Mass-spring-damper Research Articles

Physics-informed and black-box Identification of robotic actuator with a flexible joint

In robotics, precise models are critical for ensuring safety and functionality. However, acquiring a precise model characterizing a system’s dynamics can be challenging. One of the alternatives to address this issue is system Identification, which aims to obtain models through physical and experimental observations. In this manner, the developments in machine learning algorithms, such as neural networks, have significantly improved the modeling of complex and nonlinear phenomena. In this work, a mass-spring-damper (MSD) system and a low-cost original elastomer-based Series Elastic Actuators (eSEA) assembly are used to evaluate the performance of system Identification models. The black-box models selected are variations of the AutoRegressive Moving Average with eXogenous input (ARMAX) algorithm. The gray-box model aims to estimate the parameters of 4 friction models; the optimization is done utilizing physics-informed neural networks (PINNs). For both case studies, the PINNs outperformed the black-box models. In the didactic example, the parameters obtained are close to the ground truth, and the highest determinant coefficient obtained is 0.99. The friction model that best represents the robotic actuator is the LuGre model, with the parameters obtained using the PINNs, outperforming the best black-box model by lowering the mean absolute error (MAE) by 30.83%. With a determinant coefficient of 0.94, the model shows a high capacity for describing the multiple nonlinearities present in the system.

Unveiling advanced modelling and analysis: the integrated system and formula for mass–spring–damper with hydraulic damper systems

Unveiling advanced modelling and analysis: the integrated system and formula for mass–spring–damper with hydraulic damper systems

Vibration Suppression of a Flexible Beam Structure Coupled with Liquid Sloshing via ADP Control Based on FBG Strain Measurement

In this study, an adaptive dynamic programming (ADP) control strategy based on the strain measurement of a fiber Bragg grating (FGB) sensor array is proposed for the vibration suppression of a complicated flexible-sloshing coupled system, which usually exists in aerospace engineering, such as launch vehicles with a large amount of liquid propellant as well as a flexible beam structure. To simplify the flexible-sloshing coupled dynamics model, the equivalent spring-mass-damper (SMD) model of liquid sloshing is employed, and a finite-element method (FEM) dynamic model for the beam structure coupled with the liquid sloshing is mathematically established. Then, a strain-based vibration dynamic model is derived by employing a transformation matrix based on the relationship between displacement and strain of the beam structure. To facilitate the design of a strain-based control, a tracking differentiator is designed to provide the strains’ derivative signals as partial states’ estimations. Feeding the system with the strain measurements and their derivatives’ estimations, an ADP controller with an action-dependent heuristic dynamic programming structure is proposed to suppress the vibration of the flexible-sloshing coupled system, and the corresponding Lyapunov stability of the closed-loop system is theoretically guaranteed. Numerical results show the proposed method can effectively suppress coupled vibration depending on limited strain measurements irrespective of external disturbances.

Stability optimization of time-delay systems with zero-location constraints applied to non-collocated vibration suppression

We present an active control method for achieving non-collocated vibration absorption, implemented on a lumped mass system. The task of suppressing vibrations at a number of frequencies, at a target mass is recast into a problem of assigning zeros of the transfer function from the external excitation force to the position of the target mass. The controller design involves computing the controller parameters by utilizing the available output feedback to directly assign a set of zeros on the imaginary axis while simultaneously preserving the overall stability with a sufficient margin. The design also takes into account delays in the feedback path. The presence of delays leads to an infinite dimensional system for which, there exists in general, infinitely many characteristic roots. The requirement of achieving sufficient damping in the resulting closed-loop system together with suppressing vibrations of the desired frequency leads to a constrained optimization problem of minimizing the spectral abscissa function subject to zero-location constraints. These constraints exhibit polynomial dependence on the controller parameters. We present two approaches for controller parameterization for the single and multiple input case. Both approaches are based on constraint elimination and exploit the property that the constraints are affine in a selection of the gain parameters. Finally, the methodology is verified by simulating on a spring–mass–damper system following which, it is then validated experimentally on a laboratory setup.

Adaptive stable backstepping controller based on support vector regression for nonlinear systems

In this paper, a novel adaptive stable backstepping controller(BSC) based on support vector regression (SVR) has been introduced for nonlinear dynamical systems. Stable BSC is designed over Lyapunov stability of the closed-loop system. The nonlinear system dynamics required to constitute the BSC architecture are identified via SVR. The prediction competency of SVR and the stable behavior of BSC are aggregated in this architecture for nonlinear systems. The performance evaluation of the proposed adaptive BSC has been examined on a nonlinear inverted pendulum(IP) and a nonlinear mass–spring–damper(NMSD) system. The acquired results provide a successful and stable BSC control performance for both nonlinear systems.

An improved van der Pol wake oscillator model for a 1DOF circular cylinder undergoing vortex induced vibrations

An optimization procedure is presented for prescribing the empirical coupling parameters of a modified coupled van der Pol (VDP) wake oscillator and mass–spring–damper system for a one degree of freedom (1DOF) circular cylinder system undergoing vortex induced vibrations (VIV). This approach allows for the generalization of the traditional VDP model to cover a large VIV parameter range. The procedure is developed using experimental data covering the parameter space: 3<U∗<11, 8<m∗<20, 0.003<ζ<0.063, ReD=4000. The VDP model contains empirical coupling coefficients that describe the regime specific dynamic damping, ϵ, and wake coupling, β. Two calibration procedures are considered and the improvements over the traditional VDP model in predicting the VIV response are discussed. Subsequently, a regression fit is used to prescribe the two empirical coupling coefficients as functions of m∗, ζ and U∗. The generalized model results in a better prediction of the VIV response in terms of identification of lock-in regimes, lock-in frequency and amplitude response over several regimes when compared to the classical models. The use of a functional representation replaces the need for using tables and reduces user input for determining regime-specific coupling coefficients.

Event-based performance guaranteed tracking control for constrained nonlinear system via adaptive dynamic programming method

An optimal tracking control problem for a class of nonlinear systems with guaranteed performance and asymmetric input constraints is discussed in this paper. The control policy is implemented by adaptive dynamic programming (ADP) algorithm under two event-based triggering mechanisms. It is often challenging to design an optimal control law due to the system deviation caused by asymmetric input constraints. First, a prescribed performance control technique is employed to guarantee the tracking errors within predetermined boundaries. Subsequently, considering the asymmetric input constraints, a discounted non-quadratic cost function is introduced. Moreover, in order to reduce controller updates, an event-triggered control law is developed for ADP algorithm. After that, to further simplify the complexity of controller design, this work is extended to a self-triggered case for relaxing the need for continuous signal monitoring by hardware devices. By employing the Lyapunov method, the uniform ultimate boundedness of all signals is proved to be guaranteed. Finally, a simulation example on a mass–spring–damper system subject to asymmetric input constraints is provided to validate the effectiveness of the proposed control scheme.

Design of proportional integral observer‐based resilient control for periodic piecewise time‐varying systems: The finite‐time case

AbstractThe prime intent of this study is to scrutinize the state estimation and finite‐time (FT) stabilization problems for a class of periodic piecewise time‐varying systems (PPTVSs) in the presence of external disturbances, immeasurable states and gain fluctuations by dint of resilient control. Moreover, the proportional integral observer system is built in line with the output of the PPTVSs to reconstruct the states of the PPTVSs, since the state dynamics of the system are often not measurable. Meanwhile, the output signals are quantized with the use of a logarithmic quantizer prior to being fed into the constructed observer systems. Further, the fluctuations in the controller and observer gain matrices are considered to occur in a random manner and they pursue the pattern of a Bernoulli distributed white sequence. Subsequently, by means of Lyapunov stability theory and the matrix polynomial lemma, the adequate requirements for ascertaining the FT boundedness and IO‐FT stabilization of the addressed system are procured in the context of linear matrix inequalities. Thereupon, on the premise of the established criteria, the gain matrices of the controller and observer are obtained. Ultimately, the simulation results have been showcased including the mass spring damper system to exemplify the efficacy and usefulness of the theoretical conclusions drawn.

Deep learning-based soil compaction monitoring: A proof-of-concept study

The dynamic behavior of the vibratory drum of a soil compactor for earthworks is known to be affected by soil stiffness. Real-time monitoring techniques measuring the acceleration of vibratory drums have been widely used for soil compaction quality control; however, their accuracy can be affected by soil type and conditions. To resolve this problem, a novel deep learning-based technique is developed. The method allows the regression estimation of soil stiffness from vibration drum acceleration responses. By expanding the range of applicability and improving accuracy, the proposed method provides a more reliable and robust approach to evaluate soil compaction quality. To train the estimation model, numerous datasets of noise-free waveform data are numerically generated by solving the equations of motion of the mass–spring–damper system of a vibratory roller. To validate the effectiveness of the proposed technique, a field experiment is conducted. A good correlation between the estimated and measured values is demonstrated by the experimental results. The correlation coefficient is 0.790, indicating the high potential of the proposed method as a new real-time monitoring technique for soil compaction quality.

Composite fault reconstruction and fault-tolerant control design for cyber-physical systems: An interval type-2 fuzzy approach

This article scrutinizes the stabilization and fault reconstruction issues for interval type-2 fuzzy-based cyber–physical systems with actuator faults, deception attacks and external disturbances. The primary objective of this research is to formulate the learning observer system with the interval type-2 fuzzy technique that reconstructs the actuator faults as well as the immeasurable states of the addressed fuzzy based model. Further, the information of reconstructed actuator faults is incorporated in the developed controller with the imperfect premise variables for ensuring the stabilization of the system under consideration. At the same time, the H∞ technique is employed to reduce the impact of external disturbances in the considered model. In addition to that, the deception attacks are represented as a stochastic variable that satisfies the Bernoulli distributions. On the ground of this, a set of sufficient criteria is deduced in the context of linear matrix inequalities to affirm the stability of the addressed systems. Furthermore, the requisite gain matrices are computed by resolving the obtained linear matrix inequality based stability criteria. At last, two simulation examples, including the mass–spring–damper system are exhibited to demonstrate the usefulness of analytical findings of the developed strategy.

Treadmill Deck Performance Optimization Design Based on Muscle Activity during Running

In previous research on treadmills, the main focus has been on comparing the physiological differences induced by running on treadmill decks and other exercise surfaces, with relatively little research on the mechanical properties of treadmill decks. Reducing sports injuries is a common desire of runners, which may be closely related to muscle activity. Obviously, the mechanical properties of the treadmill play an important role in this process. Muscle activity was evaluated based on a mass-spring-damper (MSD) model that provides a simulated signal of the ground reaction forces (GRF) and vibration of the lower-limb soft tissues (LLST) during the landing of the human body during running. We improved the original human motion model by considering the stiffness and damping effect of the treadmill deck. In addition, based on the theory of muscle activity regulation, the dimensionless objective function is established, and the particle swarm optimization algorithm is used to find the best range of treadmill deck parameters under pre- and post-fatigue conditions. The results show that the hardness of the treadmill deck can affect the regulation of muscle activity. Based on this, the parameters of the specific safe area of the treadmill deck are obtained, and the size of the safe area after fatigue is significantly reduced compared to that before fatigue. By studying the physiological effects of the mechanical properties of the treadmill deck on runners, the research results are expected to provide references for the design of treadmill deck parameters and reduce the risk of runners’ sports injuries, which has practical application value for treadmill design and runners’ health.

An analytical solution of the maximum response of the coupled multiple parallel modulated pedestrian-bridge system

This paper studies analytically the steady state response of the generalised coupled modulated pedestrian-beam system. Pedestrians are considered as modulated spring–mass–damper systems with and without a partially attached mass to the structural system. The governing non-dimensional parameters that defined the system’s behaviour are found in an analytical framework. An application of the expression is developed, and a parametric analysis is presented considering a reference pedestrian. The frequency shift effect and the dynamic amplification factor, DAFHSI, of the coupled pedestrian-beam system are characterised. The main contribution of this paper is to provide a closed-form solution of the generalised coupled modulated pedestrian-beam system. This solution can consider any distribution of pedestrians and any definition of the pedestrian SDOF properties. A simple and representative example is presented to demonstrate the utility of the found expression in the context of footbridge dynamics.

Robust feedback models with structured uncertainties for human–structure interaction

Structural engineering developments have implemented new design techniques that allow the construction of large structures with lighter and high-strength elements. Structures such as grandstands, slabs and pedestrian bridges may be prone to excessive vibrations due to human activities. Traditional dynamic modeling approaches to represent the human–structure interaction (HSI) are often used to predict the dynamic behavior of the coupled system based on deterministic mass–spring–damper (MSD) mechanical components. However, these models cannot precisely represent variabilities and uncertainties in human and structural systems. In this study, three HSI robust models are developed and experimentally validated on a test structure. First, a traditional MSD coupled model is proposed where the human subsystem includes structured uncertainties in its biomechanical parameters. Then, two models are developed and synthesized from robust control theory using H∞ and μ-synthesis, implementing structured uncertainties in the frequency and damping ratio of the plant to capture the intra- and inter-variability in the biomechanical parameters. The results demonstrate that the developed robust HSI models provide effective and reliable approaches to predict the range of probable responses of the coupled human–structure system for different occupants.

Development of Empirical Model for Electromagnetic Damping Coefficient Damper

The significance of the electromagnetic damper in vibration systems has attracted considerable interest from researchers, making it a prominent area of research. Various papers have been consulted to explore the vibration concept associated with electromagnetic dampers and their practical applications. A vibration test rig with a simple electromagnetic damper has been designed and tested to investigate the effect of electromagnetic force. An experimental study on the response of the electromagnetic damper was conducted. A logarithmic decrement method was deployed to find the damping coefficient, c, of a one-degree freedom system (mass spring damper system). A test rig and electromagnetic damper element were introduced as a damper in the system. Design factors included the type of geometry, type of material and the current supply to the system. The testing was conducted using the in-house developed vibration test rig. The data obtained from the experiment has been analysed to determine the electromagnetic damping performance. A factorial analysis was performed to identify the significant factors influencing the damping coefficient of the system. Two empirical models obtained through regression analysis of Excel and Minitab. It was found that the influential effects for the response are the type of material (aluminum), slotted geometry and a bigger amount of current (3 A). The application of a cylindrical conductor and magnet as a damper reduced the vibration response of spring mass damper.

Quantification of the human–structure interaction effect through full-scale dynamic testing: The Folke Bernadotte Bridge

An analytical expression for the frequency response function of a coupled pedestrian-bridge system is presented and evaluated using an experimental measurement campaign performed on the Folke Bernadotte Bridge in Stockholm, Sweden. A finite element model and the modal models that consider the human–structure interaction effect are calibrated with respect to the measurements. The properties of the spring–mass–damper model representing the pedestrians were identified, considering the different structural modes of the system. Good agreement was obtained between the experimental and theoretical frequency response functions. A sensitivity analysis of the obtained solution was performed, validating the determined analytical expression for the frequency response function of the coupled pedestrian-bridge system that takes into account the human–structure interaction effect.

Single body sensor for calibration of Spring-Mass-Damper parameters in biodynamic pedestrian modelling

For structural design and vibration monitoring purposes, several simplified equivalent-force models or more complex computational strategies are available to describe Human-Structure Interaction (HSI) phenomena on pedestrian systems, and in particular the vertical reaction forces induced by walking occupants. Among others, various Spring-Mass-Damper (SMD), Single Degree of Freedom (SDOF) biodynamic models of literature can be used to mechanically describe a single pedestrian in the form of equivalent body mass m, spring stiffness k and viscous damping coefficient c. Basically, existing SMD formulations are characterized by specific theoretical assumptions and (often complex) experimental methods for the calibration of m, k, c. Usually, SMD parameters can be optimally quantified when multiple sensors (on pedestrian’s body and on the structure) are used to capture motion features and the corresponding reaction force. In this paper, body accelerations of a pedestrian are tracked by means of a single Centre of Mass (CoM) sensor and are elaborated to derive basic input parameters for an alternative, newly optimized SMD model. Experimental registrations from a total of 30 random walks and more than 300 gaits (on rigid floor) are taken into account, and fitting expressions for m, k, c are proposed. The present SMD formulation (SMD-0) is validated towards a selection of literature proposals (SMD-1 to SMD-4), based on parametric numerical dynamic analyses (100 in total), which are carried out by taking into account various pacing frequencies (fp = 1.5–2 Hz the explored range) and four different pedestrian structures / floors (F#1 to F#4). The comparison of classical performance indicators for human-induced structural vibrations proves the efficiency and potential of current SMD-0 approach, and suggests further investigations in support of optimized protocols.

Earthquake data augmentation using wavelet transform for training deep learning based surrogate models of nonlinear structures

Machine learning methods have gained traction in the civil engineering community for analysis of civil infrastructures. A major field of application is in development of surrogate models for non-linear dynamic analysis for civil infrastructures. These data-driven models generally consist of a large number of parameters which need to be estimated from available data. However, due to limited availability of data in the field of earthquake engineering, often, it is difficult to develop stable models that can predict structural response by taking into account both material and earthquake uncertainty. To this end, the authors propose a discrete wavelet transform (DWT) enabled surrogate modeling framework for prediction of nonlinear structural response while accounting for material and earthquake uncertainty. DWT is used for earthquake data augmentation and feature extraction, which, along with constitutive material parameters, are used to train a deep neural network (DNN). It is shown that DWT can efficiently augment existing small earthquake datasets while the DNN can capture the non-linear structural response. The framework is validated by applying it to successfully predict structural response of 3DOF nonlinear spring–mass–damper system and non-linear three-story steel moment-resisting frame for unseen earthquakes with a median error less than 12%.

Suboptimal control law for a multi fractional high order linear quadratic regulator system in the presence of disturbance

A new method for finding a suboptimal control law for a multi fractional spring–mass system as a fractional high order linear quadratic regulator system in the presence of known disturbance is introduced. The method is presented in a general form for solving practical multi fractional linear quadratic regulator systems with disturbances. After defining a closed-loop controller, formulations for finding the suboptimal controller are derived. In the proposed method, there is no need to solve the fractional vector differential equation. Simulation studies demonstrate the applicability of the method for practical purposes. Some new types of a spring–mass–damper system are considered.

Efficient model‐correction based reliability analysis of uncertain dynamical systems

AbstractIn this contribution, a model‐correction‐based strategy is applied for efficient reliability analysis of uncertain dynamical systems based on a low‐fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of the high‐fidelity (HF) model. In the model‐correction approach utilized, the LF model is calibrated to the HF model close to the so‐called most probable point (MPP) in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single span railway bridge modelled as simply supported Euler‐Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge‐train interaction by modelling the passing train as mass‐spring‐damper (MSD) system, however increasing the computational effort of the limit state function evaluations. The failure probabilities evaluated with the model‐correction approach are contrasted and discussed with the failure probabilities of the sophisticated bridge‐train interaction model. It is shown that the model‐correction‐based approach provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

Novel aortic heart valve model parameterizing normal and pathological cases: Aortic stenosis and regurgitation

In this paper, a novel heart valve model with parameters selections enabling to illustrate some physiological patterns such as healthy, aortic stenosis, and regurgitation has been shown. The motion of the elastic leaflet body is modeled by a nonlinear mass–spring–damper and coupled with a cardiovascular lumped parameter model. As a result, our model takes in to account a single continuous time-varying fluid resistance, the aortic flow rate, left ventricular and aortic pressures, and the leaflets’ pose. This novel model is able to show the backflow, vortex and dicrotic notch phenomenon as approaching to the gold standard in physiological evaluation. The model also additionally presents reverse pressure effect on the aortic valve in the diastolic phase that is not observed in the recent literature. The physiological validation is performed by comparing two reference models and previously reported on physiological data. Nominal leaflet spring ( α) and damping (c) parameters have been determined for the healthy valve ( α = 0.1, c = 2), aortic stenosis ( α = 0.005, c = 0.1), and aortic regurgitation ( α = 0.001, c = 2). When compared to other studies, the left ventricular and aortic pressure, leaflets’ opening-closing angles are generated in similar patterns. In addition, the reduction of the blood flow rate can now be observed in aortic regurgitation, which is not presented in the literature. In conclusion, a set of critical aortic valve conditions is parameterized by our novel model. The elastic structure of the leaflet model enabling us to demonstrate not only the vortex and dicrotic notch phenomenon but also the backflow and reverse pressure effect. Furthermore, it offers a generalized framework to realize extensive aortic valve conditions.