Excitation Amplitude Research Articles

Numerical analyses of acoustic vibrational resonance in a Helmholtz resonator

AbstractIn this study, the numerical analyses of a system, which describes the motion of air particles in the cavity of a Helmholtz resonator (HR), excited by a sound wave, was conducted. The low-frequency (LF) signal in the acoustic field is amplitude-modulated by an additive high-frequency (HF) perturbation, which can enhance the detection of the low-frequency, through Vibrational Resonance (VR) phenomena. The focus was on the combined effect, of amplitude and frequency of the acoustic excitation, on the motion of particles and induction of resonance. It was demonstrated that the system exhibits several nonlinear behaviours, VR ceasing to exist for a particular motion of the particles, which is dictated by the excitation frequency in relation to the resonator’s geometry. Furthermore, the regimes in which the performance of the system can be optimized, was identified, which facilitated the design of broadband acoustic resonators, suitable for most applications.

Highly Accurate and Robust Constraint-Based Orbital-Optimized Core Excitations.

We adapt our recently developed constraint-based orbital-optimized excited-state method (COOX) for the computation of core excitations. COOX is a constrained density functional theory (cDFT) approach based on excitation amplitudes from linear-response time-dependent DFT (LR-TDDFT), and has been shown to provide accurate excitation energies and excited-state properties for valence excitations within a spin-restricted formalism. To extend COOX to core-excited states, we introduce a spin-unrestricted variant which allows us to obtain orbital-optimized core excitations with a single constraint. Using a triplet purification scheme in combination with the constrained unrestricted Hartree-Fock formalism, scalar-relativistic zero-order regular approximation corrections, and a semiempirical treatment of spin-orbit coupling, COOX is shown to produce highly accurate results for K- and L-edge excitations of second- and third-period atoms with subelectronvolt errors despite being based on LR-TDDFT, for which core excitations pose a well-known challenge. L- and M-edge excitations of heavier atoms up to uranium are also computationally feasible and numerically stable, but may require more advanced treatment of relativistic effects. Furthermore, COOX is shown to perform on par with or better than the popular ΔSCF approach while exhibiting more robust convergence, highlighting it as a promising tool for inexpensive and accurate simulations of X-ray absorption spectra.

The effects of dynamic factors inside the bubble on sono-hydrogen yield: A numerical study

The formation of H2 by introducing ultrasonic waves to liquid has been widely recognized as a way to provide a clean, efficient, and reliable source of H2, known as Sono-Hydro-Gen. H2 comes from the chemical effects of ultrasonic waves (sonochemistry) caused by the growth and collapse of acoustic cavitation bubbles. In this work, the effects of dynamic parameters (i.e., bubble temperature, the amount of water vapor trapped inside the bubble, and collapse time) in the evolution of cavitation bubbles on H2 production are studied numerically. For an oxygen bubble, computational simulations are performed for the wide range of acoustic amplitudes (1.5–3 atm), ultrasonic frequencies (140–515 kHz), and ambient radii (0.25–20 μm), considering 22 reversible chemical reactions and 10 chemical species inside the bubble. The numerical results show that the amount of water vapor has a significant effect on the bubble collapse temperature. At low excitation amplitudes, the amount of water vapor is not enough to cause the bubble to form a strong collapse. Nevertheless, at high excitation amplitudes, the amount of water vapor is too much to reduce the bubble temperature. There exist optimal values of bubble temperature and amount of water vapor for H2 production. The optimal bubble temperatures are 5267, 4813, 4626, and 3856 K, corresponding to H2 productions of 4.21 × 10−18, 1.29 × 10−18, 2.61 × 10−19, and 8.48 × 10−20 mol, respectively, at ultrasonic frequencies of 140, 213, 355, and 515 kHz. No matter what the excitation parameters are, the optimal water vapor fraction is 0.78 ± 0.04 for H2 production. The obtained results of the present work can provide guidelines for H2 production in acoustic cavitation.

A nonlinear dual-mode active vibration controller for hardening systems and its experimental application to a beam

The control of nonlinear vibrations, such as those arising from geometric nonlinearities and large amplitude, has often been limited to the first natural mode. In this work, a novel nonlinear dual-mode multiple-input multiple-output (MIMO) positive position feedback (PPF) controller is presented and tested experimentally on a clamped composite sandwich beam subject to a stepped-sine-excitation with increasing force excitation levels in the frequency neighbourhood of the first and second natural frequencies. This nonlinear controller contains two linear PPF controllers, targeting the first and second natural frequencies, each one augmented with a cubic term designed to counter the hardening behaviour of the test structure. Control parameters were chosen by optimizing the vibration reduction observed at low excitation amplitudes. The nonlinear dual-mode MIMO controller was compared to a single-input single-output version, and to a linear version in which the cubic terms were removed. The nonlinear dual-mode MIMO was shown to outperform both. A linear dual-mode MIMO controller was compared to two linear single-mode MIMO controllers, targeting the first and second mode respectively, and was found to outperform both when applied to the first mode. The effect of varying the nonlinear cubic term of the nonlinear dual-mode MIMO controller was also investigated, indicating better vibration reduction for higher cubic terms, up to some values. Overall, this study shows that a nonlinear dual-mode MIMO PPF controller is the preferred option for controlling the first two modes of the beam under study.

Comparison of the Dynamic Characteristics of Tuned Liquid Column Dampers with Different Elbow Forms

This study aims to examine experimentally the damping performance of Tuned Liquid Column Dampers (TLCD) by considering different elbow forms. Experiments carried out within the scope of the study point out the dynamic characteristics of TLCD, which has 45 (open) and 90 (closed) degree-elbow forms with theoretical angular frequencies that are the same. Experiments consist of ±5 and ±10 mm harmonic excitation amplitude to observe the relationship between the damping ratio and head loss coefficient with the mass ratio of the TLCD. These experiments with an incremental mass ratio of 0.05% from 0.80% to 1.00% and with an incremental mass ratio of 0.10% from 1.00% to 1.20% had been carried out in detail. Mass and stiffness Modification Factors (MF) are suggested to minimize the difference between numerical and experimental results. Determined MFs are used to examine the earthquake performance of TLCD on a Single-Degree-of-Freedom (SDOF) system. It is shown that the open elbow has an advantage of approximately 25% over the close elbow under earthquake performance on the SDOF system.

Characterization of Three-Mode Combination Internal Resonances in Electrostatically Actuated Flexible–Flexible Microbeams

Abstract 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.

A simple mechanical model of turbulence

This work examines the control and stabilization problems of vibrations in a hierarchical chain of oscillators with hysteresis couplings. Hysteresis coupling is formalized within the Bouc — Wen phenomenological model. The mass, stiffness, and damping properties of the oscillators are set to follow a specific scaling rule and decrease exponentially along the chain, thus forming a hierarchy. The model is verified using Kolmogorov’s hypotheses. To do this, energy spectra are constructed under hysteresis in coupling and without it at different amplitudes of the external excitation. As a result of computational experiments, it is shown that for a chain with hysteresis couplings at a high amplitude of excitation, the energy spectrum curve sufficiently corresponds to Kolmogorov’s hypotheses. The amplitude-frequency characteristics of the system are calculated under hysteresis in coupling using the frequency scanning method. In numerical experiments, frequency ranges of external excitation are identified, which correspond to the chaotic behavior of oscillators and their synchronization.

Design and shock isolation performances of two-layered ring-spring meta-isolation systems

Two-layered ring-spring-resonator (RSR) based meta-isolation system (RSR-M) has been developed to improve shock isolation effect. In the numerical calculations, shock acceleration ratio (SAR), shock displacement ratio (SDR) and relative displacement ratio (RDR) were introduced to evaluate the shock isolation performances of the conventional isolation system (CON) and the RSR-M, considering the effects of the excitation amplitude, the weight of the middle mass block and the stiffness of the lower isolators. The results demonstrate that the SAR of the RSR-M is always smaller than that of the CON, and the SDR of the RSR-M is first larger than that of CON in the shock amplification region and then smaller in the isolation region. The weight of middle mass blocks has little influence on the isolation performance of the RSR-M because of the softening behavior of the isolators. Finally, shock isolation experiments were also carried out to measure the damping ratios of the selected wire rope isolators and further analyze the isolation performance of the RSR-Ms. In general, meta-isolation system provides a new reference and idea for the design of the isolation system of engineering structures in some sense.

Non-uniformly Spaced Antenna Arrays with Overlapped Elements Constraint

In the literature, inter-element spacing antenna design methods have been widely discussed and presented as an alternative approach to element excitation amplitude and/or phase control methods that may be relied upon to achieve the required array pattern shapes. However, methods associated with non-uniformly distributed elements suffer from the element overlap problem, where some of the optimized element locations may overlap each other and cause changes in the overall array aperture length. Practically, these element overlaps cannot be implemented, due to the physical antenna element size, without omitting some of them. Consequently, the overall performance of the antenna array is degraded. Further, degradation may occur when considering phased arrays with scanned main beams. In this paper, we first illustrate the effect of the problem of overlapped element locations and then we propose two approaches based on the genetic algorithm to optimize non-uniformly spaced arrays with overlapped element locations, while simultaneously preserving the array's directivity. To solve the problem of overlapping and to determine the physical array element size, the minimum element-spacing constraints are incorporated in a simple way in the proposed approaches. Thus, the time required to perform optimization-related computations is greatly reduced. Simulation results confirm the effectiveness of the two proposed solutions, where the probability of the elements overlapping has been reduced to zero under specific conditions related to the locations of the some of the elements, while the peak sidelobe levels were always kept below -15 dB and directivity was maintained, to the extent possible, at the level of that of standard uniformly spaced arrays.

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic
wind loads, focusing on control strategies and multistability. We consider time varying wind speeds and analyze a specific case where hydrodynamic drag forces align
with the flow velocity. The stability of the model’s equilibrium points is analyzed
theoretically and numerically. Melnikov’s method is employed to identify conditions
for homoclinic bifurcation. Numerical simulations employing basin of attraction
confirm the analytical predictions. Our findings show a decrease in the threshold
for chaos with increasing amplitudes of external excitation, damping coefficient,
and parametric damping. The global dynamics are explored numerically using a
fourth-order Runge-Kutta method. When solely subjected to external excitation, the
system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode
oscillations, and chaos. Conversely, with self- and parametric drag forces, the system
displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity,
transient chaos, and chaos. Poincar´e maps analyze the geometric structure of chaotic
attractors, revealing a strong influence of dimensionless drag parameters. These
parameters can be manipulated to control or eliminate chaos. Furthermore, the system
exhibits multistability, with coexisting attractors. Beyond its application in protecting
infrastructure from wind damage, this research can contribute to ecological balance by
improving our understanding of tree wind resistance.

Importance of Orbital Invariance in Quantifying Electron-Hole Separation and Exciton Size.

A fundamental tenet of quantum mechanics is that properties should be independent of representation. In self-consistent field methods such as density functional theory, this manifests as a requirement that properties be invariant with respect to unitary transformations of the occupied molecular orbitals and (separately) the unoccupied molecular orbitals. Various ad hoc measures of excited-state charge separation that are commonly used to analyze time-dependent density-functional calculations violate this requirement, as they are based on incoherent averages of excitation amplitudes rather than expectation values involving coherent superpositions. As a result, these metrics afford markedly different values in various common representations, including canonical molecular orbitals, Boys-localized orbitals, and natural orbitals. Numerical values can be unstable with respect to basis-set expansion and may afford nonsensical results in the presence of extremely diffuse basis functions. In contrast, metrics based on well-defined expectation values are stable, representation-invariant, and physically interpretable. Use of natural transition orbitals improves the stability of the incoherent averages, but numerical values can only be interpreted as expectation value in the absence of superposition. To satisfy this condition, the particle and hole density matrices must each be dominated by a single eigenvector so that the transition density is well described by a single pair of natural transition orbitals. Counterexamples are readily found where this is not the case. Our results suggest that ad hoc charge-transfer diagnostics should be replaced by rigorous expectation values, which are no more expensive to compute.

A nonlinear and asymmetric monostable compliant ortho-planar spring piezoelectric vibrational energy harvester using a H-I structure

This paper proposes a compliant ortho-planar spring piezoelectric vibrational energy harvester (COPS-PVEH) using a H-I structure that exploits a nonlinearity and an asymmetric mono-stability to enhance the bandwidth of the device. The main structure is based on a double clamped beam (I-structure) coupled with four cantilever beams (H-structure) and an ortho-planar spring in the centre of the I-structure. Finite element analysis (FEA), analytical modelling, and experimentation were performed to analyse the dynamical and electrical behaviour of the harvester. The device was fabricated using polylactide (PLA). Six bimorph PZT-5H piezoelectric materials, electrically connected in parallel, were used as piezoelectric generators. The device was tested using an optimum load resistor of 90 kΩ under sinusoidal and sine sweep excitation in the gravity (vertical) direction. The pre-load (due to gravitation) causes a softening nonlinearity and an asymmetric mono-stability in the potential energy function. The results show that multiple peaks are observed in the output voltage of the harvester in the FEA, analytical modelling, and experimentation under harmonic excitation in the sub 15 Hz frequency range. Furthermore, analytical modelling and experimentation exhibit softening nonlinearity and chaotic behaviour, reaching a maximum amplitude power of 1.01 mW (at 8.3 Hz) and 1.07 mW (at 9.5 Hz) respectively under sine sweep excitation (amplitude of 0.6 g (g = 9.81 m/s2)). Experimentally, the harvester also generates an average power greater than 46 µW with a 6.8 Hz bandwidth under the same excitation. The softening nonlinearity and asymmetric mono-stability enhance the bandwidth of the harvester in the low frequency region where most broadband ambient vibrations are available in practical environments.

Understanding AFM-IR Signal Dependence on Sample Thickness and Laser Excitation: Experimental and Theoretical Insights.

Photothermal induced resonance (PTIR), also known as atomic force microscopy-infrared (AFM-IR), enables nanoscale IR absorption spectroscopy by transducing the local photothermal expansion and contraction of a sample with the tip of an atomic force microscope. PTIR spectra enable material identification at the nanoscale and can measure sample composition at depths >1 μm. However, implementation of quantitative, multivariate, nanoscale IR analysis requires an improved understanding of PTIR signal transduction and of the intensity dependence on sample characteristics and measurement parameters. Here, PTIR spectra measured on three-dimensional printed conical structures up to 2.5 μm tall elucidate the signal dependence on sample thickness for different IR laser repetition rates and pulse lengths. Additionally, we develop a model linking sample thermal expansion dynamics to cantilever excitation amplitudes that includes samples that do not fully thermalize between consecutive pulses. Remarkable qualitative agreement between experiments and theory demonstrates a monotonic increase in the PTIR signal intensity with thickness, with decreasing sensitivities at higher repetition rates, while signal intensity is nearly unaffected by laser pulse length. Although we observe slight deviations from linearity over the entire 2.5 μm thickness range, the signal's approximate linearity for bands of sample thicknesses up to ≈500 nm suggests that samples with comparably low topographic variations are most amenable to quantitative analysis. Importantly, we measure absorptive undistorted profiles in PTIR spectra for strongly absorbing modes, up to ≈1650 nm, and >2500 nm for other modes. These insights are foundational toward quantitative nanoscale PTIR analyses and material identification, furthering their impact across many applications.

The effects of resistance training to near volitional failure on motor unit recruitment during neuromuscular fatigue.

It is unclear whether chronically training close to volitional failure influences motor unit recruitment strategies during fatigue. We compared resistance training to near volitional failure vs. non-failure on individual motor unit action potential amplitude (MUAP) and surface electromyographic excitation (sEMG) during fatiguing contractions. Nineteen resistance-trained adults (11 males, 8 females) underwent 5 weeks (3×/week) of either low repetitions-in-reserve (RIR; 0-1 RIR) or high RIR training (4-6 RIR). Before and after the intervention, participants performed isometric contractions of the knee extensors at 30% of maximal peak torque until exhaustion while vastus lateralis sEMG signals were recorded and later decomposed. MUAP and sEMG excitation for the vastus lateralis were quantified at the beginning, middle, and end of the fatigue assessment. Both training groups improved time-to-task failure (mean change = 43.3 s, 24.0%), with no significant differences between low and high RIR training groups (low RIR = 28.7%, high RIR = 19.4%). Our fatigue assessment revealed reduced isometric torque steadiness and increased MUAP amplitude and sEMG excitation during the fatiguing task, but these changes were consistent between groups. Both low and high RIR training improved time-to-task failure, but resulted in comparable motor unit recruitment during fatiguing contractions. Our findings indicate that both low and high RIR training can be used to enhance fatiguability among previously resistance-trained adults.

The influence of lumped mass deployment on the performance of a hybrid galloping energy harvester

This article delves into a hybrid galloping energy harvester featuring a longitudinally aligned triangular-shaped bluff body. Numerically, it analyzes how the position of a lumped mass influences the average output power of the harvester under concurrent base excitation and galloping. In comparison with traditional hybrid energy harvesters, the impact of three factors—low airflow velocity, small base excitation amplitude, and low frequency—is examined for each case of lumped mass deployment. The findings reveal that optimal mass positioning markedly enhances the harvester’s average power output when there are low airflow velocities, small base excitation amplitudes, and low frequencies, constituting a novel contribution to the field. Through numerous numerical simulations, it is determined that the optimal position for deploying the lumped mass is approximately 60–70% along the length of the cantilever beam substrate.

The next-generation of metaverse embodiment interaction devices: A self-powered sensing smart monitoring system

Wearable electronic devices collect user data to provide personalized experiences and real time interactions for the metaverse, thereby driving its development. This paper introduces an innovative self-powered sensing smart monitoring system (SSSMS) designed to harvest human low-frequency inertial energy and achieve gait recognition and fall monitoring (GR-FM) capabilities. The SSSMS enhances energy harvesting efficiency through Halbach electromagnetic enhancement mechanisms, achieving self-sustainability. Additionally, the conical roller triboelectric nanosensor (CR-TENS) accurately captures various human gaits, transmitting these states in the form of electrical signals to a computer via OpenBCI in real time. These signals are integrated with a deep learning-driven diagnostic module to achieve GR-FM. The experimental results demonstrate that under optimal external excitation, the output power is 2.27 mW, with a power density of 32 W/m3. After 100,000 cycles, both the EMG and CR-TENS modules still maintain good performance, sufficiently providing stable and continuous charging for the battery and offering a stable signal for signal processing. Both the EMG and CR-TENS modules exhibit robust feature capture capabilities, effectively sensing different excitation frequencies, amplitudes, and gait patterns. The SSSMS based on the GRU deep learning model achieves recognition accuracies of 96.13 %, 96.60 %, and 92.22 % for frequency, amplitude, and gait, respectively. Deployment experiments demonstrate high accuracy and sensitivity in frequency, amplitude, gait recognition, and fall monitoring, highlighting the advantageous design of EMG and CR-TENS for perceiving subtle motion state changes. The SSSMS demonstrates outstanding capabilities in self-sustainability and real time user embodiment information capture, showcasing its potential as a next-generation interaction device for the metaverse.

Field observation and cause investigation of low-frequency cable vibrations in a cable-stayed bridge

Considerable levels of stay cable vibration in cable-supported bridges pose significant challenges for safety, operation, and maintenance. An accurate understanding of the specific causes and types of vibration is essential for effectively addressing this problem with an optimal design for mitigation measures. This study aims to investigate the cause of an extraordinary low-frequency vibration that occurred at a stay cable by employing comprehensive methodologies. The modal properties and vibrational characteristics of cable vibrations were first obtained through finite element modeling, operational modal analysis, and vision-based displacement measurements. Subsequently, the primary cause of low-frequency vibrations was investigated using field observations and wind tunnel tests. Field-monitoring data revealed that stay cables were excited by small deck vibrations. The critical wind speeds at which deck vibrations occurred in a wind tunnel experiment were aligned with the ranges of wind speeds that produce low-frequency cable vibrations, which confirmed that these two phenomena are manifested under similar wind conditions. A calculated support excitation amplitude, determined via the use of an empirical equation, exhibited consistency with vision-based measured displacement obtained in field experiments. These findings underscore the potential risk of support excitation between minor deck vibrations and cables, which provides valuable insights that could mitigate cable vibrations.

High sound pressure level sound absorption in acoustic resonant metamaterials: adapting a mass-spring model for nonlinear effects

In aerospace, conventional materials as micro-perforated panels backed by cavities are used to absorb sound at high sound pressure levels. These solutions are not optimal at low frequencies compared to resonant acoustic metamaterials, which present an acoustic absorption peak with a wavelength-to-thickness ratio greater than conventional materials, i.e., a ratio>>5. Acoustic resonant metamaterials are mainly studied at low sound pressure levels (<100dB). However, they are sensitive to increasing the amplitude of acoustic excitation. The studied metamaterial is composed of a compact array of thin single-perforation panels spaced by thin cylindrical air cavities. In a previous study, a mass-spring model was developed for low sound pressure levels. The perforations are modeled by equivalent masses, whereas cavities by equivalent springs. In the perforations, high values of the acoustic velocity lead to an increase in the acoustic resistance of the material. This effect is considered by the Forchheimer parameter in the mass-spring model. To determine this parameter and to study how it is impacted by the metamaterial geometry, the computational fluid dynamic method is used. The mass-spring model is adapted with a nonlinear term. The adapted model agrees well with simulations by finite element method at sound pressure levels up to 140dB.

The use of ensembles trees machine learning method in estimating damping of granular materials

Granular Materials (GM) employed within the mechanism of particle dampers attenuate the vibration energy due to their interparticle and particle-wall interactions. Estimating their damping effect using the analytical equivalent single mass approach overlooked the particles' individual losses that are built into the total damping. Alternatively, the numerical techniques (e.g., Discrete Element) are time-inefficient and computationally demanding. Therefore, this study explores the implementation of Machine learning (ML) algorithms to estimate the damping effect of GM. The ML model in this study will rely on a Data-Driven Modeling approach (DDM) incorporating the ensemble tress nonlinear regressor method. The models' training and testing data were obtained from an experimental setup of an acrylic beam internally integrated with Stainless Steel (SS) and glass spheres undergoing low excitation amplitude (RMS <1N). The main aim was to map between seven input features (e.g. filling ratio) and one targeted output: the beam's damped frequency response. Three ensemble trees' algorithms were used to create the DDM; Decision Tree, Random Forest, and XGBoost. The hyperparameter combination based on the Gridsearch CV function increases the prediction accuracy of each model. The developed ML models provided high accuracy (86-93%) in predicting the damping effect of the granular materials spheres.

Nonlinear vibration and parametric excitation of magneto‐thermo elastic embedded nanobeam using homotopy perturbation technique

AbstractThe present study focuses on the modeling and analyzing the nonlinear vibration patterns and parametric excitation of embedded Euler–Bernoulli nanobeams subjected to thermo‐magneto‐mechanical loads. The Euler–Bernoulli nanobeam is developed with external parametric excitation. The governing equation of motion is derived by utilizing nonlocal continuum theory and nonlinear von Karman beam theory. Subsequently, the homotopy perturbation technique is employed to determine the vibration frequencies. Finally, the modulation equation of Euler–Bernoulli nanobeams is derived for simply supported boundary condition. In order to validate our findings, we conduct a comparative analysis against existing literature, thereby underscoring the effectiveness and robustness of our proposed methodology. The influence of stress, magnetic potential, temperature, damping coefficient, Winkler coefficient, and nonlocal parameters are tested numerically on nonlinear frequency‐amplitude and parametric excitation–amplitude responses. The numerical examples indicate the significant impact of physical variables on the nonlinear frequency and parametric excitation. The primary objective of this study is to investigate the effects of external physical variables on the dynamic behavior of nanobeams, particularly in nonlinear regimes. This study provides insights into the design and control of nanostructures under complex load conditions, contributing to developing advanced materials and nanosystems.