Forced Wave Propagation in Elastic Cables With Small Curvature

1995 ◽  
Author(s):  
M. Behbahani-Nejad ◽  
N. C. Perkins
Keyword(s):  
Nonlinear Response ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Transverse Waves ◽  
Propagating Waves ◽  
Distinct Frequency ◽  
Elastic Cables

Abstract This study presents an investigation of the coupled longitudinal-transverse waves that propagate along an elastic cable. The coupling considered derives from the equilibrium curvature (sag) of the cable. A mathematical model is presented that describes the three-dimensional nonlinear response of a long elastic cable. An asymptotic form of this model is derived for the linear response of cables having small equilibrium curvature. Linear in-plane response is described by coupled longitudinal-transverse partial differential equations of motion, which are comprehensively evaluated herein. The spectral relation governing propagating waves is derived using transform methods. In the spectral relation, three qualitatively distinct frequency regimes exist that are separated by two cut-off frequencies. This relation is employed in deriving a Green’s function which is then used to construct solutions for in-plane response under arbitrarily distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves which propagate through the cable, one characterizing extension-compressive deformations (rod-type) and the other characterizing transverse deformations (string-type). These waves may propagate or attenuate depending on wave frequency. The propagation and attenuation of both wave types are highlighted through solutions for an infinite cable subjected to a concentrated harmonic excitation source.

Harmonically Forced Wave Propagation in Elastic Cables With Small Curvature

1997 ◽  
Vol 119 (3) ◽  
pp. 390-397 ◽  
Author(s):  
M. Behbahani-Nejad ◽  
N. C. Perkins
Keyword(s):  
Nonlinear Response ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Periodic Waves ◽  
Transverse Waves ◽  
Propagating Waves ◽  
Elastic Cables

This study presents an investigation of coupled longitudinal-transverse waves that propagate along an elastic cable. The coupling considered derives from the equilibrium curvature (sag) of the cable. A mathematical model is presented that describes the three-dimensional nonlinear response of an extended elastic cable. An asymptotic form of this model is derived for the linear response of cables having small equilibrium curvature. Linear, in-plane response is described by coupled longitudinal-transverse partial differential equations of motion, which are comprehensively evaluated herein. The spectral relation governing propagating waves is derived using transform methods. In the spectral relation, three qualitatively distinct regimes exist that are separated by two cut-off frequencies which are strongly influenced by cable curvature. This relation is employed in deriving a Green’s function which is then used to construct solutions for in-plane response under arbitrarily distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves which propagate through the cable, one characterizing extension-compressive deformations (rod-type) and the other characterizing transverse deformations (string-type). These waves may propagate or attenuate depending on wave frequency. The propagation and attenuation of both wave types are highlighted through solutions for an infinite cable subjected to a concentrated harmonic excitation source.

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.

Dynamic Analysis of Engine-Mount Systems

1992 ◽  
Vol 114 (1) ◽  
pp. 79-83 ◽  
Author(s):  
H. Ashrafiuon ◽  
C. Nataraj
Keyword(s):  
Circular Plate ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Infinite Plate ◽  
Forced Response ◽  
Engine Vibration ◽  
Airplane Engine ◽  
Engine Mount ◽  
Industrial Rubber ◽  
Stiffness And Damping

This paper examines the forced response of an airplane engine supported by an elastic foundation. It is assumed that the vibrations of the engine and the foundation are small enough such that the equations of motion are linear. The engine is modeled as a rigid body connected to the foundation by standard industrial rubber mounts which act as three-dimensional springs with a significant amount of hysteresis damping. Three fundamental models of the foundation are considered; rigid, statically flexible, and dynamically flexible. In the flexible cases, the foundation is modeled as a clamped circular plate, infinite plate, or any structure identified by a finite element stiffness matrix. In all cases, the mass, stiffness, and damping matrices of the engine-mount system are constructed and the frequency response to the rotating unbalance is determined. For the infinite and clamped circular plate cases, analytical methods are used to determine the real and imaginary parts of the flexibility matrix at different frequencies in response to the harmonic forces transmitted to the plate through the rubber mounts. It is shown here that the foundation elasticity may have a significant effect on the engine vibration and the mounting forces transmitted from the engine to the structure. It is also shown that only the dynamic model of the foundation is able to capture the correct response of the system at frequencies close to the foundation’s natural frequencies.

Nonlinear Vibration Analysis of Gas Turbine Bladings With Shroud Coupling

2008 ◽  
Author(s):  
Andreas Hohl ◽  
Christian Siewert ◽  
Lars Panning ◽  
Andreas Kayser
Keyword(s):  
Finite Element ◽  
Nonlinear Vibration ◽  
Gas Turbine ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Forced Response ◽  
Vibrational Amplitude ◽  
Friction Forces ◽  
Mapping Procedure

Rotating blades are subjected to vibrations caused by excitation forces due to a non-homogeneous pressure field of the fluid. Therefore, damping devices like tip shrouds are implemented which reduce the vibrational amplitude and apply additional stiffness and damping to the structure. To predict the resulting vibration response and stresses, a three dimensional contact model has been developed to determine the friction forces. The resulting equations of motion are solved in the frequency domain. The developed method has been implemented in a nonlinear forced response code called DATAR designed for the gas turbine division of Siemens Energy. In this paper, the transfer of common Finite Element models of bladings with shrouds or underplatform dampers to the DATAR code is presented. A mapping procedure based on Finite Element shape functions is used to couple the model with the regular contact grid used in the nonlinear vibration analysis performed with the DATAR code. As a practical example, the vibration behavior of a gas turbine blading with interlocked shrouds is investigated with the developed method.

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

2011 ◽  
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 un-cracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reducedorder) 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.

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

2012 ◽  
Author(s):  
Eulher Chaves Carvalho ◽  
Paulo Batista Gonçalves ◽  
Zenon J. G. N. del Prado
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Torsional Stiffness ◽  
Complex Dynamic ◽  
Runge Kutta Method ◽  
The Galerkin Method

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.

Large Amplitude Forced Vibrations of Simply Supported Thin Cylindrical Shells

1973 ◽  
Vol 40 (2) ◽  
pp. 471-477 ◽  
Author(s):  
J. H. Ginsberg
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Boundary ◽  
Circular Cylindrical Shell ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Inertia Effects ◽  
Modal Expansion ◽  
Wide Range ◽  
Nonlinear Equations Of Motion

The response of a thin circular cylindrical shell to resonant harmonic excitation is examined by a modal expansion approach. The nonlinear strain-displacement relations lead to a nonlinear boundary condition, as well as nonlinear equations of motion. The solution, which retains tangential inertia effects, is obtained by a perturbation technique that yields a consistent first approximation of the nonlinear response. The results are applicable for a wide range of parameters and to cases of excitation near any of the three lowest natural frequencies corresponding to given axial and circumferential wavelengths. For situations where shallow shell theory is valid, the results of previous studies, which were based upon such a theory, are in close agreement.

Dynamic Analysis of Engine-Mount Systems

1991 ◽  
Author(s):  
Hashem Ashrafiuon ◽  
C. Nataraj
Keyword(s):  
Circular Plate ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Infinite Plate ◽  
Forced Response ◽  
Engine Vibration ◽  
Airplane Engine ◽  
Engine Mount ◽  
Industrial Rubber ◽  
Stiffness And Damping

Abstract This paper examines the forced response of an airplane engine supported by an elastic foundation. It is assumed that the vibrations of the engine and the foundation are small enough such that the equations of motion are linear. The engine is modeled as a rigid body connected to the foundation by standard industrial rubber mounts which act as three dimensional springs with a significant amount of hysteresis damping. Three fundamental models of the foundation are considered: rigid, statically flexible, and dynamically flexible. In the flexible cases, the foundation is modeled as a clamped circular plate, infinite plate, or any structure identified by a finite element stiffness matrix. In all cases, the mass, stiffness, and damping matrices of the engine-mount system are constructed and the frequency response to the rotating unbalance is determined. For the infinite and clamped circular plate cases, analytical methods are used to determine the real and imaginary parts of the flexibility matrix at different frequencies in response to the harmonic forces transmitted to the plate through the rubber mounts. It is shown here that the foundation elasticity may have a significant effect on the engine vibration and the mounting forces transmitted from the engine to the structure. It is also shown that only the dynamic model of the foundation is able to capture the correct response of the system at frequencies close to the foundation’s natural frequencies.

scholarly journals Modeling strong motions produced by earthquakes with two-dimensional numerical codes

1988 ◽  
Vol 78 (1) ◽  
pp. 109-121
Author(s):  
Donald V. Helmberger ◽  
John E. Vidale
Keyword(s):  
Wave Equation ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Cylindrical Coordinates ◽  
Dimensional Theory ◽  
Dimensional Solution ◽  
Source Excitation ◽  
Dislocation Sources

Abstract We present a scheme for generating synthetic point-source seismograms for shear dislocation sources using line source (two-dimensional) theory. It is based on expanding the complete three-dimensional solution of the wave equation expressed in cylindrical coordinates in an asymptotic form which provides for the separation of the motions into SH and P-SV systems. We evaluate the equations of motion with the aid of the Cagniard-de Hoop technique and derive close-formed expressions appropriate for finite-difference source excitation.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

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