Special Topics in Structural Dynamics, Volume 6

This study is on the forced motions of non-homogeneous elastic beams. Euler-Bernoulli theory is employed and applied to a two-segment configuration subject to harmonic forcing. The objective is to determine the frequency response function for the system. Two different solution strategies are used. In the first, analytic solutions are derived for the differential equations for each segment. The constants involved are determined using boundary and interface continuity conditions. The response, at a given location, can be obtained as a function of forcing frequency (FRF). The procedure is unwieldy. Moreover, determining particular integrals can be difficult for arbitrary spatial variations. An alternative method is developed wherein material and geometric discontinuities are modeled by continuously varying functions (here logistic functions). This results in a single differential equation with variable coefficients, which is solved numerically, for specific parameter values, using MAPLE^®. The numerical solutions are compared to the baseline analytical approach for constant spatial dependencies. For validation purposes an assumed-modes solution is also developed. For a free-fixed boundary conditions example good agreement between the numerical methods and the analytical approach is found, lending assurance to the continuous variation model. Fixed-fixed boundary conditions are also treated and again good agreement is found.

This study focuses on a hybrid surrogate modelling technique in order to predict parameter-dependent mode coupling instabilities for uncertain mechanical systems subjected to friction-induced vibration. For this purpose, the most common strategy consists in associating a Monte Carlo procedure and/or a scanning technique together with the Complex Eigenvalue Analysis (CEA). This numerical strategy is computationally too prohibitive, particularly in an industrial context such as in the brake systems. To overcome this drawback, a novel approach is proposed. It consists in the combination of the generalized polynomial chaos (GPC) together with the kriging based meta-models. The association of both methods gives rise to a hybrid meta-model allowing taking into account two sets of uncertain parameters in the prediction of mode coupling instabilities. Moreover, it permits avoiding the use of the prohibitive MC and scanning methods. Thereby, this study analyses the feasibility of the proposed meta-model and its potential to be an efficient predictor of squeal propensity under parameter uncertainty.

In order to eliminate the moment unbalance of rotary systems, a certain type of equilibrator mechanism which is able to perfectly balance is utilized. It includes helical spring(s), a pulley and a cable attached to a certain hinge point on the rotary body. Actual implementations of the mechanism may not allow realization of the conditions for perfect balancing. Then, the problem is transformed into a multi objective constrained optimization, which includes multiple parameters and multiple objectives like minimizing the residual moment unbalance, minimizing the diameter and the length of the spring(s), maximizing the spring fatigue life as well necessity to satisfy some geometrical layout constraints and operational constraints on the springs. The system has been modeled in a quasi-static manner and optimized parametrically around the regions of operation. Pareto optimal fronts have been determined and the optimized parameters have been used as design parameters for realization of the actual system. The design of the mechanism has been algorithmically automated based on the requirements and constraints.

A finite element method (FEM) model of a lightly damped built-up structure was developed based on a two-step approach. The approach utilizes the fact that damping changes natural frequencies very little but the resonance amplitudes significantly. In the first step, stiffness property of the model is refined by correlating natural frequencies obtained from test and FEM model while neglecting the effect of damping. In the second step, damping properties are identified by matching the resonance amplitudes of the system obtained from the FEM model with those obtained from measurement. As a demonstration, the approach was applied first to a simple cantilevered beam structure, then a simplified vane assembly structure clamped into a fixture. The two-step approach was applied to the vane assembly in the free-free configuration, then the assembly clamped in a cantilever form. It was shown that the proper boundary condition of the FEM model can be identified by adjusting the contact stiffness at the clamped edge. Various different damping models were adopted to identify the damping property of the model. It was shown that different models lead to slightly different forced responses of the system. Typical applications and limitations of the correlated model are discussed with some practical examples.

The mounted resonance frequency of an accelerometer is a key parameter to ensure a sensor’s performance, its functional state, and how well the sensor has been mounted. The mounted resonance frequency leads directly to the frequency response limitation of a sensor, and how sensor sensitivity will change according to the frequency range of interest.

The mounted resonance frequency can become a quite important property, and provides hints on the mounting conditions of a sensor and related issues. Some things that could be identified are, the accelerometer is loose from the mounting surface, the sensor’s mass is too high in relation to the mass of the structure, or the mounting method lacks in stiffness. Under these various cross conditions, different resonance modes are theoretically expected from the solution of the equation of motion.

The way to measure and specify such a parameter can be challenging. It is important to understand in which conditions this value has been determined when comparing one sensor to another, from one mounting method to another, from the sensor mass to unit under test mass ratio, or from a sensor’s initial state to its response after several test cycles.

In this paper we will provide you with an overview of the different procedures to perform such a resonance frequency test. We will then provide you with an understanding on a response if the measurement is performed on a free sensor hanging in the air, on a sensor mounted on a heavy structure, and with other intermediate mounting configurations.

We will discuss different results obtained if the resonance frequency determination is performed using a shaker and performing a frequency sweep, using the pencil break test technique (similar to ASTM E976-84) or inverse piezoelectric excitation. We will then provide a rule of thumb on how to derive from those measurements an approximation of the sensor frequency response, if the damping properties of the sensor are known.

Time synchronous averaging for the extraction of periodic waveforms is a rather common processing method for rotating machinery diagnosis. By synchronizing the signal to the rotational angle of the component of interest (e.g. by using a tachometer reference signal), it is possible to perform the averaging in the angular domain, thus obtaining an angle-synchronous signal. Such an operation usually requires a resampling procedure. In this paper, the operations of averaging in time and angular domain are thoroughly discussed, referring to the main measurement and computational errors affecting the extracted waveform. The effects of uncertainty in the pulse arrival times of the reference signal on the synchronous averaging method are studied. Jittering of the reference signal affects the quality of the synchronous averaging process resulting in an attenuation of the extracted synchronous signal components, especially in the high frequency band. A theoretical analysis of the linkage between jitter model and attenuation is performed and an analytical model is proposed. The agreement of the analytical model with numerical simulations is first shown, and then an experimental case study involving the early detection of bearing damage is examined. The results are discussed in the framework of the cyclostationary signal theory.

Research involving energy harvesters to power systems are often carried out in a laboratory setting where vibration excitations are created using shakers. These excitations can be tuned to the natural frequency of the harvester to achieve the highest power output possible for the harvester design. Adapting these designs for use in real world applications introduces many additional layers of complexity. These added complications come in various forms including different input vibration and acceleration profiles, sensor energy requirements, and space constraints. Existing optimization for application-based research is limited to one set of constraints and cannot be applied to different cases.

This optimization is based around the use of a zigzag shaped device that is capable of achieving low natural frequencies in a compact design space. Piezoelectric materials bonded to the zigzag substrate are used to harvest energy from vibration excitations. The research in this paper presents an optimization for a zigzag shaped energy harvester based on varying input parameters in the design space to provide the optimum design for each application scenario. Optimization of this harvester creates the start of a design guide software that can be quickly used to implement sensor networks subject to a wide range of operating conditions typical of varied applications.

In this paper, a non-probabilistic method based on fuzzy logic is used to update finite element models (FEMs). Model updating techniques use the measured data to improve the accuracy of numerical models of structures. However, the measured data are contaminated with experimental noise and the models are inaccurate due to randomness in the parameters. This kind of aleatory uncertainty is irreducible, and may decrease the accuracy of the finite element model updating process. However, uncertainty quantification methods can be used to identify the uncertainty in the updating parameters. In this paper, the uncertainties associated with the modal parameters are defined as fuzzy membership functions, while the model updating procedure is defined as an optimization problem at each α-cut level. To determine the membership functions of the updated parameters, an objective function is defined and minimized using two metaheuristic optimization algorithms: ant colony optimization (ACO) and particle swarm optimization (PSO). A structural example is used to investigate the accuracy of the fuzzy model updating strategy using the PSO and ACO algorithms. Furthermore, the results obtained by the fuzzy finite element model updating are compared with the Bayesian model updating results.

Nowadays, many of the proposed solutions to improve automotive vehicle efficiency, such as downsized engines and advanced torque lock-up strategies (for automatic transmissions), can lead to worse noise and vibration characteristics. A typical phenomenon that occurs in such situations is low-frequency booming noise, which happens because of the irregular torque vibrations that are transferred through the flexible driveline elements. This paper presents a combined test and 1D modelling approach used to analyze and predict driveline torsional oscillations and their effect on low frequency booming noise and vibration. In this context, Model Based System Testing (MBST) can be defined as the framework that combines physical testing and simulation with the objective of validating and improving the behavior of 1D multiphysical models. Tests are carried out to obtain insight in the dynamical system behavior, as well as to obtain specific component parameters. This data is then used to create and improve 1D models of the full vehicle driveline, and to predict booming noise characteristics.

The paper addresses the methodologies required to derive a reliable and representative simulation environment for virtual shaker testing to predict the outcome of Spacecraft vibration tests numerically prior to its physical execution. The simulation environment needs to comprise the coupled dynamical models of the testing facility, structure under test and control system active during test. Especially, the paper focuses on modelling and experimental validation techniques for coupled test specimen models using the coupled head expander and beam test specimen system as validation test case. The main focus is set on the derivation of experimental test structure models which will be used in the future for modal coupling by applying Component Mode Synthesis and to assess the added value against alternative sub structuring techniques w.r.t. frequency bands and residual compensation, analysis of the measurement of required interface and connection point degrees-of-freedom and consideration of damping. Consequently, those experimental results are used to derive preliminary methodologies and guidelines to couple structural sub component models, and use them in further numerical closed-loop vibration control simulations. Consequently, the developed and experimentally derived structural dynamic models are coupled with a multi-physical electrodynamic shaker and sine controller model to virtually replicate sine control tests, followed by a validation with experimental test results and to be used in preliminary numerical investigations for vibration control performance enhancement.

In-service cables such as stay cables and suspenders of cable-stayed bridges and suspension bridges, are subjected to dynamic loads (e.g., the vehicle loads and wind excitation). Performing vibration measurements and subsequently identifying the dynamic properties and establishing a dynamic model of cable vibration are essential for their dynamic analysis, condition assessment, and performance prediction. For example, based on the taut-string theory, the cable tension, as a critical indicator of cable performance and health state, can be computed using its frequency that can be identified from the measured cable vibration responses. Traditional contact-type wired or wireless sensors, such as accelerometers and strain gauge sensors, require physically attaching to the structure for vibration measurements, which could induce the mass effect. In addition, installing these sensors on structures is costly, time-consuming, and allows instrumentations at a limited number of places. On the other hand, digital video cameras have emerged as a cost effective and agile non-contact vibration measurement method, offering high-resolution, simultaneous, measurements. Recently, digital video camera measurements processed by advanced computer vision and machine learning algorithms have been successfully used for experimental and operational full-field vibration measurement and modal analysis. This study develops a video measurement and processing based technique that can autonomously and blindly extract the full-field dynamic parameters of cable vibration from the video measurements. In addition, by exploiting the taut string theory, full-order (as many modes as possible) dynamic parameters are also extracted. Therefore, a full-field, full-order dynamic (modal) model of cable vibration is established. Laboratory experiments are conducted to validate the developed approach.

In Random Vibration environmental testing, it is a common practice to specify the requirements as acceleration power spectral densities (PSDs) that need to be reproduced at user-defined control channels. Such a test is typically performed in a single-axis setting, where the test article is subjected to vibrations in one direction only. If more than one direction is of interest sequential single-axis tests are performed after rotating the test article or using a slip table configuration. This way to perform multi-axial Random Vibration tests is out of date: there is definitely some lack of realism in sequential single-axis testing, as the stress loading and boundary conditions will significantly differ from the true three-dimensional environment. For very heavy structures, often the excitation level safely reachable by a single shaker is not even sufficient, the limitation being the risk of damaging the sometimes very expensive and fragile test articles due to high concentrated stresses. All these limitations are overcome if a Multiple-Input-Multiple-Output (MIMO) Random Vibration Control test is performed. Even though the benefits of MIMO tests are clear and accepted by the environmental engineering community, their practice still needs to grow. This is mainly due to the high degree of expertise needed to perform these tests. The challenges of MIMO Random Control start even before the actual test, in the test definition phase. The target that needs to be reached during the test is a full Spectral Density Matrix where the cross terms are as important as the diagonal ones. Defining this matrix with no a-priori knowledge of the cross-correlation between control channels is very challenging: filling in the off-diagonal terms, in fact, must guarantee that the target has a physical meaning. This is translated in the algebraic constraint that the target matrix needs to be positive (semi)-definite. On the other end the pushing driver of any Random Vibration Control test is to be able to replicate specific PSDs, given, for instance, by qualification specifications or optimal profiles (in terms of fatigue damage or comfort requirements). In defining the target matrix the main challenge is to guarantee a physically realizable full target spectral density matrix that has fixed PSD terms. Several authors tackled the problem of defining the best target possible (in terms of minimum drives energy, in terms of control performances,) even though few works can be addressed that tackle the problem of defining a realizable target first. This leads to a gap in the standards about a generally accepted and robust procedure to define the MIMO Random target matrix. The purpose of this work is to investigate different target generation procedures pointing out the advantages and the challenges in terms of physical meaning and their impact on the random control strategy. Alternative solutions based on on-going research topics will also be considered to propose alternative robust target definition routines in order to aim to a well-defined automatic procedure to include in the standard practice.

Additive manufacturing has seen a resurgence in recent years, mainly driven by its ability to produce parts with complex designs and incorporate multiple components into a single manufactured part. However, additive manufacturing still needs improvements to fabricate consistent and reliable parts. Some of the shortcomings of the process include the effects of build orientation on the material properties. Current structural health monitoring methods, such as visual photography and thermography, provide limited data to quantify the quality and reliability of printed part. In this study, 3D printed novel sensors were embedded within a part and were evaluated as an alternative method for structural health monitoring. The embedded sensors were composed of conductive filament and are intended as a non-intrusive method to monitor the structural health of the part during its service life. Filaments using novel materials, e.g. conductive polylactic acid, were evaluated for their material characteristics and suitability for sensor verification and validation. Specifically, the novel conductive filaments were evaluated for their performance in terms of mechanical and electrical properties. The Young’s modulus of the additively manufactured, polylactic acid part was determined by both tensile testing and cantilever testing; and temperature sensitivity was determined by resistance measurements during thermal cycling. Embedding of the sensor into the service part during the printing process can reduce the cost and production time for structural health monitoring and provide a new application area for additive manufacturing.

This work deals with a flexible structure modeled as a tower crane, with proper servomechanisms for load manipulation. This type of flexible structure deserves popularity due to well-known advantages as high payload and large workspace. The main challenges, however, are the undesirable endogenous and exogenous vibrations attributed to typical tasks as trajectory tracking, at high speed rates, and inherent structural behavior, of tower and arm, excited by internal as external forces. The overall mechanical structure is modeled by means of finite element methods, experimental modal analysis techniques and some specific criteria are used to select the most appropriate locations of sensors and actuators. In order to attenuate these undesirable vibrations, an active tendon system is implemented, on a specific part of the structure, attaching a PZT stack actuator, on a supporting cable, to regulate its tension and to improve the effective modal damping and closed-loop stability and, as a consequence, improve the quality of the trajectory tracking and also reduce, as possible, the structural vibrations. Two active vibration control schemes are employed during the synthesis of the controller, that is, Multiple Positive Position Feedback (MPPF). Finally, the overall dynamic performance is evaluated and validated by numerical and experimental results on a small-scale mechanical platform.

In machine learning, black box models are often used to make excellent predictions of system behaviour. They are especially useful where the physics of a system is unknown or hard to model. This paper examines whether prior knowledge of certain physical properties of a system, encoded in a white box model, can be incorporated into black box methods to improve predictive performance. A combination of genetic algorithm optimisation of the white box model and Gaussian process regression on the residual error is presented as an improved method for system identification. This approach retains physical insight into the behaviour of the system while also reducing the error. Comparisons are made between pure white and black box models and the combined grey box model for several test applications. It is found that the combined model has significant advantages in predictive accuracy. This is especially seen in the case of nonlinear models. Here the full physics of the system is often too complicated or inaccessible to be modelled accurately with a white box method, but also the state space relationships are sufficiently complicated to make black box modelling equally challenging. The use of the proposed grey box method can reduce the complexity of the relationship that the black box is attempting to represent, leading to gains in accuracy. As a bonus, training time is reduced, as less complicated techniques are required to identify the process accurately.

This paper presents an investigation into the inspection of an Automated Tape Placement (ATP) process, which automates the process of laying composite plies on a tool surface. The requirement for monitoring this process is motivated by the need to detect common defects in the process which could compromise the integrity of the structure. An experimental procedure was carried out where the tool surface, where composite is laid during the ATP, was excited with ultrasonic guided waves. Several types of defects were introduced during this process, and a methodology is presented to extract features from the signals acquired, model the process using this data and identify the defects. The main challenge in modelling this process is the cumulative trend throughout the application of the composite plies. A method is presented where wavelet analysis is used to model short-term dynamics while the long term, cumulative trends are eliminated using cointegration. The results show that one can detect various types of defects during an ATP process using the methodology introduced here. The key result is that the trends in the data due to the normal process can be removed using cointegration in order to reveal abnormalities.

Roller rigs have been built world-wide to investigate the dynamics of railway vehicles and they have particularly been applied to the development of high-speed trains.

Specifically, the present paper focuses on a roller-rig for testing full scale locomotives/coaches, which is located at the Osmannoro Centre of Experimental Dynamics (Italy). The considered roller rig allows to test vehicles having up to 6 axes and 3 bogies. Aim of the roller-rig is to verify traction systems of locomotives, to test anti-slip/anti-skid control systems, to identify braking performance and to perform electromagnetic tests (radiated emissions and immunity) on the locomotive.

At present, the design of the control strategy allowing to perform the desired tests is under development. To account for roller-rig and vehicle dynamics at a design stage, a numerical model of the complete test bench (including the locomotive/coach, the roller rig, the actuating devices and the corresponding control systems) was developed and interfaced with the control strategy of the test bench. The present paper describes the simulation tool developed on purpose and preliminary experimental results are presented.