Topics in Modal Analysis & Parameter Identification, Vol. 9

Modal parameter estimation involves solving for mode shapes, damping, and natural frequency. Modal mass can be estimated if drive point acceleration over force (A/F) frequency response functions (FRFs) are available. This paper shows that force over acceleration (F/A) drive point FRFs are also required to calculate modal mass for FRFs with acceleration as references. In practice, F/A FRFs can be obtained either by using drive point accelerations as references or from modal survey tests performed alongside force-limited vibration qualification tests, where force links measure the interface forces. The process will be demonstrated on FRFs from both a simple, lumped-parameter system and on FRF data from test articles.

Many methods for nonlinear structural dynamics testing have been developed, some of which utilize modal shakers for excitation. One such technique is force appropriation, where the structure’s excitation is maintained at resonance using closed-loop control as the forcing amplitude is increased. This technique has been shown to provide both valuable insights into the true physics of structures and rich data sets from which to perform system identification; however, there is a potential for these results to be distorted if any nonlinear physics is inadvertently introduced to the system by the modal shaker either in the measured input or structural response. This work investigates—via experimentation—the nonlinear electrodynamic effects produced by the shaker and injected into the structure. This nonlinearity is evaluated by conducting force appropriation on a linear structure up to sufficiently high levels such that harmonic distortions appear in the excitation force.

Bismuth telluride (Bi₂Te₃) is a thermoelectric material enabling efficient heat-to-electricity conversion and vice versa, finding applications in radioisotope thermoelectric generators (RTGs) for satellites (Launius, Powering space exploration: US space nuclear power, public perceptions, and outer planetary probes. In: 6th International energy conversion engineering conference (IECEC), p 5638, 2008) and solid-state cooling systems (Chu, Simons, Application of thermoelectrics to cooling electronics: review and prospects. In: Eighteenth international conference on thermoelectrics. proceedings, ICT’99 (Cat. No. 99TH8407). IEEE, Piscataway, p 270–279, 1999, August). However, the intrinsic brittleness of Bi₂Te₃ renders it susceptible to microcracking during manufacturing, leading to disruptions in its operational efficiency. Traditional methods for crack detection, reliant on manual optical inspection, prove to be time-consuming, subjective, and costly. In this study, we introduce an innovative automated crack detection technique, employing resonant acoustic excitation in conjunction with the Kirchhoff-Love plate theory and advanced image processing. While our method presents promise, our experimentation on 27 Bi₂Te₃ wafers reveals that it is a work in progress, with effectiveness not yet reaching its full potential. Although some agreement with the conventional baseline is observed, further refinement and enhancement are essential. This research holds the potential to advance the reliability and efficiency of thermoelectric materials, transcending Bi₂Te₃ wafers to offer broader applications in material science and engineering.

The capabilities of humanoid robots have greatly increased in recent years, especially when it comes to bipedal locomotion. However, there is a lack of standardized definitions of human-like gait for robots and little attention has been paid to how human-like the gaits of such robots are. In this contribution, we aim to contribute to such a standardized benchmark. We perform a Proper Orthogonal Decomposition of the kinematic gait data during one step of our humanoid robot Lola to extract the underlying gait pattern. We then use the Modal Assurance Criterion to quantitatively compare the gait pattern of Lola with the gait pattern extracted from human gait data. The results show that our approach allows for a quantitative assessment of gait similarity, and thus of human-likeness of gait. We observe some overall similarities in the gait patterns of Lola and the human subjects, however, the gait patterns between different human subjects are much more similar than when comparing them to Lola.

Rattlesnake is an open-source vibration controller designed for multiple-input, multiple-output vibration control research and development. Over the past few years, the software has become the main vibration controller for multiple-input, multiple-output testing at Sandia National Laboratories. Rattlesnake has been used on several large-scale tests, controlling nearly 100 control channels using up to 24 shakers while simultaneously streaming over 200 channels to disk. With several researchers using modal quantities to study vibration control, there has been interest in adding experimental modal analysis capabilities into the Rattlesnake software, which would allow the user to seamlessly switch between modal and vibration testing without changing software or hardware setup. This chapter will present recent upgrades to the Rattlesnake vibration controller software, including an overhauled system identification module, the addition of warning and abort limits, and its new modal testing package.

The ultimate goal of this study is to produce a data-driven monitoring system that provides advantages over current methods for identifying structural damage on an aircraft component. Two types of damage-sensitive features (DSFs) were tested in an experimental setting to evaluate their ability to detect a small change in flexural stiffness in a particular, frame-like aircraft component. The vibration-based monitoring system utilized two DSFs: one based on spectral distancing metrics and another based on spectral peak characteristics. This system was developed to discover damage classified as the loss of stiffness at connections by comparing experimental aircraft component data to a healthy baseline. The DSFs were fed to a support vector machine redundant learning model to provide a damage decision. The results of this decision were compared to the known damage case of the test to evaluate the true positive rate and true negative rate of the classification method for the model. The DSF based on spectral distancing metrics had 70.4% true positive and 81.3% true negative rates for a 2.5% reduction in bending stiffness at the location of most likely damage. The second DSF based on spectral peak characteristics had 100% true positive and 100% true negative rates for the same reduction in bending stiffness. This system shows potential for a simple, accurate, efficient, and cost-effective inspection technique for this particular type of damage, meant to simulate crack propagation.

This chapter deals with one of the main Noise, Vibration, and Harshness (NVH) problems on low-cost motorcycles, that is, the handlebar submitted to exogenous and endogenous vibrations generated by the engine operation and transmitted to rider and passengers. Therefore, it is proposed the application of passive vibration absorbers on both sides of the handlebar, whose designs and synthesis are based on the Acoustics Black Hole methodology. In particular, a low-cost motorcycle with a 150 cc, unbalanced single-cylinder and air-cooled engine is considered as a case study for the numerical and experimental modal analysis of the handlebar and all the structural dynamics involved on the human–machine–structure interaction. Both vibration absorbers are sharp-end beam-like counterweights placed at the ends of the handlebar, which are simple and inexpensive solutions. Some numerical modeling using finite element methods and experimental modal analysis techniques are described to evaluate the dynamic performance of the implementation on the motorcycle, for different operating conditions.

This chapter explores the application of modal analysis to assess the impact of making process on the structural properties of maple wood used as sides of quartet family instrument. Specifically, we investigate the effects of a bending process using hot iron and water on the sides of the violin. The mechanical properties are compared before and after bending process at same relative humidity condition. Finite Element Model Updating (FEMU) is employed to identify changes in the longitudinal modulus of elasticity and damping. Our results indicate a weakly significant decrease in longitudinal rigidity, but a significant reduction in longitudinal specific modulus, and a highly significant increase in longitudinal damping, shedding light on the implications for instrument makers.

The demand for global electricity is expected to increase steeply in the coming years. In the worldwide quest for more renewable energy sources, the rapid development of offshore wind power suggests this power source will be crucial to meet climate and energy targets. The exponential growth of offshore wind energy can be attributed to several factors, including abundant space and greater, consistent wind resources, as well as technological advantages for offshore wind turbines with high availability and capacity factors.

The dimensions of (offshore) wind turbines are continuously growing, and modern turbines reach a power-generating capacity of 14 MW and more with blade lengths exceeding 100 m. Slender structures such as these blades have interesting dynamics, and it is important to characterize these by modal testing. Hereto the blades are dynamically excited, and the vibration responses are measured with accelerometers. A data-driven system model is identified from these measurements, yielding the modal parameters of the blades: eigenfrequencies, damping ratios, and mode shapes. These blade properties are, for instance, used to validate blade structural models (finite element models).

This paper reports on recent testing efforts with as objective to establish practical industrial guidelines for blade modal testing. Following topics will be discussed: sensor choice (ICP seismic, ICP regular modal, micro-electromechanical system [MEMS]), blade excitation techniques, data pre-processing techniques, and modal analysis techniques (experimental modal analysis [EMA] and operational modal analysis [OMA]).

Metamaterial structures are characterized by a periodic arrangement of the so-called unit cells. These can be of a wide variety of types, e.g., (alternating) geometry or material variations. A characteristic feature of these structures is that a bandgap can occur in a relatively wide frequency range. In this chapter, we investigate a metamaterial structure which consists of unit cells which are themselves resonators. When considering infinite unit cell repetitions, complex boundary conditions (Bloch–Floquet theorem) may be applied to a single unit cell to reduce the computational cost. By modeling an infinite structure, boundary reflections are not taken into account. However, these reflections play an important role regarding a real structure with a finite number of repetitions of the unit cell.

In the present work we analyze the vibration behavior of beam-like 2D periodic metamaterial. The focus is on wave reflections and corresponding edge effects, which can be particularly pronounced in metamaterial structures. In the bandgap, there are vibration modes in which almost apparently just the edge regions of the entire structure oscillate, while the central part hardly shows any deflection. The local response at the edges occurs due to superposition of waves with different traveling directions. These vibration phenomena prevent direct transfer of metamaterial design based on infinite wave propagation analysis to finite structures. For metamaterial design, we propose a fast dynamic analysis of finite structures using state-of-the-art model order reduction techniques. This way the local vibration effects are included in the steady-state response and allow for vibration analysis of the real structure. Furthermore, we tackle the edge reflections in the real structure by local dissipation measures.

Space flight hardware (SFH) experiences intense vibratory loading during flight, which only lasts a few minutes. When determining the appropriate size of these components to withstand such loading without becoming damaged, standard design practice is to assume that the peak dynamic loads are applied statically. In doing so, the resulting stress is compared against a material strength parameter obtained from a quasi-static experiment. Since the near-peak stresses are only experienced over a small fraction of time in reality, this approach leads to design conservatism that unnecessarily increases structural mass as well as the associated inefficiency and financial cost.

In an effort to modernize engineering design standards to appropriately consider the higher practical strength of dynamically loaded structures, this overarching research project seeks to develop an experimental test procedure for quantifying the dynamic strength of metallic alloys as a function of excitation frequency. In the ideal case, the characterization test would include an in situ method for monitoring the onset and progression of plastic deformation of the test specimen undergoing vibratory loading. These new tests are designed to be high-intensity (forcing amplitude), short-term (60 s at full amplitude), and cyclic in nature (sinusoidal excitation via attached stinger to an otherwise cantilevered beam). In addition, the initial alloy under investigation is 6061 aluminum, due to its wide use and applicability for SFH.

Thus far, the primary candidate under development with live capacity is to track hysteresis behavior of the beam from power dissipation trends, calculated via force (from the transducer on excitation stinger) and velocity (measurements from the laser Doppler vibrometer) data and work to distinguish between elastic and plastic features. As a key component of this ongoing method development, any pseudo-live indication of plasticity could be corroborated against the outcome of a pre-post assessment of damage via macroscopic evaluation of beam geometry (i.e., assessing any permanent change in the beam’s tip deflection). Support for experimental design decisions as well as dynamic strength data from tests with excitation frequencies of 10, 40, and 55 Hz will be discussed. This work contributes to the foundation for a new type of vibration-based characterization experiments and generates initial data on the functional strength of 6061 aluminum under the conditions considered.

Gears are commonly used components in a variety of mechanical applications to transmit rotary motion. Understanding the dynamic behavior is important for both gear design and operation to optimize performance and vibration behavior. In this experimental study, we investigate the effects of particle damping on the dynamic behavior by comparing additively manufactured gears with and without particle damping. The gears are excited using an automatic impulse hammer to subject them to almost identical excitation conditions. A free suspension of the gears allows free vibrations. The response amplitudes are measured using a laser vibrometer. The focus of this experimental study is to investigate the nonlinear effect of amplitude-dependent damping characteristics. For this reason, a variation of excitation force levels was performed. By analyzing the collected data using structural identification methods, an expressive comparison between the gears can be drawn. The presence of particle damping within the gears leads to a significant reduction in vibration amplitudes. The research shows that gears with particle damping exhibit a more pronounced dependence on excitation amplitude than those without. The results of this study contribute to the understanding of the benefits of particle damping in gear systems. By reducing vibration amplitudes, particle damping offers a promising approach to mitigating gear-induced vibrations and their associated issues.

Several examples of structure-borne traveling waves (SBTW) are found in nature as a form of locomotion such as in rays, snails, and snakes. While researchers have been able to replicate these traveling waves in one- and two-dimensional structures, this work has been primarily focused on SBTW propagating along a single axis. In recent years, researchers have begun investigating methods of steering the propagation direction of traveling waves in two-dimensional structures. One method of steering these waves is to superimpose orthogonal traveling waves (OTW) and adjust the parameters of the individual SBTW to change the direction of the overall wave. However, until now, work on these superimposed OTWs relies only on simulations and a linear superposition to replicate multiple actuators under simultaneous excitation.

This chapter seeks to expand on the previous work by experimentally examining the behavior of superimposed orthogonal traveling waves. Previous theoretical work has sought to explain the overall propagation behavior resulting from these SBTW combinations using the structural intensity (SI) incited by the wave. This explanation has so far been examined using simulations, assuming that the SI induced by each contributing SBTW can be superimposed to produce an overall SI field on the surface. This work will challenge this assumption by experimentally updating the FE model before calculating the SI field for individual and combined traveling wave cases. To this end, a modal test will be performed on a real square plate to update a finite element model. This model will then be used to determine the excitation conditions necessary for suitable SBTW in the plate. These superimposed orthogonal SBTW will then be measured experimentally individually and then simultaneously. First, it will be examined if the superimposed excitation conditions result in an overall superimposed waveform. While this has been shown to hold for SBTW propagating along a single axis, it is yet to be shown for SBTW operating orthogonal until now. Finally, the SI will be calculated for the individual SBTW and the case with superimposed excitation conditions.

The frequency response function (FRF) is the most important measurement in the field of experimental modal analysis. Theoretically, for linear systems, the natural frequencies stay constant irrespective of using various numbers, types, and locations of inputs. In practice, there is some difference between natural frequencies estimated using various input configurations due to different numbers/locations of exciters (shakers) and force transducers. Historically, these anomalies have been attributed to local changes in mass, stiffness, and damping at the connection points. In this chapter, FRF estimates obtained with various single-input multi-output (SIMO) and multi-input multi-output (MIMO) configurations using pneumatic excitation have been compared. While testing a circular steel plate and a rectangular steel plate using pneumatic exciters with force transducer on structure (FTS) mounting, it was discovered that natural frequencies vary slightly in different input configurations. Unlike shaker excitation, pneumatic excitation decouples the exciter from the structure, which results in identical mass, stiffness, and damping matrices of the structure in different configurations. In addition, all the force transducers were kept mounted throughout these tests, even when force was not applied/measured through them, to ensure the invariability of the structure. Despite taking these precautions, it was found that the natural frequencies obtained in various configurations are different. With further investigation, it was discovered that the cause of variation in natural frequencies is the variable mass loading effect of force transducers in different input configurations.

Brake discs play a crucial role in the safe operation of passenger cars and commercial vehicles. The standards and guidelines that guarantee this safe operation specify which measurement sensors can be used when testing brake discs in development. In this chapter, a novel measurement method is used to analyze vibrations on brake discs based on the German guideline VDA 301. Using a high-speed camera and image-based vibration analysis, modes are determined by means of experimental modal analysis and compared with the results of the standard sensor specified in the guideline. Preparation of the brake disc via speckle pattern is not required since an AI-based full-field optical flow algorithm is utilized to extract displacement data from recorded images. A smart pulse hammer is used for automatic, reproducible excitation.

Resonant fixture testing techniques, particularly resonant plate, bar, and beam, have been used for mechanical shock testing for several decades. The resonant bar shock test is a dynamic test of a mid-field pyroshock environment where a test article is mounted on one end of the bar and a projectile is struck against the opposite end along the longitudinal axis. This causes the bar to resonate at the natural frequency of the extensional mode of the fixture, while the transverse response remains low. Model-driven test design of resonant bars suggests that the transverse response can achieve similar orders of magnitude to that of the axial response with an off-center projectile impact. A model of a legacy resonant bar fixture shows that impacts not located at the centroid of the cross-section cause transverse motion of the test article. The transverse motion can even be in the same order of magnitude as the in-axis motion. This paper shows that the response of the resonant bar fixture is strongly influenced by boundary conditions and input force representation for the projectile impact. Integrating these findings, a new resonant bar fixture is designed with a square cross-section to deliver a multi-axis response, all at similar levels of magnitude.

Turbomachinery blisks are essential components in gas turbine compressors. Blisks are structures manufactured as one single piece, exhibit low damping, and operate under high forcing magnitudes, which makes them subject to high-cycle fatigue. Therefore, it is important to identify and implement methods that reduce blisk vibrations. The state-of-the-art blisk dampers include techniques that use energy transfer from the host structure (i.e., blisk) to an attached resonant vibration absorber (RVA) to reduce the vibration of the host. In addition, RVAs can benefit from energy dissipation via nonlinear contacts (e.g., friction and/or impacts). Previous literature includes the authors’ investigation of RVAs applied to a cantilever beam. This paper analyzes the effects of an RVA attached to a blisk sector. The RVA is placed under the blade, in the disk, and is designed as a small cantilever beam with a hemispherical end to allow for both friction and impact contacts. The energy dissipated depends on the excitation forcing magnitude, material properties, contact surface, and normal force at contact surfaces. The blisk sector and the RVA are additively manufactured from titanium to allow for future exploration of tuning options through microscopic material changes rather than structural geometric modifications. Results reveal that for fixed excitation amplitude and material selection, there is a range of normal forces at the friction contact interface where there is a significant blade tip response reduction. Furthermore, blade tip responses shift to higher frequencies when impacts are introduced due to the stiffening effect present at contacts. Additionally, the gap (at equilibrium) between the RVA and the impact interface significantly affects the energy dissipation. It was further observed that RVAs with both friction and impact contacts can significantly reduce blade tip vibration in blisks for optimal impact gap sizes and friction contact normal loads.