An interference fit is a common joining technique used to connect a shaft and a hub. In the presence of dynamic loads and vibrations, characteristic variables such as contact pressure and slippage are load and state-dependent quantities. Such effects have either not been investigated using previous simulation methods or have only been addressed in a simplified manner. The reason for this is the nonlinear contact between the shaft and the hub, which makes a Finite Element simulation with fine meshing, while taking all dynamic effects into account, very demanding and significantly increases the computational effort. This work also offers a new and alternative view of contact modes. This perspective is particularly advantageous for structures with initial stresses that occur in the presence of an interference fit. In this paper, so-called contact modes are applied to interference fits with some modifications. This closes the previously mentioned gap in the simulation landscape because it allows nonlinear, accurate, and fast numerical time integration of finely meshed Finite Element models with interference fits, without the need for simplifications regarding contact, friction, and dynamics due to vibrations and nonlinear rigid body motion. Local plasticity and temperature fields were not taken into consideration.
An equality-based weighted residual formulation is proposed for the periodic responses of vibrating systems subject to two-dimensional dry friction on a plane. Coulomb's law is expressed as two coupled nonsmooth equality conditions which augment the equations of motion, resulting in a mixed displacement-friction force formulation whose periodic solutions are sought using a standard Ritz-Galerkin procedure in time. The shape functions considered are the classical Fourier functions, and a quasi-analytical expression for the Jacobian of the friction terms is derived in a piecewise linear fashion and computed in a weighted residual sense. The method is based on an exact equality representation of Coulomb's law for interfaces with mass, thus avoiding common hypotheses such as regularization, penalization, or massless interfaces. It is entirely carried out in the frequency domain, contrary to existing frequency-time methods which require the calculation of contact forces in the time domain at each iteration of the nonlinear solver. The method is compact and found to be robust and accurate. It is void of convergence or other numerical issues up to very large numbers of harmonics of the response in all cases considered. Periodic responses featuring complex two-dimensional interface motions and multiple stick-slip transitions are calculated accurately at various resonant and sub-resonant excitation frequencies, at a reasonable computational cost. Since the only approximation in the procedure is the finite number of terms in the Ritz-Galerkin expansion, the intricate behavior of the two-dimensional friction force dictated by Coulomb's law can be captured with a high degree of accuracy.
Additive manufacturing has gained popularity for its ability to produce complicated geometries that distribute material optimally and allow several parts to be consolidated into one.
Vibration testing of nonlinear structures presents various challenges that are not only related to the nonlinear structure itself but also to the exciter. As the exciter and the structure are coupled, the structure's nonlinear behaviour distorts the exciter force amplitude and frequency content, making the test results more difficult to compare with numerical predictions or even less accurate when isolating and identifying nonlinear normal modes (NNMs). This paper uses an adaptive feedforward cancellation (AFC) real-time control algorithm to track multi-harmonic force reference signals accurately. The controller estimates the disturbance generated by the structure and adjusts the exciter's input to compensate for it, recovering the desired force signal. The advantage of this approach is that it does not rely on the knowledge of the exciter and structure models. It only requires knowledge of the phase at the output of the exciter, which is straightforward to estimate experimentally from the force measurements. The effectiveness of the AFC for exciter force control is demonstrated numerically and experimentally on a cantilever beam structure excited by an electromagnetic shaker. The beam's free tip is attached to a spring mechanism, giving rise to geometric nonlinearities and mode interactions resulting in strong harmonic distortions in the excitation force. The method is further demonstrated on a compressor blade from Roll-Royce's aero-engine.
Non-metallic materials display frequency and temperature dependent dynamic properties which must be characterised for use in computer simulations. The characterisation of such properties is important as most modern structures utilise these materials. Hence, a novel test method has been developed, which combines vibration testing with finite element analysis, to yield dynamic modulus of elasticity and damping. Material properties have been measured in the frequency range 2Hz - 2000Hz. The test method involves a cantilever beam. Two samples of the test material sandwich the root of the beam and are held in place between inertial masses. Experimental modal analysis techniques, where an instrumented hammer vibrates the beam, are used to exercise the material. The modulus of elasticity of the material is found by constructing a finite element model of the test setup and tuning the simulated response with that of the experiment. Damping properties are extracted by applying data fitting techniques to the time histories and spectrum. These are then converted into a material damping property by simulating and accounting for the energy balance between the samples and test setup. Doing so also improves damping simulation accuracy by eliminating the need to estimate Rayleigh damping values at each frequency, as now the material damping property at each frequency can be directly used. For test accuracy, it is important that the sample is gripped in a manner that exercises it effectively and is simple to simulate. Eliminating slipping, and thus the difficult to model friction, is a key concern and has been investigated in depth. The best solution found is an axisymmetric bolting arrangement which holds the samples in place. Additionally, the test setup utilises suspension, together with inertial masses and an orthogonal layout to isolate against external vibrations. Polymers and rubbers, which exhibit complex frequency dependent behaviour, have been reliably characterised using this method. The damping material Sorbothane has also been re-characterised and produced results that aligned with manufacturer specifications. This method proves to be a reliable procedure for dynamic material property testing.
This paper addresses the problem of optimal tuning of a tuned mass damper (TMD) attached to a complex structure that is dynamically excited by its base. It proposes new analytical formulae which are based on the reduction of the multiple degree of freedom (MDOF) model of the host-structure into an equivalent single degree of freedom (SDOF) model. As it has been recognized in the literature that the traditional single mode approximation used to perform this reduction is not valid for base-excited systems, we propose an improved version that leads to the definition of two mass ratios instead of one in the traditional approach. Taking into account this new mass ratio, the equal peak method is used to derive analytically the optimal values of stiffness and damping of the TMD for a given mass ratio of the device. The introduction of a second mass ratio leads to the existence of two sets of equations for the optimal parameters, depending on the relative values of the two mass ratios. It is shown, however, that only the first set of equations is of practical use. The application of these new tuning rules is illustrated using a MDOF model of a high-rise building. It demonstrates the efficiency of the approach when the first mode of vibration is targeted. When higher modes are of interest, modal interactions are important, which cause a slight to moderate unbalance of the peaks.
Brake squeal is an instability that generates self-excited limit cycles which vary with time and operating conditions in real experiments. To analyze test results, it is proposed to use a Harmonic Balance Vector (HBV) signal model. It combines Harmonic Balance Method and analytic signal methodologies. From the Harmonic Balance Method, one uses the space-time decomposition where spatial distribution of each harmonic is described by a complex vector and frequency is common to all sensors. From analytic signal, one keeps the assumption that quantities are slowly varying in time. Synchronous demodulation and principal coordinate definitions are combined in a multistep algorithm that provides an HBV estimation. On an industrial brake test matrix, HBV estimation is shown to be robustly applicable. The HBV signal being slowly varying, time sub-sampling reduces the volume of test data by two orders of magnitude. Limit cycle frequency, amplitude and shapes can thus be added to the parallel coordinates that associate to each time sample the operating parameters: pressure, velocity, temperature, torque, disk position, disk/bracket distance, ... This opens a path to a range of analyzes otherwise difficult to perform. Classification of squeal occurrences is first discussed showing pressure and amplitude dependence. The effect of amplitude on both frequency and shape is next demonstrated. The entry and exit of instability when parameters change are then analyzed by proposing a transient root locus built from test. Thus squeal test results are related to the classical complex eigenvalue analysis. Intermittent growth/decay events are shown to be correlated with wheel position. Furthermore, distance measurements indicate that disk shape variations of a few microns play a clear parametric role. Parametric testing and clustering are then used to map the instability region and its edges. Pressure is shown to have an effect dominating other variations. Prospective uses of these results to combine test results and finite element models are discussed last.
Energy dissipation (i.e., damping) is a critical quantity to identify in order to understand the dynamic performance of a mechanical design as the dissipation directly influences response amplification near resonance. In linear systems, dissipation is often modeled with constant viscous damping for each mode and can be extracted directly from modal tests in the form of damping ratio. For nonlinear systems there have been many proposed techniques for characterizing damping from experimental measurements, however researchers have yet to reach a consensus on a unified approach. This work investigates three damping identification methods and evaluates each of their limitations. The context for the identification is the nonlinear force appropriation testing technique. The study conducts virtual experiments utilizing multi-harmonic balance solutions where phase resonance is enforced on single-degree-of-freedom models with different nonlinearities – both conservative and non-conservative. In this way, the calculated damping ratios are compared directly to corresponding analytical approximations from the models to enable a critical assessment of their accuracy and of any limitations of each damping identification technique. Additionally, the effect of higher harmonics (both in phase resonance and uncontrolled) on the damping ratio estimates is explored by including an electro-mechanical model of a shaker. In addition to identifying limitations for each damping identification technique considered, this work shows that the trend in the damping ratio does not necessarily reflect the true nature of the nonlinear damping restoring force. Moreover, the damping ratio trend identified from nonlinear force appropriation experiments is sensitive to higher harmonics in the excitation force regardless of whether they are maintained in phase resonance or uncontrolled.
Multi-input, multi-output (MIMO) testing is used in component qualification to reproduce operational responses in the laboratory. It is often preferred to single-input and base-shake testing because of the potential for equivalent or better tests using smaller actuators and shorter test suites. Given a target response, two key steps in MIMO test design are selecting actuator locations and solving for input loads. Actuator locations are often manually selected using expert judgment. If an automatic method is used, locations are usually determined by simulating the vibration control problem and minimizing a combination of the input energy and control residuals. To select a configuration, the relative importance of input energy and residuals must be specified. Specifying relative weights is, in general, a manual and subjective process. This paper develops an objective function that compares actuator configurations based on control accuracy and required input energy without any manual parameter tuning. The objective function uses an optimally selected tradeoff parameter for each candidate configuration. To choose actuator locations using the new objective function, a pivoting algorithm for integer programming problems is developed. Starting with an initial configuration (such as the one generated by a greedy algorithm), the pivoting algorithm guarantees an objective function decrease in each iteration until convergence is reached. In a simulation featuring a structure excited by a diffuse acoustic field, electrodynamic shaker locations and regularized inputs are solved for without any analyst-specified parameters. Simulations are performed in MIMO configurations where the number of target responses is less than, equal to, and greater than the number of actuators.
This research presents a novel analysis method for calculating nonlinear Frequency Response Functions from a nonlinear frequency response surface (NFRS). The research aims to provide engineers with a user-friendly technique to evaluate the nonlinear frequency responses when the modal parameters as a function of the vibration amplitude are available. The Frequency Response Functions (FRFs) are the most widely used functions to characterise the dynamic behaviour of structures. The experimental modal analysis stands on four pillars 1) measurement, 2) identification, 3) regeneration, and 4) comparison, and these four steps must be ensured under linear and nonlinear vibrations. However, the nonlinear vibrations are challenging for identifying, regenerating, and comparing nonlinear FRFs. This research postulates that a nonlinear FRF solves a geometrical intersection between the nonlinear frequency response surface and any constant amplitude force surface. The paper demonstrates the hypothesis with ONE- and TWO-DoF systems with a cubic stiffness nonlinearity by showing how to generate a nonlinear frequency response surface when the force-displacement relationship is calculated. The verification of the proposed formulation is yielded by comparing nonlinear FRFs generated by the new analysis method (NM) to the ones generated by the Harmonic Balance Method and numerical integration. Furthermore, the paper presents a new identification method, based on the Dobson formulation, for extracting amplitude-dependent modal parameters. These parameters generate an NFRS, from which synthesised nonlinear FRFs are evaluated and compared to the experimental ones. The most important innovation of this research is that the four steps listed earlier can quickly be implemented with the proposed technique.
A very compact weighted residual formulation is proposed for the construction of periodic solutions of oscillators subject to frictional occurrences. Coulomb's friction is commonly expressed as a differential inclusion which can be cast into the complementarity formalism. When targeting periodic solutions, existing algorithms rely on a procedure alternating between the frequency domain, where the dynamics is solved, and the time domain, where friction is dealt with. In contrast, the key idea of the present work is to express all governing equations including friction as equalities, which are then satisfied in a weak integral sense through a weighted residual formulation. The resulting algebraic nonlinear equations are solved numerically using an adapted trust region nonlinear solver and basic integral quadrature schemes. To increase efficiency, the Jacobian of the friction forces is calculated analytically in a piecewise linear fashion. The shape functions considered in this work are the classical Fourier functions. It is shown that periodic solutions with clear multiple sticking and sliding phases can be found with a high degree of accuracy. The equality-based formulation is shown to be effective and efficient, convergence being achieved in all cases considered with low computational cost, including for large numbers of harmonics. Importantly, this new friction formulation does not suffer from the typical limitations or hypotheses of existing frequency-time domain methods for non-smooth systems, such as regularization, penalization, or massless frictional interfaces.
This paper is the completion of a previous work in which an algorithm was developed for direct calculation of the hysteresis cycle of Underplatform Dampers (UDs) used for vibration damping of turbine blades. While the previous paper examined the scalability of a given damper shape as a function of its size, platform size and contact parameters, this paper instead shows how the method can be used to evaluate the advantages and disadvantages of different damper shapes. Only the asymmetric contact damper is studied, in its simplest version with three pad-located contacts, and its merits are discussed in comparison with the wedge or cottage-roof type dampers, whose crucial difficulties are pointed out. A family of five dampers obtained by distortion of an isosceles damper of 60° vertex aperture is studied as an example of application of the method. A criterion is defined for positioning the single contact to avoid lifting when the coefficient of friction is maximum at =0.7, then performance is studied for two values in the range expected for normal operation, =0.5 and =0.3. For each of the five dampers, the different problems that arise in calculating the hysteresis cycle at the onset of the total full-slip, called Base-Cycle, are examined, first on the damper in terms of the “force Base-Cycle,” otherwise known as the “equilibrium trace diagram,” then of the “moment Base-Cycle” on the platform. The “moment Base-Cycle” is employed in the context of the Platform Centered Reduction (PCR) technique, and diagrams of the real and imaginary components of the complex moment-rotation stiffness of the platform representing the mutual damper-platform actions are defined. Finally, the values of the energies dissipated on the contact pads of the dampers are determined, the sum of which is checked against the total energy dissipated by the moment of the contact forces acting on the platform, consistent with the Platform Centered Reduction. The parameters and diagrams that characterize the shape of a damper, regardless of its subsequent coupling with a specific turbine blade, are useful tools for characterizing its performance with greater insight than can be obtained from the usual purely numerical approach.
The tightness of a flange depends on the current deformation state of the structure. For obvious reasons, this could be a critical and important information. In this paper, numerical preliminary investigations are presented on the question whether the deformation state inside a contact surface can be estimated in a very short time with a few strain measurements outside the contact surface.The presented theory is numerically evaluated using a flange with 12 bores. It turns out, that the deformation inside the contact area, and therefore the tightness, can be computed out of strain data in milliseconds. The magnitude of the strains are in a measurable range and the numerical results show robustness against noise. Even if the presented approach may not be applicable one-to-one for measurements, this numerical preliminary investigation shows that mechanical strains could in principal be used to observe the state or the tightness within a joint.
Experimental and hybrid substructuring allow the virtual analysis of complex structures that are rather represented by experimental models than simulations. Coupling the experimental models with other structures requires adequate driving points generally described by six degrees of freedom with three translations and rotations with forces and torques as inputs as well as translational and angular accelerations as outputs. However, the measurement of these driving points is challenging. In particular, experimentally acquiring all six inputs causes the main effort. For example, excitation devices that directly exert a torque are not widely applicable or three translational directions are not accessible. Therefore, adapter structures are frequently used in practice, which increase the experimental effort or can be impractical due to a lack of space and reachability. The proposed strategy is the replacement of active excitation devices, i.e. impulse hammers or shakers, on the driving points, by passive, rigid bodies, simply referred to as masses. Compared to the existing mass uncoupling method which is also based on this fundamental idea, the main improvement is a simplified notation that enables the incorporation of data from multiple masses and parametric state-space systems. The main finding is that the proposed strategy is suited to estimate driving point dynamics that allow to predict the effect of assembled structures on channels that were not used in the estimation of the driving point, indicating the physical relevance of the estimated dynamics.
Experiments and physics-based modeling efforts both show that the features within a jointed interface can have an outsized influence on the nonlinear dynamics of a large-scale structure. The interfacial features, including asperities and meso-scale topology, are often six to ten orders of magnitude smaller in scale than the structure itself, yet can significantly change the natural frequencies and damping of a structure and can lead to the premature failure due to wear if not properly designed. A significant amount of recent research has been invested in understanding and predicting the nonlinear dynamics of structures with jointed interfaces; however, there are many challenges that still remain before accurate predictions of a jointed structure's nonlinear dynamics and wear properties becomes accessible to design engineers. This article is a reflection of the outcomes of the 2023 International Workshop on the Mechanics of Jointed Structures in which the state of the art of joints modeling was assessed and future directions for research on jointed structures were identified. As such, this paper makes several recommendations for new research thrusts to improve the understanding of jointed structures in addition to highlighting the current state of the art and recent advances in modeling and experimentally characterizing jointed structures.
Many examples of dynamic analysis require modelling of dry friction. Often this is represented by a so-called Jenkins element or multiple Jenkins elements in parallel, which is sometimes termed a parallel-series Iwan element. This study considers the case where a system that includes these representations of dry friction are loaded dynamically with a combination of deterministic harmonic and random excitation. This paper presents a new efficient method for predicting the response of systems subject to combined deterministic and random excitation. The method is based on equivalent linearisation and involves averaging across both an ensemble of random responses as well as over a harmonic excitation period. The key novelty in the approach is the use of an auxiliary harmonic term to facilitate an analytical representation of the nonlinear force. This overcomes the challenge of analysing a discontinuous hysteretic nonlinearity in the presence of random excitation. Analytical solutions are presented and it is shown that the proposed approach can predict the approximate response at a significantly reduced computational cost.
It is known that isolated frequency response branches (isolas) can occur near primary resonances under modal interactions or nonlinear damping. The present work demonstrates how emergence and vanishing of such isolas can be systematically analyzed in an experiment. Feedback control of the phase is employed to track the phase-resonant backbone curve. In addition, the amplitude of the response is controlled, as the excitation level undergoes turning points in the presence of an isola. The acquired data indicates what excitation levels lead to the formation of an isola / its merging with the main branch. Some further analysis of the data permits to characterize possible internal resonances and amplitude-dependent damping. To assess the proposed method, a test rig is considered which involves two similar cantilevered beams undergoing soft collisions via a unilateral spring. A simplified model, which relies on linear modal damping and a massless unilateral spring, suggests that isolas should appear as a result of internal resonances. In contrast, the experiment shows an isola due to nonlinear damping. More specifically, the damping ratio first increases substantially due to the frictional dissipation in inevitable joints. Subsequently, the unilateral interactions scatter energy to other modes (in a non-resonant way), which have much lower damping. This leads to a sudden drop of the effective damping ratio with the amplitude, and leads to the formation of an isola.
The Tribomechadynamics Research Challenge (TRC) was a blind prediction of the vibration behavior of a thin plate clamped on two sides using bolted joints. The first bending mode's natural frequency and damping ratio were requested as function of the amplitude, starting from the linear regime until high levels, where both frictional contact and nonlinear bending-stretching coupling become relevant. The predictions were confronted with experimental results in a companion paper; the present article addresses the experimental analysis of this benchmark system. Amplitude-dependent modal data was obtained from phase resonance and response controlled tests. An original variant of response controlled testing is proposed: Instead of a fixed frequency interval, a fixed phase interval is analyzed. This way, the high excitation levels required outside resonance, which could activate unwanted exciter nonlinearity, are avoided. Consistency of testing methods is carefully analyzed. Overall, these measures have permitted to gain high confidence in the acquired modal data. The different sources of the remaining uncertainty were further analyzed. A low reassembly-variability but a moderate time-variability were identified, where the latter is attributed to some thermal sensitivity of the system. Two nominally identical plates were analyzed, which both have an appreciable initial curvature, and a significant effect on the vibration behavior was found depending on whether the plate is aligned/misaligned with the support structure. Further, a 1:2 nonlinear modal interaction with the first torsion mode was observed, which only occurs in the aligned configurations.
Honeycomb sandwich panels are widely used in the industry because of their outstanding stiffness to mass ratios. The dynamic response of such structures is known to be relatively complex especially in the mid-frequency domain where a strong orthotropy can be observed. Several equivalent models are available in the literature to understand and predict this behavior, all relying on an assumption that the geometry of the honeycomb core is periodic and perfectly known. The effect of imperfection inevitably introduced at the manufacturing stage are characterized in this work using X-ray measurements. A simple homogenization approach is proposed, allowing the prediction of the shearing properties of the core from statistics extracted from its geometry. These properties are used to predict the dynamic behavior of the structure using a multi-layer analytical model. Results are compared to Laser Doppler Vibrometer measurements, showing a very good agreement with the predictions based on X-ray pictures.
In Lamb Waves based Structural Health Monitoring (LWSHM) of composite aeronautic structures, Deep Learning (DL) methods have proven to be promising to monitor damage using the signals collected by piezoelectric sensors (PZTs). However, those data driven algorithms are strongly problem dependent: any structural change dramatically impacts the accuracy of the predictions and the generalization of the learnt algorithms to other structures within the fleet is impossible. Transfer Learning (TL) promises to face that issue by capitalizing on the knowledge acquired on a given structure to transfer it on another from the fleet. An original TL approach based on the Optimal Transport (OT) theory is proposed here to handle this issue. OT provides a rigorous mathematical framework for TL that can be practically implemented using Input Convex Neural Networks modelling Kantorovich potentials but that has never been used for LWSHM. Using OT, the knowledge acquired on a rich LW database is transferred to poorer LW databases collected on different structures with rising structural divergences. A Structural Index (SI) is defined and used to compute the gap between those different structures and can be used to estimate a priori the necessity of the use of TL methods. The proposed OT based TL method for LWSHM manages to reduce by almost 50\% the predictions errors between numerical structures with strong differences (bias in mechanical properties and erroneous PZT position) in comparison with standard approaches. That leads to a promising approach to combine rich numerical database with poorer database in order to build robust algorithms for LWSHM of a fleet of aeronautical composite structures.
