Topics in Modal Analysis & Parameter Identification, Volume 9

This chapter is concerned with the automated extraction of modal parameters (frequency and damping estimates) from accelerometer data measured on a full-scale wind turbine tower without its nacelle and blades installed. The proposed algorithm is based on an existing Operational Modal Analysis research software employing the Stochastic Subspace Identification algorithm for manual selection and extraction of modal parameters. The automatization of the algorithm is discussed in terms of the choices made and their consequences with respect to sensitivity and robustness. The algorithm is finally tested on a large experimental dataset consisting of 10 days of signals sampled at 25 Hz from two accelerometers mounted at the top of the tower in orthogonal directions. The automated algorithm is successful in time tracking the development of the first two modes with respect to frequency and damping despite the challenge posed by the fact that due to the high degree of symmetry in the setup the frequencies of the two modes are very similar.

Accelerometer data is the most commonly used data for experimental modal analysis of structures. Together with measuring applied force, it provides the basis for FRF estimation and subsequent modal parameter estimation and validation. As discussed in the paper by Dr. Coppolino (Experimental modal analysis using non-traditional response variables. In: IMAC Proceedings, 2021), there are situations where test analysis cross orthogonality is difficult to determine on inaccessible key regions of a test article. In that chapter, it is contended that it is in theory possible to augment data from accelerometers with data from other sensor sources at these key regions that have a proportionality to acceleration or displacement. This is important as strain and pressure have been shown to be useful measurements for modal analysis (Zienkiewicz et al., The finite element method: its basis and fundamentals, 6th edn. Butterworth-Heinemann, Oxford, p 563–584, 2005; Kranjc et al., J Sound Vib 332:6968, 2013; Kranjc et al., J Vib Control 22(2):371–381, 2016; Dos Santos et al., Strain-based experimental modal analysis: new concepts and practical aspects. In: Proceedings of ISMA. IEEE, Piscataway, p 2263–2277, 2016; Dos Santos et al., An overview of experimental strain-based modal analysis methods. In: Proceedings of the international conference on noise and vibration engineering (ISMA), Leuven, p 2453–2468, 2014). But they have not been used in augmentation with acceleration. Two specific examples discussed are fluid pressure and strain. Experimentally, this presents several problems. For example, in the most simple structures it is expected to have maximum acceleration at locations of 0 strain and vice versa. This makes it difficult to relate the modal information contained in acceleration variable to the strain variable at the location of maximum acceleration. Given that the FRF information will have to be uniform in units, this is another cause of concern when combining pressure, strain, and acceleration. Use of strain, pressure, and acceleration data all together for modal analysis purposes would reduce the need to place accelerometers in locations that are difficult to access. This chapter aims to present experimental results of strain and pressure FRF-based modal analysis on a rectangular steel plate and attempts to propose ways to combine these variables in the modal parameter estimation process.

Over the past 70 years, the US aerospace community has maintained a standard for verification and validation of experimentally determined, real structural dynamic modes and mathematical models based on mass-weighted orthogonality criteria. This standard fundamentally contradicts observable physical aspects associated with the mechanical behavior of structures. Specifically, (a) energy dissipation (damping) forces are most often concentrated in joints, rather than nearly uniformly distributed throughout the structure; (b) structural modes are mathematically complex, yet often approximately real except when successive modal frequencies are closely spaced; and (c) complex structural modes, while often are approximately real, do not strictly satisfy mass-weighted orthogonality criteria. A bottom-up approach, based on the Simultaneous Frequency Domain Technique (SFD-2018), employs left-hand eigenvectors to (1) isolate individual complex measured modes and (2) guarantee mathematical orthogonality of complex measured modes (completely independent of a theoretical mass matrix and model expectations). In addition, (3) complex modes deduced from virtually all experimental modal analysis techniques are classified in terms of a complex mode index parameter that indicates each mode’s level of “complexity,” and (4) conventional experimental mode orthogonality and experimental-to-theoretical mode cross-orthogonality metrics are adapted via replacement of the transform operator with the Hermitian operator permitting direct employment of complex experimental modes. A welcome surprise, due to review of a specific real experimental mode approximation extends the useful life of established US aerospace community standards for verification and validation of modal test data and correlation and reconciliation of modal test results and mathematical model predictions.

When it comes to multi-dataset Operational Modal Analysis (OMA) of large civil structures, the main goal is to clearly identify the global vibration modes in a very accurate and robust manner. In this chapter, a new frequency domain identification technique is applied to the vibration responses of an outstanding 363-m high television and radio transmission tower, i.e., the Riga TV Tower located at the city of Riga, Latvia. The vibration data was collected during a 3-day testing campaign where a total of 6 datasets were collected at different storeys of the structure by means of two independent measurement systems consisting of two 3D vibration sensors each. The main advantage of the identification technique applied to the tower’s vibration data with regard to the existing approaches is that the former provides clearer stabilization diagrams, facilitating the identification of the physical modes of the tested structures. The application of the novel modal identification technique to tower’s response records led to a clear diagram, from which the most important global vibration modes were easily identified.

In Experimental and Operational Modal Analysis (EMA and OMA), the main challenge is to extract the experimental modal properties of the tested structures from the vibration measurements in a very accurate and robust manner. The EMA and OMA can be carried out in time or frequency domain by fitting a parametric model to time or frequency domain measurements, so that the modal properties can be subsequently extracted for the obtained normal matrices. In this chapter, a novel poly-reference modal identification technique formulated in the frequency domain is proposed. The advantage of the proposed approach with regard to the existing techniques is that it yields numerical (non-physical) modes with negative damping, making it easier to distinguish the physical modes. The accuracy and robustness of such an approach is demonstrated by means of an application example at last part of the chapter.

A good piece of advice for those who engage in modal parameter estimation is “Don’t use data that doesn’t help your cause.” Selecting the frequency range of interest and down-selecting the references and responses are common features in all commercial modal analysis software packages. While these procedures are usually sufficient for producing acceptable results, there are some supplemental techniques available that go beyond just defining the temporal and spatial boundaries of the dataset. This chapter discusses some of these more obscure features that could be helpful in improving your modal parameter estimation results.

In structural health monitoring, wireless sensor networks are favorable for their minimal invasiveness, ease of deployment, and passive monitoring capabilities. Wireless vibration sensor nodes have been implemented successfully for frequency domain analysis in ambient vibration detection. To leverage advances in structural damage quantification techniques, which require modal information, nodes in a wireless sensor network must operate with a near-synchronous clock to enable the collection of the signal phase. The non-deterministic timing nature of wireless systems raises a significant challenge when trying to accurately determine the phase of a signal. In particular, the trigger time delay of the various nodes on the structure cannot be differentiated from a true phase caused by the examined system. This study investigates the reliability and error-handling capabilities of the ShockBurst 2.4 GHz wireless protocol in triggering and data transfer. Building on an open-source UAV-deployable sensor node, mode shapes from a 2-meter test specimen are experimentally determined. An optimization technique that enhances time domain accuracy for non-deterministic wireless triggers is presented. This work quantifies latency and error management effects that contribute to enhancing the modal extraction capabilities of wireless systems in structural health monitoring applications.

It has become evident that climate change is an escalating situation requiring societies to act to prevent a nearby climate crisis. Preservation of existing infrastructures to withstand material deterioration and the current increase in loadings does not lead only to reducing dismantling debris but also less consumption of resources, leading to the protection of the environment. Space frame structures were and are still a widely used structural solution for many infrastructures due to their considerable lightweight and ease of erection. The structural stress status of space frame structures is an important step in the preservation process. Assessing the load-bearing capacity of space frame structures gives an insight into the stress status, which is possible through identifying axial forces inside various members. The determination of axial forces of a single tension cable has been intensively investigated based on vibration methods. In addition, as a part of a two-dimensional truss, the identification of stress status in tension members has been studied. However, far less attention has been put so far on the consideration of compression members and more complex systems. This chapter adopts a previously developed methodology to identify tension forces in a two-dimensional truss and extends it to include compression forces identification and application to three-dimensional systems. The approach includes the identification of members’ axial forces based on a numerical model and experimentally determined modal parameters, particularly natural frequencies and mode shapes. This chapter uses a relatively simple space frame structure as a case study. Numerical models of the simple space frame structure give an insight into the natural frequencies and mode shapes to be expected from the physical model. The closely spaced local and global modes of vibration obtained from the numerical model are discussed and explained, and a criterion to distinguish between them is introduced. From the experimental side, applying parametric modal identification is a reliable tool to extract modal parameters by considering different model orders concerning the identification of local and global modes. With the help of experimental results, axial forces are derived based on a numerical model of the structure. The numerical model does not give only a priori information about the modal parameters but also about the minimal model order needed for the identification of global and local modes.

Robot-driven on-orbit servicing, assembly, and manufacturing (OSAM) promises to enable and enhance a wide range of space technologies in the coming decades. Supporting technologies, however, are still nascent and must still be developed to manage the unique characteristics of this upcoming construction paradigm. Robotic operation on OSAM structures will introduce modal excitations that, if not controlled, may damage or fatigue the structure. This chapter develops a method for planning robot motions such that modal excitations are minimized, thereby reducing induced loads and risk to the structure. An example is presented using a relevant exemplary OSAM structure.

For vibration mitigation and impact resistance, energy dissipation is desired in mechanical and aerospace structures to ensure their required loading capacity and dynamic performance. However, designing a structure with simultaneously high dissipation and high stiffness simultaneously is challenging due to the inherent tie between the material damping and stiffness. In this work, a tetra-chiral metamaterial beam design, which possesses enhanced damping thanks to local resonance, is proposed and optimized for simultaneously high stiffness, high dissipation, and high mass efficiency. Various designs with different geometric configurations are explored, and a design optimization framework is developed with stiffness, damping, and mass as the main metrics. Finally, the optimized dissipative tetra-chiral metamaterial beam designs that achieve high dissipation performance with consistent stiffness and mass efficiency are presented and compared to the initial one.

The need for using extendable booms to deploy payloads, solar sails, and antennas is ever increasing. Their inclusion in future space missions can provide a viable alternative to massive truss-like structures as well as significantly reduce fuel mass requirements for attitude control systems. Over the last few decades, various designs of these have been developed and studied. However, at the present, flight heritage booms are considered inadequate in terms of both bending and torsional stiffness. Booms with conic cross sections have not been studied in the past and we believe they might have certain advantages in these aspects in their deployed configuration. This chapter aims to investigate the dynamic characterization of a composite tape spring boom with a parabolic cross section. This boom sample is also planned to be the primary payload on a 3 U cubesat at Virginia Tech, Ut ProSat-1. The cubesat mission aims to collect acceleration data from the tip of the deployed boom. This data will then be validated using the test data from the ground experiments. Finally, this chapter discusses the setup of a finite element analysis in Abaqus as well as future plans for the ground vibration experiments.

Wave-driven motion is a phenomenon that has been observed in nature as a method for propulsion. Earlier studies replicated these traveling waves in finite structures to propel the structures itself or particles across the surface of the structure. Two-mode excitation has been introduced as an effective method to generate steady-state structure-borne traveling waves (SBTW). Two-mode excitation generates SBTW in finite structures by superimposing two standing waves in a structure with a prescribed phase offset. While the generation of STBW for propulsion and particle motion has been a topic of study for some time, most research has examined SBTW propagating along a single axis.

This chapter expands on this method by using two pairs of actuators to simultaneously excite two SBTW orthogonal to one another. The superposition of these orthogonal waves results in a net SBTW that propagates in any desired direction. By controlling the direction of SBTW using only two pairs of actuators, an active surface could be made to drive motion in any direction without the need for a large number of actuators. It is shown that while the excitation frequency will determine the mode shapes that dominate the behavior of the SBTW, it is the geometry, boundary conditions, and the locations of actuators that will determine the possible combinations of SBTW that can be excited in the plate. It is found that adjusting the relative amplitude and phase between the two SBTW will influence the quality of the wave, in addition to affecting the overall direction of the superimposed SBTW. This motion is studied using simulations in a 2D Finite Element model that represents a plate using first-order shear deformation theory. The quality of the SBTW is calculated using a traveling index defined by the complex orthogonal decomposition of the wavefront. The direction of the wavefront in this study is interpreted by calculating a weighted average of the structural intensity (SI) field over the plate.

Dynamic testing of systems is most realistic of real-world conditions when multiple input and multiple output (MIMO) techniques are used. To replicate measured environmental conditions, a series of desired outputs on the system must be realized by inverting the frequency response functions (FRF) matrix for force estimation. Depending on the number of inputs and outputs, the type and number of solutions can vary significantly. Depending on the objective of the test, defining the target response can also vary, such as matching accelerations, stresses, or strains at various locations. In MIMO testing, both the auto spectra and cross spectra of the outputs affect the auto spectra of the inputs. Defining the most relevant outputs and the optimal input spectra is an ongoing challenge and has attracted recent attention.

This research evaluates the errors associated with these variables on a linear multi-story frame fixed at the base. Three inputs are provided uniaxially with the use of suspended electrodynamic shakers, and up to nine output locations are instrumented with accelerometers. Results of tests are analyzed to guide improved MIMO testing.

High-rate structural health monitoring of active structures operating in high-rate dynamic environments empowers the execution of preventative behavior in response to structural degradation or external stimuli. Examples of structures operating in high-rate dynamic environments include hypersonic vehicles, space crafts, and ballistic packages. The effective selection of reactive actions to be taken in real time requires an up-to-date model of the structure’s state. Importantly, the short timescale of relevance to these structures means that the model must be continuously updated with a time step of 1 millisecond or less. However, traditional frequency-based methods for updating the finite element model online require solving the generalized eigenvalue problem, which becomes more complex as the number of nodes or FEA model increases, thereby increasing computational time. In this work, the local eigenvalue modification procedure is put forward to accelerate the extraction of natural frequencies from finite element models updated online. The local eigenvalue modification procedure works by precomputing the eigenvalue solution to a reference state of the system and then computes the single (i.e., local) change in the modal domain from the reference state to the current state online. The modal domain update in the local eigenvalue modification procedure bypasses the general eigenvalue problem, which is the most expensive computational step. For the online implementation of the state estimation, the Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research testbed is used. The testbed allows for the controlled movement of a pinned condition attached to an otherwise free cantilever beam. In previous studies, the testbed has been used with the generalized eigenvalue solver in a frequency-based model assimilation approach to infer the most likely position of the pinned condition (i.e., state of the structure). This work reports the effectivity of the local eigenvalue modification procedure compared to the generalized eigenvalue solution of the system for inference accuracy while varying the nodes dedicated to the analysis. The optimal efficiency of the system’s approach is explored for the testbed-based health assessments. Timing results and effects of sensor noise on the system are discussed in detail.

The average aerospace structural analyst is very familiar with performing various types of dynamics analysis (sine vibration, random vibration, transient) but much less familiar with dealing with shock analysis defined by SRS (shock response spectrum) dealing with very high acceleration levels (1000–2000 g’s) and high frequency content (10 kHz). Likewise, the average aerospace engineer performs a lot of sine and random vibration testing at the system and subsystem levels and feedback from those tests are readily compared to analysis predictions.

In contrast, typically very little SRS-related analysis and subsequent shock testing is performed either at the system or subsystem level and rarely would shock analysis predictions be compared with shock test results. Typically going into sine and random vibration test, a structural analyst will know expected responses at accelerometer locations. For a typical shock test, many times a dynamics engineer is a bystander and watches the test lab use their shock test method to achieve the SRS tolerance levels in each axis and typically no shock analysis predictions of the shock test setup are performed.

Sine and random vibration tests at the system level are common, but shock testing at the system level is much rarer (sometimes the customer will waive the system shock test requirement – that cannot happen!) as it is much more difficult test to perform. Shock loads are much more complex and more difficult to analyze than sine and random vibration loads and thus more engineering is required to get more accurate predictions. Thus, to reduce uncertainty in shock analysis predictions:

Reliable, safe, and sustainable form of mobility is an important issue in their engineering designs. Ropeways have been used for century for different purposes: their advantages in terms of cost, footprint, and energy efficiency make it as one of the solutions to meet the need for urban mobility. Nevertheless, because of the presence of moving cables and the time-varying modification of the mass repartition and stiffness properties of the system, ropeways are sensitive to the effects of dynamic perturbation and self-sustaining oscillations. So far, the analytical models are generally restricted to a cable span between two supports and are too simple to account for different existing phenomena. From this observation, the present chapter proposes a dynamic analysis based on an original definition of the vibration modes of the ropeway considered in its entirety according to an analytical–numerical approach. The modes are defined for several successive positions of the vehicles along the cable established on a linearized dynamics around a nonlinear static equilibrium state, using an equivalent parametric vision of the modal evolution of the system. Then, a model reduction by projection on a mode identified as problematic for the vibratory behavior is conducted. From the results given by the calculation of the quasi-static evolutions, a Mathieu–Hill type equation with a parametric excitation is obtained. A stability analysis of the reduced model is carried out in order to identify design rules that allow to avoid unstable operating zones likely to generate strong vibrations.

Modal analysis of rotating structures is a challenging field of structural analysis research. In the case of a rotating structure, the major difficulty is applying a known excitation to the system, which is crucial for frequency domain modal analysis. Another obstacle is installing and connecting enough sensors (such as strain gauges or accelerometers) to the rotating blades in order to fully characterize the system. Slip rings are used to transmit electrical signals between the rotating and non-rotating frames, complicating the installation. They can also change the actual response by shifting the natural frequencies and introducing external damping. As a result, the number of experimental modal analyses of rotating structures under operating conditions is limited. DIC and other optical methods are appealing for measuring deformations because they do not require electrical wiring or slip rings and can be easily configured to measure large test articles. While setting up an experimental setup with DIC is easier than setting up a contact measurement test setup, there are several challenges to overcome when using DIC with rotating structures. The main challenge is compensating for rigid body motion caused by structure rotation motion to evaluate deformation displacements rather than absolute displacements. This chapter uses DIC to measure deformations on a rotating structure to characterize its modal behavior in operational conditions.

Testing devices multi-axially can be a better approximation of a device’s operational conditions compared to single-axis testing. However, multi-axis environmental tests involve a more complicated test setup that is often determined by a test engineer’s judgment. Some of the complexities the engineer must determine include where to locate shakers on intricate geometries, how much input is required to excite the structure, how many control channels are necessary, and how well the environmental test boundary conditions replicate the operational environment. Any one of these things can lead to the device under test’s (DUT) response differing greatly from the desired response. Often there is no indication of whether the environmental test setup appropriately replicates field conditions prior to the start of the test. The ability to simulate a test on a finite element model (FEM) would allow the engineer to have insight into the predicted response of the DUT prior to performing an environmental test. This study uses Sandia National Laboratories Rattlesnake control software to conduct virtual tests on an FEM of the base section from a Box Assembly with Removable Component (BARC) and aims to determine if virtual testing can be used to predict optimal environmental test setup accurately and achieve the desired response from the DUT. Including a virtual test as a step in the procedure for multi-axis vibration testing could provide test engineers with the necessary information for a reliable test such as input controls and locations, equipment requirements, and sensor placement prior to performing a test, reducing the time required in the lab, and improving test results.

The field of vibration-based Structural Health Monitoring (SHM) relies on the evolution of the dynamic characteristics to identify the current health state of a monitored structure. In reality, these parameters are influenced either by a potential degradation of the structural integrity or by the varying environmental conditions. Therefore, a robust SHM scheme must clearly distinguish the different sources of changes in the monitored modal parameters. Along these lines, this study aims to identify structural changes in a monitored structure subjected to ambient vibrations and varying environmental conditions. The investigated structure consists of a wooden mast with a steel frame topside and is clamped to a concrete block at the bottom. During the monitoring campaign, the vibration response is acquired by a measurement system with four 3D sensors placed at strategic locations on the topside.

The steady-state response of harmonically excited structures can exhibit a significant traveling wave ratio. Local excitation of structures with locally increased damping or even structures that are proportionally damped, for example, lead to wave propagation phenomena. Since damping distribution plays a key role in formation of traveling waves, it needs to be considered in the dynamic analysis. In this chapter, we analyze the steady-state vibration behavior of 3D printed metamaterial structures. The investigated parts are made of resin and steel by laser sintering. The dynamic analysis with special attention to traveling wave effects is simulated based on finite element method and experimentally validated. Due to the complex geometry of the metamaterial structure, fine meshing is necessary for accurate results, making reduction techniques inevitable. A combination of modal reduction and dynamic condensation is used to obtain the simulated results. In the laboratory, laser scanning vibrometry is used to measure the entire structure and validate the simulations. We show in both simulation and experiment that the studied structures exhibit both standing waves with locally fixed nodal lines and traveling nodal lines with significant traveling wave content, depending on the excitation frequency.

Bridge monitoring projects based on vibration data analysis tend, since the early 2010s, to process vibration acceleration data using a time-domain system identification method use an automatic operational modal analysis (AOMA) algorithm to extract the modal properties, without the involvement of an operator. This work highlights the impact that the acceleration data sampling frequency has on the outcome of some of the AOMA algorithms. Acceleration datasets, downsampled to different frequencies, from the Hardanger Bridge are processed by the Magalhaes 2009, Neu 2017, and Kvåle 2020 AOMA algorithms. It is shown that that the best results in terms of modal detection rates and number of errors are obtained for sampling frequencies between 10 and 20 Hz. Additionally, no algorithm is more impacted than another by the different sampling frequencies.

Several gear dynamics models are available in the literature, each with its unique application. In this chapter, a comparison between two models will be discussed. While the source of excitation in both models will come from the time-varying mesh stiffness caused by the static transmission error, the first model aims to estimate the dynamic overloads due to the meshing with a simple one-dimensional approach, neglecting other flexibilities. The second one instead includes the flexibilities of the shafts as well as bearings and the gearbox housing, with the possibility of being extended to multistage transmissions. Both models are quick and can give useful insight in the design process of a geared transmission. A simple test case will be detailed highlighting the different capabilities of the models and providing several key results such as dynamic forces and displacements which can help designers to define better components.

Geared transmissions are prone to harmful vibrations and annoying noise emissions. The sources of excitation of those vibrations are many and different in nature, starting from torque fluctuations from the engine or the unsteady aerodynamics in wind turbines, for example. However, the main source of excitation comes from an intrinsic characteristic of meshing gears and is the time-varying mesh stiffness which is generated by the transmission error. Several flexibilities can be accounted for the calculation of the transmission error depending on the complexity of the model employed. In this chapter, the importance of including shaft flexibilities is highlighted. A nonlinear semi-analytical model is applied to the study of a simple test case with and without the misalignment caused by the inflection of the shafts under load, and several results, such as the static transmission error and the contact pressure maps, are shown and discussed.

Experimental methods are used across all engineering domains to study a system’s response to some specified input or forcing function. This work explores the response of a structural member in a wind tunnel containing a cantilever plate attached to a cylinder, under an impact excitation. The computer-aided design (CAD) model and finite element analysis (FEA) modal simulation of the experimental setup are introduced. Two experimental techniques used in this research are discussed – an accelerometer-based experimental modal analysis (EMA) method and a non-contact, full-field digital image correlation (DIC) operational deflection shape (ODS) analysis method. The FEA and experimental results of the first five mode shapes and natural frequencies of the cantilever plate are presented and compared. The modal assurance criterion (MAC) between FEA-based mode shapes and EMA-based mode shapes is at least 0.935 for each mode. Absolute values of percent errors between EMA-based and FEA-based natural frequency results are less than 5% for each mode, while absolute values of percent errors between ODS analysis and FEA-based natural frequency results are less than 11% for each natural frequency. When comparing the EMA method with the ODS analysis method, there is less than 2% difference between each mode’s natural frequency.