scholarly journals Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120474 ◽  
Author(s):  
Marco Amabili
Keyword(s):  
Cylindrical Shell ◽  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Shear Deformation Theory ◽  
Natural Mode ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Natural Modes ◽  
Admissible Functions

Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a two-step analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

A New Nonlinear Higher-Order Shear Deformation Theory for Nonlinear Vibrations of Laminated Shells

2010 ◽  
Author(s):  
M. Amabili ◽  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Curvilinear Coordinates ◽  
Geometrically Nonlinear ◽  
Geometric Imperfections ◽  
Laminated Shells

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape; taking geometric imperfections into account. The geometrically nonlinear strain-displacement relationships are derived retaining full nonlinear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only nonlinear terms of the von Ka´rma´n type. Results show that inaccurate results are obtained by keeping only nonlinear terms of the von Ka´rma´n type for vibration amplitudes of about two times the shell thickness for the studied case.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

scholarly journals A Reduced Order Model of Mistuning Using a Subset of Nominal System Modes

1999 ◽  
Author(s):  
M.-T. Yang ◽  
J. H. Griffin
Keyword(s):  
Degrees Of Freedom ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Order Model ◽  
Disk System ◽  
Element Analysis ◽  
Reduced Order ◽  
Nominal System ◽  
Harmonic Response ◽  
Bladed Disk

Reduced order models have been reported in the literature that can be used to predict the harmonic response of mistuned bladed disks. It has been shown that in many cases they exhibit structural fidelity comparable to a finite element analysis of the full bladed disk system while offering a significant improvement in computational efficiency. In these models the blades and disk are treated as distinct substructures. This paper presents a new, simpler approach for developing reduced order models in which the modes of the mistuned system are represented in terms of a sub-set of nominal system modes. It has the following attributes: the input requirements are relatively easy to generate; it accurately predicts mistuning effects in regions where frequency veering occurs; as the number of degrees of freedom increases it converges to the exact solution; it accurately predicts stresses as well as displacements; and it accurately models the deformation and stresses at the blades’ bases.

Some General Considerations on the Natural Mode Technique

1969 ◽  
Vol 73 (700) ◽  
pp. 361-368 ◽  
Author(s):  
J. H. Argyris ◽  
D. W. Scharpf
Keyword(s):  
Degrees Of Freedom ◽  
General Procedure ◽  
Practical Significance ◽  
Natural Mode ◽  
Curved Beams ◽  
Normal Forces ◽  
Natural Modes ◽  
Straight Beam ◽  
Mode Technique ◽  
Made In

We continue our fundamental discourse on the natural modes in an element and extend our considerations to large displacements. First, we present a general procedure for establishing the so-called geometrical stiffness k G , when the natural modes of the complete element are given. The theory is a substantial generalisation and clarification of the method initially given in refs. 2 and 3 and shows that in establishing the modification to the transformation matrix a N it is necessary to ignore the contribution of the natural modes, which cause rotation at the nodal points but no displacement there. The point is subtle and was not made in ref. 2, although the applications given there are correct. As an example, the geometrical stiffness of a straight beam in space with all degrees of freedom is established. There follows the extension of the geometrical stiffness concept to a sub-element. This is of great practical significance for two reasons. First, it allows to derive the geometrical stiffness of elements of complex shape and behaviour, e.g. curved beams in space subjected to normal forces, bending moments and torque.

Geometric Mistuning Reduced-Order Models for Integrally Bladed Rotors With Mistuned Disk–Blade Boundaries

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
Joseph A. Beck ◽  
Jeffrey M. Brown ◽  
Alex A. Kaszynski ◽  
Charles J. Cross ◽  
Joseph C. Slater
Keyword(s):  
Degrees Of Freedom ◽  
Principal Component ◽  
Element Model ◽  
Forced Response ◽  
Solid Model ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Small Areas ◽  
Eigen Analysis ◽  
Connection Type

New geometric mistuning modeling approaches for integrally bladed rotors (IBRs) are developed for incorporating geometric perturbations to a fundamental disk–blade sector, particularly the disk–blade boundary or connection. Reduced-order models (ROMs) are developed from a Craig–Bampton component mode synthesis (C–B CMS) framework that is further reduced by a truncated set of interface modes that are obtained from an Eigen-analysis of the C–B CMS constraint degrees of freedom (DOFs). An investigation into using a set of tuned interface modes and tuned constraint modes for model reduction is then performed, which offers significant computational savings for subsequent analyses. Two configurations of disk–blade connection mistuning are investigated: as-measured principal component (PC) deviations and random perturbations to the interblade spacing. Furthermore, the perturbation sizes are amplified to investigate the significance of incorporating mistuned disk–blade connections during solid model generation from optically scanned geometries. Free and forced response results are obtained for each ROM and each disk–blade connection type and compared to full finite element model (FEM) solutions. It is shown that the developed methods provide accurate results with a reduction in solution time compared to the full FEM. In addition, results indicate that the inclusion of a mistuned disk–blade connection deviations are small or conditions where large perturbations are localized to a small areas of the disk–blade connection.

Detection of Cracks in Mistuned Bladed Disks Using Reduced-Order Models and Vibration Data

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Chulwoo Jung ◽  
Akira Saito ◽  
Bogdan I. Epureanu
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Savings ◽  
Forced Response ◽  
Response Analysis ◽  
Reduced Order Models ◽  
Reduced Order

A novel methodology to detect the presence of a crack and to predict the nonlinear forced response of mistuned turbine engine rotors with a cracked blade and mistuning is developed. The combined effects of the crack and mistuning are modeled. First, a hybrid-interface method based on component mode synthesis is employed to develop reduced-order models (ROMs) of the tuned system with a cracked blade. Constraint modes are added to model the displacements due to the intermittent contact between the crack surfaces. The degrees of freedom (DOFs) on the crack surfaces are retained as active DOFs so that the physical forces due to the contact/interaction (in the three-dimensional space) can be accurately modeled. Next, the presence of mistuning in the tuned system with a cracked blade is modeled. Component mode mistuning is used to account for mistuning present in the uncracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reduced-order) nonlinear equations of motion are solved by applying an alternating frequency/time-domain method. Using these efficient ROMs in a forced response analysis, it is found that the new modeling approach provides significant computational cost savings, while ensuring good accuracy relative to full-order finite element analyses. Furthermore, the effects of the cracked blade on the mistuned system are investigated and used to detect statistically the presence of a crack and to identify which blade of a full bladed disk is cracked. In particular, it is shown that cracks can be distinguished from mistuning.

scholarly journals Comparative Study of Blades Reduced Order Models With Geometrical Nonlinearities and Contact Interfaces

2020 ◽  
Author(s):  
Elise Delhez ◽  
Florence Nyssen ◽  
Jean-Claude Golinval ◽  
Alain Batailly
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Performance Indicator ◽  
Element Model ◽  
Reduced Order Models ◽  
Computationally Efficient ◽  
Reduced Order ◽  
Reduction Methods ◽  
Blade Tip ◽  
Reduced Space

Abstract This paper investigates the use of different model reduction methods accounting for geometric nonlinearities. These methods are adapted to retain physical degrees-of-freedom in the reduced space in order to ease contact treatment. These reduction methods are applied to a 3D finite element model of an industrial compressor blade (NASA rotor 37). In order to compare the different reduction methods, a scalar indicator is defined. This performance indicator allows to quantify the accuracy of the predicted displacement both locally (at the blade tip) and globally. The robustness of each method with respect to variations of the external excitation is also assessed. The performances of the reduction methods are then compared in the case of frictional contact between the blade tip and the surrounding casing. This work brings evidence that reduced order models provide a computationally efficient alternative to full order finite element models for the accurate prediction of the time response of structures with both distributed and localized nonlinearities.

Reduced Order Models of Viscous Flows in Turbomachinery Using Proper Orthogonal Decomposition

1999 ◽  
Author(s):  
Bogdan I. Epureanu ◽  
Earl H. Dowell ◽  
Kenneth C. Hall
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Frequency Domain ◽  
Degrees Of Freedom ◽  
Orthogonal Decomposition ◽  
Full Potential ◽  
Reduced Order Models ◽  
Full System ◽  
Integral Boundary ◽  
Reduced Order ◽  
Proper Orthogonal

Abstract The proper orthogonal decomposition technique is applied in the frequency domain to obtain reduced order models (ROM) of the flow in a cascade of airfoils. The flow is described by a inviscid-viscous interaction model where the inviscid part is described by the full potential equation and the viscous part is described by an integral boundary layer model. The fully nonlinear steady flow is computed and the unsteady flow is linearized about the steady solution. A frequency domain model is constructed and validated showing to provide similar results when compared with previous computational and experimental data presented in the literature. A cascade of airfoils forming a slightly modified Tenth Standard Configuration is numerically investigated. We show that the ROMs with only 10 to 40 degrees of freedom predict accurately the unsteady response of the full system with approximately 10,000 degrees of freedom for the subsonic case. We also show that the ROMs with 15 to 75 degrees of freedom predict accurately the unsteady response of the full system with approximately 17, 500 degrees of freedom for the transonic case. The ROMs are shown to be accurate both for a broad range of reduced frequencies and a full spectrum of interblade phase angles.

Multiscale, Multiphenomena Modeling and Simulation at the Nanoscale: On Constructing Reduced-Order Models for Nonlinear Dynamical Systems With Many Degrees-of-Freedom

2003 ◽  
Vol 70 (3) ◽  
pp. 328-338 ◽  
Author(s):  
E. H. Dowell ◽  
D. Tang
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Total System ◽  
Reduced Order Models ◽  
Efficient Computation ◽  
Linear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Value Decomposition

The large number of degrees-of-freedom of finite difference, finite element, or molecular dynamics models for complex systems is often a significant barrier to both efficient computation and increased understanding of the relevant phenomena. Thus there is a benefit to constructing reduced-order models with many fewer degrees-of-freedom that retain the same accuracy as the original model. Constructing reduced-order models for linear dynamical systems relies substantially on the existence of global modes such as eigenmodes where a relatively small number of these modes may be sufficient to describe the response of the total system. For systems with very many degrees-of-freedom that arise from spatial discretization of partial differential equation models, computing the eigenmodes themselves may be the major challenge. In such cases the use of alternative modal models based upon proper orthogonal decomposition or singular value decomposition have proven very useful. In the present paper another facet of reduced-order modeling is examined, i.e., the effects of “local” nonlinearity at the nanoscale. The focus is on nanoscale devices where it will be shown that a combination of global modal and local discrete coordinates may be most effective in constructing reduced-order models from both a conceptual and computational perspective. Such reduced-order models offer the possibility of reducing computational model size and cost by several orders of magnitude.

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