Mode shape expansion using perturbed force approach

A new approach for expanding incomplete experimental mode shapes is presented which considers the modelling errors in the analytical model and the uncertainties in the vibration modal data measurements. The proposed approach adopts the perturbed force vector that includes the effect of the discrepancy in mass and stiffness between the finite element model and the actual tested dynamic system. From the developed formulations, the perturbed force vector can be obtained from measured modal data and is then used for predicting the unmeasured components of the expanded experimental mode shapes. A special case that does not require the experimental natural frequency in the mode shape expansion process is also discussed. A regularization algorithm based on the Tikhonov solution incorporating the generalized cross-validation method is employed to filter out the influence of noise in measured modal data on the predictions of unmeasured mode components. The accuracy and robustness of the proposed approach is verified with respect to the size of measured data set, sensor location, model deficiency and measurement uncertainty. The results from two numerical examples, a plane frame structure and a thin plate structure, show that the proposed approach has the best performance compared with the commonly used existing expansion methods, and can reliably produce the predictions of mode shape expansion, even in the cases with limited modal data measurements, large modelling errors and severe measurement noise.     

Added mass matrix estimation of beams partially immersed in water using measured dynamic responses

An added mass matrix estimation method for beams partially immersed in water is proposed that employs dynamic responses, which are measured when the structure is in water and in air. Discrepancies such as mass and stiffness matrices between the finite element model (FEM) and real structure could be separated from the added mass of water by a series of correction factors, which means that the mass and stiffness of the FEM and the added mass of water could be estimated simultaneously. Compared with traditional methods, the estimated added mass correction factors of our approach will not be limited to be constant when FEM or the environment of the structure changed, meaning that the proposed method could reflect the influence of changes such as water depth, current, and so on. The greatest improvement is that the proposed method could estimate added mass of water without involving any water-related assumptions because all water influences are reflected in measured dynamic responses of the structure in water. A five degrees-of-freedom (dofs) mass-spring system is used to study the performance of the proposed scheme. The numerical results indicate that mass, stiffness, and added mass correction factors could be estimated accurately when noise-free measurements are used. Even when the first two modes are measured under the 5 percent corruption level, the added mass could be estimated properly. A steel cantilever beam with a rectangular section in a water tank at Ocean University of China was also employed to study the added mass influence on modal parameter identification and to investigate the performance of the proposed method. The experimental results demonstrated that the first two modal frequencies and mode shapes of the updated model match well with the measured values by combining the estimated added mass in the initial FEM.     

On exact and approximated formulations for scaling-mode shapes in operational modal analysis by mass and stiffness change

When operational modal analysis (OMA) is used to estimate modal parameters, mode shapes cannot be mass normalized. In the past few years, some equations have been proposed to scale mode shapes using the mass-change method, which consists of repeating modal testing after changing the mass at different points of the structure where the mode shapes are known. In this paper, the structural-dynamic-modification theory is used to derive a set of equations, from which all the existing formulations can be derived. It is shown that the known equations can be divided into two types, the exact and the approximated equations, where the former type does in fact fulfill the equations derived from the theory of structural modification, whereas the remaining equations do not, mainly because the change of the mode shapes of the modified structure is not properly taken into account. By simulations, the paper illustrates the large difference in accuracy between the approximate and the exact formulations. The paper provides two new exact formulations for the scaling factors, one for the non-modified structure and – for the first time in the literature – one for the modified structure. The simulations indicate the influence of errors from the measured natural frequencies and mode shapes on the estimation of the scaling factors using the two exact formulations from the literature and the new exact formulation proposed in this paper. In addition, the paper illustrates statistics of the errors on mode-shape scaling. All simulations were carried out using a plate with closely spaced modes.     

A modal test method using sound pressure transducers based on vibro-acoustic reciprocity

A modal test method that uses sound pressure transducers at fixed locations and an impact hammer roving over a test structure is developed in this work. Since sound pressure transducers are used, the current method deals with a coupled structural–acoustic system. Based on the vibro-acoustic reciprocity, the method is equivalent to one, where acoustic excitations at fixed locations are given and the resulting acceleration of the test structure is measured. The current method can eliminate mass loading due to use of accelerometers, which can destroy existence of repeated or close natural frequencies of a symmetric structure. It can also avoid effects of a nodal line of a mode and an inactive area of a local mode, and measure all the out-of-plane modes within a frequency range of interest, including global and local ones. The coupling between the structure and the acoustic field in a structural–acoustic system introduces asymmetry in the model formulation. An equivalent state space formulation is used for a damped structural–acoustic system and the associated eigenvalue problem is derived. The biorthonormality relations between the left and right eigenvectors and the relations between the structural and acoustic components in the left and right eigenvectors are proved. The frequency response functions associated with the current method are derived and their physical meanings are explained. The guidelines for using the current method, including the types of structures that are suitable for the method, the positions of the sound pressure transducers, and the orientation of the test structure relative to the transducers, are provided. Modal tests were carried out on an automotive disk brake using the traditional and current methods, where multiple accelerometers and microphones were used to measure its dynamic responses induced by impacts, respectively. The differences between the measured natural frequencies using the current method and those from the finite element model of the disk brake are less than 3 percent for the first 18 elastic modes, and the modal assurance criterion values of the associated mode shapes are all above 90 percent. The current method was also used to measure the natural frequencies, damping ratios, and mode shapes of a light circuit board.     

Extended Kalman filter for modal identification of structures equipped with a pendulum tuned mass damper

Pendulum tuned mass dampers (PTMDs) have been employed in several full-scale applications to attenuate excessive structural motions, which are mostly due to wind. Conducting periodic condition assessments of the devices to ascertain their health is necessary to ensure their continued optimal performance, which involves identifying the modal parameters of the underlying (bare) structure to which they are tuned to. Such an identification is also necessary for the design of control systems such as adaptive tuned mass dampers. Existing methods of arresting the motion of the damper to estimate the modal properties are expensive, time-consuming, and not suitable for online tuning. Instead, in this paper, parameter estimation using the Extended Kalman Filter (EKF) is proposed to undertake this task. The central task accomplished here is to estimate the dynamic characteristics of the bare structure (structure without the PTMD) from response measurements of the coupled main structure and PTMD system. The proposed methodology relies on ambient acceleration measurements of TMD-attenuated responses to estimate the bare structural modal frequencies, damping, and mode shapes, which can then be used either for condition assessment or for control. The application of EKF to modal parameter estimation is not new. However, a methodology to address the problem in wind engineering arising from stochastic disturbances present in both the measurement and state equations, and unknown process and noise covariances arising due to ambient excitations, is presented for the first time. Extensively studied for synthetic data, these two challenges have limited thus far the application of Kalman filtering to practical wind engineering parameter estimation problems using experimentally obtained measurements. In this paper, a detailed methodology is presented to address these challenges by using a modified form of the standard EKF equations, together with an algorithm to estimate the unknown disturbance and measurement noise covariances. Numerical simulations and an experimental study are both presented. Results demonstrate that the method proposed provides reliable estimates for the modal parameters required to perform condition assessment and control tasks for pendulum tuned mass dampers.     

Application of the maximum likelihood principle to separate exponential terms in T2 relaxation of nuclear magnetic resonance

The method of maximum likelihood has been implemented for the estimation of multiple exponential components of T2 decay curves in spin echo NMR measurements on biologic tissues. Each Each component contributes an exponential term described by two parameters (initial amplitude and T2) to the T2 decay curve. The maximum likelihood method estimates the parameters and their standard errors for all terms simultaneously, avoiding the subjectivity inherent in methods such as graphical peeling. In the model used, it was assumed that water protons are compartmentalized and that the measured spin echo signals from the protons undergoing relaxation obey the Poisson distribution. A system of non-linear equations was derived and solved iteratively for the values of the exponential parameters which maximize the likelihood of obtaining the observed data under these assumptions. The approach was implemented for bi- and tri-exponential models on a MicroVAX II computer (Digital Equipment Corporation, Maynard, MA). Simulations of bi- and tri-exponential data, with and without system noise, were analyzed to assess the accuracy and reproducibility of the method. A subset of the simulations was repeated with non-linear least squares techniques and was compared to the results obtained with maximum likelihood. Rabbit muscle and gerbil brain samples were measured and analyzed with the maximum likelihood method. The simulations showed that within specific limits on relative sizes and relaxation rates of components, these parameters can be estimated with errors less than 5%. The comparison to non-linear least squares analysis showed that the maximum likelihood method is generally superior in estimating the parameters in difficult cases. The results from tissue measurements demonstrate that the method is effective even in cases where graphical peeling would clearly not yield reliable results.     

The possibility of estimating the values of a function at given points of the measurement results of a finite number of its linear functionals

The possibility of estimating the values of a function at given points of its domain is investigated. The estimation uses results of the measurements of a finite number of linear functionals; the results of the measurements are distorted by an error. It is shown that only the component of the functions from the linear finite-dimensional subspace can be estimated with a finite error. A method for estimating this component with accuracy control is proposed. The mathematical methods of measurement reduction proposed by Yu.P. Pyt’ev are used. An example of the estimation of an emission spectrum that is measured by a double-slit spectrometer is described.     

Modeling three-dimensional elastic wave propagation in circular cylindrical structures using a finite-difference approach

Wave propagation along circular cylindrical structures is important for nondestructive-testing applications and shocks in tubes. To simulate elastic wave propagation phenomena in such structures the governing equations in cylindrical coordinates are solved numerically. To reduce the required amount of computer memory and the computational time, the stress components are eliminated in the equilibrium equations. In the resulting coupled partial differential equations, in which only the three displacement components are involved, the derivatives with respect to spatial coordinates and time are approximated using second order central differences. This leads to the present new approach, which is both accurate and efficient. In order to obtain a stable scheme the displacements must be allocated on a staggered grid. The von Neumann stability analysis is performed and the result is compared with an existing empirical criterion. Mechanical energies are observed in order to validate the finite-difference code. Since no material damping or energy dissipation is taken into account in the equations of motion, the total energy must remain constant over time. Only negligible variations are observed during long-term simulations. Dispersion relations are used to check the physical behavior of the waves calculated with the proposed finite-difference method: Theoretically calculated curves are compared with values obtained by a spectrum estimation method, applied to the results of a simulation.     

Structural model updating using incomplete transfer function of strain data

In this paper a model updating algorithm is presented to estimate structural parameters at the element level utilizing frequency domain representation of the strain data. Sensitivity equations for mass and stiffness parameters estimation are derived using decomposed form of the strain-based transfer functions. The rate of changes of eigenvectors and a subset of measured natural frequencies are used to assemble the sensitivity equation of the strain-based transfer function. Solution of the derived sensitivity equations through the least square method resulted in a robust parameters estimation method. Numerical examples using simulated noise polluted data of 2D truss and frame models confirm that the proposed method is able to successfully update structural models even in the presence of mass modeling errors.     

Modal parameter identification through Gabor expansion of response signals

A new method of identifying modal parameters by decomposing response signals with Gabor transform is presented in this paper to estimate natural frequencies, damping ratios and mode shapes of linear time invariant systems. According to Gabor expansion theory, responses of a multi-degree-of-freedom system can be decomposed into uncoupled signal components, each vibrating at a single natural frequency. From these uncoupled signals, modal parameters are subsequently extracted with common methods. The proposed method can process stationary and non-stationary responses and requires no input signal except for the response signals generated by unknown excitation acting on a system. In the sense of less restriction on the in-out signals, the approach based on time-frequency decomposition is very general. A simulation study on a simply supported beam under non-stationary excitation has demonstrated that the proposed method is effective in parameter estimation.     

Impact of atomic relaxation on the breaks of constant force surfaces in AFM

A calculation model for determination of the shapes of the constant force surfaces and profiles of lateral forces for the case of the AFM tip scanning the closely packed lattice in contact mode is proposed. Atomic relaxation is taken into account in this model. The existence of breaks on constant force surfaces, which was predicted earlier in an approximation of the fixed lattice, is confirmed. It is shown that due to non-zero atomic mobility, breaks appear for smaller scanning forces than assumed earlier. The shapes of the continuous constant force surfaces and profiles of lateral force components are computed. These results may be used for diagnostics of point defects on the surface.     

Spectroscopic method for measuring plasma magnetic fields having arbitrary distributions of direction and amplitude

An approach for measurements of magnetic fields, based on the comparison of the magnetic-field-induced contributions to the line shapes of different fine-structure components of an atomic multiplet, is proposed and experimentally demonstrated. Contrary to the methods based on detecting an anisotropy in either the emitted radiation or in the dispersion properties of the medium, the present method is applicable when the field direction or amplitude vary significantly in the region viewed or during the time of observation. The technique can be used even when the line shapes are Stark or Doppler dominated. It has potential applications in laser-matter interactions, plasmas driven by high-current pulses, and astrophysics.     

Analysis of nonlinear steady state vibration of a multi-degree-of-freedom system using component mode synthesis method

In this paper, an analytical approach for nonlinear forced vibration of a multi-degree-of-freedom system is proposed using the component mode synthesis method. The whole system is divided into some components and a nonlinear modal equation of each component is derived using the free-interface vibration modes. The modal equations of all components and the conjunction conditions are solved simultaneously, and then the modal responses of components are derived. Finally, the dynamic responses of the whole system can be obtained. The degrees of freedom of modal equations can be reduced when the lower vibration modes are only adopted in each component. As a numerical example, a nine-degree-of-freedom system is considered, in which all spring have cubic type nonlinearity. As a result, it is shown that when there are no rigid modes in components, the compliance by the proposed method agrees very well with the exact one even if the lower vibration modes of components are only adopted. The other hand, in the case with rigid modes in components, the compliance has a little error compared with the exact result. It is recognized that the method proposed is very effective in the case without rigid modes in components for the actual application.     

Shape and size dependence of the surface plasmon resonance of gold nanoparticles studied by Photoacoustic technique

We report on the optical absorption properties of as prepared gold naoparticles of different shapes and sizes measured by photoacoustic (PA) method. The gold nanoparticles of two different shapes (dots, rods) have been prepared using the seed mediated growth method. The shape and the size of these different nanoparticles were determined by STM measurements. PA spectra show the splitting of the surface plasmon resonance (SPR) into two modes (transverse and longitudinal) in case of gold nanorods. The increase in the aspect ratio of the nanorods leads to clear redshifts of the longitudinal SPR. These shifts were used to determine the dielectric constant of the surrounding medium and its variation with the aspect ratios.     

Transverse vibrations of a non-uniform rectangular plate on an elastic foundation

Free transverse vibrations of an isotropic rectangular plate of variable thickness resting on an elastic foundation has been studied on the basis of classical plate theory. The fourth-order differential equation governing the motion is solved by using the quintic spline interpolation technique. Characteristic equations for plates of exponentially varying thickness have been obtained for three combinations of boundary conditions at the edges. Frequencies, mode shapes and moments have been computed for different values of the taper constant and the foundation moduli for the first three modes of vibration.     

A simple empirical model for the collisional spectral shift of air-broadened CO2 lines

A simple empirical model is proposed for the pressure-induced shifts of air-broadened CO₂ absorption lines. It is based on the introduction of components symmetric and antisymmetric with respect to the rotational quantum number m, and on a separation of vibrational and rotational contributions. These components are modelled though empirical analytical laws with parameters that are obtained from fits of the measured shifts in only two different bands. It is then shown that the model leads to satisfactory results for a number of other vibrational transitions in which the shifts take significantly different values. This provides a simple way to generate values for unmeasured transitions and this tool can thus be used to complement spectroscopic databases.     

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

A parametric approach to spectral-spatial EPR imaging

Continuous wave electron paramagnetic resonance imaging for in vivo mapping of spin distribution and spectral shape requires rapid data acquisition. A spectral-spatial imaging technique is presented that provides an order of magnitude reduction in acquisition time, compared to iterative tomographic reprojection. The proposed approach assumes that spectral shapes in the sample are well-approximated by members from a parametric family of functions. A model is developed for the spectra measured with magnetic field modulation. Parameters defining the spin distribution and spectral shapes are then determined directly from the measurements using maximum a posteriori probability estimation. The approach does not suffer approximation error from limited sweep width of the main magnetic field and explicitly incorporates the variability in signal-to-noise ratio versus strength of magnetic field gradient. The processing technique is experimentally demonstrated on a one-dimensional phantom containing a nitroxide spin label with constant g-factor. Using an L-band EPR spectrometer, spectral shapes and spin distribution are accurately recovered from two projections and a spectral window which is comparable to the maximum linewidth of the sample.     

Experimental and numerical investigations of resonant vibration characteristics for piezoceramic plates.

Electronic speckle pattern interferometry (ESPI) is a full field, non-contact technique for measuring the surface displacement of a structure subjected to static loading or, especially, to dynamic vibration. In this article we employ an optical system called the amplitude-fluctuation ESPI with out-of-plane and in-plane measurements to investigate the vibration characteristics of piezoceramic plates. Two different configurations of piezoceramic plates, namely the rectangular and the circular plates, are discussed in detail. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Because the clear fringe patterns will be shown only at resonant frequencies, both the resonant frequencies and the corresponding mode shapes are obtained experimentally at the same time by the proposed AF-ESPI method. Excellent quality of the interferometric fringe patterns for both the in-plane and out-of-plane vibration mode shapes is demonstrated. The resonant frequencies of the piezoceramic plates are also measured by the conventional impedance analysis. From experimental results, we find that the out-of-plane vibration modes (type A) with lower resonant frequencies cannot be measured by the impedance analysis and only the in-plane vibration modes (type B) will be shown. However, both the out-of-plane (bending) and in-plane (extensional) vibration modes of piezoceramic plates are obtained by the AF-ESPI method. Finally, the numerical finite element calculations are also performed, and the results are compared with the experimental measurements. It is shown that the numerical calculations and the experimental results agree fairly well for both the resonant frequencies and the mode shapes.     

Analysis of a monolithic crystal plate acoustic wave filter

We study thickness–shear and thickness–twist vibrations of a finite, monolithic, AT-cut quartz plate crystal filter with two pairs of electrodes. The equations of anisotropic elasticity are used with the omission of the small elastic constant c₅₆. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and the vibration confinement due to the electrode inertia are calculated and examined.