Damped Mechanical Systems Research Articles

Automated construction of effective potential via algorithmic implicit bias

We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in Rd, where d⩾1. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent [44], which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs.

On the dynamics of a 2-DOF nonlinear vibratory system with bistable characteristic and circulatory forces

We consider an autonomous damped 2-DOF mechanical system in which the two DOFs are coupled by a linear spring. A circulatory force is introduced into the system that may result in self-excited vibrations. A nonlinearity is added in the form of a cubic spring so that there are three fixed points, two of them are stable without the circulatory force, i.e., a bistable behavior. The basins of attraction for different values of the circulatory forces are numerically studied to see how the patterns of them change. Using a Poincaré map, the basins of attraction have three dimensions and they are cut with further cross-sections for the ease of visualization. The results are usually complicated maps with non-smooth boundaries, except when the circulatory force uncouples one DOF from the other. Special patterns of the maps are seen when in the nonlinear case a stable limit cycle is about to occur. Even in the case without stable limit cycle, initial conditions within the special range may lead to numerous cycles of periodic-like transient motion before asymptotically converging to a stable fixed point. Phase planes and bifurcation diagrams are also used, and multiple coexisting periodic solutions are found. Some interesting phenomena are also found in the supercritical region, including a restabilized motion that coexists with the associated period-doubled motion, and strong symmetric and asymmetric autonomous bursting-like motions.

Robust integrated sequential covariance intersection fusion Kalman filters and their convergence and stability for networked sensor systems with five uncertainties

AbstractFor networked sensor systems (NSSs) with hard and soft sensors including five uncertainties, two universal approaches of solving the robust fusion estimation problems are presented. It includes an integrated sequential covariance intersection (SCI) fusion minimax robust Kalman filtering approach with cross‐covariance information and a generalized Lyapunov equation approach with four pairs of Lyapunov equations. Applying them, the robust local and SCI fused time‐varying and steady‐state Kalman filters are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds. The equivalent batch SCI fusers are also presented. Their robustness and accuracy relations are proved, and the sensitivity of the SCI fuser with respect to the fused orders of sensors is analyzed. Applying the dynamic error system analysis method and the dynamic variance error system analysis method, a new convergence and absolute asymptotic stability theory of robust fusion Kalman filtering is presented. The classical Kalman filtering convergence and stability theory is developed. Compared with the original covariance intersection fuser, they significantly reduced the computational complexity and burden. Compared with the optimal and conservative SCI fusers, they significantly improved the robust accuracies. They are suitable to deal with asynchronous or random delayed data and are suitable for real‐time applications. A simulation applied to the two‐mass spring damper mechanical system shows their effectiveness.

Coulomb friction effect on the forced vibration of damped mass–spring systems

This paper aims at assessing the effect of dry friction on the dynamic behaviour of a damped mechanical system subject to harmonic forcing. Previous work on friction damped systems highlighted that not including other forms of damping in the dynamic analysis can lead to unrealistic results such as the presence of infinite resonances. In this contribution, an exact solution is derived for the continuous steady-state response of multi-degree-of-freedom systems with a contact between one of the masses and an external wall, using Coulomb’s law to model the friction force and a modal damping model to account for the system’s damping. Closed-form expressions are also derived for the amplitude and phase of the continuous responses, while stick–slip responses are investigated by using an ad-hoc numerical approach. In addition, analytical and numerical results are used for exploring the features and the motion regimes of the dynamic response, leading to the following conclusions: (i) system’s damping has a limited effect on low- and high-frequency behaviours, on the presence of invariant points and inversions across the transmissibility curves and can therefore be neglected, in non-resonant conditions, in the analysis of structures where dry friction is the main source of dissipation; (ii) when the damping of the system is accounted for in the mechanical models along with Coulomb damping, finite resonant peaks are also obtained in continuous sliding regime and their amplitude decreases linearly with the friction force generated in the contact.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Synchronous vs asynchronous switching-based output-feedback control for switched nonlinear systems with measurement noise sensitivity

This paper investigates the problem of output-feedback control in the presence of measurement noise sensitivity for a class of switched nonlinear systems with unknown control coefficients under synchronous vs asynchronous switching. A constructive output-feedback control technique is provided by exploiting the dual-domination method and the average dwell time (ADT) method, which permits removal of a common restriction for output-feedback control in which the measurement sensitivity in switched systems is always a known constant. Also, output-feedback controllers of subsystems with a switched linear observer are designed to guarantee asymptotic stability of the closed-loop system under a class of switching signals with ADT, which effectively handle the influence of different sensor sensitivities for different subsystems in both synchronous vs asynchronous switching. Finally, simulation examples, including a mass–spring–damper mechanical system as a practical example, are given to illustrate the effectiveness of the technique proposed.

Fractional differential equation modeling of viscoelastic fluid in mass-spring-magnetorheological damper mechanical system

The mass-spring-damper system has the minimum complexity scenario which characterizes almost all the mechanical vibration phenomena. Also it is well known that a second-order differential equation can model its dynamics. However, if the damper has a magnetorheological fluid, then it shows viscoelastic properties in the presence of a magnetic field. Hence the mathematical model that best reflects the dynamics of this system is a fractional order differential equation. Naturally, here the Mittag–Leffler function appears in the analytical solution. Mathematical modeling of the mass-spring-magnetorheological damper mechanical system has been presented here. The main focus of the investigation is to show how the fractional order $$\gamma $$ changes by varying the viscosity damping coefficient $$\beta $$ . These observations have been made by varying current intensity in the range of 0.2–2 A. A Helmholtz coil has been used to produce the magnetic field. It is worth mentioning that, this work has a high pedagogical value in the connection of fractional calculus to mechanical vibrations as well as it can be used as a starting point for a more advanced treatment of fractional mechanical oscillations.

Conformity and stability analysis of a modified spring–mass–damper system dynamics-based car-following model

Modeling the dynamics of a traffic system involves using the principles of both physical and social sciences since it is composed of vehicles as well as drivers. A novel car-following model is proposed in this paper by incorporating the socio-psychological aspects of drivers into the dynamics of a purely physics-based spring–mass–damper mechanical system to represent the driver–vehicle longitudinal movements in a traffic stream. The crux of this model is that a traffic system can be viewed as various masses interacting with each other by means of springs and dampers attached between them. While the spring and damping constants represent the driver behavioral parameters, the mass component represents the vehicle characteristics. The proposed model when tested for its ability to capture the traffic system dynamics both at micro, driver, and macro, stream, levels behaved pragmatically. The stability analysis carried out using perturbation method also revealed that the proposed model is both locally and asymptotically stable.

Термомеханічна задача про коливання в'язкопружного гумоподібного стержня при динамічному навантаженні

The problem on vibration of a viscoelastic rod under dynamic load at one of its ends is considered. The external loading has a signfficant influence on the dynamic characteristics of the material. By using the complex moduli, the problem on vibration of the viscoelastic rod was solved. The complex shear and Young's moduli of a rubberlike material should exhibit the same dependence on frequency. The properties of a rubberlike material was applied. The temperature influence is associated both with the Newton boundary conditions and dissipative heating. The dissipative function is expressed in terms of deformations. The frequencies of high-damping materials occur at or near frequencies that are normally of interest in vibration problems at room temperature. For solving the problem a finite element model was applied. Using this model, qualitative analysis of the influence of dynamic load and dissipative heating on the resonant vibrations of viscoelastic rod is performed. According to the theory of viscoelasticity an analysis of the results was done. The reliability of the values of frequencies for the first resonances was checked. The numerical results qf the problem on vibration of a viscoelastic cylindrical rod under dynamic load at one of its end by the general thermomechanical laws on vibration in damped mechanical systems were obtained and investigated. Distribution of the temperature of dissipative heating along the rod axis is analyzed.

Interval type-2 fuzzy tracking control for nonlinear systems via sampled-data controller

This paper addresses an interval type-2 (IT2) fuzzy sampled-data tracking control problem of nonlinear systems subject to uncertain parameters. The controlled plant is modeled as Takagi–Sugeno (T–S) fuzzy structure, where the parameter uncertainties are captured effectively by the lower and upper membership functions. By means of the linear matrix inequalities (LMIs) and free-weighting matrices, an IT2 fuzzy sampled-data control scheme is developed such that the system states are driven to track those of the reference model. Finally, an inverted pendulum system and a mass–spring–damper mechanical system are used to show the efficiency of the suggested tracking control design.

Novel Dynamic Model Updating Technique for Damped Mechanical System

Abstract Response surface method and Derringer’s function approach have been combined together to develop novel structural dynamic model updating technique for a damped mechanical system. Response surface models have been incorporated instead of finite element models, in order to increase computational efficiency of proposed technique. Derringer’s function approach is useful in dealing successfully with multi-objective optimization type model updating problems. Few such undamped techniques have been recently developed for undamped mechanical systems by combining the benefits of response models with Derringer’s function approach. This paper presents the theory and numerical application of a damped updating technique, which is based upon response models and Derringer’s function approach. In this technique, updating process is formulated as an optimization problem, wherein desirability functions are formulated, based on natural frequencies, modal-assurance-criterion values, resonance and anti-resonance points of frequency response functions. Desirability functions are then optimized to evaluate updated elastic parameters in stage one and updated damping constants in stage two of proposed technique. By using this technique, total absolute errors in natural frequencies, modal-assurance-criterion values, elastic parameters and frequency response functions have been reduced to 0.02%, 0.00%, 3.69% and 0.11%, respectively.

Parametric study of the mode coupling instability for a simple system with planar or rectilinear friction

In this paper, the study of a damped mass–spring system of three degrees of freedom with friction is proposed in order to highlight the differences in mode coupling instabilities between planar and rectilinear friction assumptions. Well-known results on the effect of structural damping in the field of friction-induced vibration are extended to the specific case of a damped mechanical system with planar friction. It is emphasised that the lowering and smoothing effects are not so intuitive in this latter case. The stability analysis is performed by calculating the complex eigenvalues of the linearised system and by using the Routh–Hurwitz criterion. Parametric studies are carried out in order to evaluate the effects of various system parameters on stability. Special attention is paid to the understanding of the role of damping and the associated destabilisation paradox in mode-coupling instabilities with planar and rectilinear friction assumptions.

Quantization of a uniform gravitational field with a damping quadratic force

A quadratic damped mechanical system is quantized using the WKB formalism. In particular, a uniform gravitational field with a damping force proportional to the square of its speed has been treated in this paper. We construct the Hamiltonian of the system and use it to obtain the Hamilton-Jacobi equation. The action function S has been found using the separation of variables and the chain rule. This function enables us to formulate the wave function and to quantize the system using the WKB approximation. Finally, we are able to prove that the quantum results are in exact agreement with the classical results.

Amplitude Controlled Adaptive Feedback Resonance in a Single Degree-of-Freedom Mass-Spring Mechanical System

Many mechanical and micro-mechanical applications utilize artificially generated self-excited oscillations either as the working principle or as the performance enhancer. The present paper investigates a switched damping adaptive feedback control method for inducing artificial controlled self-excited oscillation in a single degree-of-freedom damped mass-spring mechanical system. The adaptive feedback control can generate oscillation at the natural frequency of the system with controlled amplitude and maintain the frequency of oscillation at the natural frequency of the system under parametric variations. The desired amplitude of oscillation even under parametric variations can be achieved when the controller is designed utilizing a frequency detector (used to adapt the natural frequency of the system). Numerical simulations confirm the analytical results.

Вынужденные крутильные колебания при наличии сил сопротивления

The work proposed differential equations describing the torsional oscillations of one- and two-mass mechanical systems taking into account the dissipative losses of various kinds and nature. The dependences for determining the equivalent rigidity of the elastic ties. Using the results of these studies can be realized rational selection of inertial and elastic properties of materials and components damper mechanical system

Parametric model order reduction of damped mechanical systems via the block Arnoldi process

This paper proposes a block Arnoldi method for parameterized model order reduction. This method works when design parameters have only low-rank impacts on the system matrix. The method preserves all design parameters in the reduced model and is easy to implement. Numerical results show that the block Arnoldi process outperforms some existing methods up to a factor of ten.

On the oscillation death phenomenon in a double pendulum system with autoparametric interaction

This study is concerned with autoparametric interaction in a four degree of freedom damped mechanical system consisting of two identical pendula fitted onto a horizontal massive rod which can oscillate vertically and rotationally. One pendulum is harmonically excited. The equations of motion indicate that autoparametric interaction is possible by means of typical external and internal resonance conditions involving the system natural frequencies and excitation frequency. An intriguing phenomenon is demonstrated when the unforced pendulum is decoupled and no energy goes into it, as a result of which it stops oscillating. Numerical simulations are carried out to determine when and why this phenomenon occurs for a different excitation magnitude as well as for zero and non-zero initial conditions of the unforced pendulum.

On a generalized port-Hamiltonian representation for the control of damped underactuated mechanical systems

A well-known problem in controller design for underactuated mechanical systems using the Interconnection and Damping Assignment (IDA-PBC) technique is friction in unactuated degrees of freedom. For certain equilibria the definiteness requirements on the virtual energy of the port-Hamiltonian (pH) target system and the closed-loop dissipation matrix can not be satisfied simultaneously. In this contribution a modification of the pH target system is proposed, where particularly the total energy function is augmented by a cross term between coordinates and momenta. The approach stems from the fact that, although IDA-PBC may fail, unstable equilibria of underactuated mechanical systems are stabilized by linear state feedback, if the linearization is stabilizable. Then the solution of a Lyapunov equation for the linearized closed-loop system is not block diagonal, which gives rise to the proposed structure of the energy.

Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems

Floquet–Bloch theorem is widely applied for computing the dispersion properties of periodic structures, and for estimating their wave modes and group velocities. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and well-posed spectral problem representing wave dispersion in undamped media. Most studies employ the Floquet–Bloch approach for the analysis of undamped systems, or for systems with simple damping models such as viscous or proportional damping. In this paper, an alternative formulation is proposed whereby wave heading and frequency are used to scan the k-space and estimate the dispersion properties. The considered approach lends itself to the analysis of periodic structures with complex damping configurations, resulting for example from active control schemes, the presence of damping materials, or the use of shunted piezoelectric patches. Examples on waveguides with various levels of damping illustrate the performance and the characteristics of the proposed approach, and provide insights into the properties of the obtained eigensolutions.

The optimal form of the fractional-order difference feedbacks in enhancing the stability of a sdof vibration system

Many practical systems including the slightly damped mechanical systems, even they are already stable, are required to be controlled, in order to get better performance or better stability. In this paper, the concept of fractional-order difference feedback that generalizes the displacement difference feedback, velocity difference feedback and acceleration difference feedback, is proposed for improving the stability of a sdof vibration system. It is found that among the various state difference feedbacks, some fractional-order difference feedbacks including fractional-order integrators and fractional-order differentiators improve the stability of vibration systems best. Fractional-order integrator/differentiator is a controller with memory for the whole time history, its implementation is usually more complicated than the classical PID control and acceleration control. Thus, proper classical controller is suggested for improving the stability of the vibration system with small damping and small delay. If a displacement sensor is used, then the optimal form of state difference feedbacks for enhancing stability is the displacement difference feedback with k > 0 . If an acceleration sensor is used, then the optimal form of state difference feedbacks for enhancing stability is the acceleration difference feedback with k < 0 . Moreover, on the basis of the principal of stability switch, the admissible feedback gains and delay governing the asymptotical stability and γ -stability are studied in detail, and illustrated with numerical experiments.