Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5

Impact localization refers to the problem of estimating the location of a spatially concentrated impulsive force acting on a mechanical structure, using vibration measurements from a sensor network (in this chapter, accelerometers). Obtaining accurate location estimates is challenging due to three incident wave distortion factors; namely: (1) wave reflections from boundaries, (2) wave dispersion (frequency-dependent wave propagation speeds), and (3) frequency-dependent damping. In this chapter, a new frequency-domain impact localization method is introduced, where the aforementioned distortion factors are eliminated by prior collection of a few frequency response functions (FRFs), corresponding to a sparse number of known impact locations (calibration points (CPs)). This calibration phase is done offline as a prerequisite to localization. Furthermore, the calibration phase includes a spline interpolation step, which adds more FRF impact points via interpolating between the initial CPs, effectively increasing the density of the FRF calibration grid. Upon detection of an impact, recorded sensor signals are used—within an optimal search framework—to estimate the impact’s location. The search for the impact location is done using the interpolated FRFs from the calibration phase. The method is evaluated using a finite element (FE) beam model. Results show promising accuracy and justify further development of the approach.

In the early days of modal testing, when data acquisition channels were precious and few, time data was often tape-recorded and then postprocessed into the frequency domain. Nowadays, our data acquisition hardware can measure a large number of channels and the software can perform data processing concurrently such that the measurements are conveniently available as soon as the test run is complete. However, there are some advantages to the old ways. This chapter extols the virtues of keeping the time response data measured during a modal test and discusses some obscure signal processing features that could be helpful in improving the quality of your frequency response functions.

Biomechanical studies of the spine have traditionally employed either cadavers or finite element modelling techniques, but both of these approaches have inherent cost and time limitations. Cadaveric studies introduce significant variability in measurements across patients, while finite element modelling can be computationally demanding depending on the required level of detail. 3D printable analogue spine models have been proposed in this research as a low-cost and high-fidelity approach to support these conventional methods. An L1–S1 analogue spine model was developed, incorporating vertebral bodies, intervertebral discs, and intertransverse and interspinous ligaments. Stereolithography 3D printing was used exclusively to manufacture the model. A custom bending jig was employed to apply load-controlled pure bending moments of up to 7.5 Nm in flexion–extension, lateral bending, and axial rotation to the model. The results were then compared to historical ex vivo L1–S1 data to evaluate model accuracy. The use of viscoelastic materials in the construction of the analogue model contributed to a hysteretic response during loading and unloading, simulating the mechanical behaviour of a human spine. Specifically, the model’s range of motion (ROM) during flexion was estimated to be \(12.92\pm 0.11^\circ \) when subjected to a load of \(+\)7.5 Nm, as compared to an ex vivo ROM of 16.58\({ }^\circ \). The model exhibited bilateral symmetry during lateral bending (LB) and axial rotation (AR). Notably, the ROM in LB was measured at \(13.55\pm 0.11^\circ \) under \(+\)7.5 Nm (ex vivo: \(13.32^\circ \)), and \(-13.79\pm 0.19^\circ \) under \(-\)7.5 Nm (ex vivo: \(-17.1^\circ \)), whereas the ROM in AR was recorded as \(19.82\pm 0.19^\circ \) under \(+\)7.5 Nm (ex vivo: \(14.44^\circ \)) and \(-15.57\pm 0.12^\circ \) under \(-\)7.5 Nm (ex vivo: \(-13.81^\circ \)). The 3D printed analogue model of the lumbar spine presented in this study demonstrated rotational stiffness within 20% of the ex vivo responses, with a high degree of repeatability. Further testing incorporating a wider range of materials is necessary to improve model stiffness control and statistical significance.

Video-based structural dynamics are becoming increasingly popular on account of their high spatial resolution and relatively low costs. These techniques show promise for applications such as structural health monitoring. However, video-based approaches to measuring structural dynamics are not as sensitive as on-contact sensor-based techniques. Furthermore, because video-based techniques involve capturing a 2D perspective projection of the 3D environment, there is an added complication in extracting calibrated quantitative measurements from these systems. In many applications, it would be advantageous to deploy networks, or arrays of imagers for monitoring large-scale infrastructure from different perspectives. Often, computer vision techniques, such as multi-view stereo, benefit from knowledge of the position and orientation of the cameras. Unfortunately, accurately knowing this information can lead to increased setup costs. Furthermore, it is not always clear what arrangement of cameras would be sufficient for any given infrastructure monitoring applications. In addition, the performance of arrays of imagers is affected by factors such as lighting conditions and weather. There may also be significant restrictions on the physical locations where imagers are allowed to be placed. For these reasons, it is advantageous to have tools to aid in the design and development of imager-based sensor networks for structural dynamics applications. In this work, we demonstrate the use of open-source computer graphics software for simulating the behavior of imagers observing dynamic structures. We demonstrate the ability to use these tools to plan the deployment of imager-based sensor networks to rapidly test and develop algorithms and techniques for video-based structural dynamics.

This chapter presents the development and modal characterization of a compliant wing for an underwater marine hydrokinetic (MHK) kite. MHK kites are systems engineered to efficiently harness current and tidal energy from bodies of water, employing cross-current flight patterns to augment velocity and, thus, power production. Given the substantial fluid dynamic loading that MHK kites undergo, which scales with the square of the apparent flow speed at the kite, comprehensive structural analysis is critical for the success of the system design. The speed-dependent loading of the wing can result in wing bending and twisting, which alters the flight dynamics of the kite and requires adaptive control strategies to maintain performance across all flow regimes. The previous North Carolina State University (NCSU) MHK scaled experimental vehicle features a rigid wing, fabricated using aluminum ribs and spars and carbon-fiber laminate skin, which does not appreciably flex during underwater flight. However, when sized for full-scale deployment, MHK kite wings typically cannot remain rigid without excessive internal structural material resulting in high vehicle weight, or without decreasing the wing aspect ratio, which negatively impacts kite performance. To observe and demonstrate wing flexure and twist under small-scale experimental flight conditions, a new compliant wing is devised and built, making use of a foam core and a thin fiberglass skin. To integrate with the NCSU experimental vehicle, the wing is fabricated as two individual left- and right-hand mirrored halves and is approximately of the same wingspan of 0.842 meter. The damped natural frequency of the first bending mode of one compliant half-wing is measured using a laser Doppler vibrometer, and the wing’s effective elasticity modulus was identified. The characterized compliant wing was then simulated in the dynamic model of a small-scale MHK kite and was predicted to experience a maximum tip deflection of 4.05% of the kite wingspan during cross-current flight. Such modeling could prove useful for optimal kite flight controller tuning and appropriate design of the kite wing for structural fatigue life considerations and efficient material usage.

The dynamic behavior of a turbopump inducer is investigated in this study, with a particular focus on the added mass. Inducers are commonly used in turbopump applications to prevent cavitation. Since inducers have no shroud on the blades, blades of inducers are more prone to vibration. Moreover, it is known that the natural frequencies of the inducer immersed in liquid will decrease due to the added mass effect. Since inducers have complex shapes, there is no analytical formulation to compute natural frequencies. This chapter represent a study that calculates natural frequencies and mode shapes of the inducer immersed in water. Initially, a literature review identified test-verified analytical solutions for natural frequency reductions in simple geometries such as cantilevers and simply supported plates. The ANSYS fluid/structure coupling approach was subsequently used to obtain numerical solutions, which demonstrated good agreement with published results. The techniques derived from this analysis were then applied to the more complex inducer model, and the ANSYS numerical results for both natural frequency and mode shape were obtained. Results of this study reveal that not only natural frequencies but also mode shape of the inducer have changed due to added mass effect. The findings of this study provide valuable insights into the design on the dynamic behavior of turbopump inducer. The results can be utilized by engineers and designers in the optimization to enhance inducer performance, minimize vibration-induced failures, and improve overall system reliability.

During the structural analysis of micro- and nanostructures, classical elasticity theory fails to capture the small-scale size effects that are present in the experimental results. To incorporate the small-scale size effects into the equation of motion, either intermolecular forces over relatively long distances or higher-order derivatives of displacement field need to be considered. Two such theories employing the mentioned models are nonlocal elasticity theory and strain gradient theory, respectively. In this work, nonlocal strain gradient theory, which includes small-scale size effects, is used to model carbon nanotubes with the assumptions of Timoshenko beam theory. Natural frequencies are found with the differential quadrature method, and genetic algorithm is used to determine the nonlocal and length scale parameters present in the theory using molecular dynamic results for different aspect ratios to obtain the best overall fit. Instead of obtaining different small-scale parameters for each mode number, optimization of all modes at once is carried out, which is extended to also include the classical material properties. The results are inclined toward nonlocal elasticity theory for the problem at hand.

Finite element analysis (FEA) is used in many root cause analyses of field failures for blower silencers and locomotive/engine silencers. The purpose of the FEA dynamic frequency response is to determine any high stress regions in the silencer at normal operating conditions, from blower idling to maximum speed. Typically, the compressor’s gas pulsation is high and could be a potential source of strong vibration force. Additionally, a rotational excitation from the driven engine of the compressor could add to the silencer vibrations, resulting in higher stresses around the silencer’s shell along the internal baffle plate. The Creo Simulate was used in the FEA analysis. The FEA forcing functions should be uniform across the interested frequencies. The frequency range and amplitude should be based on the measured data.

The recently proposed theory of Population-Based Structural Health Monitoring (PBSHM) aims at improving diagnostic inferences, by sharing damage-state knowledge across a population of structures via transfer-learning algorithms—specifically domain adaptation. Before applying these algorithms, the similarity between structures, or substructures, should be evaluated. This assessment helps prevent negative transfer, ensuring better performance and higher robustness of data-based SHM.

When structures are sufficiently similar, different transfer-learning strategies can be applied, according to the original features and the specific case study. In this framework, structural attributes play a crucial role, especially for heterogeneous populations in which the main differences can be caused by material properties, geometry, or dimensions. Therefore, investigating how to consider the influence of these properties in distance metrics became necessary, and new similarity metrics have been adopted to focus on geometric features and dimensions. However, to gain a comprehensive understanding of attribute relevance and to address it at the similarity-evaluation phase, it is necessary to evaluate the performance of transfer-learning algorithms as these structural features vary.

The present work extends this research by examining the effect of material and dimension attributes on the performance of a domain-adaptation method—the Transfer Component Analysis (TCA). This analysis is applied to an experimental population of laboratory-scale aircraft, comprising structures with different materials and dimensions, and similar topology. A confusion matrix is employed to compare the findings and show how these properties can influence the transfer-learning performance, especially for localised damage, thus highlighting the importance of their evaluation in the context of PBSHM.

Accurate measurement of frequency response functions is essential for system identification, model updating, and structural health monitoring. However, sensor noise and leakage cause variance and systematic errors in estimated FRFs. Low-noise sensors, windowing techniques, and intelligent experiment design can mitigate these effects but are often limited by practical considerations. This chapter is a guide to implementation of local modeling methods for FRF estimation, which have been extensively researched but are seldom used in practice. Theoretical background is presented, and a procedure for automatically selecting a parameterization and model order is proposed. Computational improvements are discussed that make local modeling feasible for systems with many input and output channels. The methods discussed herein are validated on a simulation example and two experimental examples: a multi-input, multi-output system with three inputs and 84 outputs and a nonlinear beam assembly. They are shown to significantly outperform the traditional H\({ }_1\) and H\({ }_{\text{SVD}}\) estimators.

Centrifugal pendulum vibration absorbers are order-tuned masses oscillating on purposefully designed cutouts with the support of cylindrical rollers. The complementary geometric shapes of cutouts and rollers, in combination, form the path that absorbers take during the centrifugal pendulum vibration absorber (CPVA) operating cycle. The path of each absorber can be different from the ideal baseline design due to manufacturing errors or wear as a result of prolonged use under heavy operating conditions. These aberrations of the path geometry may lead to serious loss in the vibration reduction performance due to asynchronous and localized response of absorbers. This work considers periodic waviness-type path imperfections, which are usually caused by machining errors and investigates their impact on CPVA’s forced vibration response. Waviness imperfections are analyzed in different designs, including cyclically symmetric and asymmetric configurations by evaluating vibration reduction capability and absorber oscillation amplitudes. The nonlinear dynamic model of the CPVA developed and utilized in this study includes translational and torsional dynamics of the rotor and oscillations of absorbers. Details of the dynamic model and the definition of waviness-type path imperfection are presented initially. The impact of number of absorbers, number of nonideal absorbers and influence of translational dynamics of the rotor are investigated in terms of their interaction with the waviness order and amplitude. As a result, a geometric tolerance recommendation is made on the circularity of the cutouts to circumvent the significant performance degradation linked to absorber path waviness errors.

Traditional spine biomechanical studies involve cadavers or finite element models, both of which have cost and time limitations. Validated analogue spine models have been proposed as a low-cost and high-fidelity approach in biomechanical testing. To develop them it is essential to understand the effect of underlying soft tissues on model’s biomechanical response. This chapter aims to assess the impact of intertransverse and interspinous ligaments on the rotational stiffness of lumbar spine by systematically constructing and testing an analogue spine model. A fully 3D-printed analogue lumbar spine was developed with L1-S1 vertebrae, intervertebral discs, intertransverse and interspinous ligaments. Stereolithography 3D printing was used to construct the model. The model construction involved three phases (P1, P2, P3). After each phase, model rotational stiffness was assessed by applying cyclic pure bending loads up to 5Nm in flexion–extension, lateral bending, and axial rotation, and the rotation of motion (ROM) was recorded. P1 involved assembly of vertebrae and intervertebral discs, while P2 involved addition of interspinous ligament to the P1 model. P3 concluded model construction by adding intertransverse ligaments on either side of the P2 model. The analogue spine model demonstrated a hysteretic behavior during load–unload cycles much like the human spine owing to its viscoelastic components. In flexion–extension motion, addition of ligaments significantly decreased the model ROM (\(Flexion - P1=9.9\pm 0.1^\circ \) \(P2=9.4\pm 0.1^\circ \) \(P3=8.3\pm 0.1^\circ \) and extension \(- P1=9.0\pm 0.1^\circ \) \(P2=6.7\pm 0.1^\circ \) \(P3=7.1\pm 0.1^\circ \)). In LB, there was no discernable ROM change from P1 to P2 but a decrease from P2 to P3 \((left - P1=9.0\pm 0.1^\circ \) \(P2=8.9\pm 0.1^\circ \) \(P3=8.7\pm 0.1^\circ \) and right \(- P1=9.1\pm 0.1^\circ \) \(P2=9.2\pm 0.1^\circ \) \(P3=8.6\pm 0.1^\circ \)). No trends were observed in AR as construction progressed (\(left - P1=12.8\pm 0.1^\circ \) \(P2=14.3\pm 0.2^\circ \) \(P3=12.6\pm 0.2^\circ \) and right \(- P1=12.1\pm 0.3^\circ \) \(P2=10.8\pm 0.1^\circ \) \(P3=11.4\pm 0.2^\circ \)). The model exhibited bilateral symmetry in lateral bending and axial rotation. Inclusion of interspinous and intertransverse ligaments significantly increased flexion and lateral bending stiffness, respectively, in agreement with the current understanding of lumbar biomechanics. This novel model holds potential as an effective research and educational tool to further investigate damage modes in the lumbar spine structure.

As industries strive for enhanced reliability and efficiency in product optimization, virtual prototyping has gained prominence over its physical counterpart. Advanced engineering simulation tools have thus become essential for addressing the complexities of today’s technology-driven world. The field of rotordynamics plays a critical role in the design and analysis of rotating machinery as they are widely found in various engineering devices for, e.g., energy transmission. Over the years, numerous techniques have been developed to capture the dynamic behavior of rotors. Among these, continuum finite element formulations have emerged as powerful and generic tools. Nevertheless, rotordynamics analysts often rely on simple models based on beam and rigid elements. Moreover, a variety of different formulations and implementations in commercial software exist, but their relationship to one another is oftentimes not clear. This chapter, therefore, presents an overview of rotordynamics finite element formulations based on continuum elements for generic geometries from a structural and multibody dynamics perspective. A derivation of the equations of motion for rotordynamic analyses according to a nodal-based floating frame of reference formulation is provided. Throughout the contribution, emphasis is placed on the comparison and evaluation of different modeling techniques and solution strategies, especially with respect to standard FE commercial software approaches. The discussion includes modal and steady-state analyses.

In the field of structural dynamics, tasks related to data acquisition and processing are commonplace. Despite the growing popularity of Python as a tool for data manipulation, its application in data acquisition has been either limited or difficult, requiring significant programming effort from individuals. The lack of a comprehensive Python-based system for data collection is an obstacle to the effectiveness and simplicity of data collection and preservation processes. To address this problem, LDAQ, a pure Python-based data acquisition package, was developed.

LDAQ provides a user-friendly platform for data acquisition and signal generation from a variety of sources, including but not limited to National Instruments, serial communication hardware such as Arduino and ESP, and camera devices such as FLIR thermal imaging cameras. It also facilitates easy integration of new data acquisition and signal generation sources that are not currently supported. LDAQ enables live visualization of both measured and processed signals, including FFT and arbitrary functions, and provides further customization capabilities of these real-time graphs. As an open-source package, LDAQ is freely available and open for collaboration on improvements.

LDAQ offers users the opportunity to integrate this package into their data acquisition processes and contribute to its further development. It paves the way to combine different data collection systems, organize data collection processes, and create a foundation for further refinement and personalization. The advantages of LDAQ are its ease of use, its Python compatibility for additional signal processing, and its potential to simplify data collection systems, increasing efficiency and productivity in data collection.

This chapter presents a novel approach for system identification in the presence of incomplete output information available and with limited knowledge of the input forces. The correct identification of the dynamic system is a challenging task, and it becomes more problematic when the input information is unavailable. To overcome this limitation, this work integrates a set of system measurements with computational model responses, enabling recovery of dynamic system states and subsequent analysis of the model through an inverse problem formulation based on Bayesian model updating. The difference between the computational model response and the measurements is described with Gaussian Process discrepancy model that uses time-based kernel covariance function for the inference on model parameters. Such an assumption mitigates the effect of the measurement noise on parameter estimation, leading to improved fidelity in parameter estimation and uncertainty quantification. To find the forces applied to the system, an optimization strategy is used that aims to minimize the residuals of the input forces at the locations where the knowledge of forces is available. The inputs are identified using a combination of the known system measurement with pseudo-measurements, followed by an inference on the structural model parameters. The proposed technique shows promising results, offering a methodology for input and parameter estimation. The practical implications of the work include its potential application in real-world scenarios requiring consistent system identification and force estimation.

Data-driven system identification is essential for understanding the behavior of real-world systems. Since the dynamics of such systems are often complicated, complex, or intractable, estimating them is essential to developing a model and applying control. This chapter compares the performance of various data-driven identification methods on a candidate system, the multi-axis cart system (MACS). The MACS is a mass–spring–damper system consisting of four discrete masses arranged in two axes. These two axes are then coupled using a rigid massless link. While the MACS is simple in construction, it can be configured to be linear or nonlinear and exhibits modes with components in two axes. Understanding the dynamics of this simple system can help gain insight into the behavior of more complicated systems.

This chapter compares the performance of four popular data-driven identification methods, SINDy, SINDy-PI, DMD, and Hankel DMD (HDMD). The models were trained using many trajectories tracking the states and derivatives over time. The data was generated by simulating the governing equations for the system which were derived using Lagrange’s equations. To make a more appropriate comparison to real-world measurements, which innately have sensor noise, proportional random Gaussian noise was added. In addition to the trajectory error, several implicit properties of the models were analyzed such as model sparsity and model stability.

In recent years, modal parameter-based techniques have gained particular interest in the development of structural health monitoring (SHM) strategies. Natural frequencies are the most commonly used dynamic parameter for damage detection in vibration-based SHM as damages manifest as a change in structural properties (stiffness or mass). Unfortunately, environmental changes, such as temperature drift, can cause modal frequency variations even more predominant than possible incoming damages.

In this chapter, a laboratory truss girder made with steel beams and aluminum rods subjected to environmental variability has been analyzed. Modal parameters and ambient conditions have been evaluated through the employment of accelerometers and temperature transducers. Natural frequencies and ambient temperature have been processed to develop damage detection techniques with robust immunity to environmental interference.

A principal component analysis (PCA)-based approach has been performed to identify the features most sensitive to temperature changes. Once the environmentally sensitive features were excluded, the remaining features containing information about the health of the structure were processed to obtain the baseline of the system in healthy conditions. Then predetermined damages, such as mass variation or loss of bolt preload at rod intersections, have been induced in the structure. Condition-sensitive features of the system have been used to develop damage identification techniques through statistical indicators (i.e., Mahalanobis squared distance).

The results prove that a damage index (DI) evaluated based on these statistical indicators characterizes the incoming damage in the examined structure without the influence of environmental variations.

Friction-induced vibrations are one of the main mechanical stimuli at the origin of tactile perception, allowing perception and discrimination of surface textures. While acoustic waves and electromagnetic waves are successfully reproduced for mimicking the auditive (loudspeakers) or visual (monitors) stimuli, the stimuli at the origin of tactile perception are still not fully understood and are still not reproduced. This work presents the development and the dynamic analysis of a device, allowing the reproduction of vibrations induced by the sliding of the finger on a surface.

The overall bio-electro-mechanical transfer function, including the fingertip, mechanical device and its control electronics, has been first characterized. The system is highly nonlinear, due to the contact nonlinearities and the nonlinearities proper of the tissues of the fingertip, and a parametrical analysis has been developed for investigating the effect of the contact parameters (contact force, subject, etc.) on the transfer function of the overall biomechanical system. Then, the vibrations measured on the nail of the subject, during the exploration task of different surfaces, have been reproduced by the tactile device.

A first validation is obtained by the comparison of the original and the simulated vibration spectra. Then, a discrimination campaign has been developed to verify the ability in discriminating different textures, both during the exploration of real textures and when mimicking the respective vibrational stimuli. The obtained spectra can be correlated with both correct discrimination results and discrimination errors, allowing the identification of the spectral features responsible for texture perception and discrimination.