Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Traveling Waves ◽  
Random Noise ◽  
Mode Shapes ◽  
Wave Speeds ◽  
One Dimensional ◽  
Wave Numbers ◽  
Displacement Profile ◽  
The Time Domain ◽  
Euler Bernoulli Beam ◽  
Complex Modal

A method of complex orthogonal decomposition is summarized for the time-domain, and then formulated and justified for application in the frequency-domain. The method is then applied to the extraction of modes from simulation data of sampled multimodal traveling waves for estimating wave parameters in one-dimensional continua. The decomposition is first performed on a transient nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The decomposition is finally applied to a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler–Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of modal components (i.e., mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. Modal signal energies are also quantified. The results are robust to random noise.

Complex Modal Extraction for Estimating Wave Parameters in One-Dimensional Media

2010 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Group Velocity ◽  
Traveling Waves ◽  
Random Noise ◽  
Mode Shapes ◽  
Transient Wave ◽  
Wave Speeds ◽  
One Dimensional ◽  
Wave Numbers ◽  
Displacement Profile ◽  
Complex Modal

A method of complex orthogonal decomposition is applied to the extraction of modes from simulation data of multi-modal traveling waves in one-dimensional continua. The decomposition of a transient wave is performed on a nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The group velocity is also extracted directly from a decomposition of the traveling envelope of the waveform. The observations from the first two examples are used to help interpret the decomposition of a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler-Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of wave components (i.e. mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. The frequency distribution is then approximated. The results are robust to random noise.

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Rickey A. Caldwell ◽  
Brian F. Feeny
Keyword(s):  
Orthogonal Decomposition ◽  
Bernoulli Beam ◽  
Beam Experiment ◽  
Wave Numbers ◽  
Elastic Cord ◽  
The Impact ◽  
Euler Bernoulli Beam ◽  
Modal Coordinates ◽  
Wave Properties ◽  
Complex Modal

Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

Straked Riser Design With VIVA

2010 ◽  
Author(s):  
Dhyanjyoti Deka ◽  
Paul R. Hays ◽  
Kamaldev Raghavan ◽  
Mike Campbell
Keyword(s):  
Traveling Waves ◽  
Oil And Gas ◽  
Cross Flow ◽  
Oil And Gas Industry ◽  
Response Prediction ◽  
Design Tool ◽  
Mode Shapes ◽  
Fe Model ◽  
Viv Suppression ◽  
Modal Solution

VIVA is a vortex induced vibration (VIV) analysis software that to date has not been widely used as a design tool in the offshore oil and gas industry. VIVA employs a hydrodynamic database that has been benchmarked and calibrated against test data [1]. It offers relatively few input variables reducing the risk of user induced variability of results [2]. In addition to cross flow current induced standing wave vibration, VIVA has the capability of predicting traveling waves on a subsea riser, or a combination of standing and traveling waves. Riser boundary conditions including fixed, pinned, flex joint or SCR seabed interaction can be modeled using springs and dashpots. VIVA calculates riser natural frequencies and mode shapes and also has the flexibility to import external modal solutions. In this paper, the applicability of VIVA for the design of straked steel catenary risers (SCR) and top tensioned risers (TTR) is explored. The use of linear and rotational springs provided by VIVA to model SCR soil interaction and flex joint articulation is evaluated. Comparisons of the VIV fatigue damage output with internal and external modal solution is presented in this paper. This paper includes validation of the VIVA generated modal solution by comparing the modal frequencies and curvatures against a finite element (FE) model of the risers. Fatigue life is calculated using long term Gulf of Mexico (GoM) currents and is compared against the industry standard software SHEAR7. Three different lift curve selections in SHEAR7 are used for this comparison. The differences in riser response prediction by the two software tools are discussed in detail. The sensitivity of the VIVA predicted riser response to the absence of VIV suppression devices is presented in this paper. The riser VIV response with and without external FE generated modal input is compared and the relative merits of the two modeling approaches are discussed. Finally, the recommended approach for VIVA usage for SCR and TTR design is given.

Complex Modal Analysis of Non-Self-Adjoint Hybrid Serpentine Belt Drive Systems

2000 ◽  
Vol 123 (2) ◽  
pp. 150-156 ◽  
Author(s):  
Lixin Zhang ◽  
Jean W. Zu ◽  
Zhichao Hou
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mode Shapes ◽  
Belt Drive ◽  
Serpentine Belt ◽  
Complex Modal Analysis ◽  
Drive Systems ◽  
Complex Modal

A linear damped hybrid (continuous/discrete components) model is developed in this paper to characterize the dynamic behavior of serpentine belt drive systems. Both internal material damping and external tensioner arm damping are considered. The complex modal analysis method is developed to perform dynamic analysis of linear non-self-adjoint hybrid serpentine belt-drive systems. The adjoint eigenfunctions are acquired in terms of the mode shapes of an auxiliary hybrid system. The closed-form characteristic equation of eigenvalues and the exact closed-form solution for dynamic response of the non-self-adjoint hybrid model are obtained. Numerical simulations are performed to demonstrate the method of analysis. It is shown that there exists an optimum damping value for each vibration mode at which vibration decays the fastest.

Vibration of Beam with Elastically Restrained Ends and Rotational Spring-Lumped Rotary Inertia System at Mid-Span

2015 ◽  
Vol 15 (02) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Mojtaba Hozhabrossadati ◽  
Ahmad Aftabi Sani ◽  
Masood Mofid
Keyword(s):  
Technical Note ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Mass System ◽  
Rotational Spring ◽  
Beam System ◽  
Sample Frequency ◽  
Elastically Restrained ◽  
Euler Bernoulli Beam

This technical note addresses the free vibration problem of an elastically restrained Euler–Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses.

scholarly journals Comparison of Two Numerical Algorithms for Computing the Effects of Mistuning of Fan Flutter

2016 ◽  
Author(s):  
L. Salles ◽  
M. Vahdati
Keyword(s):  
Numerical Models ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Mode Shapes ◽  
Minimal Amount ◽  
Aeroelastic System ◽  
Cpu Time ◽  
Influence Coefficients ◽  
Non Linear ◽  
The Time Domain

The aim of this paper is to study the effects of mistuning on fan flutter and to compare the prediction of two numerical models of different fidelity. The high fidelity model used here is a three-dimensional, whole assembly, time-accurate, viscous, finite-volume compressible flow solver. The Code used for this purpose is AU3D, written in Imperial College and validated for flutter computations over many years. To the best knowledge of authors, this is the first time such computations have been attempted. This is due to the fact that, such non-linear aeroelastic computations with mistuning require large amount of CPU time and cannot be performed routinely and consequently, faster (low fidelity) models are required for this task. Therefore, the second model used here is the aeroelastic fundamental mistuning model (FMM) and it based on an eigenvalue analysis of the linearized modal aeroelastic system with the aerodynamic matrix calculated from the aerodynamic influence coefficients. The influence coefficients required for this algorithm are obtained from the time domain non-linear Code by shaking one blade in the datum (tuned) frequency and mode. Once the influence coefficients have been obtained, the computations of aero damping require minimal amount of CPU time and many different mistuning patterns can be studied. The objectives of this work are to: 1. Compare the results between the two models and establish the capabilities/limitations of aeroelastic FMM, 2. Check if the introduction of mistuning would bring the experimental and computed flutter boundaries closer, 3. Establish a relationship between mistuning and damping. A rig wide-chord fan blade, typical of modern civil designs, was used as the benchmark geometry for this study. All the flutter analyses carried out in this paper are with frequency mistuning, but the possible consequences of mistuned mode shapes are briefly discussed at the end of this paper. Only the first family of modes (1F, first flap) is considered in this work. For the frequency mistuning analysis, the 1F frequency is varied around the annulus but the 1F mode shapes remain the same for all the blades. For the mode shape mistuning computations, an FE analysis of the whole assembly different mass blades is performed. The results of this work clearly show the importance of mistuning on flutter. It also demonstrates that when using rig test data for aeroelastic validation of CFD codes, the amount mistuning present must be known. Finally, it should be noted that the aim of this paper is the study of mistuning and not steady/unsteady validation of a CFD code and therefore minimal aerodynamic data are presented.

Vibrations of beams with a breathing crack and large amplitude displacements

2015 ◽  
Vol 230 (1) ◽  
pp. 34-54 ◽  
Author(s):  
Gonçalo Neves Carneiro ◽  
Pedro Ribeiro
Keyword(s):  
Time Domain ◽  
Equations Of Motion ◽  
Shape Functions ◽  
Breathing Crack ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Time Histories ◽  
Linear Effects ◽  
Fourier Spectra ◽  
The Time Domain

The vibrations of beams with a breathing crack are investigated taking into account geometrical non-linear effects. The crack is modeled via a function that reduces the stiffness, as proposed by Christides and Barr (One-dimensional theory of cracked Bernoulli–Euler beams. Int J Mech Sci 1984). The bilinear behavior due to the crack closing and opening is considered. The equations of motion are obtained via a p-version finite element method, with shape functions recently proposed, which are adequate for problems with abrupt localised variations. To analyse the dynamics of cracked beams, the equations of motion are solved in the time domain, via Newmark's method, and the ensuing displacements, velocities and accelerations are examined. For that purpose, time histories, projections of trajectories on phase planes, and Fourier spectra are obtained. It is verified that the breathing crack introduce asymmetries in the response, and that velocities and accelerations can be more affected than displacements by the breathing crack.

Sign in / Sign up

Export Citation Format

Share Document