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 The Exact Frequency Equations for the Euler-Bernoulli Beam Subject to Boundary Damping

2020 ◽  
Vol 25 (2) ◽  
pp. 183-189
Author(s):  
Angela Biselli ◽  
Matthew P. Coleman
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Methods ◽  
Bernoulli Beam ◽  
Boundary Damping ◽  
Numerical Examples ◽  
Wave Numbers ◽  
Complex Wave ◽  
Energy Conserving ◽  
Vibrating Beams ◽  
Euler Bernoulli Beam

The Euler-Bernoulli (E-B) beam is the most commonly utilized model in the study of vibrating beams. The exact frequency equations for this problem, subject to energy-conserving boundary conditions, are well-known; however, the corresponding dissipative problem has been solved only approximately, via asymptotic methods. These methods, of course, are not accurate when looking at the low end of the spectrum. Here, we solve for the exact frequency equations for the E-B beam subject to boundary damping. Numerous numerical examples are provided, showing plots of both the complex wave numbers and the exponential damping rates for the first five frequencies in each case. Some of these results are surprising.

scholarly journals Complex Orthogonal Decomposition Applied to Nematode Posturing

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
B. F. Feeny ◽  
P. W. Sternberg ◽  
C. J. Cronin ◽  
C. A. Coppola
Keyword(s):  
Orthogonal Decomposition ◽  
Brief Assessment ◽  
Complex Modes ◽  
Characteristic Wavelength ◽  
Time Record ◽  
Forward Locomotion ◽  
Locomotion Mode ◽  
Short Time ◽  
Modal Coordinates ◽  
Complex Modal

The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of Caenorhabditis elegans, an intensively studied nematode. The decomposition indicates that the worm has a multimodal posturing behavior, involving a dominant forward locomotion mode, a secondary, steering mode, and likely a mode for reverse motion. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode is estimated in the complex plane. The frequency is obtained from the complex modal coordinate's complex whirl rate of the complex modal coordinate, and from its fast Fourier transform. Short-time decompositions indicate the variation of the wavelength and frequency through the time record.

scholarly journals Complex Modal Decomposition Applied to Nematode Posturing

2009 ◽  
Author(s):  
B. F. Feeny ◽  
P. W. Sternberg ◽  
C. J. Cronin
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Orthogonal Decomposition ◽  
Brief Assessment ◽  
Ensemble Data ◽  
Complex Modes ◽  
Characteristic Wavelength ◽  
Locomotion Mode ◽  
Modal Coordinates ◽  
Complex Modal

The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of a wild Caenorhabditis elegans nematode. The decomposition indicates that the worm has a multi-modal posturing behavior, involving at least a dominant locomotion mode and a secondary, steering mode. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode was estimated in the complex plane. Frequency was obtained from the complex modal coordinate’s complex whirl rate of the complex modal coordinate, and from its fast Fourier transform.

Free Vibrations of Beams with Arbitrary Boundary Conditions Based on Wave Propagation Method

2011 ◽  
Vol 66-68 ◽  
pp. 753-757
Author(s):  
Wan You Li ◽  
Hai Jun Zhou ◽  
Jun Dai ◽  
Bing Lin Lv ◽  
Dong Hua Wang ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Beam Theory ◽  
Free Vibrations ◽  
Bernoulli Beam ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Wave Propagation Method ◽  
Wave Numbers ◽  
Euler Bernoulli Beam

Under the Euler-Bernoulli beam theory, the wave propagation method is used for the vibration analysis of beams with arbitrary boundary conditions. The boundary conditions end the beam could be arbitrary that all the conventional homogeneous beam boundary conditions can be included by setting the stiffnesses of the springs be infinity or zero. In this paper, the flexural displacement of the beam is expressed in the wave propagation form including wave numbers. The wavenumber could be obtained in a known form for conventional boundary conditions. So the results are obtained through the boundary conditions and the known wavenumbers and compared with the numerical results. In order to validate the correctness, results with different stiffness are compared with those obtained by previous published papers.

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.

LMI-based hybrid temporal-spatial differential control design for a class of Euler-Bernoulli beam system

2021 ◽  
pp. 095440622110036
Author(s):  
Jiaqi Zhong ◽  
Xiaolei Chen ◽  
Yupeng Yuan ◽  
Jiajia Tan
Keyword(s):  
Vibration Suppression ◽  
Direct Method ◽  
Recursive Algorithm ◽  
Linear Matrix ◽  
Bernoulli Beam ◽  
Hybrid Controller ◽  
Beam System ◽  
Active Vibration ◽  
Differential Control ◽  
Euler Bernoulli Beam

This paper addresses the problem of active vibration suppression for a class of Euler-Bernoulli beam system. The objective of this paper is to design a hybrid temporal-spatial differential controller, which is involved with the in-domain and boundary actuators, such that the closed-loop system is stable. The Lyapunov’s direct method is employed to derive the sufficient condition, which not only can guarantee the stabilization of system, but also can improve the spatial cooperation of actuators. In the framework of the linear matrix inequalities (LMIs) technology, the gain matrices of hybrid controller can obtained by developing a recursive algorithm. Finally, the effectiveness of the proposed methodology is demonstrated by applying a numerical simulation.

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