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The nonlinear effects of spinning on the dynamics of a pitching cantilever

JOURNAL OF SOUND AND VIBRATION 2024年 569卷

作者： Clarabut, David Robinson, Brandon Poirel, Dominique Sarkar, Abhijit Carleton Univ Dept Civil & Environm Engn Ottawa ON K1S 5B6 Canada Royal Mil Coll Canada Dept Mech & Aerosp Engn Kingston ON K7K 7B4 Canada

We investigate the structural dynamics of a pitching cantilever as it spins about a fixed axis, emulating the motion of the rotor blades of wind turbines or rotorcraft to some extent. From first principles, we derive a system of equations consisting of a nonlinear partial differential equation (PDE) coupled with a nonlinear ordinary differential equation (ODE) that describes the motion of the system in bending and pitch as it spins according to a prescribed rotor velocity and acceleration. The equations of motion consider the interaction between dynamics of a flexible body in bending, rigid body motion in pitch, and the nonlinear effects of spinning, concurrently. This simplified model of a rotor blade permits an intuitive understanding of the various nonlinear terms that arise in the equations of motion, and their relative influence on the system dynamics. In a series of numerical experiments, we study the effects of the coupling between the bending and pitch degrees of freedom, the influence of rotor velocity, and rotor acceleration on the system dynamics. We compare the full nonlinear simulation results to those of the underlying linear system, obtained by imposing a small angle assumption in pitch and excluding all remaining nonlinear terms. The numerical experiments are performed by non-dimensionalizing the equations, discretizing the non-dimensionalized equations in space by applying the Galerkin method, discretizing the resulting system of ODEs in time using the Houbolt method and finally solving the resulting system of implicit algebraic equations using Newton's method.

关键词： nonlinear structural dynamics Geometric nonlinearity Rotor blades Spinning cantilever

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The discrete null space method for the energy-consistent integration of constrained mechanical systems. Part III: Flexible multibody dynamics

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MULTIBODY SYSTEM dynamics 2008年第1-2期19卷 45-72页

作者： Leyendecker, Sigrid Betsch, Peter Steinmann, Paul Univ Kaiserslautern Dept Mech Engn Chair Appl Mech Kaiserslautern Germany Univ Siegen Dept Mech Engn Chair Computat Mech Siegen Germany

In the present work, the unified framework for the computational treatment of rigid bodies and nonlinear beams developed by Betsch and Steinmann (Multibody Syst. Dyn. 8, 367-391, 2002) is extended to the realm of nonlinear shells. In particular, a specific constrained formulation of shells is proposed which leads to the semi-discrete equations of motion characterized by a set of differential-algebraic equations (DAEs). The DAEs provide a uniform description for rigid bodies, semi-discrete beams and shells and, consequently, flexible multibody systems. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. The present approach thus circumvents the use of rotational variables throughout the whole time discretization, facilitating the design of energy-momentum methods for flexible multibody dynamics. After the discretization has been completed a size-reduction of the discrete system is performed by eliminating the constraint forces. Numerical examples dealing with a spatial slider-crank mechanism and with intersecting shells illustrate the performance of the proposed method.

关键词： conserving time integration constrained mechanical systems flexible multibody dynamics nonlinear structural dynamics differential-algebraic equations

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Development of a family of explicit algorithms for structural dynamics with unconditional stability

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nonlinear dynamics 2014年第4期77卷 1157-1170页

作者： Gui, Yao Wang, Jin-Ting Jin, Feng Chen, Cheng Zhou, Meng-Xia Tsinghua Univ State Key Lab Hydrosci & Engn Beijing 100084 Peoples R China San Francisco State Univ Sch Engn San Francisco CA 94132 USA

A new family of explicit integration algorithms is developed based on discrete control theory for solving the dynamic equations of motion. The proposed algorithms are explicit for both displacement and velocity and require no factorisation of the damping matrix and the stiffness matrix. Therefore, for a system with nonlinear damping and stiffness, the proposed algorithms are more efficient than the common explicit algorithms that provide only explicit displacement. Accuracy and stability properties of the proposed algorithms are analysed theoretically and verified numerically. Certain subfamilies are found to be unconditionally stable for any system state (linear elastic, stiffness softening or stiffness hardening) that may occur in earthquake engineering of a practical structure. With dual explicit expression and excellent stability property, the proposed family of algorithms can potentially solve complicated nonlinear dynamic problems.

关键词： Explicit algorithm nonlinear structural dynamics Stability Computational efficiency Discrete transfer function

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A novel predictor-corrector explicit integration scheme for structural dynamics

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STRUCTURES 2021年 34卷 2735-2745页

作者： Liu, Wei Guo, Wenhua Hunan Inst Sci & Technol Sch Civil Engn Yueyang 414006 Peoples R China Cent South Univ Sch Civil Engn Changsha 410075 Peoples R China

A novel predictor-corrector explicit scheme is presented to solve structural dynamic problems. It is a three sub-steps method, in which the previous two sub-steps are set as predictors and the last sub-step is regarded as correctors. The explicit scheme is third-order accuracy and can achieve forth-order accuracy in the absent of physical damping. The stability limit of the proposed scheme is much larger than the existing methods. Also, the numerical dissipation and dispersion of the explicit scheme can be controlled through the different algorithm parameters. The explicit scheme not only possesses adequate numerical dissipation and dispersion in high-frequency responses, but also gets small numerical errors in the whole frequency domain for dynamic system. These performances of the proposed explicit scheme are further highlighted in comparison with other typical explicit schemes.

关键词： Explicit time integration Desirable numerical dissipation Small numerical errors Linear structural dynamics nonlinear structural dynamics

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dynamics of nonlinear structures with multiple equilibria: A singular perturbation-invariant manifold approach

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ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK 1999年第6期50卷 892-924页

作者： Georgiou, IT Corless, MJ Bajaj, AK USN Res Lab Special Project Nonlinear Sci Washington DC 20375 USA Purdue Univ Sch Mech Engn W Lafayette IN 47907 USA Purdue Univ Sch Aeronaut & Astronaut W Lafayette IN 47907 USA

This work analyzes the motions of a stiff linear oscillator coupled to a soft nonlinear oscillator and subject to a forcing term. This system is a representative of a large class of structural dynamical systems with stiff and soft substructures and with multiple equilibrium states. Using the methodology of singular perturbations and the theory of invariant manifolds, we describe globally (in time) the motions in a finite neighborhood of the origin in phase space. It is shown that, every motion of the system depends on a slowly varying component and a component which rapidly decays with time. The long term behavior of the system is, thus, described by a lower order slow system, which is the restriction of the system to an invariant manifold that contains ail the slow motions of the system. For slow periodic forcing, the slow manifold is periodic. The dynamics of the slow system on the invariant manifold, as well as the effects of the singular perturbation parameter on the validity of the slow manifold approximation are also explored.

关键词： nonlinear structural dynamics singular perturbations slow invariant manifold

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Response of Jointed-Structures in a Shock Tube: Simultaneous PSP and DIC with Comparison to Modeling

Response of Jointed-Structures in a Shock Tube: Simultaneous...

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AIAA Aviation Forum and Exposition

作者： Lynch, Kyle P. Jones, Elizabeth M. Brink, Adam R. Rohe, Daniel P. Kuether, Robert J. Wagner, Justin L. Mathis, Allen Quinn, D. Dane Sandia Natl Labs Tech Staff Engn Sci Ctr Albuquerque NM 87185 USA Univ Akron Dept Mech Engn Akron OH 44325 USA

ISBN: (数字)9781624105890

ISBN: (纸本)9781624105890

Experiments and modeling were used to study the nonlinear dynamics of a jointed-structure in a shock tube. The structure was a full-span square cylinder with internal bolted connections excited by fluid loading. The width-based Reynolds number was approximate to 10(5). The cylinder was exposed to an impulsive force associated with the incident shock followed by transverse loading imposed by vortex shedding. In the experiment, aerodynamic loading was characterized with high-speed pressure sensitive paint (PSP), while digital image correlation (DIC) concurrently measured the structural response. The maximum displacement occurred when the vortex shedding frequency most closely matched the structural mode of the beam associated with rocking motion at the joint. A finite element model was developed using Abaqus, where the nonlinear contact dynamics of the joint were simulated using Coulomb friction. The PSP data loaded the model and the interaction was treated as one-way coupled. The simulations well-matched the trends observed in the experiment. Overall, the root-mean-square values of the transverse displacement agreed to within 24% of the experiment. The modeling showed rocking about the joint during vortex shedding was critical to the nonlinear damping and energy dissipation in the structure. This campaign highlights the importance of jointed-connections to energy dissipation in structures under aerodynamic loading.

关键词： Aerodynamic Loads Pressure Sensitive Paint Strouhal Number Finite Element Analysis structural Displacement Abaqus nonlinear dynamics Contact dynamics nonlinear structural dynamics Frequency Response Functions

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structural dynamics and 3D Flow Characteristics Analysis in High Aspect RatioWings within the Post-Flutter Regime

Structural Dynamics and 3D Flow Characteristics Analysis in ...

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AIAA SciTech Forum

作者： Kuhpar, Mostafa Khazaee Samsam-Khayan, Hadi Mousavisani, Seyedmohammad Seyed-Aghazadeh, Banafsheh Univ Massachusetts Dept Mech Engn 285 Old Westport Rd Dartmouth MA 02747 USA

ISBN: (数字)9781624107115

ISBN: (纸本)9781624107115

This study explores the complex fluid-structure interactions in high-aspect-ratio wings, particularly focusing on their behavior after reaching the flutter threshold under various operational conditions. While such wings offer aerodynamic benefits, they also face aeroelastic challenges like flutter, divergence, and control reversal. The research primarily investigates post-flutter dynamics, through a series of experimental tests, where a flexible airfoil's response in its post-critical phase is studied, considering different angles of attack and flow velocities. Key findings include the significant influence of the angle of attack on the limit cycle oscillations, oscillation frequencies, and mode shapes of the response. Additionally, higher flow velocities tend to increase tip deflection, especially at high angles of attack, while smaller angles of attack show lesser deflection. The study also examines the three-dimensional flow around the airfoil using Particle Tracking Velocimetry (PTV). This analysis unveils that variations in attack angle and flow velocity affect the flow separation and vortex dynamics in the wake of the airfoil. These, in turn, impact their interaction with the structural modes of the wing. The research highlights how the wing's deflection alters the behavior of leading and trailing edge vortices.

关键词： High Aspect Ratio nonlinear structural dynamics Flow Characteristics Aerodynamic Performance Vortex Structure Flow Induced Vibrations NACA 0012 Particle Tracking Velocimetry Fluid Structure Interaction Flexible Wings

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A Case Study for Integrating Comp/Sim Credibility and Convolved UQ and Evidence Theory Results to Support Risk Informed Decision Making 36th

A Case Study for Integrating Comp/Sim Credibility and Convol...

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36th International Modal Analysis Conference and Exposition (IMAC) on structural dynamics

作者： Orient, G. Babuska, V Lo, D. Mersch, J. Wapman, W. Sandia Natl Labs POB 5800 Albuquerque NM 87185 USA

ISBN: (纸本)9783319747934;9783319747927

A case study highlighting the computational steps to establish credibility of a solid mechanics model and to use the compiled evidence to support quantitative program decisions is presented. An integrated modeling and testing strategy at the commencement of the CompSim (Computational Simulation) activity establishes the intended use of the model and documents the modeling and test integration plan. A PIRT (Phenomena Identification and Ranking Table) is used to identify and prioritize physical phenomena and perform gap analysis in terms of necessary capabilities and production-level code feature implementations required to construct the model. At significant stages of the project a PCMM (Predictive Capability Maturity Model) assessment, which is a qualitative expert elicitation based process, is performed to establish the rigor of the CompSim modeling effort. These activities are necessary conditions for establishing model credibility, but they are not sufficient because they provide no quantifiable guidance or insight about how to use and interpret the modeling results for decision making. This case study describes a project to determine the critical impact velocity beyond which a device is no longer guaranteed to function. Acceleration, weld failure and deformation based system integrity metrics of an internal structure are defined as QoIs (Quantities of Interest). A particularly challenging aspect of the case study is that predictiveness of the model for different QoIs is expected to vary. A solid mechanics model is constructed observing program resource limitations and analysis governance principles. An inventory of aleatory, computational and model form uncertainties is assembled, and strategies for their characterization are established. Formal UQ (Uncertainty Quantification) over the aleatory random variables is performed. Validation metrics are used to evaluate discrepancies between model and test data. At this point, the customers and the CompSim team agree

关键词： Computational simulation credibility nonlinear structural dynamics Uncertainty quantification Evidence theory Risk informed decision

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Parametric nonlinearity Identification of a Gearbox from Measured Frequency Response Data 1

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32 International Modal Analysis Conference on structural dynamics

作者： Aykan, Murat Altuntop, Elif ASELSAN Inc Def Syst Technol Div TR-06172 Ankara Turkey

ISBN: (数字)9783319045221

ISBN: (纸本)9783319045221;9783319045214

structural nonlinearity is commonly encountered at mechanical connections such as bearings and gears under dynamic loading. Usually, linear approximations of the nonlinearity will yield acceptable results. However, when the nonlinearity is dominant, the nonlinear analysis becomes unavoidable. Most of the time, in engineering assemblies the whole design is too complex to include the nonlinearity in the model. Then it becomes necessary to simplify the structure in order to analyze the nonlinear element separately. In this study, a method developed in an earlier work is implemented on a test rig containing gearbox. The method is capable of parametrically identifying nonlinearities from measured frequency response functions. In this paper, it is aimed to present the validity of the method by applying it to a real test structure and thus parametrically identifying the nonlinear element in the system to obtain a mathematical model, and then employing the model in harmonic response analysis of the system in order to compare predicted responses with measured ones.

关键词： nonlinear structural dynamics Parametric nonlinearity identification nonlinear FRF Experimental verification nonlinear vibration testing

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Numerically Assessing the Relative Significance of nonlinear Normal Modes to Forced Responses 36th

Numerically Assessing the Relative Significance of Nonlinear...

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36th International Modal Analysis Conference and Exposition (IMAC) on structural dynamics

作者： Hill, T. L. Neild, S. A. Cammarano, A. Univ Bristol Dept Mech Engn Bristol Avon England Univ Glasgow Sch Engn Glasgow Lanark Scotland

ISBN: (纸本)9783319742809;9783319742793

nonlinear normal modes, which describe the unforced, undamped responses of nonlinear systems, are often used for understanding the dynamic behaviour of nonlinear structures in engineering. As such, a key property of nonlinear normal modes (NNMs) is that they relate to the dynamic behaviour of the system when forcing and damping are applied. Previous work has shown that an extremely large number of NNMs may be predicted for relatively simple systems;however only a small number of these NNMs have a meaningful influence on the forced and damped dynamics. As such, a method for determining which NNMs are significant (i.e., strongly relate to the forced dynamics) and which are not is crucial for the use of NNMs as a tool for understanding nonlinear systems. As shown in the literature, NNMs only relate to the forced dynamics in the presence of a mechanism for energy transfer between resonating components of the system. Whilst this provides an understanding of the mechanism, computing which NNMs exhibit this property requires detailed analytical analysis. In this paper, we discuss a numerical approach for determining the degree of significance, with respect to the forced responses, of the NNMs. Such an approach removes the need for analytical investigation and provides an efficient and practical approach to determining the significance of the NNMs.

关键词： nonlinear normal modes nonlinear structural dynamics Backbone curves Energy balancing nonlinear beam

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