Nonlinear Mechanical Systems Research Articles

Data-free non-intrusive model reduction for nonlinear finite element models via spectral submanifolds

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.

Latent-Energy-Based NNs: An interpretable Neural Network architecture for model-order reduction of nonlinear statics in solid mechanics

Nonlinear mechanical systems can exhibit non-uniqueness of the displacement field in response to a force field, which is related to the non-convexity of strain energy. This work proposes a Neural Network-based surrogate model capable of capturing this phenomenon while introducing an energy in a latent space of small dimension, that preserves the topology of the strain energy; this feature is a novelty with respect to the state of the art. It is exemplified on two mechanical systems of simple geometry, but challenging strong nonlinearities. The proposed architecture offers an additional advantage over existing ones: it can be used to infer both displacements from forces, or forces from displacements, without being trained in both ways.

An innovative finite-time leader-follower consensus control for a nonlinear mechanical disturbed system with actuation saturation

ABSTRACT This paper presents a novel leader-follower consensus control method for a special class of nonlinear multi-agent mechanical systems in the presence of control input saturation and external disturbances, and this class includes robot system dynamics with a wide range of potential applications in industry. In the innovative technique, one of the agents is selected as the leader and the other agents are guided in such a way that the entire states of the system can reach a consensus within the transient limits of the determined performance. Due to the presence of disturbance in most practical systems, the effect of limited disturbance has been evaluated in the consensus control method and actuator saturation is included in the design process of the control technique. Finally, a terminal sliding mode control method is adapted to ensure the stability of the overall system and fast finite-time leader-follower consensus control. The simulation results of the multi-agent nonlinear robot system in the MATLAB environment, in different scenarios by simultaneously considering the actuator saturation and external disturbance, show the effectiveness of the proposed control method.

A new numerical path to retrieve isolated branches on large scale nonlinear mechanical systems

A new numerical path to retrieve isolated branches on large scale nonlinear mechanical systems

Stability analysis for viscoelastic fluid with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.

A low-complexity solution for LMIs of hardly constrained nonlinear MIMO tracking control systems

This research is dedicated to the discrete-time tracking control of nonlinear multi-input multi-output (MIMO) Newtonian mechanics systems with an equal number of inputs and outputs, in the presence of input saturation and high-order dynamics of the actuation system. Firstly, the continuous-time model of the plant and actuation system has been transformed into the discrete-time domain using Adams–Bashforth and Adams–Moulton discretization techniques. In the next step, considering saturation boundaries, the modified position and velocity references are derived using the discrete-time reaching law-based control approach. These modified references are obtained through an optimization algorithm that calculates the closest achievable position and velocity trajectories with respect to the desired reference trajectory. Modifying desired trajectories in MIMO systems results in the formation of a system of linear matrix inequalities (LMIs). Solving such systems of inequalities requires the use of complex methods, and this difficulty increases with the increase in the system’s degrees of freedom (DOFs). The proposed strategy offers a more straightforward approach to solving these inequalities. Once the modified references are calculated, the corresponding position and velocity control signals can be obtained using the mentioned reaching law strategies. The final step of the proposed algorithm is to combine the resulting position and velocity control signals using a Gaussian-weighted averaging method to form the main control signal. A desirable performance for the control system can be achieved by appropriately adjusting the standard deviations of the Gaussian weighing functions. To evaluate the performance characteristics of the proposed method, it has been applied in the tracking control procedure of a nonlinear 3-DOF mechanical system. The numerical simulation results demonstrate the effectiveness of the method.

Optimal Control of a Roll-to-Roll Dry Transfer Process With Bounded Dynamics Convexification

Abstract For efficient roll-to-roll (R2R) production of flexible electronic components, a precise R2R transfer peeling process is essential, requiring accurate modeling and control. This paper introduces a novel approach to confining the dynamics of a nonlinear R2R mechanical peeling system within a convex set known as a norm-bounded linear differential inclusion (NLDI). This method utilizes constraints on uncertain system variables to create a tighter NLDI representation compared to other convexification techniques. Moreover, it offers drastically reduced computational cost compared to previous methods applied to convexify the R2R peeling system. The NLDI is employed to generate an H∞-optimal controller for the R2R peeling system, and both simulations and experiments demonstrate better dynamic performance compared to other controllers for R2R transfer.

Analytical Study of Random Oscillations of Vibration Protection Devices with an Elliptical Hysteresis Loop for Absorbing Harmful Energies by Non-Linear Mechanical Systems and their ElasticDamping Elements

The paper discusses the analysis of forced random oscillations of a non-linear mechanical system, namely a heavy drilling machine, taking into account the damping ability of the operator (human body) as a mechanical system and the elliptical hysteresis loop of the absorption of harmful energies by the elastic-damping element.

Velocity observer design for a class of uncertain nonlinear mechanical systems: a self-adaptive fuzzy logic-based approach

This study focused on designing a smooth velocity observer (VO) for mechanical systems whose mathematical model is uncertain. The uncertainties that appear in the observer dynamics are, via utilising their universal approximation property, modelled with fuzzy logics. A novel self-adaptive fuzzy logic (SAFL)-based term in which control representative value matrix (CRVM), centres and widths of membership function are all dynamically updated is used as part of the observer design. Through the application of Lyapunov-type stability analysis techniques, the practical stability of the observed velocity error was guaranteed. The outcomes derived from experimentation on a planar robotic manipulator are showcased to illustrate the performance of the devised VO design.

Application of Dissipative Contact Force Model for Contact–Impact Analysis in Deep Groove Ball Bearing

This work proposes a new dynamic model for evaluating the nonlinear characteristics of deep groove ball bearing considering the contact–impact characteristics. With clearance taken into account, the elements of the deep groove ball bearing are modeled as the contact bodies. The normal force in the contact bodies is established by the dissipative contact force model, which includes elastic deformation and energy loss. Based on the modified Coulomb friction model, the tangential force can be established. The dynamic analysis model of deep groove ball bearing with contact–impact characteristics is obtained. Moreover, the experimental platform is established and the dynamic performance experiment is conducted. Finally, the influence of structure features on the dynamic behavior of deep groove ball bearing is discussed in detail. The simulation results could represent that clearance plays an important element in evaluating accurately the dynamic behavior of deep groove ball bearing, which would cause strong nonlinear mechanical systems.

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Solving an electrically conducting nanofluid over an impermeable stretching cylinder problem with a spectral reproducing kernel method

In this paper, a nonlinear mechanical system of ordinary differential equations (ODEs) with multi-point boundary conditions is considered by a novel type of reproducing kernel Hilbert space method (RKHSM). To begin, we define the unknown variables in terms of the reproducing kernel function. The roots of the Shifted Chebyshev polynomials (SCPs) are then utilized to collocate the resulting system. Finally, Newton’s iterative method is employed to find the unknown expansion coefficients. The solutions of this system of equations, which arise from the flow of an electrically conducting nanofluid over an impermeable stretching cylinder, are numerically analyzed, and convergence analysis is discussed to demonstrate the reliability of the presented method (PM). Tables and figures are provided to further discuss the solutions and assess the effectiveness of the method in comparison to other techniques in the literature.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

Backbone curve tailoring via Lyapunov subcenter manifold optimization

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system’s parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient.

Overview of Design Considerations for Data-Driven Time-Stepping Schemes Applied to Nonlinear Mechanical Systems

Overview of Design Considerations for Data-Driven Time-Stepping Schemes Applied to Nonlinear Mechanical Systems

Building functional networks for complex response analysis in systems of coupled nonlinear oscillators

Some aspects of engineering dynamics, such as nonlinearities and transient motion of many interconnected parts, remain difficult to handle today. To comply with increasing demands on resilience and safety, the dynamics of large machines need to be better understood. Complex network methods, already present in many scientific disciplines, provide a tool set complementary to conventional methods of system analysis. This work aims at providing a new, function-based view on mechanical systems by generating functional networks. To this end, a network algorithm is applied to sets of cyclically coupled Duffing oscillators as a common example of a complex nonlinear mechanical system. In the functional network, each node represents an oscillator while the direction of the network edges represents a functional coupling. Results show that the network method is capable of identifying dynamical transitions and synchronization between components, as well as determining the number of different states present within a system. Additionally, the time evolution of the component interactions, especially in response to a disturbance, is studied via a sliding-window approach. The results of this analysis might hopefully open new ways for a more efficient system analysis through optional sensor placement, and for effective countermeasures against unwanted dynamics through improved analysis of transient dynamics.

Application of Numerical Bifurcation Tracking Strategy to Blade-Tip/Casing Interactions in Aircraft Engines

Abstract Building on the regularized-Lanczos harmonic balance method (RL-HBM), a previously developed frequency method, this paper presents a numerical bifurcation tracking strategy dedicated to high-dimensional nonlinear mechanical systems. In order to demonstrate its applicability to industrial applications, it is here used to obtain original results in the context of blade-tip/casing interactions in aircraft engines. The emphasis is put specifically on the tracking of predicted limit point (LP) bifurcations as key parameters—such as the amplitude of the aerodynamic forcing applied on the blade, the friction coefficient or the operating clearances—vary. Overall, presented results underline that the employed frequency method is well-suited to tackle the numerical challenges inherent to such computations on high-dimensional systems. For the mechanical system of interest, the industrial fan blade National Aeronautics and Space Administration (NASA) rotor 67, it is shown that the application of the presented strategy yields an efficient way to identify isolated branches of solutions, which may be of critical importance from a design standpoint.

Nonlinear vibrations of a ballistic vehicle with friction joints subjected to vibroacoustic excitation - experiments, modeling and simulations

This study presents the experiments, modeling and numerical simulations of an industrial aeronautical structure subjected to vibroacoustic excitation. In flight, especially during atmospheric re-entry, an aeronautical structure is exposed to major pressure fluctuations on its external surface. These fluctuations are described as a dynamic excitation which generates structural vibrations of the outer surface and of the components inside the aeronautical structure. Due to some of these internal components such as joints, the assembly may exhibit a nonlinear vibration response. The simulation of these nonlinear vibrations requires reliable modeling of the wall pressure fluctuations and a nonlinear vibration simulation method adapted to the nonlinear modeling of the structure. In this study, a modeling and simulation method is developed to compute the nonlinear vibration response of an industrial assembly to such a surface, random, correlated dynamic excitation. More specifically, numerical simulations are performed by proposing an extension of the well-known Harmonic Balance Method for nonlinear mechanical systems subjected to complex vibroacoustic excitation. The method is validated using a dedicated ground experiment. A metallic industrial assembly representing a ballistic vehicle and including friction joints is used. This structure is subjected to controlled vibroacoustic excitation: diffuse acoustic loading in a reverberant chamber. The structure exhibits a nonlinear vibration response due to friction. The finite element model is validated through an experimental modal analysis with an electrodynamic shaker. The vibroacoustic modeling of the excitation is then validated through a test-simulation comparison using diffuse acoustic field testing at low excitation level. Then the global nonlinear simulation process is validated using diffuse acoustic field testing at increasing excitation levels. Test and simulation results exhibit the same nonlinear behavior: increase of dissipation and softening effect at the main resonances. This work thus represents a new step towards the use of nonlinear vibration simulation methods with real industrial structures and real-life loading.

A novel finite-time leader-follower consensus control for a disturbed mechanical nonlinear system in presence of actuator saturation

This paper presents a novel leader-follower consensus control for a particular class of nonlinear multi-agent mechanical systems in the presence of control input constraints and external disturbances which includes robot system dynamics with a wide range of potential applications in industry. In this case, one of the agents is selected as the leader to direct the other agents in such a way that the whole system can reach consensus within certain prescribed performance transient bounds. Due to the presence of disturbances in most practical systems, the effect of limited disturbance in the consensus control method has been investigated, and actuator saturation is included in the design process. A terminal sliding mode control method has been adapted to ensure the stability of the overall system and fast finite-time leader-follower consensus control. The simulation results of the multi-agent nonlinear robot system in MATLAB environment, in different scenarios with simultaneous consideration of actuator saturation and external disturbance, will show the efficiency of the proposed control method.

Identification of linear flat outputs using neural networks—Examples of two-degree-of-freedom underactuated mechanical systems

This paper proposes a neural networks-based approach of finding flat output of linearized underactuated mechanical systems (UMS). Given that differential flatness and controllability are equivalent for linear systems, the problem is equivalent to finding the Brunovsky canonical form of linearized UMSs. We use a two degree-of-freedom (2DOF) system to illustrate the theoretical development. The proposed method identifies the local flat output of nonlinear mechanical systems from the measurements only, without a detailed mathematical model. The identification allows us to combine the well-known active disturbance rejection control (ADRC) and differential flatness control. A time-domain direct identification (TDDI) algorithm and its variant based on the algebraic method are proposed for flat output identification (FOID). The neural networks for implementing the TDDI algorithm called FOID-NN are created to evaluate flat output candidates and to use the flat output to reconstruct the system states in terms of a linear mapping. The neural networks are trained with the loss functions defined by reconstruction errors. Two special layers, namely tracking differentiator (TD) and algebraic layer, are inserted in the FOID-NN to handle noisy signal differentiations. Simulations of a cart–pole system and a stable 2DOF nonlinear UMS are carried out to show the range of applications of the FOID-NN. The experimental data of an underactuated rotary crane are used to validate the identified flat output.