional-derivative controller design.

Purpose This paper investigates into the existence of the second frequency spectrum of an uniform Timoshenko beam, which has been debated in the literature. The equivalence of the fourth order partial differential equation to two second-order partial differential equations of an uniform Timoshenko beam from the perspective of boundary conditions, another topic of debate in the literature, is examined by using the concept of observability of a state space model. Methods Natural frequencies and mode shapes are computed using the new approach developed by the author in his previous papers, which is based on the spatial state space analysis. Results Numerical results are presented for pinned-pinned, clamped-free, clamped-clamped and clamped-pinned Timoshenko beams. Conclusions It has been found that the second frequency spectrum exists irrespective of boundary conditions. Also, boundary conditions for the fourth-order partial differential equation can be derived from the boundary conditions for two original second-order partial differential equations.

No AccessEngineering NotesEffects of Structural Flexibility on Flight Control of a QuadrotorChristopher S. Smith and Alok SinhaChristopher S. SmithPennsylvania State University, University Park, Pennsylvania 16802*Researcher, Applied Research Lab at Penn State; . Member AIAA (Corresponding Author).Search for more papers by this author and Alok SinhaPennsylvania State University, University Park, Pennsylvania 16802†Professor of Mechanical Engineering; . Associate Fellow AIAA.Search for more papers by this authorPublished Online:18 Jul 2023https://doi.org/10.2514/1.G007550SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Ahuja V., Hartfield R. and Chakraborty I., “Gust Response Analysis for Early Design of Advanced Air Vehicle Concepts,” APISAT 2019: Asia Pacific International Symposium on Aerospace Technology, Engineers Australia, Gold Coast, Australia, 2019. 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A. and Paglione P., “Aircraft Control Based on Flexible Aircraft Dynamics,” Journal of Aircraft, Vol. 54, No. 1, 2017, pp. 262–271. https://doi.org/10.2514/1.C033834 LinkGoogle Scholar[11] Dillsaver M., Cesnik C. and Kolmanovsky I., “Gust Load Alleviation Control for Very Flexible Aircraft,” AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2011-6368, 2011. LinkGoogle Scholar[12] Qi P. and Zhao X., “Flight Control for Very Flexible Aircraft Using Model-Free Adaptive Control,” Journal of Guidance, Control, and Dynamics, Vol. 43, No. 3, 2020, pp. 608–619. https://doi.org/10.2514/1.G004761 LinkGoogle Scholar[13] Gevarter W. B., “Basic Relations for Control of Flexible Vehicles,” AIAA Journal, Vol. 8, No. 4, 1970, pp. 666–672. https://doi.org/10.2514/3.5739 LinkGoogle Scholar[14] Stevens B. L., Lewis F. L. and Johnson E. N., Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems, Wiley, Hoboken, NJ, 2015, Chaps. 1–2. CrossrefGoogle Scholar[15] Bangura M., Melega M., Naldi R. and Mahony R., “Aerodynamics of Rotor Blades for Quadrotors,” arXiv preprint, arXiv:1601.00733, 2016. Google Scholar[16] Bristeau P.-J., Martin P., Salaün E. and Petit N., “The Role of Propeller Aerodynamics in the Model of a Quadrotor UAV,” 2009 European Control Conference (ECC), IEEE, New York, 2009. Google Scholar[17] Powers C., Mellinger D. and Kumar V., “Quadrotor Kinematics and Dynamics,” Handbook of Unmanned Aerial Vehicles, Springer, Dordrecht, 2015, pp. 307–328. CrossrefGoogle Scholar[18] Leishman J. G., Principles of Helicopter Aerodynamics, Cambridge Univ. Press, Cambridge, England, U.K., 2002, Chaps. 2–8. Google Scholar[19] Franklin G. F., Powell J. D. and Emami-Naeini A., Feedback Control of Dynamic Systems, Prentice–Hall, Upper Saddle River, NJ, 2019, Chap. 10. Google Scholar[20] Corke P., Robotic Vision and Control, Springer, Berlin, 2017, Chap. 4. Google Scholar[21] Pounds P. E. I., “Design, Construction and Control of a Large Quadrotor Micro Air Vehicle,” Ph.D. Thesis, Australian National Univ., Canberra, Australia, 2007. Google Scholar[22] Bouabdallah S., Noth A. and Siegwart R., “PID vs LQ Control Techniques Applied to an Indoor Micro Quadrotor,” 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(IEEE Cat. No. 04CH37566), No. 3, IEEE, New York, 2004, pp. 2451–2456. Google Scholar[23] García R. A., Rubio F. R. and Ortega M. G., “Robust PID Control of the Quadrotor Helicopter,” IFAC Proceedings Volumes, Vol. 45, No. 3, 2012, pp. 229–234. https://doi.org/10.3182/20120328-3-IT-3014.00039 CrossrefGoogle Scholar[24] O’Brien R. T. and Howe J. M., “Optimal PID Controller Design Using Standard Optimal Control Techniques,” Proceedings of the American Control Conference, IEEE, New York, 2008. Google Scholar[25] Miranda-Colorado R. and Aguilar L. 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All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAircraft Components and StructureAircraft ControlAircraft DesignAircraft Flight Control SystemAircraft Operations and TechnologyAircraft Stability and ControlAircraftsControl TheoryFlexible and Active StructuresGuidance, Navigation, and Control SystemsMechanical and Structural VibrationsQuadcopterStructural Design and DevelopmentStructural EngineeringStructural Kinematics and DynamicsStructures, Design and TestUnmanned Aerial Vehicle KeywordsQuadcopterAircraft Flight Control SystemLinear Time Invariant SystemStructural Kinematics and DynamicsFlexible and Active StructuresAircraft Components and StructureAerodynamic AnalysisMechanical and Structural VibrationsFlight DynamicsDigital Received16 February 2023Accepted9 June 2023Published online18 July 2023

This paper has illustrated the potential accuracy of often-used Blade Element Momentum Theory (BEMT) model in hovering thrust predictions and a limiting feature of the Blade Element Theory (BET) model during forward flight. When employed in hover, BEMT with the linear lift curve slope assumption and small angle approximation is shown to accurately predict the hover thrust. However, during forward flight an underprediction of the thrust and torque is likely to occur because the lift boost from dynamic stall and the extra torque due to post stall drag are missing. Additionally, without a dynamic stall model the rotor flapping angles for a quadrotor aircraft can contain higher frequencies that are likely fictitious. A state space dynamic stall model has been integrated into BET and it has been shown that reasonable predictions of lift and rotor flapping angle are produced. Finally, a reduced order model amenable to inclusion in 6DOF flight simulations is presented. The reduced order model seeks to make inclusion of the often-neglected time varying thrust due to blade flapping. It also includes the effects of dynamic stall on the thrust generated by the rotor.

No AccessTechnical NotesFlutter Response of a One-Dimensional Panel with Arbitrary Nonuniformities and DiscontinuitiesAlok Sinha and Caleb J. BarnesAlok SinhaPennsylvania State University, University Park, Pennsylvania 16802*Professor of Mechanical Engineering; . Associate Fellow AIAA.Search for more papers by this author and Caleb J. BarnesU.S. Air Force Research Laboratory, Wright–Patterson Air Force Base, Ohio 45433†Aerospace Systems Directorate. Associate Fellow AIAA.Search for more papers by this authorPublished Online:25 Sep 2023https://doi.org/10.2514/1.J062823SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Dowell E. H., Aeroelasticity of Plates and Shells, Noordhoff International, Leyden, The Netherlands, 1974. Google Scholar[2] Dowell E. H., A Modern Course in Aeroelasticity, 5th ed., Springer, Cham, Switzerland, 2015, Chap. 4. 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R., “Development of a Three-Dimensional Viscous Aeroelastic Solver for Nonlinear Panel Flutter,” Journal of Fluids and Structures, Vol. 16, No. 4, 2002, pp. 497–527. https://doi.org/10.1006/jfls.2000.0434 CrossrefGoogle Scholar[7] McNamara J. and Friedmann P. P., “Aeroelastic and Aerothermoelastic Analysis in Hypersonic Flow: Past, Present, and Future,” AIAA Journal, Vol. 49, No. 6, June 2011, pp. 1089–1122. LinkGoogle Scholar[8] Huang D., Rokita T. and Friedmann P.P., “Integrated Aerothermoelastic Analysis Framework with Application to Skin Panels,” AIAA Journal, Vol. 56, No. 11, Nov. 2018, pp. 4562–4581. LinkGoogle Scholar[9] Amato M., Elishakoff I. and Reddy J. N., “Flutter of a Multicomponent Beam in a Supersonic Flow,” AIAA Journal, Vol. 59, No. 11, Nov. 2021, pp. 4342–4353. https://doi.org/10.2514/1.J060631 LinkGoogle Scholar[10] McHugh K. A. and Deaton J. D., “Modal Design Optimization for Panel Flutter and Buckling,” International Forum on Aeroelasticity and Structural Dynamics (IFASD), Curran Associates, Red Hook, NY, June 2022. Google Scholar[11] Kuo C.C., Morino L. and Dugundji J., “Perturbation and Harmonic Balance Methods for Nonlinear Panel Flutter,” AIAA Journal, Vol. 10, No. 11, 1972, pp. 1479–1484. LinkGoogle Scholar[12] Sinha A., “A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures with Arbitrary Nonuniformities,” Journal of Computational and Nonlinear Dynamics, Vol. 15, Nov. 2020, Paper 111004. https://doi.org/10.1115/1.4048360 Google Scholar[13] Sinha A., “Free Vibration of an Euler-Bernoulli Beam with Arbitrary Nonuniformities and Discontinuities,” AIAA Journal, Vol. 59, No. 11, 2021, pp. 4805–4808. https://doi.org/10.2514/1.J060745 LinkGoogle Scholar[14] Sinha A., “Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam,” Journal of Computational and Nonlinear Dynamics, Vol. 17, April 2022, Paper 041001. Google Scholar[15] Newmark N. M., “A Method of Computation for Structural Dynamics,” Journal of the Engineering Mechanics Division, Vol. 85, No. 3, 1959, pp. 67–94. CrossrefGoogle Scholar[16] van Dyke M., “A Study of Second-Order Supersonic Flow Theory,” NACA TR 1081, 1951. Google Scholar[17] MATLAB, Software Package, R2021b, The MathWorks, Natick, MA, 2019, www.mathworks.com. Google Scholar[18] Meijer M. and Dala L., “Generalized Formulation and Review of Piston Theory for Airfoils,” AIAA Journal, Vol. 54, No. 1, 2016, pp. 17–27. LinkGoogle Scholar Previous article FiguresReferencesRelatedDetails What's Popular Volume 61, Number 12December 2023 CrossmarkInformationThis material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. Topics Aerodynamics Aerospace Sciences Computational Fluid Dynamics Continuum Mechanics Finite Element Method Fluid Dynamics Fluid Mechanics Fluid Structure Interaction Material Properties Materials and Structural Mechanics Mechanical Properties Numerical Analysis Solid Mechanics Structural Dynamics and Characterization Structural Mechanics KeywordsComputational Fluid DynamicsStructural Dynamics and CharacterizationPiston TheoryAerodynamic ForceContinuum MechanicsFluid Structure InteractionNumerical IntegrationMechanical PropertiesComputer Programming and LanguageFinite Element MethodAcknowledgmentsAlok Sinha gratefully acknowledges the 2022 U.S. Air Force Summer Faculty Fellowship. This fellowship was administered by System Plus, Inc. under the U.S. Air Force Office of Scientific Research. The work of C. Barnes was supported in part by the U.S. Air Force Office of Scientific Research under a laboratory task monitored by G. Abate. The authors are also grateful for several helpful conversations with Kevin McHugh at the U.S. Air Force Research Laboratory and Jack McNamara at The Ohio State University. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government.PDF Received16 January 2023Accepted24 August 2023Published online25 September 2023

Natural frequencies and mode shapes of a Timoshenko beam with arbitrary non-uniformities, discontinuities, discrete spring/mass constraints and boundary conditions are computed by developing a new method based on the state transition matrix for spatially varying state equations. Algorithms to treat discontinuities in material and geometrical properties and discontinuities due to discrete spring constraints are clearly presented. Equations for natural frequencies and mode shapes are derived for following boundary conditions: clamped-free, attached springs/masses at both ends and pinned–pinned. Numerical results are presented and compared to those in the existing literature.

Profiling and cancer risk assessment on the polycyclic aromatic hydrocarbons (PAHs) content of coal ashes produced by the major coal combustion plants from the eastern coalfield region in India was conducted. Thirteen PAHs were detected on coal ashes collected from ash deposition sites of major thermal power plants and the profiling of the PAHs was done. Benzo[a]pyrene equivalents (BaPeq) for individual PAHs were calculated and applied to the probabilistic assessment model from US EPA (1989). Monte Carlo simulations were conducted to assess the risk of inhabitants exposed to PAHs through the dust of the coal ash deposition site. In fly ash, the range of total amount of carcinogenic PAHs was from 3.50 to 6.72 µg g-1 and for the bottom ash, the range was 8.49 to 14.91 µg g-1. Bottom ashes were loaded with ample amounts of 5- and 6-ring carcinogenic PAHs, whereas fly ashes were dominated by medium molecular weight PAHs. The simulated mean cancer risks from fly ashes were 2.187 E-06 for children and 3.749 E-06 for adults. For the case of bottom ash, the mean risks were 1.248 E-05 and 2.173 E-05 respectively for children and adults. Among all the three exposure routes, dermal contact was the major and caused 81% of the total cancer risk. The most sensitive parameters were exposure duration and relative skin adherence factor for soil, which contributed the most to total variation. The 90% risks calculated from the bottom ashes (2.617 E-05 for children and 4.803 E-05 for adults) are marginally above the acceptable limit (>1.000 E-06) according to US EPA. In this study, a comprehensive risk assessment on carcinogenic PAHs present in coal ashes was done for the first time that may be helpful to develop potential strategies against occupational cancer risk.

The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam's cross section.

No AccessTechnical NotesSolving Fundamental Partial Differential Equation for Mistuned Bladed Disk VibrationAlok SinhaAlok SinhaThe Pennsylvania State University, University Park, Pennsylvania 16802*Professor, Mechanical Engineering; . Associate Fellow AIAA.Search for more papers by this authorPublished Online:11 Feb 2022https://doi.org/10.2514/1.J061150SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Sinha A., Vibration of Nearly Periodic Structures and Mistuned Bladed Rotors, Cambridge Univ. Press, New York, 2017. CrossrefGoogle Scholar[2] Mignolet M. P., Hu W. and Jadic I., “On the Forced Response of Harmonically and Partially Mistuned Bladed Disks. Part I: Harmonic Mistuning,” International Journal of Rotating Machinery, Vol. 6, No. l, 2000, pp. 29–41. https://doi.org/10.1155/S1023621X0000004X Google Scholar[3] Feiner D. M. and Griffin J. H., “A Fundamental Model of Mistuning for a Single Family of Modes,” ASME Journal of Turbomachinery, Vol. 124, No. 4, Oct. 2002, pp. 597–605. https://doi.org/10.1115/1.1508384 CrossrefGoogle Scholar[4] Huang W.-H., “Vibration of Some Structures with Periodic Random Parameters,” AIAA Journal, Vol. 20, No. 7, 1982, pp. 1001–1008. https://doi.org/10.2514/3.51159 LinkGoogle Scholar[5] Kenyon J. A. and Griffin J. H., “Forced Response of Turbine Engine Bladed Disks and Sensitivity to Harmonic Mistuning,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 125, Jan. 2003, pp. 113–120. https://doi.org/10.1115/1.1498269 CrossrefGoogle Scholar[6] Kovacic I., Rand R. and Sah S. M., “Mathieu’s Equation and Its Generalizations: Overview of Stability Charts and Their Features,” ASME Applied Mechanics Review, Vol. 70, No. 2, 2018, Paper 020802. https://doi.org/10.1115/1.4039144 Google Scholar[7] Sinha A., “A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures with Arbitrary Nonuniformities,” ASME Journal of Computational and Nonlinear Dynamics, Vol. 15, Nov. 2020, Paper 111004. https://doi.org/10.1115/1.4048360 Google Scholar[8] Sinha A., “Free Vibration of Euler-Bernoulli Beam with Arbitrary Non-Uniformities and Discontinuities,” AIAA Journal, Vol. 59, No. 11, 2021, pp. 4805–4808. https://doi.org/10.2514/1.J060745 LinkGoogle Scholar[9] Kailath T., Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. Google Scholar[10] Sinha A., “Computing Natural Frequencies and Mode Shapes of a Non-Uniform Circular Membrane,” Mechanics Research Communication, Vol. 107, July 2020, Paper 103553. https://doi.org/10.1016/j.mechrescom.2020.103553 Google Scholar[11] Coisson R., Vernizzi G. and Yang X., “Mathieu Function and Numerical Solutions of the Mathieu Equation,” IEEE International Workshop on Open-Source Software for Scientific Computations, Guiyang, P. R. China, Sept. 2009. https://doi.org/10.1109/OSSC.2009.5416839 Google Scholar[12] Ruby L., “Applications of the Mathieu Equation,” American Journal of Physics, Vol. 64, No. 1, Jan. 1996, pp. 39–44. https://doi.org/10.1119/1.18290 CrossrefGoogle Scholar[13] Mathieu E., “Memoire sur Le Mouvement Vibratoire d’une Membrane de forme Elliptique,” Journal de Mathématiques Pures et Appliquées, Vol. 13, 1868, pp. 137–203. Google Scholar[14] MATLAB, The MathWorks, Inc., Natick, MA, 2019, https://www.mathworks.com/ Google Scholar[15] MATLAB PDE Toolbox, MathWorks, Inc., Natick, MA, 2017, https://www.mathworks.com/ Google Scholar Previous article FiguresReferencesRelatedDetails What's Popular Volume 60, Number 6June 2022 CrossmarkInformationCopyright © 2022 by the author. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsApplied MathematicsComputational Fluid DynamicsFinite Element MethodFluid DynamicsGeneral PhysicsMathematical AnalysisMechanical and Structural VibrationsStructural EngineeringStructures, Design and Test KeywordsDifferential EquationsComputer ProgramsFinite Element AnalysisFree VibrationsPDF Received6 August 2021Accepted9 January 2022Published online11 February 2022

A general approach of the multi-input shaping for an underdamped flexible structure is developed using the deadbeat control theory, which states that a discrete-time system can be made to reach the desired position in a minimum time by placing all the poles at the origin of z-plane. It is shown that this theory can also be used to minimize sensitivities to uncertainties in natural frequencies and modal damping ratios. Numerical results are presented for a spring mass model, and a finite element model of a tetrahedral truss structure. These examples deal with cases of number of inputs being equal to number of modes, and also non-integer ratio of numbers of states and inputs.

No AccessTechnical NotesFree Vibration of an Euler–Bernoulli Beam with Arbitrary Nonuniformities and DiscontinuitiesAlok SinhaAlok SinhaThe Pennsylvania State University, University Park, Pennsylvania 16802*Professor of Mechanical Engineering; . Associate Fellow AIAA.Search for more papers by this authorPublished Online:14 Jul 2021https://doi.org/10.2514/1.J060745SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Taleb N. J. and Suppiger E. W., “Vibration of Stepped Beams,” Journal of Aerospace Sciences, Vol. 28, No. 4, 1961, pp. 295–298. https://doi.org/10.2514/8.8962 LinkGoogle Scholar[2] Sinha A., “A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures with Arbitrary Nonuniformities,” ASME Journal of Computational and Nonlinear Dynamics, Vol. 15, Nov. 2020, Paper 111004. https://doi.org/10.1115/1.4048360 Google Scholar[3] Elishakoff I., Amato M., Ankitha A. P. and Mazarani A., “Rigorous Implementation of the Galerkin Method for Stepped Structures Needs Generalized Functions,” Journal of Sound and Vibration, Vol. 490, Jan. 2021, Paper 115708. https://doi.org/10.1016/j.jsv.2020.115708 Google Scholar[4] Elishakoff I., Amato M. and Marzani A., “Galerkin’s Method Revisited and Corrected in the Problem of Jaworski and Dowell,” Mechanical Systems and Signal Processing, Vol. 155, June 2021, Paper 107604. https://doi.org/10.1016/j.ymssp.2020.107604 Google Scholar[5] Jang S. K. and Bert C. W., “Free Vibration of Stepped Beams: Exact and Numerical Solutions,” Journal of Sound and Vibration, Vol. 130, No. 2, 1989, pp. 342–346. https://doi.org/10.1016/0022-460X(89)90561-0 CrossrefGoogle Scholar[6] Jaworski J. W. and Dowell E. H., “Free Vibration of a Cantilevered Beam with Multiple Steps: Comparison of Several Theoretical Methods with Experiment,” Journal of Sound and Vibration, Vol. 312, Nos. 4–5, 2008, pp. 713–725. https://doi.org/10.1016/j.jsv.2007.11.010 CrossrefGoogle Scholar[7] Auciello N. M. and Ercolano A., “Exact Solution for the Transverse Vibration of a Beam a Part of Which is a Taper Beam and Other Part is a Uniform Beam,” International Journal of Solids and Structures, Vol. 34, No. 17, 1997, pp. 2115–2129. https://doi.org/10.1016/S0020-7683(96)00136-9 Google Scholar[8] MATLAB, The MathWorks, Inc., Natick, MA, 2019, www.mathworks.com. Google Scholar Previous article Next article FiguresReferencesRelatedDetailsCited bySolving Fundamental Partial Differential Equation for Mistuned Bladed Disk VibrationAlok Sinha11 February 2022 | AIAA Journal, Vol. 60, No. 6 What's Popular Volume 59, Number 11November 2021 CrossmarkInformationCopyright © 2021 by the author. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsBeam (Structures)Computational Fluid DynamicsFlexible and Active StructuresFluid DynamicsMaterial PropertiesMaterials and Structural MechanicsMechanical PropertiesMechanical and Structural VibrationsNumerical AnalysisNumerical InterpolationSolid MechanicsStructural AnalysisStructural EngineeringStructural MechanicsStructures, Design and Test KeywordsEuler Bernoulli BeamsTransverse VibrationsFlexural RigidityYoung's ModulusLinear InterpolationCantilever BeamComputer ProgramsDirac Delta FunctionGalerkin MethodPDF Received26 March 2021Accepted2 June 2021Published online14 July 2021

This paper presents algorithms to compute natural frequencies and mode shapes of a circular membrane with concentric or eccentric non-uniformity as found in many applications such as Indian musical drums. Algorithm for concentric non-uniformity is based on the theory of linear time varying systems whereas that for eccentric non-uniformity is developed by expressing each eigenfunction as a linear combination of eigenfunctions of the corresponding circular membrane with concentric non-uniformity. Numerical results from these approaches are corroborated by comparison with those from a finite element method and existing results in the literature. (C) 2020 Elsevier Ltd. All rights reserved.

No AccessTechnical NotesNew Approach to Compute Buckling Loads and Buckled Shapes of Nonuniform BeamsAlok SinhaAlok SinhaPennsylvania State University, University Park, Pennsylvania 16802*Professor of Mechanical Engineering; . Associate Fellow AIAA.Search for more papers by this authorPublished Online:2 Apr 2020https://doi.org/10.2514/1.J059113SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Timoshenko S. P., Theory of Elastic Stability, McGraw–Hill, New York, 1936. Google Scholar[2] Sri Ram G. and Rao G. B. R., “Buckling of an N-Section Column,” Journal of the Aeronautical Sciences, Vol. 19, No. 1, Jan. 1952, pp. 66–67. https://doi.org/10.2514/8.2153 Google Scholar[3] Sri Ram G., “Euler Load of a Stepped Column—An Exact Formula,” AIAA Journal, Vol. 1, No. 1, 1963, pp. 211–212. https://doi.org/10.2514/3.1493 Google Scholar[4] Gatewood B. 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T., Theory of Vibration with Applications, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1992. Google Scholar[14] Kailath T., Linear Systems, Prentice–Hall, Englewood Cliffs, NJ, 1980. Google Scholar[15] Sinha A., “A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration,” Proceedings of the ASME International Design Engineering Technical Conference, ASME Paper DETC2019-97263, New York, Aug. 2019, pp. 1–10. Google Scholar[16] Bowden N., Brittain S., Evans A. G., Hutchinson J. W. and Whitesides G. M., “Spontaneous Formation of Ordered Structures in Thin Films of Metals Supported on an Elastomeric Polymer,” Nature, Vol. 393, No. 6681, 1998, pp. 146–149. https://doi.org/10.1038/30193 Google Scholar[17] Matlab, MathWorks, Natick, MA, 2019, www.mathworks.com. Google Scholar[18] Darbandi S. M., Firouz-Abadi R. D. and Haddadpour H., “Buckling of Variable Section Columns Under Axial Loading,” Journal of Engineering Mechanics, Vol. 136, No. 4, 2010, pp. 472–476. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000096 Google Scholar Previous article FiguresReferencesRelatedDetailsCited byBuckling and Prestressed Vibrations of Periodic Nonprismatic Beams Using Integral Equation ApproachJitish Miglani and Rakesh K. Kapania22 November 2021 | AIAA Journal, Vol. 60, No. 2 What's Popular Volume 58, Number 5May 2020 CrossmarkInformationCopyright © 2020 by Alok Sinha. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsBeam (Structures)Buckling AnalysisComputational Fluid DynamicsFlexible and Active StructuresFluid DynamicsMaterial PropertiesMaterials and Structural MechanicsMechanical PropertiesNumerical AnalysisNumerical InterpolationSolid MechanicsStress-Strain AnalysisStructural AnalysisStructural EngineeringStructural MechanicsStructures, Design and Test KeywordsBuckling ProblemYoung's ModulusSimply Supported BeamsFlexural RigidityEuler Bernoulli BeamsCritical LoadingMATLABLinear InterpolationBessel FunctionsGalerkin MethodPDF Received3 October 2019Accepted14 February 2020Published online2 April 2020

This paper deals with the estimation of forcing function, modal damping, and mistuned modal stiffnesses in a bladed rotor. Previous research on parameter estimation in a mistuned bladed rotor relies on the knowledge of the forcing function as well as the vibration data. This paper presents two novel approaches. The first approach relies on knowledge of both the forcing and vibration data. The parameters are treated as states of the system and an augmented state space model is created. Unscented Kalman filter (UKF) is then used on the steady-state data to estimate the parameters. The second approach eliminates the dependence on forcing data. Both the forcing and parameters are now treated as states of the system to construct an augmented state space model. UKF is then used on transient vibration data for estimation. Numerical results are presented for a simple model of a mistuned bladed rotor which considers a single mode of vibration per blade.