This paper presents a nonlinear dynamic model for high-speed motorized spindles that accounts for multi-layer interface effects (interference fit, clearance, and gap configurations) coupled with thermomechanical interactions. A nonlinear stiffness function is formulated to derive the governing equations of motion. The influences of assembly parameters on critical speeds, stability boundaries, and bifurcation characteristics are systematically investigated using the Lyapunov exponent criterion and numerical simulations. The results demonstrate that the system exhibits pronounced Duffing-type softening nonlinearity, with the primary resonance occurring at 6200 rpm—representing a 36.5% reduction compared to the linear prediction of 9765 rpm. This resonance frequency reduction is primarily attributed to bearing load-zone narrowing and thermo-mechanical coupling effects combined with geometric nonlinearities. The nonlinear effects reach their maximum within an intermediate tolerance range. At high speeds (>15,000 rpm), the system exhibits quasi-linear behavior due to centrifugal forces and thermal expansion that enlarge the effective clearance. Based on parametric stability analysis, optimal assembly parameter ranges are identified: bearing radial clearance ( C 3 ) of 10–12 μm, outer ring-housing clearance ( C 1 ) of 12–18 μm, and inner ring-shaft interference ( C 2 ) of 14–18 μm. These findings provide a theoretical basis for optimizing assembly parameters and enhancing the operational stability of high-speed motorized spindles.

The anterolateral ligament (ALL) has gained interest due to ongoing debates on its clinical significance. This retrospective study aimed to determine the ALL-injury reporting rate on primary magnetic resonance imaging (MRI) assessment compared to a secondary review by a fellowship-trained musculoskeletal radiologist. We hypothesized that ALL injuries were underreported on routine MRI assessments. Inclusion criteria consisted of patients ≤25 years with sport-related anterior cruciate ligament (ACL) tears and concomitant injuries in the lateral compartment treated with primary ACL reconstructions from 2015 to 2019. ACL revisions and non-sport-related injuries were excluded. Secondary ALL visualization was defined as full, partial, or not visualized and characterized as Grade I (mild/intermediate sprain), II (severe sprain/partial tear), or III (complete tear/avulsion). Sample size was determined using chi-squared test of independence (P < .05). A total of 370 patients met the inclusion criteria and 200 were sequentially enrolled (average age: 21.1 ± 6.0 years, 34% women, 66% men). Injuries occurred during recreational (n = 86), high school (n = 83), college (n = 29), and professional sports (n = 2). The primary ALL-visualization rate was 0% (0/200) compared to 99.5% (199/200) during secondary review (133 fully, 66 partially). Of visualized ALLs, 67.5% (n = 135) demonstrated a concomitant injury that was missed on the initial MRI assessment: 41.5% Grade I, 44.4% Grade II, and 14.1% Grade III. Concomitant ALL injuries are frequently missed on routine MRI in young patients with primary sports-related ACL tears. Education and heightened awareness of the ALL may improve diagnostic accuracy on routine MRIs and provide guidance for surgical management.

Primary resonance suppression in spur gear systems using hybrid proportional and fractional-order derivative displacement feedback control

This paper evaluates the perspective of a spring-coupled double pendulum with strict vibration control applied using a positive position negative derivative feedback (PPNDF) controller. The approach used in this case involved applying one of the most well-known perturbation techniques to discover the approximate mathematical solution of the equations of motion of the spring-coupled double pendulum with PPNDF damper. Our approach to obtaining the analytical results was to use the multiple-scale perturbation technique (MSPT) of the first-order approximation. Through the use of frequency response equations close to the simultaneous primary and internal resonance scenarios, the stability of the system has been studied and monitored. The MATLAB and Maple software were used to complete and clarify all the numerical discussions. It was very interesting and important to compare the effects of negative derivative feedback (NDF) only, positive position feedback (PPF) only, and PPNDF simply on the framework using the phase plane and time history. Moreover, a comparison was conducted between the numerical and analytical solutions. Ultimately, the study of the frequency response curves in three dimensions revealed the stable and unstable regions in each picture, illustrating the impact of altering the parameter values on the amplitude.

Floating structures are composite systems comprising multiple anchor cables and a floating body. The dynamic coupling among these components can induce complex nonlinear responses, which cause the system’s large amplitude vibration. Currently, the research focus is limited at the dynamic model with single cable, while neglecting the indirect coupling between multiple cables such as the modal resonance. Aiming to investigate the coupling mechanism between the components, this study established a dual-cable floating-body model with consideration of the 1:1:1 internal resonance among two anchor cables and the floating body, incorporating primary resonance induced by external excitation applied to either the floating body or one of the cables. The anchor cable and floating body are coupled through the mechanical and displacement boundary conditions at the junction. Galerkin modal truncation is utilized to obtain the reduced ordinary differential equations. Asymptotic perturbation solutions are obtained by the method of multiple scales and verified by Runge-Kutta numerical method. The results show that the vibrational energy between the cables is transmitted through the floating body; the frequency response curve of the cable exhibits multi-peak characteristics due to the modal resonance mechanism; the symmetry of the system and key structural parameters, such as the cable inclination angle and mass ratio, significantly influence the dynamic response of the cable, while having a comparatively minor effect on the floating body.

Abstract In this paper, the dynamical properties of a hybrid Rayleigh-van der Pol-Duffing system subjected to two external periodic excitations are investigated, focusing on primary resonance, primary-superharmonic combined resonance, and primary-subharmonic resonance, as well as the chaotic dynamic. The method of multiple scales is employed to derive the first-order approximate analytical solution of system. The stability conditions for steady-state periodic solutions are obtained, and the influence of system parameters on the amplitude and stability of steady-state periodic solutions is analyzed through the frequency–amplitude equations. Furthermore, the Melnikov method is applied to conduct a global analysis of the system and yield the threshold of chaos in different cases of harmonic excitation frequencies. Numerical simulations of the system are conducted to obtain frequency–amplitude curves, time histories, phase trajectories, and maximum Lyapunov exponent, thereby analyzing the influence of parameters on the system's dynamic characteristics and verifying the theoretical analysis results.

This paper investigates the nonlinear vibration characteristics of spiral bevel gear transmission systems. A dynamic model was established by incorporating key nonlinear factors, including bearing support forces, time-varying mesh stiffness, gear backlash, mesh damping, and static transmission error. The system’s vibration differential equations were solved using the Runge–Kutta method. The nonlinear dynamic behavior under varying parameters was analyzed through time-domain waveforms, frequency spectra, phase portraits, Poincaré sections, bifurcation diagrams, wavelet time-frequency plots, and Lyapunov exponent spectra. Employing the multiscale method, the primary resonance equation was derived to examine the influence of mesh damping, mesh stiffness, and external load fluctuations on the system’s primary resonance characteristics. The results indicate that spiral bevel gear systems exhibit significant nonlinearity. As the excitation frequency or static transmission error increases, the system transitions from periodic to chaotic motion. Conversely, increasing the driving torque can shift the system from chaotic to periodic motion. To ensure transmission stability and reliability, relevant parameters must be carefully optimized.

An elastically coupled beam system is a complex mechanical system composed of two or more beam-like structures interconnected by elastic elements or directly through a continuous medium. The current research related to coupled beam systems demonstrates that the use of reasonable nonlinear devices can effectively prevent vibration in elastically coupled beam systems. Considering that actual engineering structures are subjected to multiple excitations, the excitation frequencies of the external excitations often differ. The lack of relevant research has restricted the application of nonlinear vibration control theory in the vibration control of elastically coupled beam systems. This work establishes the model of a coupled horizontal beam system (CHBS) stimulated by complex excitations enabled with a boundary nonlinear oscillator (BNO). The model formulas of the CHBS stimulated by complex excitations enabled with a BNO are derived and solved. According to the analysis and discussion, the current method is good at predicting the lateral responses of the CHBS stimulated by complex excitations enabled with a BNO. Introducing the BNO effectively suppresses the peak lateral responses in the multi-order primary resonance regions of horizontal beam 1 and horizontal beam 2, where the nonlinear parameters of the BNO should be reasonably designed based on the actual situation. The BNO is not merely a vibration suppression device but also a vibration transmission device. Reasonable use of BNO can effectively suppress the vibration of the CHBS under different-frequency excitations.

In this study, a rolling mill’s nonlinear vibration under entry strip-thickness fluctuation with multi-parametric excitation and multi-frequency external excitation is investigated. A time-delay feedback proportional-derivative controller was applied to a dynamic rolling force model, and primary resonance with two different time-delays was examined. Based on the multiple scales perturbation method, the approximate analytical solution of the nonlinear vibration equations is obtained and stability under the primary resonance condition is analyzed. Finally, with results consistent with other research, effects of time-delay control parameters and other system parameters on nonlinear vibration are presented. The results indicate that although strip-thickness fluctuation amplitude, multi-parameter and multi-frequency external excitation can cause system instability, adjusting time-delay control parameters, especially the velocity time-delay parameter, can suppress the system’s nonlinear vibration characteristics. In addition, after time-delay regulation, nonlinear vibration phenomena like jumps and multi-valued from the original system parameters weaken or disappear. Moreover, both displacement and velocity time-delay suppress the system chaotic motion, converting it to periodic motion. The purpose is to provide a theoretical basis for vibration control of dynamic rolling force models.

Over the years, wind turbine blades have become increasingly larger and more flexible to achieve higher efficiency and lower energy costs, which brings more issues related to forced resonance and aeroelastic instabilities because of higher dynamic loads and complex inflow conditions. The manuscript systematically reviews the literature, covering experimental versus computational studies, reduced-order models (ROM), machine learning (MI) based aeroelastic models, Euler-Bernoulli beam models, Timoshenko beam models, geometrically exact beam formulations, and computational fluid dynamics (CFD) models coupled with structural dynamics models to investigate primary and internal resonances and dynamic stalls for National Renewable Energy Laboratory (NREL) 5-MW and International Energy Agency (IEA) 15-MW wind turbines. The article also addresses vibration mitigation techniques, including passive, active, and semi-active control, to resolve aeroelastic instabilities. Most studies have assumed linear aeroelastic models and isotropic blade material for initial structural dynamics analysis. The higher mode frequencies computed using the Euler-Bernoulli model differ by approximately 5.23%, those using the Timoshenko model by 3.13%, and those through the Rayleigh model by 3.4% from the geometrically exact formulations employed. Euler-Bernoulli models significantly overestimated flutter speeds compared to the geometrically exact beam model for the NREL 5-MW blades. A key takeaway is that modern, prolonged, flexible blades are sensitive to flutter instabilities, where aerodynamic damping can drop significantly at certain operational speeds. The Euler-Bernoulli beam model proved to be a valuable tool at the initial design stage due to its simplicity and computational efficiency. Future research on managing forced resonance and dynamic stalls in ultra-large blades should focus on integrating nonlinear modeling, cutting-edge materials and structures, artificial intelligence (AI)-powered digital twins, and exploring targeted active control techniques.

Primary Noncontrast Magnetic Resonance Imaging for Prostate Cancer Screening: A Randomized Clinical Trial (PROSA).

This paper investigates the nonlinear horizontal vibration of a cold rolling system induced by the gyroscope precession effect-a critical yet underexplored issue affecting strip quality and rolling stability. A nonlinear dynamic model is developed by incorporating the axial excitation force and the elastic deformation of the work roll based on d’Alembert principles. The primary parametric resonance response, corresponding to the first Arnold tongue, is analyzed using the multi-scale method and validated experimentally. To further understand the systematic dynamic behavior, the homotopy analysis method is employed to trace the evolution of energy orbits, revealing bifurcation and jump phenomena as the frequency ratio varies. A devil’s staircase pattern emerges, indicating multiple frequency-locked regions. These nonlinear features are further validated through cell mapping techniques, which depict the transformation of modal energy manifolds. Moreover, by introducing active control inputs, a constraint space for control parameters is designed to induce amplitude death within the maximum Arnold tongue region. The findings contribute to a deeper understanding of the resonance mechanism and offer a theoretical basis for stabilizing precision cold rolling systems via nonlinear control strategies.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-27552-2.

As a fundamental structural form, beam structures are widely used in fields such as architecture, navigation, and aerospace. When analyzing complex structures in engineering, they are typically divided into several substructures, and the coupling relationships between these substructures are established to simulate real-world conditions. With the development of research, scholars have gradually paid attention to nonlinear factors in engineering and introduced them into the elastically coupled structures. However, the existing literature mainly discussed the potential influence of coupling nonlinearities on the dynamic behavior of elastic coupled systems, where such coupled structures were typically subjected to one excitation. It is of great engineering significance to study the dynamic behavior of elastic structures with nonlinear coupling factors affected by multi-source excitations. This work presents a vibration analysis model of a coupled beam system with a nonlinear coupling term at the end, which is subjected to double excitations. The numerical lateral magnitudes of CB 1 and CB 2 can be precisely calculated using the Galerkin truncation method (GTM). Within a specific range, increasing k NW introduces additional resonance magnitudes into the numerical lateral dynamic response curve of CB 1 and CB 2. When k NW rises to a certain threshold, multi-solution dynamic behavior emerges in the system’s numerical lateral magnitude curve. Blindly increasing k NW leads to an abnormal elevation of numerical response magnitudes at certain primary resonance regions of CB 1 and CB 2, which is detrimental to the vibration control of CB 1 and CB 2. Besides, the numerical lateral magnitudes of CB 1 and CB 2 corresponding to different α values all fall within a fixed region. Due to differences in the phase angles of the external excitations, the numerical lateral magnitudes of each sub-beam in a coupled beam system fluctuate within the numerical lateral magnitude interval.

Abstract Abstract Aim This work analyzes a two-degree-of-freedom (DOF) auto-parametric system for concurrent vibration control and enhanced energy harvesting at resonance. The main objective is to achieve simultaneous vibration mitigation and energy extraction by integrating mechanical and electrical subsystems through electromechanical coupling. Methods The investigated system consists of a damped oscillator with an attached pendulum and two distinct energy harvesting (EH) devices: a piezoelectric harvester and an electromagnetic harvester. The primary structure, composed of a vertical damped oscillator coupled with a suspended pendulum, is equipped with a negative-velocity-feedback (NVF) controller to reduce unwanted vibrations that impair system performance, especially under resonance conditions. The governing equations of motion are derived using Lagrange’s equations (LE) and solved approximately by the perturbation method, namely the multiple scales method (MSM). Validation of the obtained approximate solutions (AS)is performed through comparison with the computed numerical solutions (NS) via the Runge–Kutta fourth order (RK-4) method. Results After classifying the resonance cases, the analysis focuses on the worst-case scenario, namely the primary external resonance. Under this condition, the influence of various system parameters is thoroughly examined. The system’s stable and unstable responses are analyzed using frequency responses (FR), time histories, Poincaré maps (PM), and bifurcation diagrams. The results show that the NVF controller effectively reduces the amplitude of unwanted vibrations while maintaining stable system behavior. Both the piezoelectric and electromagnetic harvesters successfully extract electrical energy from the system’s oscillations. The comparison between analytical and numerical results demonstrates excellent agreement, confirming the reliability of the derived solutions. Applications This dynamical model offers a dual advantage by harvesting energy from vibrating motion through both electromagnetic and piezoelectric transducers, which convert mechanical vibration into electrical power via magnetic induction and material strain, respectively, while actively reducing unwanted oscillations using an NVF controller. It is particularly useful in systems operating near resonance, such as in vehicles, buildings, and industrial machinery. The model enhances both energy efficiency and structural reliability, making it highly suitable for smart infrastructure and self-powered monitoring applications.

The Bell–Bloom magnetometer is promising for mobile applications, but its accuracy is limited by heading errors. A recently identified source of such error is a phase shift in the magnetic resonance, which arises from the superposition of two signals, i.e., the primary resonance from synchronous pumping and a secondary resonance, 90° out-of-phase, driven by the AC light shift of the pump laser. Through Bloch equation modeling and experiment, we uncover a counter-intuitive mechanism: although initiated by the AC light shift, the phase shift’s magnitude is determined solely by the pump light’s average power (DC component) and is independent of its AC modulation. This occurs because the amplitude ratio of the two resonances depends exclusively on the DC-power-induced atomic polarization. Based on this insight, we propose a novel DC compensation scheme by adding a continuous counter-polarized beam to cancel the net DC pumping. Theoretically, this simultaneously suppresses both the AC-light-shift-induced phase shift and the DC-light-shift-induced frequency shift. The scheme’s advantage is its simplified approach to polarization control, avoiding the need for high-speed polarization modulation or major hardware changes as the beams share the same optical path. This makes it highly suitable for demanding mobile applications like aerial magnetic surveying and wearable bio-magnetic sensing in unshielded environments.

Geometrically nonlinear higher-order shear deformable model of TiO2/GNP/polymer nanocomposite rectangular plates: A numerical study on mechanical properties and nonlinear primary resonance features

Pyrazinamide (PZA), an FDA-approved antibiotic, was investigated for potential use as a hyperpolarized MRI contrast agent. PZA was readily hyperpolarized via parahydrogen-based NMR Signal Amplification By Reversible Exchange (SABRE) at ∼5.5 mT, allowing 1H polarization and relaxation dynamics to be characterized. More importantly, performing SABRE hyperpolarization in the microtesla regime (i.e., SABRE in SHield Enables Alignment Transfer to Heteronuclei, SABRE-SHEATH) allowed direct detection of enhanced 15N NMR and corresponding polarization/relaxation dynamics to be characterized in this system for the first time, despite the low natural abundance of 15N. Initial experiments detected at 9.4 T following SABRE-SHEATH at 0.2 μT permitted observation of a single 15N resonance at 332.7 ppm (tentatively assigned to the ring N site meta to the amide group, based on the 1H enhancement pattern, suggesting the less sterically hindered N site dominates this substrate's interaction with the Ir-based SABRE catalyst). Direct SABRE-SHEATH 15N hyperpolarization using a more optimized setup (with respect to mixing field, temperature, p-H2 flow rate, etc.) that is directly coupled to a 1.4 T benchtop NMR system resulted in improved detection sensitivity, including 15N enhancements of >140,000-fold for the primary 15N resonance (corresponding to a polarization of ∼7% with substrate concentration of over 100 mM). This effort also yielded a 15N T1 measurement for PZA of over 2 min at 1.4 T. Finally, the potential utility of hyperpolarized PZA as a contrast agent was also demonstrated via quantitative 1H MRI studies performed using a low-field (64 mT) portable "point-of-care" clinical scanner.

This study investigates the vibrational stability and torsional vibration characteristics of an integrated electric drive system composed of a permanent magnet synchronous motor (PMSM) and a two-stage gear pair under parametric excitation. An electromechanically coupled nonlinear torsional dynamic model is established, incorporating electromagnetic effects and time-varying mesh stiffness. The method of multiple scales is employed to analyze the parametric excitation-induced vibrational stability of the system, and the Runge-Kutta method is used to solve the vibrational differential equations and examine the dynamic response characteristics. The results indicate that the system exhibits significant coupled vibrational behavior: the spectrum of the dynamic meshing force contains not only the meshing frequency of the current gear pair but also the system’s natural frequencies and meshing frequency components from other gear stages. Under conditions without external excitation, the system is found to exhibit not only primary resonance responses due to time-varying mesh stiffness excitation but also various nonlinear vibrational phenomena such as subharmonic resonance, superharmonic resonance, and combination resonance. The response is particularly pronounced near twice the first-order natural frequency.

We have explored the nonlinear dynamics of six common, driven, diatomic molecules using a shifted Tietz-Wei (sTW) model of their molecular potential functions. We focused on the variations in their resonances, bifurcations and multistability with changes in the spectroscopic and driving force parameters, namely the dissociation energy (V_{0}), the potential function optimization parameters (b_{h} and c_{h}), driving frequency (ω), and amplitude (F_{0}). We used the method of multiple timescales to obtain frequency response curves for the primary and secondary superharmonic resonances. The primary resonances were larger for I_{2} and Cl_{2} than for CO or O_{2}. Variations in F_{0},b_{h}, and c_{h} had profound impacts on the primary resonance features, with higher F_{0} and lower V_{0} enhancing the response amplitude. Evidence for hysteresis in the frequency-response-a signature of multistability-is demonstrated. Superharmonic resonances are marked by increased amplitudes and significant hysteresis, especially for I_{2} and Cl_{2}, driven by large F_{0} at low V_{0}. Bifurcation diagrams, maximal Lyapunov exponents, and Poincaré maps were used to unravel the transitions between periodic and chaotic states. Period-doubling bifurcations, sudden chaos, and an abundance of crisis events, viz boundary, interior, and attractor-merging crises, were identified as the routes to a range of different chaotic states. Symmetry-breaking, attractor-bubbling, and multistability were all found and are reported. Coexisting attractors and their basins of attraction showed striped, fractal, and Wada-like basin structures. The results highlight the complex dynamics stemming from the interaction between spectroscopic properties and external excitations of the sTW oscillator in diatomic molecules. They carry significant implications for experimental applications.

Primary and Internal Resonances of a Novel Cable-Stayed Plate System with Shear Effects