N'Gbo N'Gbo, Changpin Li, and Min Cai
J. Comput. Nonlinear Dynam. Mar 2025, 20(3): 031002.
https://doi.org/10.1115/1.4067471

​This article focuses on investigating fractional Lyapunov exponents for generalized ψψ-fractional differential systems. By employing a new and more suitable definition, we derive an expression for the fractional Lyapunov exponents using the inverse of the Mittag-Leffler function, which depends on the kernel, weight, and order of the considered fractional derivative. We also provide an upper bound for the obtained fractional Lyapunov exponents that is tighter than the one available in existing literature. Finally, experiments conducted on a hyperchaotic 5D system and the well-known Lorenz system serve to illustrate and verify our main results.

Yoshitaka Shizuno, Shuonan Dong, Ryo Kuzuno, Taiki Okada, Shugo Kawashima, Kanjuro Makihara, and Keisuke Otsuka
J. Comput. Nonlinear Dynam. March 2025, 20(2): 021002.
https://doi.org/10.1115/1.4067201

Owing to their low induced drag, high-aspect-ratio wings are often applied to aircraft, particularly high-altitude long-endurance (HALE) aircraft. An analytical method that considers geometrical nonlinearity is necessary for the analysis of high-aspect-ratio wings as they tend to undergo large deformations. Nonlinear shell/plate or solid finite element methods are widely used for the static analysis of wing strength. However, an increase in the number of elements drastically increases the computational costs owing to the complexity of wing shapes. The modal rotation method (MRM) can avoid this additional expense by analyzing large deformations based on modes and stiffness matrices obtained from any linear or linearized model. However, MRM has only been formulated as a static analysis method. In this study, a novel modal-based dynamic analysis framework, referred to as dynamic MRM (DMRM), is developed to analyze slender cantilever structures. This paper proposes a method to discretize dynamics by capitalizing on the fact that MRM considers geometrical nonlinearity based on deformed shapes. The proposed method targets slender structures with small strains and large displacements and considers geometrical nonlinearity, but not material nonlinearity. Additionally, a formulation method for the work performed by a follower force is proposed. The energy stored in the structure agreed with the work performed by an external force in each performed simulation. DMRM achieved a 95% reduction in the calculation time compared with a nonlinear plate finite element method in a performed simulation.

Jamal-Odysseas Maaita and Dimitrios Prousalis
J. Comput. Nonlinear Dynam. January 2025, 20(1): 011008.
https://doi.org/10.1115/1.4067010

A nonregular oscillation is not enough to define a system as chaotic. A more in-depth investigation is required to prove the existence of chaotic behavior, which is challenging. Although many scientists use the Lyapunov Characteristic Exponents to detect chaos, it is not the only method. Several scientists have introduced different methods that utilize various properties of dynamical systems. Hidden Attractors may be chaotic or regular. The fact that they have small basins of attraction introduces difficulties in locating and characterizing them. The paper presents four different chaotic indicators based on the evolution of the deviation vectors: the maximal Lyapunov Exponent, the Lyapunov Characteristic Exponents, the Fast Lyapunov Index (FLI), and the Small Alignment Index. It includes their properties and the advantages and disadvantages of each method. Also, it includes the algorithms to calculate them and their implementation in Python. The paper closes with a comparison between the four indices applied to a system with hidden attractors.

Praveen Kumar
J. Comput. Nonlinear Dynam. December 2024, 19(12): 121010.
https://doi.org/10.1115/1.4066728

Nonlinear intermodal coupling based on internal resonances in MEMS resonators has advanced significantly over the past two decades for various real-world applications. In this study, we demonstrate the existence of various three-mode combination internal resonances between the first five flexural modes of electrostatically actuated flexible–flexible beams and dynamic modal interaction between three modes via internal resonance. We first calculate the natural frequencies of the beam as a function of the stiffnesses of the transverse and rotational springs of the flexible supports, utilizing both analytical formulation and finite element analysis (FEA). Following this, we identify six combination internal resonances among the first five modes and use applied DC voltage to validate the exactness of one commensurable internal resonance condition (⁠ω2=ω5−ω4⁠). Subsequently, we studied a detailed forced vibration analysis corresponding to this resonance condition by solving the five-mode coupled governing equations through numerical time integration and the method of multiple scales. The results compellingly exhibit three-mode intermodal coupling among the second, fourth, and fifth modes as a function of excitation amplitude and frequency. Alongside this, intriguing nonlinear phenomena such as threshold behavior, saturation phenomena, and autoparametric instability are observed. Finally, this paper provides a systematic methodology for investigating three-mode combination internal resonances and related nonlinear dynamics, offering significant insights that could be used in observing phonon or mechanical lasing phenomena in MEMS resonators.

Mohammed F. Daqaq
J. Comput. Nonlinear Dynam. January 2025, 20(1): 014501.
https://doi.org/10.1115/1.4066659

​The symmetric monostable Duffing oscillator exhibits a superharmonic resonance of order three when excited harmonically at an excitation frequency that is one third its linear natural frequency. In this letter, it is shown that a certain class of periodic excitations can inherently quench the superharmonic resonance of order three. The Fourier series expansion of such excitations yields a harmonic component at the natural frequency whose magnitude can be properly tuned to completely quench the effect of the superharmonic component. Based on this understanding, the parameters of a piecewise periodic function and the modulus of the cosine Jacobi elliptic function are intentionally designed to passively suppress the superharmonic resonance. Such periodic functions can be used to replace single-frequency harmonic excitations whenever the effects of the superharmonic resonance are to be passively mitigated.

Jean M. Souza, Luciana Loureiro S. Monteiro, Marcelo A. Savi
J. Comput. Nonlinear Dynam. December 2024, 19(12): 121005.
https://doi.org/10.1115/1.4066469

​Magnetic shape memory alloys (MSMAs) constitute a class of smart materials capable of exhibiting large magnetic field induced strain (MFIS) when subjected to magnetomechanical loadings. Two distinct mechanisms are responsible for the induced strain: martensitic variant reorientation and phase transformation. The martensitic reorientation is the most explored mechanism presenting the advantage to potential provide high-frequency actuation since it does not rely on phase transformation cycles. Despite its capabilities and potential dynamical applications, the dynamical behavior of MSMAs is not extensively explored in the literature that is usually focused on quasi-static behavior. Thereby, the objective of this work is to analyze the nonlinear dynamics of MSMAs. In this regard, an MSMA nonlinear oscillator is investigated, exploiting the system response under different bias magnetic field levels and actuation frequencies. A phenomenological model is employed to describe the MSMA magnetomechanical behavior. Numerical simulations are carried out using the operator split technique together with an iterative process and the fourth-order Runge–Kutta method. Results show that the application of a bias magnetic field can reduce the mean displacement of the system, increasing the oscillation amplitude. Furthermore, the period of oscillation can be modified, even achieving complex behaviors, including chaos. The potential use of MSMAs to dynamical systems is explored showing the possibility to provide adaptive behaviors.

R. Ju, S. M. Yang, H. Ren, W. Fan, R. C. Ni, and P. Gu
J. Comput. Nonlinear Dynam. Dec 2024, 19(12): 121001
https://doi.org/10.1115/1.4066221

​Steady-state rotary periodic responses of mechanisms lead to stress cycling in flexible structures or connecting joints, which in turn can result in structural fatigue. A general approach is developed to study rotary periodic solutions of rigid and flexible mechanisms with large spatial rotations based on the incremental harmonic balance (IHB) method. The challenge in analyzing such dynamic systems emanates from the noncommutativity of the spatial rotation and the nonsuperposition nature of the rotational coordinates. The generally used rotational coordinates, such as Euler angles, cannot be expanded into Fourier series, which prevents direct usage of the IHB method. To overcome the problem, the natural coordinates method and absolute nodal coordinate formulation (ANCF) are used herein for the dynamic modeling of the rigid and flexible bodies, respectively. The absolute positions and gradients are used as generalized coordinates, and rotational coordinates are naturally avoided. Equations of motions of the system are differential-algebraic equations (DAEs), and they are solved by the IHB method to obtain the steady-state rotary periodic solutions. The effectiveness of the proposed approach is verified by the simulation of rigid and flexible examples with spatial rotations. The approach is general and robust, and it has the potential to be further extended for other extensive multibody dynamic systems.

Xu Dai, József Kövecses, and Marek Teichmann
J. Comput. Nonlinear Dynam. Nov 2024, 19(11): 111005 
https://doi.org/10.1115/1.4066329

​Contact simulation is essential in modeling mechanical systems. The contact models require accurate geometric information, which is determined through collision detection methods. When the mechanical system includes flexible bodies such as structural components, the dynamic formulation and collision detection can be more challenging, as the geometric boundaries of such components keep changing during the simulation. The floating frame of reference (FFR) formulation is suitable for flexible systems with small deformation. In this work, a stable and efficient dynamic simulation method is introduced for flexible systems with contact based on the FFR formulation. In addition, a curve-based collision detection method is proposed, which is more consistent with the dynamic formulation and more efficient than common existing collision detection methods. Case studies of flexible beams and multibody systems are employed to demonstrate the performance of the proposed dynamic simulation and collision detection methods.

Paolo Magri, Marco Gadola, Daniel Chindamo, and Giulia Sandrini
ASME. J. Comput. Nonlinear Dyn. Oct 2024, 19(10): 101006
https://doi.org/10.1115/1.4066092​

Suspension and steering design play a major role in ensuring the correct dynamic behaviour of road vehicles. Passenger cars are especially demanding from this point of view: NVH and ride comfort requirements often collide with active safety-related requirements such as road holding in steady-state conditions and stability in transients. Driving pleasure is also important for market success, therefore accurate steering feedback and predictable handling properties are additional priorities.
Since flexible bushings are used as interface between the suspension arms and the chassis, extra degrees of freedom make the design process a complex task. While the use of a multibody software is common practice in the industry, a dedicated computational tool can be more practical and straightforward, especially when undertaking the design of a new suspension concept ground-up.
The paper presents a computational methodology for the design of an independent suspension with the associated kinematic and compliance attributes. Typical elastokinematic properties like toe, camber, wheelbase, and track variations vs tyre forces and moments can be computed by means of a dedicated software tool. A sort of validation was performed either by means of a comparison with a MathWorks Simscape® Multibody based model. Finally, a sensitivity analysis is given as an example.
Computationally, the method proposed is intuitively based on the equilibrium equations. The nonlinear equations are then solved with Newton-Raphson algorithm. The method can be also optimized for computational efficiency and is thoroughly described so that the reader can easily replicate it in the desired programming environment.

Sankalp Tiwari, Junaidvali Shaik, and C. P. Vyasarayani
J. Comput. Nonlinear Dynam. Feb 2024, 19(2): 021002 
​https://doi.org/10.1115/1.4064251

​Delayed systems are those in which the present dynamics is governed by what happened in the past. They are encountered in manufacturing, biology, population dynamics, control systems, etc. Determining stability of such systems is an important and difficult problem. In the existing works, stability is determined by assuming the governing differential equation. However, the equation may not be known or difficult to obtain. Unlike existing works, our method determines the linear stability of a delayed system using its response to a few known inputs. In particular, our method does not require or assume the differential equation governing that system. The only system information we use is its largest delay time, and the only assumption we make about the underlying equation is that its coefficients are either constant or time-periodic. Our approach involves giving the first few functions of an orthonormal polynomial basis as input and measuring/computing the corresponding responses to generate a state transition matrix, whose largest eigenvalue determines the stability. We demonstrate our method's correctness, efficacy, and convergence by studying four candidate DDEs with differing features. Importantly, we show that our approach is robust to noise in measurement, thereby establishing its suitability for practical applications.

Tomoyuki Suzuki, Kenji Hirohata, Yasutaka Ito, Takehiro Hato, Akira Kano
J. Comput. Nonlinear Dynam. November 2023, 18(11): 111001. 
https://doi.org/10.1115/1.4063224

Eduardo Okabe, Victor Paiva, Luis Silva-Teixeira, Jaime Izuka
J. Comput. Nonlinear Dynam. Oct 2023, 18(10): 104501 
https://doi.org/10.1115/1.4063222

Junaidvali Shaik, Sankalp Tiwari, and C. P. Vyasarayani
J. Comput. Nonlinear Dynam. Sep 2023, 18(9): 091005, https://doi.org/10.1115/1.4062633​

Delayed systems are those in which the present dynamics is governed by what happened in the past. They are encountered in manufacturing, biology, population dynamics, control systems, etc. Determining linear stability of such systems is an important and difficult problem. In this work, we propose a new method to determine stability of time-periodic delay differential equations (DDEs). In the usual approaches, the DDE is converted into an approximate system of time-periodic ordinary differential equations (ODEs). Later, standard ODE techniques are employed. In this paper, we develop a method that is more direct and general. Our approach is analogous to the well-known Floquet theory for ODEs: we consider one polynomial basis function at a time as the input function and stack the coefficients of the corresponding DDE solutions to construct a matrix whose largest magnitude eigenvalue determines linear stability. We demonstrate the correctness, efficacy and convergence of our method by studying several candidate DDEs with time-periodic parameters and/or delays, and comparing the results with those obtained from other standard methods. Our approach has the following additional advantages: (a) it is parallelizable, (b) it converges quickly, and (c) it requires knowledge of only elementary linear algebra and numerical computation of DDE solutions. ​

Josh H. P. Henry and Jacqueline Bridge
ASME. J. Comput. Nonlinear Dynam. August 2023; 18(8): 081012.
https://doi.org/10.1115/1.4062660

​A major issue with advancing prosthetics and robotics technology is mimicking natural motion. Super-Coiled Polymer (SCP) actuators are an economical and high-yield alternative to traditional actuators. However, the challenge lies in identifying suitable applications due to their hysteresis. By comparing SCP actuators with varying thread sizes, this paper suggests potential directions for their utilization. The actuators were created using coiling and heat training, controlled by Arduino. MATLAB was used to analyze the recorded data, focusing on the motion to assess the hysteresis characteristics. The observed characteristics inform recommendations for diverse applications in sensitive and coarse control systems.