scholarly journals Obtaining Mode Shapes through the Karhunen-Loève Expansion for Distributed-Parameter Linear Systems

2002 ◽  
Vol 9 (4-5) ◽  
pp. 177-192 ◽  
Author(s):  
Claudio Wolter ◽  
Marcelo A. Trindade ◽  
Rubens Sampaio
Keyword(s):  
Linear Systems ◽  
Random Noise ◽  
Mode Shapes ◽  
Distributed Parameter ◽  
Spectral Technique ◽  
Physical Experiments ◽  
Analysis And Synthesis ◽  
Impulsive Forces ◽  
Mass Spring ◽  
Proper Orthogonal

The Karhunen-Loève expansion is a powerful spectral technique for the analysis and synthesis of dynamical systems. It consists in decomposing a spatial correlation matrix, which can be obtained through numerical or physical experiments. The decomposition produces orthogonal eigenfunctions or proper orthogonal modes, and eigenvalues that provide a measurement of how much energy is contained in each mode. The relation between KL modes and mode shapes of linear vibrating systems has already been derived and demonstrated for two and three dofs mass-spring-damper systems. The purpose of this paper is to extend this investigation to more complex distributed-parameter linear systems. A plane truss and a simply supported plate subjected to impulsive forces, commonly used in modal analysis are studied. The resulting KL modes are compared to the analytical mode shapes. Damping and random noise effects in the procedure performance are evaluated. Two methods for indirectly obtaining natural frequencies are also presented.

scholarly journals Reduced Mass-Weighted Proper Decomposition for Modal Analysis

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Venkata K. Yadalam ◽  
B. F. Feeny
Keyword(s):  
Modal Analysis ◽  
Mass Distribution ◽  
Mass Matrix ◽  
Linear Interpolation ◽  
Modal Identification ◽  
Distributed Parameter ◽  
Arbitrary Mass ◽  
Spring System ◽  
Mass Spring ◽  
Proper Orthogonal

A method of modal analysis by a mass-weighted proper orthogonal decomposition for multi-degree-of-freedom and distributed-parameter systems of arbitrary mass distribution is outlined. The method involves reduced-order modeling of the system mass distribution so that the discretized mass matrix dimension matches the number of sensed quantities, and hence the dimension of the response ensemble and correlation matrix. In this case, the linear interpolation of unsensed displacements is used to reduce the size of the mass matrix. The idea is applied to the modal identification of a mass-spring system and an exponential rod.

Analysis of Unsteady Flow Structures in a Radial Turbomachine by Using Proper Orthogonal Decomposition

2018 ◽  
Author(s):  
Matthias Witte ◽  
Benjamin Torner ◽  
Frank-Hendrik Wurm
Keyword(s):  
Flow Field ◽  
Unsteady Flow ◽  
Proper Orthogonal Decomposition ◽  
Orthogonal Decomposition ◽  
Mode Shapes ◽  
Flow Structures ◽  
Periodic Flow ◽  
Noise Emission ◽  
Turbulent Flow Field ◽  
Proper Orthogonal

Tonalities in hydro and airborne noise emission are a known problem of turbomachines, wherein the tonalities in the noise spectrum are associated with the different orders of the blade passing frequency (BPF). The proper orthogonal decomposition (POD) method was utilized to find the relationship between the fluctuations in the pressure field at the BPF orders which are the origin of the noise emission and the correlated fluctuations in the turbulent velocity field in terms of coherent, periodic flow structures. In order the provide the input data for the POD analysis, a URANS k-ω-SST scale adaptive simulation (SAS) of the turbulent flow field in a single stage radial pump under part load conditions was performed. Compared to traditional two equation turbulence models this approach is less dissipative and allows the development of small scale turbulence structures and is therefore an appropriate method for this study. In order to compute the POD correlation matrix Sirovich’s “Methods of Snapshots” was applied to the unsteady pressure and velocity fields from the CFD simulation. The discrimination of coherent, periodic flow structures and the incoherent, chaotic turbulence was carried out by analyzing the POD eigenvalue distributions, the POD mode shapes and the spectral properties of the POD time coefficients. Five coupled POD mode pairs were identified in total, which were strictly correlated with the 1st, 2nd, 3rd, 4th and 5th order of the BPF and therefore responsible for the noise emission at these discrete frequencies. The coherent structures were explored on the basis of the spatial POD velocity und pressure mode shapes and in terms of vortical structures after an additional phase averaging. The scope of this study is to introduce an enhanced collection of post processing techniques which are capable of analyzing highly unsteady flow fields from numerical simulations in a better way than is possible by just using traditional techniques like the evaluation of integral or time averaged quantities. The identified coherent flow structures and their associated pressure fluctuations are key elements for a proper comprehension of the internal dynamics of the turbulent flow field in a turbomachine and therefore essential for the understanding of the noise generation processes and the optimization of such machines.

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.

A Subharmonic Resonance-Based MEMS Filter

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Multiple Scales ◽  
Subharmonic Resonance ◽  
Mode Shapes ◽  
Center Frequency ◽  
Distributed Parameter ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Reduced Order ◽  
Filter Performance ◽  
Mems Filter

We present a novel micromechanical filter exploiting the subharmonic resonance of order one-half to obtain a center frequency twice the fundamental frequency of the primary resonators, an ideal stopband, and a sharp roll-off. The filter is made up of two clamped-clamped microbeam resonators connected by a coupling beam. We discretize the distributed-parameter system using the Galerkin procedure to obtain a reduced-order model composed of two nonlinear coupled ODEs. It accounts for geometrical and electrical nonlinearities as well as the coupling between these two fields. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We use these equations to determine closed-form expressions for the static and dynamic deflections of the filter. The basis functions in the discretization are the linear undamped global mode shapes of the unactuated filter. The filtering mechanism is based on the exploitation of the interval where the trivial response to subharmonic excitations is unstable. We found criteria to tune the effective nonlinearities of the filter to realize a bandpass filter of an ideal stopband rejection and a sharp roll-off. When these criteria are not met, multivalued responses appear and distort the filter performance.

A Conceptual Flutter Analysis of a Packet of Vanes Using a Mass-Spring Model

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Roque Corral ◽  
Juan Manuel Gallardo ◽  
Carlos Martel
Keyword(s):  
Mode Shape ◽  
Mode Shapes ◽  
Spring Model ◽  
Aerodynamic Forces ◽  
Coupled Mode ◽  
Mass Spring Model ◽  
Stabilizing Effect ◽  
New Type ◽  
Frequency Split ◽  
Mass Spring

The linear aeroelastic stability of a simplified mass-spring model representing the basic dynamics of a packet of Na airfoils has been used to uncover a new type of coupled mode flutter. This simple model retains an essential dynamical feature of the vane packet: the presence of a cluster of Na−1 nearly identical purely structural natural frequencies due to the much larger stiffness of the lower platform as compared to that of the airfoil. Using this model it may be seen that this degeneracy makes the Na−1 associated mode shapes extremely sensible to the addition of small perturbations such as the aerodynamic forces. Since the determination of the aerodynamic vibrational correction (damping and frequency) requires knowing the mode shape, the aerodynamic corrections of the Na−1 cluster modes are now unavoidably coupled together. Moreover, the computation of the aerodynamic correction independently for each structural mode shape leads typically to dangerously overpredicting the stabilizing effect of vane packing. It is shown that the expected stabilizing effect due to the packets may be negligible, depending on the relative frequency split associated with the strength of the aerodynamic forces and realistic structural effects such as the finite stiffness of the lower platform. It is also shown that in these cases, the most unstable mode may be, in a first approximation, very similar to that obtained modeling the stator as a continuous ring.

Vibratory Parameters of Blades From Coordinate Measurement Machine (CMM) Data

2005 ◽  
Author(s):  
Alok Sinha ◽  
Benjamin Hall ◽  
Brice Cassenti ◽  
Gary Hilbert
Keyword(s):  
Finite Element ◽  
Proper Orthogonal Decomposition ◽  
Natural Frequencies ◽  
Orthogonal Decomposition ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Coordinate Measurement ◽  
Coordinate Measurement Machine ◽  
Bladed Rotor ◽  
Proper Orthogonal

This paper deals with the development of a procedure to model geometric variations of blades. Specifically, vibratory parameters of blades are extracted from CMM data on an integrally bladed rotor (IBR). The method is based on proper orthogonal decomposition (POD) of CMM data, solid modeling and finite element techniques. In addition to obtaining natural frequencies and mode shapes of each blade on an IBR, statistics of these modal parameters are also computed and characterized. Numerical results are validated by comparison with experimental results.

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