Efficient Model Correction Method with Modal Measurement

Correction of stiffness and mass matrices utilizing simulated measured modal data

Parameter identification is the process of constructing the mathematical model of a dynamic system based on measured dynamic responses. A non-classical search method namely genetic algorithm (GA) is employed in this study, which has several advantages over classical system identification techniques. Nevertheless, direct application of GA does not necessarily work well, particularly with regards to computational efficiency in fine-tuning when the solution approaches the optimal value. Thus, a local search (LS) method is introduced into the GA approach to improve this capability. Numerical example of a cantilever beam is presented to show the efficiency of the combined GA and LS method (GA-LS). The FE model of the cantilever beam is established with 10 beam elements. In this example, only the element properties associated with flapping motion are identified. The GA-LS method gives much better results than the GA method does. The mean error is reduced to 5.3% from 14.9% while to 14.5% from 47.2% for the maximum individual error. It is concluded that the GA-LS method is a more efficient search method than the GA method. The GA-LS method is then applied to identify the mass and stiffness parameters of a model helicopter blade using measured excitations and accelerations in this paper. Since the mass and stiffness of the model helicopter blade are not available, the only way to verify the identified parameters is to compare the ffequencies and mode shapes calculated from the identified parameters and the measured frequencies and modal shapes of the actual blade. The mean eiror of the frequencies is 0.9% and the maximum individual error 2.4%. The sequence of identified mode shapes is the same with measured mode shapes. Transactions on Engineering Sciences vol 43, © 2003 WIT Press, www.witpress.com, ISSN 1743-3533

Study on flexural vibration of triangular composite plates influenced by fibre orientation

Accelerometers on a bridge can be used to extract dynamic properties such as mode shape which are known to be damage-sensitive. However, as bridges are often physically large, this can require a significant number of sensors. In this paper, the concept of re-deploying a small number of sensors is proposed, building up the mode shape from a series of field acceleration measurements. Frequency Domain Decomposition is used to find segments of mode shape in each measurement, using a different train as the source of excitation in each case. In numerical simulations of a simply supported beam, it is shown that a minimum of two accelerometers can be used to construct two points of a mode shape, the first mode being the one most sensitive to excitation. Mode shape amplitudes are calculated in segments which are then stitched together to obtain the global mode shape. In numerical simulations, failure of the support bearings is represented as an increase in rotational spring stiffnesses. This is shown to have a significant effect on the shape of the first mode. The concept was tested in a field trial, before and after the bridge was repaired. The measured mode shape, represented as an envelope of possible shapes, was seen to change significantly when the support system was rehabilitated, and new bearings installed.

The loss of orthogonality in the Gram-Schmidt orthogonalization process

This research proposed and verified an innovative method to identify and locate structural damage using only the response of operational vibration, that is the displacement acquired by a non-contact optical method. The most efficient and economical way to detect damage within the structure is to monitor its structural health while in operation. However, the uncertainties and the randomness of ambient vibrations due to the operation and environments cause a challenge in conducting the operational analysis. Current technology limits the ability to collect data on the properties of the structure without the interruption of operation. Frequencies and mode shapes have been widely used in structural damage detection, but they are not sensitive enough and cannot provide sufficient information for identifying damage locations and their quantification. Therefore, the goal of this research is to design and verify a method to detect the damage, as well as its location and severity, of structures in operation without any physical contacts for data acquisition (i.e., non-contact based structural health monitoring (SHM)). Three algorithms are integrated into this SHM process. The first algorithm is the determination of structural characteristics (frequencies and mode shapes) of a vibrating structure from output-only data. Stochastic Subspace Identification (SSI) method is applied to measured displacements over time to extract the structural characteristics. The second algorithm is to estimate the scaling factor. The mode shapes obtained from the output-only model analysis are unscaled due to the absence of the information of input excitation forces. Mass Change Modal Scale (McMS) algorithm is used to estimate the modal scaling factors and determine the scaled mode shapes. The third algorithm is to estimate the structural system matrices (i.e., mass and stiffness matrices) and assess the damages. A Finite Element Model Updating (FEMU) is applied and the system matrices are updated from frequencies and scaled mode shapes. The damage within the structure can then be detected by analysing changes in mass and stiffness matrices. All three phases are verified by numerical simulation and laboratory experiments with deflections acquired by non-contact optical methods through video system. At last, to achieve the non-contact based SHM, a modal scaling method based on temperature change is proposed and verified by numerical simulation. Experimental program reveals that the proposed algorithm using McMS method is applicable to detect damage locations and their mass losses. With proposed non-contacted based SHM, the limitations of contact based sensor can be addressed, and the structural damage

The mode shape of bending waves in thin silicon and silicon carbide membranes is measured as a function of space and time, using a phase shift interferometer with stroboscopic light. The mode shapes hold information about all the relevant mechanical parameters of the samples, including the spatial distribution of static prestress. We present a simple algorithm to obtain a map of the lateral tensor components of the prestress, with a spatial resolution much better than the wavelength of the bending waves. The method is not limited to measuring the stress of bending waves. It is applicable in almost any situation, where the fields determining the state of the system can be measured as a function of space and time. I. INTRODUCTION Mechanical waves are a powerful tool to study material properties over a broad range of length scales. The applications range from seismic-wave methods to studying the structure of the Earth [1], SONAR for underwater location and navigation [2], ultrasonic testing of materials in engineering [3,4] and medical ultrasonography [3] ,d own to methods applicable to micro- and nanostructures like acoustic microscopy [3], Brillouin scattering [5], and picosecond pump-probe spectroscopy [6,7]. For all these methods, the length scale of the inhomogeneities of the material properties is supposed to be larger than the wavelength. In a previous work [8], we show how to extract the dispersion relation of bending waves of prestressed thin plates from measured mode shapes. Stress and bending stiffness of the membrane are determined as fit parameters using a theoretical model of the dispersion relation. Since the existence of a well-defined dispersion relation requires a homogeneous system, this method has the same shortcomings as the methods listed above. Here, we present an approach using full knowledge of the wave function to determine the mechanical properties of the sample. We measure the shape of a bending-wave mode in direct space and compare it to the equation of motion. The parameters found in the equation are the components of the stress tensor divided by the density of the membrane σij=ρ ,a s well as a constant proportional to the bending stiffness. With our simple algorithm, these parameters are obtained as a function of space using a linear fit. Since the full mode shape is available and we are not limited to far-field information, the wavelength of the bending waves is no limit for the spatial resolution. For the measurement of the stress-tensor components of semiconductors, all optical methods can be applied with spatial resolution in the order of a few micrometers [9,10] to roughly 100 μm [11], depending on the respective methods. However, these methods make use of particular optical properties and cannot straightforwardly be generalized to insulators or metals.

The paper deals with an experimental and a numerical investigation of dynamic properties of the structure. Object of interest is a steel rod with circle section. The modal parameters, such as mode shapes and natural frequencies, are obtained by two different ways: experimentally by using experimental modal analysis and numerically by finite element analysis which were computed in NX Nastran solver. The results from both methods are compared in NX through the correlation function. A relation between measured mode shapes and computed mode shapes is expressed by Modal Assurance Criterion. This process of comparing is appropriate for verification of experimental approaches.

This paper presents an experimental and numerical modal analysis for an external gear pump considering its mounting on a test rig in a laboratory setting. Most of the previous studies on experimental modal analysis (EMA) of hydraulic pumps focused on the modal frequencies to allow model validation. However, the mode shapes of pump bodies have not extensively been discussed. Furthermore, the nature of the pump components assembly and mounting poses some modeling challenges, such as the uncertain material properties of each component, the behavior of the bolted joints, and some critical modeling boundary conditions related to pump mounting. In this regard, the experimentally obtained vibration modes of a reference pump using the least-square complex exponential (LSCE) method are analyzed with an emphasis on the characteristics of the mode shapes. Then, simple modeling strategies are proposed and validated by performing the analysis from the component level to the full assembly. As a result, the mode shapes are categorized depending on the type of motions that the modes exhibit. It is observed that the pump casing does not show any substantial deformation but is close to the rigid body motion. Moreover, without considerably increasing model complexities, the proposed numerical approach provides reasonable accuracy with average errors in modal frequencies of 6%, as well as good agreement in terms of mode shapes. The vibration reduction strategy is briefly discussed based on the measured mode shapes, and the proposed modeling approaches can be useful to study external gear pumps with minimal model complexities yet allowing reasonable result accuracy.

This study presents a modal-based damage detection technique for framed structures. A framed structure is tested using a vibration-based method and the mode shapes corresponding to undamaged and damaged states of the structure are obtained. The damage to the framed structure is induced by reducing the stiffness of columns at a particular storey and tested using the shaking table. The corresponding finite element (FE) model was verified for its natural frequency with the experimental model. Moreover, a series of studies were performed by concentrating the damage to a smaller scale by including spot damage on the framed structure and varying the damage location and intensity. The model was analysed using finite element software for these tests and mode shape slope differences and mode shape curvature differences were generated. Derivatives of mode shape slopes were plotted and correlated with the damage location. Mode shape slope is discontinuous and becomes a maximum value close to the damaged location. Mode shape curvatures also show a maximum difference near the damaged location. Finite element model, as well as the experimental investigation, shows the competence of the proposed approach in structural damage identification.

We report measurements of vibrational mode shapes of mechanical resonators made from ultrathin carbon nanomembranes (CNMs) with a thickness of approximately 1 nm. CNMs are prepared from electron irradiation induced cross-linking of aromatic self-assembled monolayers and the variation of membrane thickness and/or density can be achieved by varying the precursor molecule. Single- and triple-layer freestanding CNMs were made by transferring them onto Si substrates with square/rectangular orifices. The vibration of the membrane was actuated by applying a sinusoidal voltage to a piezoelectric disk on which the sample was glued. The vibrational mode shapes were visualized with an imaging Mirau interferometer using a stroboscopic light source. Several mode shapes of a square membrane can be readily identified and their dynamic behavior can be well described by linear response theory of a membrane with negligible bending rigidity. By applying Fourier transformations to the time-dependent surface profiles, the dispersion relation of the transverse membrane waves can be obtained and its linear behavior verifies the membrane model. By comparing the dispersion relation to an analytical model, the static stress of the membranes was determined and found to be caused by the fabrication process.

Various non-destructive testing (NDT) methods have been developed to extract information about state of a structure. Two of them: vibration-based and guided wave-based techniques are one of the most commonly used and well developed. Both approaches can be implemented using Scanning Laser Doppler Vibrometer measurements and excitation by means of piezoelectric transducer. In this paper authors present a combined approached for NDT using successive and simultaneous measurement of both mode shapes and guided waves. Vibration-based damage detection is focused on detection of mode shape singularity, created by material discontinuity. This method utilizes wavelet transform and Teager energy operator for damage indication. Guided wave-based damage detection uses propagating elastic wave energy variation on the specimen surface as well as any changes in wave propagation pattern due to its interaction with material discontinuity as a tool for structural health assessment. Combining this two different techniques can give higher accuracy in defect detection. At the same time any additional specimen preparation are necessary, any set-up changes are required and the all the data can be registered in the same amount of time (simultaneous excitation). To confirm proposed technique a honeycomb core sandwich aluminum plate with debonding is tested. A results obtained with both techniques and combined approach are presented.

For nearly ten years Volkswagen has been applying scanning laser vibrometry for analyzing vibrations and structure-born noise of auto parts, like engines, auxiliaries, gearboxes, and the whole body as well. Compared with holographic interferometry the scanning vibrometry offers the great advantage of measuring operation mode shapes. Operating modes or 'real motions' are unharmonic motions, consisting of more than one frequency component. By applying an FFT- algorithm it is possible to analyze the frequency spectrum of the vibrating structure. This measurement of multiple mode shapes is obtained by only one scanning process, a very time saving procedure. In this paper we present (1) applications of this technique in automotive development, (2) a detailed description of the measurement procedure, (3) a discussion of advantages and disadvantages, and a (4) comparison of specific features of commercially available vibrometers and our own prototype, called SOVAS.

T HE importance of mode shapes is undeniable in contemporary flutter analyses. Not only are modes used to transform the governing equations from spatial to modal coordinates, but they are also used to transform flutter solution eigenvectors back to physical space to facilitate viewing of the critical flutter mechanism components. Mode shapes have even been used to gain insight into nonlinear aeroelastic behavior, such as limit cycle oscillations [1]. Mode shapes are so commonplace and easy to generate that their ability to overcome weaknesses inherent in flutter solution methods is not appreciated. In essence, modes are powerful tools that are not consistently exploited to their full potential. One cause for the underutilization of mode shapes could be the difficulty in quantitatively comparing them in an efficient manner with other modes. Quantitative comparison of modal frequencies is convenient, but a subjective assessment of the deflection mode shapes is the convention. Because of this subjectivity, the results of modal comparisons in this fashion could be regarded as less reliable or of limited utility. The situation is further impaired for complex or large models that are often cumbersome and sometimes obscure key shape features of interest. Anyone who has evaluated large quantities of flutter analyses is aware of the benefits of being able to compare mode shapes of different dynamical systems. Thesemode shapes can be indicative of energy-extraction and damping pathways within the structural system. They can provide insight into the similarity of predicted solution results, thereby reducing the number of critical analysis cases that may require flutter flight testing. They can also aid in the identification of critical flutter modes during flight testing, which is necessary for damping determination [2] and flutter margin [3,4] methods. Many methods are available to identify modes, but methods to explicitly compare one mode shape with another are not well documented in the literature. Themethods that are available formode shape comparisons are predominantly visual-based, such as 3-D still plots, animations, and node lines, although some work to extend these methods to quantitative approaches is available. Of particular interest to the flutter engineer is Maxwell’s work (as reported in [5]) to depict the wingtip deformation characteristics of the flutter mode shape and Northington’s [6] basis vector approach. There are also strain energy methods [7]. In short, it appears that many in-house methods for mode shape comparison likely exist, but they are only passed from generation to generation internally and are not disseminated widely to the engineering public. The development and dissemination of concise and distinctive mode shape description methodologies would allow the flutter engineering community to take better advantage of the inherent utility of mode shapes. Some applications that would become more practicable are sorting, tracking, and screening of modes for large quantities of analyses and using modes in alternative solution methodologies, such as artificial neural networks [8]. This Note describes a simple method in which curve-fit equations are used to depict the mode shapes of the wing. Some examples will be presented that compare the conventional 3-D still plots of the mode shape to the fitted curves as well as the coefficients of the curve-fit equations.

Many industrial machines nowadays are sold based on analysis performed on mathematical models of the rotors, bearings, substructures, and other components. The validity of the analysts therefore depends on the accuracy of the models themselves. When the rotor is available, modal testing may be used to validate the model of the rotor by comparing the calculated and measured free-free natural frequencies and mode shapes. This work presents additional tools for the verification of analytical models against experimental data. These tools use models of the rotor constructed from the measured data and the analytical model. A comparison of the first six calculated and measured free-free natural frequencies and mode shapes for a multi-mass flexible rotor is presented. The natural frequencies compare within 1.8%. The calculated and measured mode shapes were used to construct independent reduced order models of the rotor. These models were used to perform forced response and stability analyses. Forced response functions are presented comparing the forced response characteristics obtained from the two models. This provides a comparison between the measured and calculated forced response functions for the same number of modes. For the stability analysis, identical bearing models were added to both reduced order models. The eigenvalues were calculated using both models for a range of bearing stiffness and damping coefficients and were plotted for comparison.