Softening Characteristics Research Articles

Frequency Response of a Nonlinear Vibration Isolator Containing a Viscoelastic End-Stop Buffer

Considering that a mechanical spring can be compressed or stretched only, a protected object is prone to detach from the support spring when the system falls into the resonant region. To avoid this phenomenon, a nonlinear vibration isolator containing a viscoelastic end-stop buffer is developed, which can exhibit stiffness–hardening or stiffness–softening characteristics. Because the restoring force is asymmetrical and piecewise linear, a modified averaging method is applied to obtain steady-state solutions, which contain a bias term. A numerical method is then used to verify the analytical solutions, and their stability is determined. Finally, the effects of the structural parameters on the frequency response and force transmissibility are investigated. The jump avoidance condition is subsequently given. The results show that the modified averaging method is more accurate than the classical method. Decreasing the stiffness ratio can reduce the transmissibility peak and resonance frequency. Applying heavy damping in the end-stop buffer is beneficial for reducing transmissibility in the resonant region, whereas the isolation performance at high frequencies remains unchanged. The developed vibration system exhibits better isolation performance than the linear system when stiffness–softening characteristics are considered.

Experimental investigation on mechanical properties and strength criteria of frozen soft rock.

Excavation of underground engineering structures involving deeply buried water-rich soft rocks is generally carried out using the artificial freezing method. A series of undrained uniaxial and triaxial shear and creep tests were conducted on soft rocks under different confining pressures (0, 0.2, 0.5, and 1.0 MPa) at different freezing temperatures (room temperature, -5°C, -10°C, and -15°C). Test results indicate that the frozen soft rocks show strain softening characteristics. The stress-strain curve changes from a straight line to a curve as deviatoric stress constantly increases, while it decreases abruptly after the deviatoric stress reaches the peak and is slightly affected by the freezing temperature. At the same temperature, shear strength increases at a rate of 5.6 MPa/°C with increasing confining pressure; as freezing temperature decreases, the shear strength increases at 0.34 MPa/°C, and cohesion increases at 0.6 MPa/°C. Under the same confining pressure, the failure strain of soft rock decreases with the decrease of temperature. The Mohr-Coulomb (M-C) criterion can accurately describe the failure process of frozen soft rocks in the pre-peak stage, with a correlation coefficient greater than 0.98. Within the test stress range, soft rocks display attenuated stable creep deformation. Acoustic emission (AE) tests were conducted to further verify that the soft rocks show shear failure under load, with a shear plane showing an angle of 45° with the horizontal. The research findings provide technical support and theoretical reference for studying rock mechanical properties as well as for designing and carrying out underground freezing of rocks in a low-temperature environment.

Constitutive model of lime-stabilized laterite considering curing time and softening characteristics

Constitutive model of lime-stabilized laterite considering curing time and softening characteristics

Correction method for ionization chamber dosimetry in flattening filter free radiotherapy based on Monte Carlo simulation.

The clinical use of flattening filter free (FFF) radiotherapy has significantly increased in recent years due to its effective enhancement of dose rates and reduction of scatter dose. A proposal has been made to adjust the incident electron angle of the accelerator to expand the application of FFF beams in areas such as large planning target volumes (PTVs). However, the inherent softening characteristics and non-uniformity of lateral dose distribution in FFF beams inevitably lead to increased dosimetry errors, especially for ionization chambers widely used in clinical practice, which may result in serious accidents during FFF radiotherapy. This study constructs a comprehensive Monte Carlo model that encompasses not only conventional FFF beams but also incorporates FFF beams with varying incident electron angles, to investigate dosimetry errors and correction methods in FFF radiotherapy. We have innovatively introduced a FFF output correction factor ( ) to address dosimetry errors in various ionization chambers under different incident electron angle conditions in FFF beams. The primary variations in were analytically determined to result from changes in and the perturbation correction terms of the ionization chamber. Ionization chambers with smaller sensitive volumes typically exhibit reduced dosimetry errors. Our findings indicate that for ionization chambers with sensitive volumes ranging from 0.016 to 0.125cm3, the dosimetry error under various FFF beam conditions consistently remains below 1.15%. This study provides crucial guidance for selecting appropriate ionization chambers in FFF radiotherapy. A correlation was established between the absorbed dose to water in beams with a flattening filter (WFF) and those without (FFF), defined by the FFF output factor ( ). Using the proposed Monte Carlo model, the can be derived and applied to theoretically calculate the absorbed dose to water in FFF beams at varying incident electron angles, with a relative standard uncertainty of 0.2. This study provides a valuable reference for clinical dose measurements and crucial support for establishing dose calibration standards in FFF radiotherapy.

Crack Propagation Behavior of Single‐Crystal Titanium Under Cyclic Loading: A Molecular Dynamics Study

ABSTRACTIn this study, molecular dynamics (MD) simulations were employed to investigate the crack propagation behavior of single‐crystal titanium with various crystal orientations under cyclic loading. The analysis demonstrates that each crack model displays temporary cyclic hardening and predominant cyclic softening characteristics. The orientation of crack propagation primarily impacts the characteristics of the softening stage, with less influence on the initial hardening stage. A notable orientation correlation is evident in the mechanism of crack propagation, characterized by the presence of various slip modes and deformation twinning (DT) systems. The crack tip deformation behavior obtained from the simulation aligns with the theoretical predictions of linear elastic fracture mechanics (LEFM). The crack growth rate (CGR) and ΔJ for different crack models show good correlation, and both the crack propagation direction and crack plane orientation affect the characteristics of the ΔJ–da/dN curves.

Deformation Behaviors and Microstructure Evolution of Mg-Zn-Y-Zr Alloys During Hot Compression Process

This study investigated the thermal compression deformability of the low-alloyed Mg-Zn-Y-Zr magnesium alloy temperatures ranging from 300 to 450 °C, and strain rates between 0.01 s−1 and 1 s−1. A hot processing map was established using a novel constitutive model. The results demonstrate that the flow stress of the low-alloyed Mg-Zn-Y-Zr alloy is markedly affected by the deformation temperature and strain rate, predominantly manifesting characteristics of work hardening (WH) and dynamic recrystallization-induced softening. The high-temperature rheological behavior of the alloy is accurately portrayed with a constitutive model, with an activation energy measured at 287 kJ/mol. The mechanism of dynamic recrystallization (DRX) gradually shifts from twinning dynamic recrystallization (TDRX) to continuous dynamic recrystallization (CDRX) and discontinuous dynamic recrystallization (DDRX). At 400 °C, as the strain rate decreases, the I-phase in the microstructure gradually transforms into the W-phase, weakening the inhibitory effect on DRX grain growth.

A new semi–analytical method for elastic–strain softening circular tunnel with hydraulic–mechanical coupling

This study presents a novel coupled hydraulic–mechanical algorithm for analysing nonlinear seepage in circular tunnels with elastic strain softening characteristics. The model integrates porosity changes with permeability coefficients to formulate a set of nonlinear seepage equations. The Mohr–Coulomb criterion is used to determine the of rock stress yielding state, and the plastic strain εθp is used as a softening parameter to assess the degradation of rock strength. The proposed model is validated through a comparison with established numerical and analytical solutions. The pore water pressure distribution exhibits a distinctive three-stage curve under coupling conditions, with the seepage behaviour transitioning to a classical Laplace-type equation as the coupling coefficient weights diminish. Parametric analysis reveals that the brittleness index β is a pivotal factor governing the extent of the softening region. A decrease in residual strength σc∗ intensifies strain softening, leading to a more extensive softening zone. Conversely, a reduction in the angle of internal friction φ affects only the rock's strength without altering the softening intensity. The study also demonstrated that the pore water pressure diminishes the effective stress within the rock mass, resulting in a larger plastic zone than that under dry conditions. Finally, internal reinforcement and external support can effectively mitigate the effects of strain softening and seepage on the stability of the surrounding rock.

Shape recovery effect and energy absorption of reusable honeycomb structures

When using auxetic honeycomb structures to create repeatable energy-absorbing components, a key challenge is selecting the appropriate unit configuration for effective functional integration. In this work, four typical honeycomb structures were prepared, and the mechanical behaviors, shape recovery effects, and energy absorption properties of three types of auxetic honeycomb structures re-entrant honeycomb (RH), arrow honeycomb (AH), and star honeycomb (SH) were compared with those of hexagonal honeycomb (HH) through quasi-static loading–unloading tests. The findings indicate that the 3D printed polyurethane (TPU) honeycomb structures demonstrate robust shape recovery, stable energy absorption, notable stress softening characteristics. The recovery behaviors can be characterized by three distinct phases, namely hyperelastic, transitional, and viscoelastic. The unit configuration significantly influences the shape recovery capability, with apparent elastic modulus and stability of the energy absorption efficiency determining the overall shape recovery capability. The loading method also affects the energy absorption and dissipation patterns in different honeycomb structures. In terms of specific energy absorption (SEA), AH has the highest rating, with RH and SH at 86 % and 50 % of the SEA of AH respectively. The number of reusable cycles is primarily dictated by the specific configuration of the unit type. In scenarios involving reusability, the energy absorption capacity of the TPU honeycomb can only reach 70 % of its original energy absorption capacity. This study may inform the application of auxetic materials in reusable energy absorbers.

Enhancing efficiency in hot stamping of polymethyl methacrylate foils: computer analysis and modeling

ABSTRACT This work is leading the way in developing a new method to evaluate the efficacy of hot stamping on polymethyl methacrylate (PMMA) foils. The art of hot stamping, which involves stamping, heat, and pressure to produce elaborate images on flexible foil surfaces, has long been prized for its artistry. A thorough examination of the embossing mechanism is complicated by the fact that different process factors might result in different levels of quality for embossed patterns. This study stands out for its creative approach to hot stamping process optimisation. Traditionally, resource-intensive and time-consuming empirical studies are used to determine the ideal process conditions. By using computer analytic techniques, this research, on the other hand, presents a novel way for quickly and precisely determining the most effective process parameters. Furthermore, by exploring the evolution of the embossing phenomena utilising cutting-edge methodologies like the finite element method (FEM) and creative reproduction approaches, this work explores the unexplored area. A thorough analytical model is carefully created, accounting for temperature-specific stress characteristics, strain rates, stress distribution, and the softening characteristics of the elastic foil. The research makes a unique contribution by precisely characterising the stress hardening, deformation, and thermal expansion of PMMA materials. This paves the way for extrusion analysis utilising customised processes.

Fatigue damage and life prediction for AISI H13 steel under cyclic thermomechanical loading

Based on strain-controlled thermomechanical fatigue (TMF) experiments, this study conducts a comprehensive analysis of the TMF behavior of AISI H13 hot work die steel. Moreover, a life prediction model for the TMF behavior of AISI H13 steel has been developed and validated. The experimental results reveal that, under the in-phase (IP) and out-of-phase (OP) TMF conditions, the stress–strain response curves of AISI H13 steel under different mechanical strain amplitudes exhibits the similar evolution tendency. However, it is worth noting that in the stable TMF cycle, the hysteresis loop area is enlarged with the increase of the number of cycles, which can be attributed to the cyclic softening characteristics of the AISI H13 steel under cyclic thermomechanical loading. When examining different TMF conditions, it is found that at higher strain amplitudes and under OP TMF conditions, the hysteresis loop area significantly expands, leading to a substantial reduction in the TMF life of AISI H13 steel. From a microstructural perspective, the thermal–mechanical coupling effect makes the recovery of martensitic matrix and the coarsening of carbide precipitation, which substantiates the deterioration of mechanical properties of AISI H13 steel. Finally, a modified Ostergren model by integrating the hysteresis loop area has been developed to assess the TMF life of AISI H13 steel under complex thermomechanical loading conditions, and this refined model exhibits strong agreement with experimental data. An evaluation using scatter band shows that the predicted TMF life of AISI H13 steel are within 1.2 times the experimental values, which illustrates a high reliability and validity.

Improving hawthorn fruit pulp processing and quality through optimized freeze-thaw cycles

Hawthorn fruit is firm, making it difficult to process into fruit pulp. To improve the processing suitability of hawthorn fruit pulp, one, two, and three freeze-thaw cycles (FTCs, frozen at −18 °C for 24 h, then thawed at 5 °C or 25° C for 6 h) were performed on hawthorn fruit in this study. Using unfrozen specimens as the controls, the study assessed fruit softening characteristics and rheological properties, taste traits, and nutritional components of the fruit pulp. Freeze-thaw treatments decreased fruit firmness and cell-wall stability, increased juice yield during pulp processing, and reduced the viscosity, sourness, and astringency of hawthorn fruit pulp, likely by enhancing pectin solubility. The treatment effect was influenced by both FTC number and thawing temperature. Two FTCs with a thawing temperature of 5 °C were most effective, promoting the transformation of protopectin into water-soluble pectin (WSP) to the fullest extent. The treatment reduced firmness to about 25 % of the initial value, increased juice yield by about 22 %, increased the sugar-acid ratio to 11.6, and reduced the sourness and astringency of the fruit pulp. It increased the contents of total flavones and anthocyanidins without affecting the contents of vitamin C. Therefore, the use of two FTCs with a thawing temperature of 5 °C can facilitate the preparation of hawthorn fruit pulp while preserving its nutritional properties and improving its taste profile.

Fracture mechanism and constitutive model considering post-peak plastic deformation of marble under thermal–mechanical action

Temperature plays an important impact on rock mechanical properties. In this paper, the mechanical properties, fracture mechanism and constitutive model of marble under 20–120 °C and 15 MPa are studied. The results show marble deformation can be divided into four stages: compaction, linear elasticity, crack propagation and post-peak failure. Stress–strain curve is not obviously affected by temperature. Macroscopic fracture characteristics change from shear failure to tensile mixed failure with temperature increasing. With the increase of temperature, the strength of marble tends to decrease, indicating that temperature increase has a weakening effect on marble, and there are temperature-sensitive areas of 20–60 °C and temperature sub-sensitive areas of 60–120 °C. The elastic modulus of marble decreases and Poisson’s ratio increases with increasing temperature. The energy evolution law of marble under different temperature is basically the same, which shows that before crack initiation, the energy dissipation is less, and after the damage and yielding occurs, the energy dissipation increases quickly. The energy dissipation in the failure process is mainly used for crack initiation-connection-penetration, as well as plastic deformation caused by friction and slip of cracks, and the plastic deformation and energy dissipation have good linear characteristics. The statistical damage constitutive model based on three-parameter Weibull distribution function can effectively reflect the characteristics of post-peak plastic deformation and strain softening. The weakening effect of marble at 20–120 °C is related to its internal moisture excitation. With the increase of temperature, water is stimulated to absorb and attach to the original relatively dry interface, which plays a role in lubrication. The relative motion friction resistance between solid particles or crack surfaces decreases, which leads to crack initiation and friction energy consumption reduction, changes the specific surface energy of rocks and weakens the strength of marble. The results provide a theoretical basis for predicting and evaluating the long-term stability and safety of surrounding rock of underground deep engineering in complex environment with high ground temperature and high geo-stress.

Study on water softening characteristics and multi-stage sliding zone reactivation mechanism of old clay landslides in the Three Gorges Reservoir area subjected to groundwater

Study on water softening characteristics and multi-stage sliding zone reactivation mechanism of old clay landslides in the Three Gorges Reservoir area subjected to groundwater

Modified approach for predicting seismic-induced deformation of landfills considering strength parameters of GMB-GCL interface within the liner system

This study presents an innovative methodology for predicting seismic-induced sliding displacement, a key determinant in evaluating the seismic stability of landfills. The novelty of this research lies in the incorporation of the softening characteristics of the geosynthetic interface within the liner system, a factor that has been largely overlooked in previous studies. A dynamic stability analysis was performed on a landfill using the ABAQUS software, with an emphasis on the impact of coupled parameters, particularly the strength of the interface. The results highlight PGA (Peak Ground Acceleration), PGV (Peak Ground Velocity), Ia (Arias Intensity), and residual interface shear strength (μ) as effective predictors. The study further identifies the combination of PGA and μ as the optimal parameter pairing for predicting the seismic permanent deformation of the landfill. A multi-dimensional data regression approach was employed to propose a calculation formula for seismic permanent deformation, taking into account liner damage. To enhance the seismic design methodology for landfills, the probability density function was integrated into the study. This innovative approach provides a new perspective on seismic stability assessment in landfill engineering and designs.

Effect of ultrasonication and temperature on hydration process and hardness of two cowpea types

AbstractUltrasonication deployment provides a green and non‐thermal option to traditional hydrothermal treatment. This study presents the impact of ultrasonication and soaking temperatures (30 and 50°C) on the water uptake and hardness of two cowpea types under increasing soaking times (15–240 min). Moisture content and hardness of the studied samples were measured using standard test methods and instruments. An increase in soaking temperature and the use of ultrasonication enhanced water uptake and reduced hardness. Ultrasonication improved mass transfer, which enhanced the diffusion of water uptake. The samples' water uptake and softening characteristics were significantly modeled with high accuracy (R2 = 0.99) using sigmoidal and first‐order kinetics equations, respectively. The impact of sonication was found to be more significant at 30°C soaking of the studied cowpeas as the soaking time increased. This work justified using ultrasonication as a green technique to enhance the softening of cowpeas.

Elastoplastic analysis on deformation and failure characteristics of surrounding rock of soft-coal roadway based on true triaxial loading and unloading tests

Since accidents such as roof caving, rock fragmentation, and severe deformation are particularly likely to occur during roadway excavation in soft and thick coal seams, grasping the range and distribution of deformation and fracturing of surrounding rock is of crucial for evaluating roadway stability and optimizing support design in such coal seams. In this study, based on the stress paths encountered during roadway excavation, true triaxial loading and unloading tests were carried out on soft coal, and the deformation and strength evolutions of soft coal under different intermediate principal stress conditions were analyzed. The test results show that the stress–strain relationship in the pre-peak plasticity-strengthening and post-peak plasticity-weakening stages follows a quadratic function, and the strengeth evolution conforms to the Mogi–Coulomb criterion. Moreover, analytical solutions for the displacement of surrounding rock, the radius of the broken zone, and the radius of the plastic zone of soft-coal roadways under excavation stress paths were derived after taking the nonlinear hardening and softening characteristics of the strain of soft coal, the Mogi–Coulomb criterion, the intermediate principal stress, and the dilatancy characteristics of surrounding rock into comprehensive consideration. Finally, in accordance with a practical engineering case, the influences of the intermediate principal stress coefficient, the lateral pressure coefficient, and the support force on the deformation and failure characteristics of the soft-coal roadway were analyzed. The analysis reveals that an increase in intermediate principal stress aggravates the deformation of surrounding rock and enlarges the plastic and broken zones; variations in the lateral pressure coefficient alter the shape of the broken zone and the distribution of surface displacement; and an increase in the support force effectively reduces the plastic zone, broken zone, and surface displacement of the roadway. The research results can provide valuable theoretical basis for the stability evaluation and support design of soft-coal roadways.

Compressive behaviour of concrete columns confined with textile reinforced concrete composites

The enhancement of strength and deformation capacity in confined concrete members could be achieved through the application of advanced composite materials. Textile reinforced concrete (TRC), a novel cement-based composite material, is considered as a viable alternative to fiber reinforced polymer (FRP) sheets, addressing the limitations associated with organic resins. This study delved into the impact of various parameters, such as cross-section shapes, concrete strengths, and the number of TRC layers, on the confinement effect. The results reveal a consistent augmentation in both axial compressive strength and deformation capacity of confined concrete with an escalation in the number of TRC layers, irrespective of the cross-sectional configurations. Notably, the maximum enhancements in axial peak stress for rectangular, square, and circular specimens are recorded at 45.96 %, 49.98 %, and 90.06 %, respectively, accompanied by corresponding increases in axial strains of 56.28 %, 53.74 %, and 86.00 %, respectively. The effectiveness of TRC confinement exhibits a substantial correlation with the geometry of the cross-sections. The circular ones have higher utilization rates followed by the square ones, and the rectangular ones with section aspect ratio greater than 1.0 have the lowest utilization rates. Furthermore, an evaluation of existing confinement identification models indicates that both the Mirminan Model and Wu Model could effectively delineate the hardening and softening characteristics of circular and rectangular specimens. However, there exists scope for improving the accuracy of these models. Therefore, a novel modified identification model, predicated on the Mirminan model, was proposed to refine the confinement assessment process. The revised modified confinement ratio (MCR) is recommended at a value of 0.045, aiming to elevate the precision and reliability of confinement evaluations in concrete structures.

Mechanical Response Characteristics and Tangent Modulus Calculation Model of Expansive-Clay Unloading Stress Path

As a special type of clay, expansive clay is widely distributed in China. Its characteristics of swelling and softening when meeting water and shrinking and cracking when losing water bring many hidden dangers to engineering construction. Expansive clay is known as “engineering cancer”, and in-depth research on the unloading mechanical response characteristics and the unloading constitutive relationships of expansive clay is a prerequisite for conducting geotechnical engineering design and safety analysis in expansive-soil areas. In order to obtain the unloading mechanical response characteristics and the expression of the unloading tangent modulus of expansive clay, typical expansive clay in the Hefei area was taken as the research object, and triaxial unloading stress path tests were conducted. The stress–strain properties, microstructures, macro failure modes, and strength indexes of the expansive clay were analyzed under unloading stress paths. Through an applicability analysis of several classical soil strength criteria, an unloading constitutive model and the unloading tangent modulus expression of the expansive clay were constructed based on the Mohr–Coulomb (hereinafter referred to as “M-C”) criterion, the Drucker–Prager (hereinafter referred to as “D-P”) criterion, and the extended Spatial Mobilized Plane (hereinafter referred to as “SMP”) criterion theoretical frameworks. The following research results were obtained: (1) The stress–strain curves of the three stress paths of the expansive clay were hyperbolic. The expansive clay showed typical strain-hardening characteristics and belonged to work-hardening soil. (2) Under the unloading stress paths, the soil particles were involved in the unloading process of stress release, and the failure samples showed obvious stretching, curling, and slipping phenomena in their soil sheet elements. (3) Under both unloading stress paths, the strength of the expansive clay was significantly weakened and reduced. Under the lateral unloading paths, the cohesive force (c) of the expansive clay was reduced by 32.7% and the internal friction angle (φ) was increased by 19% compared with those under conventional loading, while under the axial unloading path, c was reduced by 63.5% and φ was reduced by 28.7%. (4) For typical expansive clay in Hefei, the conventional triaxial compression (hereinafter referred to as “CTC”) test, the reduced triaxial compression (hereinafter referred to as “RTC”) test, and the reduced triaxial extension (hereinafter referred to as “RTE”) test stress paths were suitable for characterization and deformation prediction using the M-C strength criterion, D-P strength criterion, and extended SMP strength criterion, respectively. (5) The derived unloading constitutive model and the unified tangent modulus formula of the expansive clay could accurately predict the deformation characteristics of the unloading stress path of the expansive clay. These research results will provide an important reference for future engineering construction in expansive-clay areas.

Water-rock interfacial softening model of cemented soft rock: An experimental and numerical study

Cemented soft rock is prone to softening when exposed to water, and its mechanical properties deteriorate rapidly, which is an important reason to induce the instability and failure of engineering soft rock mass. The softening mechanism of soft rock is essentially the deterioration of microstructure caused by the change of water-rock interface. Considering the general characteristics of soft rock, the microstructural evolution characteristics and element loss over varying immersion time are then compared and analyzed. The Interfacial Cemented Bonding (ICB) structure is proposed to describe the evolution of water-rock interface. Based on the diffusion theory, the theoretical equation describing the evolution of water-rock interface is derived and compared with the experimental results. The meso-softening damage factor (MSDF) of soft rock is proposed and introduced into a numerical method to establish a strength degradation discrete element method (SD-DEM) model. The results show that the softening process of soft rock is accompanied by the shedding and suspension of particles and the dissolution of soluble substances. Besides, the softening of soft rock has obvious nonlinear dynamic characteristics. The fitting degree between the theoretical values and the experimental results is relatively high, which verifies the reliability and rationality of the dissolution-diffusion interfacial softening model. The numerical results are in good agreement with the experimental ones. This study provides a new idea for understanding the mechanism of water-induced softening characteristics and strength degradation of soft rock.

Nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers based on complex nonlinear modes

In this study, the nonlinear dynamics of a dual-rotor system with active elastic support/dry friction dampers (ESDFDs) are investigated based on complex nonlinear modes (CNMs). The finite element method (FEM) combined with a full-3D friction model is introduced to construct the governing equation for the system. Additionally, the Craig–Bampton technique is applied to downscale the finite element model of the system. Based on the reduced order model (ROM), the nonlinear modal damping ratio of the target mode is employed to measure the dry friction damping performance of active ESDFD. The effects of the active ESDFD position, normal force, and tangential contact stiffness on the nonlinear modal damping ratio and modal frequency are analysed. Moreover, the softening characteristics of the active ESDFD are revealed, and the critical speed intervals of the active ESDFD/dual-rotor system are determined. Furthermore, by using the harmonic balance–alternating frequency/time domain (HB–AFT) method, the steady-state response of the system under unbalanced excitation is calculated. The accuracy and effectiveness of nonlinear modal analysis are validated based on the relationships between nonlinear modes and steady-state unbalanced responses. Conversely, the vibration mitigation effects of active ESDFD are determined by the unbalanced response amplitude. Additionally, the controllable region and optimal normal force for effective vibration control in the target mode are defined. Depending on the controllable region, a control strategy for turning on/off the optimal normal force is developed. The findings demonstrate that the developed control strategy enables the active ESDFD to significantly reduce the response amplitude of the dual-rotor system across various excitation levels, showing substantial potential for engineering applications.