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.     

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.     

Overview of the New OMAH Technique for Scaling OMA Mode Shapes

Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short.     

Non-linear structural dynamics characterization using a scanning laser vibrometer

This paper presents the use of a scanning laser vibrometer and a signal decomposition method to characterize non-linear dynamics of highly flexible structures. A Polytec PI PSV-200 scanning laser vibrometer is used to measure transverse velocities of points on a structure subjected to a harmonic excitation. Velocity profiles at different times are constructed using the measured velocities, and then each velocity profile is decomposed using the first four linear mode shapes and a least-squares curve-fitting method. From the variations of the obtained modal velocities with time we search for possible non-linear phenomena. A cantilevered titanium alloy beam subjected to harmonic base-excitations around the second, third, and fourth natural frequencies are examined in detail. Influences of the fixture mass, gravity, mass centers of mode shapes, and non-linearities are evaluated. Geometrically exact equations governing the planar, harmonic large-amplitude vibrations of beams are solved for operational deflection shapes using the multiple shooting method. Experimental results show the existence of 1:3 and 1:2:3 external and internal resonances, energy transfer from high-frequency modes to the first mode, and amplitude- and phase-modulation among several modes. Moreover, the existence of non-linear normal modes is found to be questionable.     

The mass normalization of the displacement and strain mode shapes in a strain experimental modal analysis using the mass-change strategy

The classic experimental modal analysis (EMA) is a well-known procedure for determining the modal parameters. The less frequently used strain EMA is based on a response measurement using strain sensors. The results of a strain EMA are the modal parameters, where in addition to the displacement mode shapes the strain mode shapes are also identified. The strain EMA can be used for an experimental investigation of a stress–strain distribution without the need to build a dynamical model. It can also be used to determine the modal parameters when, during modal testing, a motion sensor cannot be used and so a strain sensor is used instead. The displacement and strain mode shapes that are determined with the strain EMA are not mass normalized (scaled with respect to the orthogonality properties of the mass-normalized modal matrix), and therefore some dynamical properties of the system cannot be obtained. The mass normalization can be made with the classic EMA, which requires the use of a motion sensor. In this research a new approach to the mass normalization in the strain EMA, without using a motion sensor, is presented. It is based on the recently introduced mass-change structural modification method, which is used for the mass normalization in an operational modal analysis. This method was modified in such a way that it can be used for the mass normalization in the strain EMA. The mass-normalized displacement and strain mode shapes were obtained using a combination of the proposed approach and the strain EMA. The proposed approach was validated on real structures (beam and plate).     

Direct mode-shape expansion of a spatially incomplete measured mode by a hybrid-vector modification

A direct estimation method for expanding incomplete experimental mode shapes is presented. The approach adopts a hybrid vector which includes measured data at master degrees of freedom (dofs) and constant values at slave dofs. The constant values are refined by a set of mode-correction factors. Modelling errors between the analytical model and tested structure are also considered by introducing a series of model-correction factors. Initial-guess values of the mode-correction factors are used to decouple the coupled constructed equations, and an iterative technique for solving these equations is proposed. The results from a five-degree-of-freedom mass–spring system indicate that the proposed approach provided a better performance than the commonly used existing expansion methods and can reliably estimate unmeasured components of mode shapes, even in cases with limited modal measurements and severe measurement noise. The performance of the proposed method was also investigated using real measurements from a steel cantilever-beam experiment. Experimental data were measured by 20 accelerometers mounted at the cantilever beam: among these accelerometers, three of these were assumed to be measured, and the others were used to check the estimation accuracy of the proposed method. The results show that the unmeasured components in the mode shapes were properly estimated by implementing the proposed method, even for high-frequency modes.     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

On the application of blind source separation for damping estimation of bridges under traffic loading

The accurate and reliable estimation of modal damping from output-only vibration measurements of structural systems is a continuing challenge in the fields of operational modal analysis (OMA) and system identification. In this paper a modified version of the blind source separation (BSS)-based Second-Order Blind Identification (SOBI) method was used to perform modal damping identification on a model bridge structure under varying loading conditions. The bridge model was created with finite elements and consisted of a series of stringer beams supported by a larger girder. The excitation was separated into two categories: ambient noise and traffic loads with noise modeled with random forcing vectors and traffic simulated with moving loads for cars and partially distributed moving masses for trains. The acceleration responses were treated as the mixed output signals for the BSS algorithm. The modified SOBI method used a windowing technique to maximize the amount of information used for blind identification from the responses. The modified SOBI method successfully found the mode shapes for both types of excitation with strong accuracy, but power spectral densities (PSDs) of the recovered modal responses showed signs of distortion for the traffic simulations. The distortion had an adverse affect on the damping ratio estimates for some of the modes but no correlation could be found between the accuracy of the damping estimates and the accuracy of the recovered mode shapes. The responses and their PSDs were compared to real-world collected data and patterns similar to distortion were observed implying that this issue likely affects real-world estimates.     

An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia

In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal.     

Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as Re ⁴, and not as Re ³ expected from Kolmogorov’s theory, where Re is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier–Stokes equations, and some problems of principle associated with existing LES models are highlighted     

Quantification of changes in modal parameters due to the presence of passive people on a slender structure

This work proposes a method to predict changes in modal parameters of a generic structure due to the presence of passive people. The method requires to accurately know the modal model of the empty structure, i.e. natural frequencies, damping ratios and mode shapes. No restriction on the number of degrees of freedom of the structure is required. Each passive subject on the structure is modelled through his apparent mass and is introduced locally on the empty structure to obtain a model of the joint human–structure system. To include the apparent mass of the subjects in the model both measurements and models available in literature were used. The effect of using different models of the apparent mass was investigated and the effect of using apparent masses obtained with different levels of vibration was analysed.     

STRUCTURAL DAMAGE DETECTION BASED ON A MICRO-GENETIC ALGORITHM USING INCOMPLETE AND NOISY MODAL TEST DATA

This paper describes a procedure for detecting structural damage based on a micro-genetic algorithm using incomplete and noisy modal test data. As the number of sensors used to measure modal data is normally small when compared with the degrees of freedom of the finite element model of the structure, the incomplete mode shape data are first expanded to match with all degrees of freedom of the finite element model under consideration. The elemental energy quotient difference is then employed to locate the damage domain approximately. Finally, a micro-genetic algorithm is used to quantify the damage extent by minimizing the errors between the measured data and numerical results. The process may be either of single-level or implemented through two-level search strategies. The study has covered the use of frequencies only and the combined use of both frequencies and mode shapes. The proposed method is applied to a single-span simply supported beam and a three-span continuous beam with multiple damage locations. In the study, the modal test data are simulated numerically using the finite element method. The measurement errors of modal data are simulated by superimposing random noise with appropriate magnitudes. The effectiveness of using frequencies and both frequencies and mode shapes as the data for quantification of damage extent are examined. The effects of incomplete and noisy modal test data on the accuracy of damage detection are also discussed.     

Maximum likelihood identification of non-stationary operational data

In-operation modal analysis has become a valid alternative for structures where a classic input-output test would be difficult if not impossible to conduct. Due to practical considerations, measurements are sometimes performed in patches (roving sensor setups) instead of covering the entire structure at once. In practice, one is often confronted with non-stationary ambient excitation sources (e.g., wind, traffic, waves, etc.). Since the scaling of operational mode shape estimates depends on the unknown level of the ambient excitation, an extra effort is required in order to correctly merge the different parts of the mode shapes. In this contribution, two different approaches, for merging operational mode shapes from non-stationary data, are proposed. Both methods are based upon a single maximum likelihood estimation procedure. For comparison and validation, both techniques were applied to non-stationary data sets obtained by scanning laser vibrometry as well as the Z24 bridge bench mark data.     

An automatic mode pairing strategy using an enhanced modal assurance criterion based on modal strain energies

In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches.In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.     

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.     

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.     

Self-adaptive modal control for time-varying structures

For the past several years, modal controllers are widely studied and used in the field of vibration or vibro-acoustics control. They are efficient but not robust, because these methods involve a reconstructor based on a modal truncation. When the dynamic behavior of the structure change, the controller and reconstructor must be updated to cope with the changes in the structure behavior, in order to maintain both performance and robustness. A solution is adaptive control but this approach needs some specific information not generally available particularly in the case of undergone modifications. This paper deals with a self-adaptive modal control based on a real-time identifier, which avoid the need of specific information. The identifier permits to update the controller and the reconstructor according to the changes of modal characteristics of time-varying structures. A classical algorithm of identification is used to obtain a state space model with an unspecified state vector. Then, based on this model, a well adapted transformation is carried out to get the modal characteristics from the expression of complex modes, including the mode shapes. As a criterion of running identification, the value of “variance-accounted for” (VAF) is employed to carry out the identifier only when the initial or previous model is not enough exact. A Linear Quadratic Gaussian Algorithm is employed in such a way that the controller and observer can be optimized according to the updated modal model. By this way, a self-adaptive modal control is completed and can demonstrate some smart properties. The proposed methodology is carried out on a simple but representative time-varying mechanical discrete structure. An inertia modification leads not only to low modal frequency shifts but also to inversion of a mode shape which is shown to lead to unstable configuration when control system is not updated. The overall procedure will be described through simulations and performed for different operating conditions, which will prove that mode shapes have to be precisely determined and updated in the controller and observer to guarantee a robust modal control with high performance in spite of the changes of structure.     

Design of shaped piezoelectric modal sensors for cantilever beams with intermediate support by using differential transformation method

In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a cantilever beam with intermediate support is proposed by using the differential transformation method (DTM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function. By using boundary condition and continuity condition equations at intermediate support, the closed-form series solution of the second spatial derivative of the mode shapes can be determined based on DTM. Then the shapes of the designed modal sensors are obtained. Finally, numerical examples are given to demonstrate the feasibility of the proposed modal sensors for the cantilever beam with intermediate support.