Complex Mode Shapes Research Articles

Exploring the thermomechanical and dynamical mode switch transition of a reversible solid oxide cell

Reversible solid oxide cells (r-SOCs) are regarded as a solution form of clean energy generation towards a green hydrogen energy transition. However, during transportation, installation, and operation, they are required to withstand dynamic conditions as they face thermal, mechanical, and vibrational effects. Assessing the interplay of the thermomechanical and dynamical impacts on r-SOC materials during the mode-switch from electrolysis to fuel cell operation is the subject of this current comprehensive study. Advanced FEM-based homogenisation techniques have been harnessed in predicting and evaluating the detailed transients during static and dynamic loading. The intricate 3D thermal profiles throughout the transient reversible dual-modus operation from a typical r-SOC study have been derived using advanced image processing. Orthotropic material properties were incorporated into a unified macroscopic scale model by deriving the complex 3D microstructure geometry characteristics of the cell materials and component layers. The comprehensive thermomechanical and pre-stressed modal analysis results showed that the static and dynamic forces in fuel cell mode are more pronounced, whereas, in electrolysis operation, the dynamical behaviour highly depends on geometrical and operating conditions. Smaller component size shows higher natural frequencies and more intense vibration, leading to higher deformation attributed to complex mode shapes and reduced flexural rigidity. These insights are essential for optimizing r-SOC design parameters, enhancing their dynamic and structural reliability throughout their dual-mode operation lifespan.

On the complex mode shapes and natural frequencies of clamped-clamped fluid-conveying pipe

In the present study, the closed-form expression for complex mode shapes of a fluid-conveying pipe using Timoshenko beam theory is developed for the first time. By applying the method of separation of variables, the complex mode shapes and eigenvalues are obtained. To the best knowledge of the authors, there is no study carried out on the complex eigenvalue problem for vibration analysis of the Timoshenko fluid-conveying pipes. Given this oversight, in this study the effects of fluid velocity on the imaginary and real parts of the mode shapes and natural frequencies of a clamped-clamped fluid-conveying pipe are investigated. The results show that for very low fluid velocities, the first and the second mode shapes are similar to that of an equivalent beam. For high fluid velocities, the first mode shape experiences some deviation so that it looks like the second mode shape of a pipe having very low fluid velocity. The results show that fluid flow in the pipeline effectively reduces its bending moment as well as the energy of vibration. In addition, the nodal points corresponding to the second mode are replaced with quasi-node points.

Investigating the relation between complex mode shapes and local damage for structural assessment

Investigating the relation between complex mode shapes and local damage for structural assessment

Non-smooth dynamics of impacting viscoelastic pipes conveying pulsatile fluid

This paper aims to investigate the stability and nonlinear dynamics of an initially curved viscoelastic clamped–clamped pipe conveying pulsatile fluid, with special attention to the non-smooth dynamics of this gyroscopic system under the influence of initial curvature and gravity. In this study, the impacting contact is modeled as a trilinear repulsive force. Based on the Euler–Bernoulli beam theory, the nonlinear governing equations, which take into account geometric, curvature, and damping nonlinearities, are developed using Hamilton’s principle. The continuous model is discretized using the Galerkin method and then solved using the arc-length technique and a time integration method. A linear analysis of the natural frequency and complex mode shape shows the unexpected frequency intersection and modal asymmetry. The qualitative changes in the nonlinear dynamics of the impacting pipe in both sub- and super-critical regions are examined and presented in the form of bifurcation diagrams, frequency–amplitude curves, force–amplitude curves, time histories, and phase portraits. Numerical results show that some complex dynamical behaviors can occur in different impact constraint cases, including a grazing bifurcation, sticking, period doubling, and chaos at the impact location, may occur. The initial curvature and gravity result in different behaviors, such as the asymmetry change of the equilibrium configuration and the occurrence of the flutter instability. The current analysis is significant to understand the non-smooth dynamic mechanisms and also provide design and technology guidance for the pipe system with motion-limiting constraints.

Finite element analysis for free vibration of pipes conveying fluids–physical significance of complex mode shapes

A finite element formulation is presented for the natural vibration analysis of pipes conveying fluids. The solution of the resulting quadratic eigenvalue problem generally yields complex eigenvalues and eigenvectors. The present study then develops a robust mathematical procedure that combines the real and imaginary components of the eigenvectors to form physically attainable (i.e., real) mode shapes. The procedure yields a family of solutions that is more general than previously known solutions. The well-known classical mode shape is shown to be recoverable as a special case from the present solution. The study provides new insights on the effects of viscous damping, axial compressive force, and the flexibility of intermediate pipe supports on the response. Additionally, the study develops a novel algorithm based on Hermitian angles between eigenvectors to automate the tracing of mode evolution in the frequency-velocity plots.

Identification Uncertainties of Bending Modes of an Onshore Wind Turbine for Vibration-Based Monitoring

This study considers the identification uncertainties of closely spaced bending modes of an operating onshore concrete-steel hybrid wind turbine tower. The knowledge gained contributes to making mode shapes applicable to wind turbine tower monitoring rather than just mode tracking. One reason is that closely spaced modes make it difficult to determine reliable mode shapes for them. For example, the well-known covariance-driven stochastic subspace identification (SSI-COV) yields complex mode shapes with multiple mean phases in the complex plane, which does not allow error-free transformation to the real space. In contrast, the Bayesian Operational Modal Analysis (BAYOMA) allows the determination of real mode shapes. The application of BAYOMA presents a further challenge when quantifying the associated uncertainties, as the typical assumption of a linear, time-invariant system is violated. Therefore, validity is not self-evident and a comprehensive investigation and comparison of results is required. It has already been shown in a previous study that the significant part of the uncertainty in the mode shapes corresponds to their orientation in the mode subspace (MSS). Despite all the challenges mentioned, there is still a great need to develop reliable monitoring parameters (MPs) for Structural Health Monitoring (SHM). This study contributes to this by analysing metrics for comparing mode shapes. In addition to the well-known Modal Assurance Criteria (MAC), the Second-Order MAC (S2MAC) is also used to eliminate the alignment uncertainty by comparing the mode shape with a MSS. In addition, the mode shape identification uncertainties of BAYOMA are also considered. Including uncertainties is also essential for the typically used natural frequencies and damping ratios, which can be more appropriately used if the identification uncertainty is known.

Formulation of justifiable wind loading distributions up tall buildings derived from HFFB measurements

For the general case of non-linear and combined sway and twist mode shapes, analysis of HFFB measurements gives rise to serious challenges and various techniques have been developed using significant assumptions. This is because it is not possible to directly obtain the distribution of varying fluctuating wind loads throughout the height of tall buildings. This paper proposes a straightforward HFFB-based wind load/twist distribution approach in the time domain to predict the wind-induced dynamic response of tall buildings. Using this framework, it is possible to overcome the deficiencies of mode shape correction factor approach, and the complexity and impracticality of alternative approaches for tall buildings. Two 1:300 scale rigid models of benchmark tall buildings, IAWE Buildings A and B, have been selected to perform a detailed wind tunnel investigation as subjects to evaluate this proposed HFFB approach study. Building A has complex 3D mode shapes combining sway and twist, whereas the simpler Building B has uncoupled linear mode shapes in sway and twist. The high frequency pressure integration (HFPI) wind tunnel technique was utilised to measure surface pressure information in separate tests to obtain the reference wind loading for validation of the HFFB predictions. Furthermore, a commonly used HFFB mode shape correction factor approach was used to estimate the dynamic wind responses which were compared to those from the proposed approach. From the comparison between the aerodynamic and dynamic wind results from proposed HFFB approach and reference HFPI results, it is found that the proposed HFFB approach can predict the vertical distributions of the wind forces and torque (wind loads) with acceptable accuracy for carrying out the conceptual stage of tall building design.

Complex Modal Characteristic Analysis of a Tensegrity Robotic Fish's Body Waves.

A bionic robotic fish based on compliant structure can excite the natural modes of vibration, thereby mimicking the body waves of real fish to generate thrust and realize undulate propulsion. The fish body wave is a result of the fish body's mechanical characteristics interacting with the surrounding fluid. Thoroughly analyzing the complex modal characteristics in such robotic fish contributes to a better understanding of the locomotion behavior, consequently enhancing the swimming performance. Therefore, the complex orthogonal decomposition (COD) method is used in this article. The traveling index is used to quantitatively describe the difference between the real and imaginary modes of the fish body wave. It is defined as the reciprocal of the condition number between the real and imaginary components. After introducing the BCF (body and/or caudal fin) the fish's body wave curves and the COD method, the structural design and parameter configuration of the tensegrity robotic fish are introduced. The complex modal characteristics of the tensegrity robotic fish and real fish are analyzed. The results show that their traveling indexes are close, with two similar complex mode shapes. Subsequently, the relationship between the traveling index and swimming performance is expressed using indicators reflecting linear correlation (correlation coefficient (Rc) and p value). Based on this correlation, a preliminary optimization strategy for the traveling index is proposed, with the potential to improve the swimming performance of the robotic fish.

Three-Dimensional Plate Dynamics in the Framework of Space-Fractional Generalized Thermoelasticity: Theory and Validation

This work aims to study the dynamics of 3D plates under uniform and nonuniform temperature distributions in the framework of the space-fractional generalized thermoelasticity (S-FGT) approach. The quadratic eigenvalue problem is obtained, which means that the thermoelastic damping plays a meaningful role due to the plate’s thermal energy absorption. The plate’s complex frequency spectrum and mode shapes (free ends) under two different temperature distributions are considered for different values of the fractional continua order α and the length scale parameter l. For the first four frequencies, the fractional modes closest to the experimental results and the classical modes are presented with the absolute differences between them. For the nonuniform temperature distribution case, the mode shape analysis is performed assuming that modulus of elasticity, thermal expansion, and specific heat parameters are functions of the temperature. The primary outcomes of the paper can be stated as follows: 1) the S-FGT approach analysis gives more reliable results than the classical (local) theory; 2) the peak point of the out-of-plane mode amplitude is shifted toward the warmed zone; 3) a mode shifting is observed for the uniform temperature distribution in contrast to the nonuniform temperature distribution; 4) the fractional order derivative and length scale parameter depend on temperature, similar to other material properties such as elastic modulus, specific heat, and coefficients of thermal expansion; 5) a decrease in the fractional order is observed, while temperature increases for the fixed length scale parameter. These novelties indicate that the S-FGT approach establishes a new model for analyzing materials under heating, and the results may be beneficial for designing thermal structures.

New indicator for damage localization in a thick adhesive joint of a composite material used in a wind turbine blade

In this paper, a new indicator to localize fatigue damage in a fibre glass composite structure, i.e. spar cap to shear web thick adhesive joint of a wind turbine blade, is presented. This indicator is based on the effect of damping on the phase of the mode shapes of the structure. When fatigue damage occurs, damping increases in the defective area and this leads to an increase in the local energy dissipation. This non-uniformity in the energy dissipation throughout the structure causes the structure to vibrate with mode shapes whose structural elements no longer have the same phase creating complex mode shapes. A visco-elastic finite element (FE) vibration model is developed for a thick adhesive joint of a wind turbine blade. The mass, stiffness, and damping matrix extracted from the FE model are used to determine the complex mode shapes. The results show that the damaged area is located where the spatial derivative of the phase of the components of the mode shapes is minimum. Changes in the phase of mode shapes of the structural elements are strongly dependent on the location of damage. In the locations where the strain modal energy is greater, the change in the phase is also higher.

A Novel Strategy for Automatic Mode Pairing on the Model Updating of Railway Systems with Nonproportional Damping

Mode pairing is a crucial step for the stability of any model-updating strategy based on experimental modal parameters. Automatically establishing a stable and assertive correspondence between numerical and experimental modes, in many cases, proves to be a very challenging task, especially in situations where complex mode shapes are present. This article presents a novel formulation for the automatic mode pairing between experimental and numerical complex modes based on an Energy-based Modal Assurance Criterion (EMAC). The efficiency of the proposed criterion was demonstrated on the basis of a case study involving the pairing between numerical and experimental modes of a passenger railway vehicle. A highly complex detailed FE numerical model of the vehicle was developed involving the modeling of the carbody, bogies and axles. A numerical damped modal analysis allowed obtaining the main global rigid-body and flexural modes of the vehicle’s carbody, as well as several local modes associated to the vibration of specific components of the carbody. Due to the localized damping provided by the suspensions, these modes presented complex modal ordinates, especially for the rigid-body modes. The comparison between the results obtained from the application of the EMAC and the classical MAC criteria, on the pairing of five global mode shapes, proved that the EMAC criterion is much more assertive, avoiding mismatches between the experimental global modes and some of the local numerical modes with similar configurations, and, consequently, establishing the correct correspondences between experimental and numerical modes.

Low Reduced Frequency Theory of Airfoils Vibrating in Complex Modes

Abstract This paper extends a previous study on the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime to the case of complex modes, where the phase of the different points of the same airfoil is not constant. A complex mode-shape can be expressed as the combination of two rigid body modes shifted 90 deg. Therefore, the stability of such modes is more complex to predict than that of the so-called real modes. The aerodynamic work-per-cycle has been deconstructed into its fundamental components, and its dependence on the reduced frequency described. The three distinct contributions to the work-per-cycle of a complex mode have been analyzed theoretically and numerically. First, the mean pressure yields a constant contribution to the work-per-cycle that is independent of the interblade phase angle (IBPA) and the reduced frequency. This term scales with the aerodynamic loading of the airfoil and is null for nonlifting airfoils. Second, the contribution of the two fundamental modes, whose mean value is proportional to the reduced frequency. Finally, the cross works of a fundamental mode over the displacements of the other. The low reduced frequency theory predicts that this term is independent of the frequency and symmetric with the interblade phase angle. The latter has been confirmed using linearized Navier–Stokes simulations, but the former has a constant term, as predicted by the theory, plus a small variation with the frequency that has been associated with acoustic resonances.

Experimental investigation of a method for diagnosing wall thinning in an artificially thinned carbon steel elbow based on changes in modal characteristics

Curved cylindrical structures such as elbows have a non-uniform thickness distribution due to their fabrication process, and as a result have a number of complex mode shapes, including circumferential and axial nodal patterns. In nuclear power plants, material degradation is induced in pipes by flow accelerated erosion and corrosion, causing the wall thickness of carbon steel elbows to gradually thin. The corresponding frequencies of each mode shape vary according to the wall thinning state. Therefore, the thinning state can be estimated by monitoring the varying modal characteristics of the elbow. This study investigated the varying modal characteristics of artificially thinned carbon steel elbows for each thinning state using numerical simulation and experimental methods (MRIT, Multiple Reference Impact Test). The natural frequencies of specified mode shapes were extracted, and results confirmed they linearly decreased with increasing thinning. In addition, by comparing single FRF (Frequency Response Function) data with the results of MRIT, a concise and cost effective thinning estimation method was suggested.

Підхід до аналізу поздовжньої стійкості рідинної ракети-носія пакетної схеми компонування

The “core and strap-on boosters” layout of launch vehicle (LV) stages is quite common in heavy LV development. However, POGO oscillations in liquid-propellant LVs with this stage layout have some features. It is shown that the structure of LVs of this type as a dynamic object has a dense spectrum of natural frequencies and complex spatial mode shapes. The longitudinal oscillations of the identical elements of the LV side strap-on boosters may be in phase or in antiphase, while the longitudinal mode shapes of the LV central core and strap-on boosters may differ both in phase and in amplitude. In flight, the thrust of the engines of the side strap-on boosters may also oscillate in phase or in antiphase, as a result of which the interaction of the LV structure with the sustainer propulsion systems of the side strap-on boosters may have both a stabilizing and a destabilizing effect on the POGO stability of a liquid-propellant LV. This paper presents a mathematical model of the “liquid-propellant propulsion systems – LV structure” dynamic system. The model describes the interaction of the longitudinal vibrations of the structure of a two-stage “core and strap-on boosters” LV with the core and strap-on booster propulsion systems. The free longitudinal vibrations of the structure of a ‘core and strap-on boosters’ LV were simulated using computer-aided finite element design tools (CAE systems). The simulation was the first to account for the dissipation of the liquid propellant and LV structure oscillation energy. The paper suggests an approach to analyzing the POGO stability of liquid-propellant “core and strap-on boosters” LVs with the use of the Nyquist criterion generalized to the case of multidimensional dynamic systems. The approach is based on opening the thrust feedback loops of the “liquid-propellant propulsion systems – structure” closed-loop dynamic system and studying the stability of the one-channel systems obtained in this way. Based on the proposed approach, the interaction between the longitudinal vibrations of the “core and strap-on boosters” LV structure and low-frequency processes in the liquid-propellant sustainer propulsion systems of the LV first stage was studied numerically.

Correction to: Method of experimentally identifying the complex mode shape of the self-excited oscillation of a cantilevered pipe conveying fluid

Correction to: Method of experimentally identifying the complex mode shape of the self-excited oscillation of a cantilevered pipe conveying fluid

Method of experimentally identifying the complex mode shape of the self-excited oscillation of a cantilevered pipe conveying fluid

The dynamics of a flexible cantilevered pipe conveying fluid have been researched for several decades. It is known that the flexible pipe undergoes self-excited vibration when the flow speed exceeds a critical speed. This instability phenomenon is caused by nonconservative forces. From a mathematical point of view, the system has a characteristic of non-selfadjointness and the linear eigenmodes can be complex and non-orthogonal to each other. As a result, such a mathematical feature of the system is directly related to the instability phenomenon. In this study, we propose a method of experimentally identifying the complex mode from experimentally obtained time histories and decomposing the linear mode into real and imaginary components. In nonlinear analysis, we show that the nonlinear effects of practical systems on the mode in the steady-state self-excited oscillation are small. The real and imaginary components identified using the proposed method for experimental steady-state self-excited oscillations are compared with those obtained in the theoretical analysis, thus validating the proposed identification method.

Exploiting phase-based motion magnification for the measurement of subtle 3D deformation maps with FP + 2D-DIC

Phase-Based Motion Magnification (PBMM) is an effective methodology to visualise imperceptible phenomena based on magnifying periodic subtle movements on image sequences. Moreover, the integration of the optical techniques Fringe Projection and 2D Digital Image Correlation (FP + 2D-DIC) make it possible to obtain 3D displacement maps employing a single camera.In this study, the integration of PBMM with FP + 2D-DIC is exploited for the determination of subtle 3D displacements. An initial solid-rigid test validates of the combination of those techniques. A later test, employing a cantilever beam, demonstrates its correct performing in the determining of Operational Deflection Shapes. In both cases, the results are compared with those obtained with 3D-DIC and Scanning Laser Doppler Vibrometer (SLDV) to evaluate the relation between the magnification factor and the result obtained. Finally, the potential of the proposed integration for the determination of complex mode shapes is demonstrated by obtaining the ODSs of an industrial component.

Tracking the variation of complex mode shapes for damage quantification and localization in structural systems

Real structures mode shapes estimated by modal analysis techniques have a common feature: in most cases they are complex, and their level of complexity can be soundly influenced by the presence and extent of physical damage, which also affects the distribution of energy dissipation mechanisms within the structures. Starting from the contributions available in the literature, the present paper investigates, from a numerical and experimental point of view, the correlation existing between localized damage and variation of global modal complexity indices conventionally employed to quantify the nonproportionality of damping in structural systems. Finally, driven by the inferences made through numerical and experimental test cases by tracking the variation of complex modes over multiple and progressive damage scenarios, a new index for damage localization and quantification is formulated and validated against real data.

High-frequency voltage-driven vibrations in dielectric elastomer membranes

This article deals with modelling and experimental characterisation of the continuum dynamic response of dielectric elastomer actuators (DEAs) subject to high-frequency voltage excitation.DEAs are capable of large deformations in response to an electrostatic stimulus, and they show large operating bandwidths of up to a few kilohertz. Although DEA systems normally make use of simple deformation patterns and a well-defined main actuation mode, they show complex structural dynamics and mode shapes if subject to high frequency voltage inputs. Taking advantage of these complex structural dynamics potentially allows developing multi-function actuators, audio devices and vibration isolators capable to perform different tasks using different vibration regimes.We first present a multi-domain model for the structural dynamics of DEAs, accounting for the contribution of the air pressure loads generated the DEA membrane vibrations, which play a relevant role in the DEA dynamics.We then present an extensive experimental characterisation of DEA samples’ structural dynamics based on laser Doppler vibrometer measurements. In particular, we measure the complex structural mode shapes and the velocity spectra generated through a broadband excitation in correspondence of a wide set of different design and control parameters (namely, the DEA geometric layout, the mechanical-preload and the applied voltage bias).Based on the experimental results, we validate the proposed continuum model, and we demonstrate that the forced response of the DEA can be efficiently described using a lumped-parameter computationally efficient reduced reformulation.

Experimental characteristics and coupled nonlinear forced vibrations of axially travelling hyperelastic beams

Comprehensively investigated in this paper is the coupled nonlinear dynamics of hyperelastic beams. Experimental testing is performed to accurately model the hyperelasticity using different hyperelastic energy density models. Samples were fabricated using 3D printing technique for flexible thermoplastic polyurethane filaments following the ASTM D638 guidance and tested using a tensile testing machine. For a proper hyperelastic energy density model, coupled nonlinear equations of motion are obtained via the von Kármán geometrical nonlinearity and Hamilton’s principle. Free natural vibration response of the gyroscopic system is analysed using a generalised modal decomposition method and the influence of different parameters such as the hyperelastic stiffness term and the axial velocity on the natural frequencies is investigated. Complex mode shapes are obtained and the influence of axial velocity on altering the real and imaginary parts is discussed. Furthermore, a comprehensive analysis on the coupled nonlinear dynamics of the system is presented using a combination of Galerkin’s scheme and a dynamic equilibrium technique. The frequency response to different design parameters such as axial speed, axial tension, harmonic external load, hyperelastic parameters and geometrical imperfection is discussed in detail. It is shown that the system shows a wide range of rich nonlinear dynamics including hardening depending on the characteristics of the system.