This paper presents the modal analysis of a simple electromechanical system composed of two interacting subsystems, one mechanical and one electromagnetic, and shows how its dynamics differs fundamentally from that of purely mechanical systems. Classical mechanical oscillators exhibit motion governed by the exchange between kinetic and potential energies, whereas the system analyzed here oscillates solely through the exchange between mechanical kinetic and magnetic kinetic energies, since no elements store mechanical or electrical potential energy. Its governing equations involve only two matrices, M and G. Although traditionally referred as inertia and gyroscopic matrices, their physical meaning in this electromechanical context differs substantially: M incorporates both mechanical and electromagnetic inertial contributions, and G does not act as a conventional gyroscopic matrix but instead couples the subsystems and their energy exchange. To the best of our knowledge, this is the first modal analysis of a system composed solely of inertia and gyroscopic matrices. The computed natural frequencies and modes are intrinsically hybrid, involving both mechanical and electromagnetic variables. Hybrid modal coordinates are introduced, and resonance characteristics and frequency response functions are examined. An energetic analysis further reveals that the hybrid natural frequency corresponds to the rate at which energy is exchanged between the mechanical and electromagnetic subsystems.

In this paper, a model mass-belt system with random friction is used to study the computational costs of modelling a problem with stick–slip phenomena, stochastically. The objective is to assess the link between computational runtime and dynamical behaviour, specifically the duration and count of stick/slip phases. Analysing the resulting multi-dimensional random vector (runtime, duration, count) is complex. This work investigates random variable transformations as a method to reduce analytical complexity by creating independent variables. We also use this approach to examine the influence of the integration method on the computational costs. Three different strategies to compute approximations to the system solutions are compared in terms of computational runtime and their relation with the duration and count of stick/slip phases, the variables of greatest interest. One of the strategies used, for example, is Monte Carlo simulations coupled with a Multiple Scales analytical approximation. With this strategy, we find, for instance, that the number of stick–slip transitions, rather than their duration, primarily dictates the predictive behaviour.

This paper investigates the response of a deterministic, linear, time-invariant mass-spring-damper system subjected to loading modeled as a stationary stochastic process. The primary objective is to investigate, through numerical simulations using the Monte Carlo method, whether the system response exhibits stationary properties in the steady state. The analysis employs two metrics: engineering distance, which focuses on the proximity of distribution means, and Wasserstein distance, which provides a more robust comparison by quantifying divergence between probability distributions across different sections of the stochastic process. The methodology presented can be adapted for the analysis of other mechanical systems, including nonlinear systems.

In this paper, a comprehensive continuous Cosserat rod model is developed to study the dynamics of a drill-string. The model presents the following features: (1) it can simulate the 3-D dynamics of the system, including lateral, axial and torsional motion; (2) it includes damping effects by adopting a Kelvin–Voigt material; (3) it uses a velocity-independent bit-rock interaction formulation in which the forces and torques at the bit are obtained by considering the dynamics of the cutting blades and the evolution of the soil profile, which is simulated with the help of an advection equation; (4) lateral contact is also included. In this paper, two application cases are analysed. The first one deals with a vertical borehole. It includes four simulations that are used to further investigate the hypothesis that low-dimensional lumped formulations are not able to capture the dynamics of a drill-string and, while doing so, evaluate the effect of the damping in the system's response. The predictions are compared with those given by two other models: the first one considers a 2-degrees-of-freedom (DOF) formulation, with 1-axial and 1-torsional DOF; the second one is a semi-discrete approach, with 1 DOF for the axial dynamics and a continuous formulation for the torsional one. Divergences between the predictions are found, associated with the presence of higher frequencies in the signals obtained with the Cosserat model, which reassures the hypothesis that the new continuous approach can capture aspects of the dynamics that cannot be modelled with low-dimensional representations. The efficiency of the drilling is also assessed for the simulations obtained with the new model. The highest performance was observed in those simulations where only axial oscillations occurred, while a performance drop was seen when those vibrations were accompanied by torsional stick–slip, suggesting that an optimum damping value associated with the best drilling performance could be found, were damping controllable. Finally, the second application case is used to illustrate the full potential of the new model to tackle the 3-D lateral, axial, and torsional dynamics in a curved borehole.

This work compares the efficacy of lumped and continuous models in simulating drill-string dynamics. We aim to critically explore the inadequacy of lumped models and challenge their perpetuated use in the literature. Lumped models represent the drill-string with a pre-determined number of discrete masses, sometimes as few as one or two. This inherently restricts their ability to capture the distributed nature of drill-strings, compromising their accuracy. In this paper, these limitations are made evident by comparing simulations of lumped models with simulations of continuous models that increase in complexity. First, predictions of the axial behaviour obtained with two models, one lumped and another continuous, are analysed. Second, simulations involving a 3-D fully coupled continuous model are evaluated. The simulations demonstrate the inaccuracy of lumped models. Also, they refute the frequent assumption of representing the bottom hole assembly (BHA) as a single lumped mass. Beyond these results, theoretical arguments based on fundamental wave-reflection phenomena are employed to further support the inadequacy of lumped formulations for this purpose. Third, this paper also discusses the prevalent strategy of validating lumped models through comparisons with laboratory experiments. This approach suffers from being a makeshift validation, as similitude principles are often violated in laboratory setups, resulting in a practice that may have helped perpetuate lumped models in the literature. Consequently, using lumped models for general drill-string dynamics prediction is not appropriate. Recognising their limitations and exploring alternative approaches, such as continuous models, is crucial for accurate drill-string dynamics prediction.

A washing machine is a household appliance that has an interesting and complex dynamical behavior, which can be well described by a set of nonlinear differential equations. When analyzing the dynamics of a washing machine, the steady state motion (periodic solution) is an important response to consider and can be evaluated as a solution of a periodic boundary-value problem. The unbalance generated by the unevenly distribution of clothes during centrifugation is highly random and, therefore, a stochastic model is necessary to take this characteristic into account. The novelty of this paper consists in the analysis of a washing machine dynamics considering the uncertainty in the unbalance. Therefore, a stochastic model is proposed for the dynamics of a washing machine. The steady state solutions are calculated using the shooting method combined with a sequential continuation to evaluate it across all the spin speeds of the machine. The probability distributions of the washing machine vibration at those different spin speeds are approximated using Monte Carlo simulations. The impact of the random unbalance in the vibration amplitude of the washing machine is also investigated and consists in a key input for the fatigue design of many components.

Flexible beams are usually modeled under the assumption of large displacement, finite rotation, but with small strains. Such hypothesis allows the equation of motion to be built using co-rotational finite elements. The co-rotational formulation decomposes the total motion of a structural element into two parts: a rigid body and an elastic (small) deformation. This way, a geometric nonlinearity caused by the large displacements and rotations of the beam’s cross sections can be efficiently modeled. The novelty of this paper consists in incorporating this modeling technique inside a standard method to compute nonlinear normal modes (NNMs). The resulting method becomes a dedicated one to the analysis of complex flexible beams, including those with nonuniform cross sections and with pre-deformations. Those cases are not easily incorporated by other methods in the literature. The harmonic balance method (HBM) is used here to approximate the periodic solutions of the system. The arc-length parametrization is used to perform the continuation with respect to the energy level. The alternating frequency-time (AFT) method is used to compute the Fourier coefficients of the nonlinear elastic forces computed from the co-rotational finite elements. Two examples are used to illustrate the performance of the proposed method: bi-clamped flexible beams with nonuniform cross sections and a flexible riser (offshore oil pipes) in catenary configuration.

To present ideas, a model problem consisting of a moving mass-belt system with random friction showing the stick-slip phenomenon is treated. The dynamics is simulated. The objective of this work is to assess the behaviour of the computation cost in terms of the run-time, which is random, and its relationship with some of the output variables that define the dynamical behaviour of the mechanical system, such as the duration of the phases present in the simulation, sticks and slips, and the number of phases that occur in each realisation. All this is analysed from a stochastic perspective. However, the probabilistic model to analyse the distribution of a three-dimensional random vector, formed by the run-time, duration and number, belongs to $$R^4$$ , thus it is difficult to characterise and visualise. Hence, in this study, the use of random variable transformations to produce new independent variables is explored as an attempt to reduce the number of dimensions that need to be considered. Also, the change of variables is used to assess the link between the behaviour of the results and the chosen integration method. It is shown that the predictions obtained with the Monte Carlo method combined with a Multiple Scales analytical approximation are influenced by the number of transition phases rather than their durations.

In this paper, the spread of a general epidemic over time is modeled as a branching process. It is a stochastic process sorted as an individual-based model, which records population growth over generations with uncertainties to its size. The source of randomness is inherently related to the individual behavior of each member in a population. In this context, the transmissibility of the disease, i.e., the contagion from an infected person to susceptible ones is the root. Therefore, a discrete random variable models the number of infections per infector and rules the branching process. Given the probabilistic model of the contagion, the objective of the paper is to compare three methodologies to evaluate the mass functions of further generations of the branching process: probability generating functions (pgf), Markov chains (MC) and Monte Carlo simulations (MCS). The former gives analytical expressions, that can be symbolic computed, to evaluate the probability of an arbitrary number of infected members for a desired generation, whereas MC is a semi-numerical methodology and the latter is indeed a numerical one. The comparison between all of them relies on computational cost (runtime and storage) and limitation of applicability in relation to the mass function of the contagion. One of the characteristics of interest in the analysis is the determination of which methodologies allow the calculation of the mass function of a further generation without computing the mass functions of previous ones. This feature is referred in here as not time-dependent. Another characteristic of interest is the determination of which methodologies allow the computation of just some values of the mass function of a generation, i.e., probabilities related to the same generation can be achieved independently from the others. This is so-called a local property.

Impulse responses are functions that characterize linear systems uniquely. They can be used to predict the responses of a system once the applied excitation is known. With the input and output signals of vibration tests, the impulse responses of mechanical systems can be estimated using different methods. Once those functions have been correctly estimated, the modal parameters can be identified using time-domain methods, as for example, the eigensystem realization algorithm (ERA). Problems can arise when the acquired data consist of only a few noisy samples. Under this scenario, this paper analyzes the benefits of the Observer/Kalman filter identification method (OKID) when estimating impulse responses. As another contribution, this paper also evaluates how the quality of the estimated impulse responses interferes in the identification of the modal parameters. A numerical example is used here to show that the OKID gives a better impulse response estimation when compared to another classic method. Experimental data are also used to exemplify the benefits of OKID in the modal parameters identification using the ERA.

Flexible beams contain geometric nonlinearities emanated from the large displacements and large rotations of the cross sections. When the geometry of the beam is nonuniform, the equation of motion becomes complicated to derive, but can be eficiently approximated using the co-rotational nite element method. This paper proposes a procedure to compute nonlinear normal modes (NNM) of nonuniform exible beams. The Rosenberg's definition of NNM is applied. The periodic solutions are computed using the Harmonic Balance Method (HBM) and the continuation of the modes properties with respect to the energy level is performed using the arc-length method. Examples of clamped-clamped beams with dierent cross sections variations are presented, illustrating the respective impacts in the NNMs.

Electromechanical systems are composed by two interacting subsystems, a mechanical and an electromagnetic. This paper discusses the oscillatory response of a linear electromechanical system. The objective of the paper is to show that the oscillatory response of the chosen electromechanical system is provoked by the mutual interaction between mechanical and an electromagnetic subsystems, and to compare this oscillatory response with the response of purely mechanical systems. Natural frequencies and normal modes, are computed for the electromechanical system. The computed parameters involve mechanical and electromagnetic variables, i.e., they are hybrid, a novelty in the literature. Hybrid model coordinates and frequency responses graphs are also discussed.

System identification is the main goal when performing modal testing of mechanical structures. In cases where only the structural responses are measured, the identification technique is addressed in the literature as operational modal analysis (OMA). Applications of OMA are found in structures where the excitation from the ambient can not be removed or it is the only possible one. Since the ambient forces can not be measured, the identification of the modal parameters is possible if some hypotheses about its random nature are made. In this paper, the modal identification of a small and flexible wind turbine blade is performed under the OMA framework. Although the tested structure was in laboratory condition, traditional experimental modal analysis was not possible due to the small size and high flexibility of the blade. Therefore, this paper demonstrates the importance of OMA in the modal identification of this type of structure (simultaneously light and flexible). An artificially generated wind was used instead as source of excitation, which presented good properties necessary for OMA (stationarity and broadband frequency). The stochastic subspace identification method was used to identify the modal parameters, leading to good identification in a prescribed frequency band of interest. A procedure to analyze and validate the experimental results is also given at the end.