Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson＇s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.

Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

A Broad Range Triboelectric Stiffness Sensor for Variable Inclusions Recognition

With the development of artificial intelligence,stiffness sensors are extensively utilized in various fields,and their integration with robots for automated palpation has gained significant attention.This study presents a broad range self-powered stiffness sensor based on the triboelectric nanogenerator(Stiff-TENG)for variable inclusions in soft objects detection.The Stiff-TENG employs a stacked structure comprising an indium tin oxide film,an elastic sponge,a fluorinated ethylene propylene film with a conductive ink electrode,and two acrylic pieces with a shielding layer.Through the decoupling method,the Stiff-TENG achieves stiffness detection of objects within 1.0 s.The output performance and characteristics of the TENG for different stiffness objects under 4 mm displacement are analyzed.The Stiff-TENG is successfully used to detect the heterogeneous stiffness structures,enabling effective recognition of variable inclusions in soft object,reaching a recognition accuracy of 99.7%.Furthermore,its adaptability makes it well-suited for the detection of pathological conditions within the human body,as pathological tissues often exhibit changes in the stiffness of internal organs.This research highlights the innovative applications of TENG and thereby showcases its immense potential in healthcare applications such as palpation which assesses pathological conditions based on organ stiffness.

Variable stiffness tuned particle dampers for vibration control of cantilever boring bars

This research proposes a novel type of variable stiffness tuned particle damper(TPD)for reducing vibrations in boring bars.The TPD integrates the developments of particle damping and dynamical vibration absorber,whose frequency tuning principle is established through an equivalent theoretical model.Based on the multiphase flow theory of gas-solid,it is effective to obtain the equivalent damping and stiffness of the particle damping.The dynamic equations of the coupled system,consisting of a boring bar with the TPD,are built by Hamilton’s principle.The vibration suppression of the TPD is assessed by calculating the amplitude responses of the boring bar both with and without the TPD by the Newmark-beta algorithm.Moreover,an improvement is proposed to the existing gas-solid flow theory,and a comparative analysis of introducing the stiffness term on the damping effect is presented.The parameters of the TPD are optimized by the genetic algorithm,and the results indicate that the optimized TPD effectively reduces the peak response of the boring bar system.

Environmental Adaptive Control of a Snake-like Robot With Variable Stiffness Actuators

This work investigates adaptive stiffness control and motion optimization of a snake-like robot with variable stiffness actuators. The robot can vary its stiffness by controlling magnetorheological fluid(MRF) around actuators. In order to improve the robot's physical stability in complex environments, this work proposes an adaptive stiffness control strategy. This strategy is also useful for the robot to avoid disturbing caused by emergency situations such as collisions. In addition, to obtain optimal stiffness and reduce energy consumption, both torques of actuators and stiffness of the MRF braker are considered and optimized by using an evolutionary optimization algorithm. Simulations and experiments are conducted to verify the proposed adaptive stiffness control and optimization methods for a variable stiffness snake-like robots.

Improving the performance of conventional base isolation systems by an external variable negative stiffness device under near-fault and long-period ground motions

Recent studies have shown that base-isolated objects with long fundamental natural periods are highly influenced by long-period earthquakes. These long-period waves result in large displacements for isolators, possibly leading to exceedance of the allowable displacement limits. Conventional isolation systems, in general, fail to resist such large displacements. This has prompted the need to modify conventional base isolation systems. The current work focuses on the development of an external device, comprising a unit of negative and positive springs, for improving the performance of conventional base isolation systems. This unit accelerates the change in the stiffness of the isolation system where the stiffness of the positive spring varies linearly in terms of the displacement response of the isolated objects. The target objects of the present study are small structures such as computer servers, sensitive instruments and machinery. Numerical studies show that the increase in the damping of the system and the slope of the linear function is effective in reducing the displacement response. An optimal range of damping values and slope, satisfying the stability condition and the allowable limits of both displacement and acceleration responses when the system is subjected to near-fault and long-period ground motions simultaneously, is proposed.

A new variable stiffness absorber based on magneto-rheological elastomer

A new adaptive variable stiffness absorber was proposed based on a smart material, magnetorheological elastomer (MRE), and its vibration control performance was investigated. Before developing the proposed absorber, the MREs were firstly fabricated by curing a mixture of 704 silicon rubber, carbonyl iron particles and a small amount of silicone oil under an external magnetic field. Then the mechanical properties of the fabricated MREs were measured. On the basis of the measured mechanical characteristics, the MRE absorber was developed and its working characteristics were also tested under various input currents and excited frequencies. Finally, the control responses of a two-degree-of-freedom dynamic system with a MRE absorber were presented under a chirp input and used to evaluate the effectiveness of the MRE absorber.

A novel semi-active TMD with folding variable stiffness spring

An innovative variable stiffness device is proposed and investigated based on numerical simulations. The device, called a folding variable stiffness spring （FVSS）, can be widely used, especially in tuned mass dampers （TMDs） with adaptive stiffness. An important characteristic of FVSS is its capability to change the stiffness between lower and upper bounds through a small change of distance between its supports. This special feature results in lower time-lag errors and readjustment in shorter time intervals. The governing equations of the device are derived and simplified for a symmetrical FVSS with similar elements. This device is then used to control a single-degree-of-freedom （SDOF） structure as well as a multi-degree-of-freedom （MDOF） structure via a semi-active TMD. Numerical simulations are conducted to compare several control cases for these structures. To make it more realistic, a real direct current motor with its own limitations is simulated in addition to an ideal control case with no limitations and both the results are compared. It is shown that the proposed device can be effectively used to suppress undesirable vibrations of a structure and considerably improves the performance of the controller compared to a passive device.

A Stiffness Variable Passive Compliance Device with Reconfigurable Elastic Inner Skeleton and Origami Shell

Devices with variable stiffness are drawing more and more attention with the growing interests of human-robot interaction,wearable robotics,rehabilitation robotics,etc.In this paper,the authors report on the design,analysis and experiments of a stiffness variable passive compliant device whose structure is a combination of a reconfigurable elastic inner skeleton and an origami shell.The main concept of the reconfigurable skeleton is to have two elastic trapezoid four-bar linkages arranged in orthogonal.The stiffness variation generates from the passive deflection of the elastic limbs and is realized by actively switching the arrangement of the leaf springs and the passive joints in a fast,simple and straightforward manner.The kinetostatics and the compliance of the device are analyzed based on an efficient approach to the large deflection problem of the elastic links.A prototype is fabricated to conduct experiments for the assessment of the proposed concept.The results show that the prototype possesses relatively low stiffness under the compliant status and high stiffness under the stiff status with a status switching speed around 80 ms.

Force Control Compensation Method with Variable Load Stiffness and Damping of the Hydraulic Drive Unit Force Control System

Each joint of hydraulic drive quadruped robot is driven by the hydraulic drive unit（HDU）, and the contacting between the robot foot end and the ground is complex and variable, which increases the difficulty of force control inevitably. In the recent years, although many scholars researched some control methods such as disturbance rejection control, parameter self-adaptive control, impedance control and so on, to improve the force control performance of HDU, the robustness of the force control still needs improving. Therefore, how to simulate the complex and variable load characteristics of the environment structure and how to ensure HDU having excellent force control performance with the complex and variable load characteristics are key issues to be solved in this paper. The force control system mathematic model of HDU is established by the mechanism modeling method, and the theoretical models of a novel force control compensation method and a load characteristics simulation method under different environment structures are derived, considering the dynamic characteristics of the load stiffness and the load damping under different environment structures. Then, simulation effects of the variable load stiffness and load damping under the step and sinusoidal load force are analyzed experimentally on the HDU force control performance test platform, which provides the foundation for the force control compensation experiment research. In addition, the optimized PID control parameters are designed to make the HDU have better force control performance with suitable load stiffness and load damping, under which the force control compensation method is introduced, and the robustness of the force control system with several constant load characteristics and the variable load characteristics respectively are comparatively analyzed by experiment. The research results indicate that if the load characteristics are known, the force control compensation method presented in this paper has positive compensation effects on the load characteristics variation, i.e., this method decreases the effects of the load characteristics variation on the force control performance and enhances the force control system robustness with the constant PID parameters, thereby, the online PID parameters tuning control method which is complex needs not be adopted. All the above research provides theoretical and experimental foundation for the force control method of the quadruped robot joints with high robustness.

Semiactive variable stiffness control for parametric vibration of cables

In this paper, a semiactive variable stiffness （SVS） device is used to decrease cable oscillations caused by parametric excitation, and the equation of motion of the parametric vibration of the cable with this SVS device is presented. The ON/OFF control algorithm is used to operate the SVS control device. The vibration response of the cable with the SVS device is numerically studied for a variety of additional stiffness combinations in both the frequency and time domains and for both parametric and classical resonance vibration conditions. The numerical studies further consider the cable sag effect. From the numerical results, it is shown that the SVS device effectively suppresses the cable resonance vibration response, and as the stiffness of the device increases, the device achieves greater suppression of vibration. Moreover, it was shown that the SVS device increases the critical axial displacement of the excitation under cable parametric vibration conditions.

Theoretical and experimental research on a new system of semi-active structural control with variable stiffness and damping

In this paper,a new system of semi active structural control with active variable stiffness and damping (AVSD) is suggested.This new system amplifies the structural displacement to dissipate more energy,and in turn,effectively reduces the structural response in the case of relatively small story drifts,which occur during earthquakes.A predictive instantaneous optimal control algorithm is established for a SDOF structure equipped with an AVSD system Comparative shaking table tests of a 1/4 scale single story structural model with a full scale control device have been conducted.From the experimental and analytical results,it is shown that when compared to structures without control or with the active variable stiffness control alone, the suggested system exhibits higher efficiency in controlling the structural response,requires less energy input,operates with higher reliability,and can be manufactured at a lower cost and used in a wider range of engineering applications.

Approximate analysis method for displacement responses of structures with active variable stiffness systems

The performance of structures with active variable stiffness (AVS) systems exhibits strong nonlinearity due to the variety with time of the stiffness of each storey unit,in which the AVS system is installed.Hence,the classical dynamic analysis method for linear structures,such as the mode-superposition method,is not applicable to structures with AVS systems.In this paper,an approximate analysis method is proposed for displacement responses of structures with AVS systems.Firstly,an equivalent relationship between single-degree-of-freedom (SDOF) structures equipped with AVS systems and so-called fictitious linear structures is established.Then,an approximate mode-superposition (AMS) method is presented for multi-degree-of-freedom (MDOF) structures equipped with AVS systems.The accuracy of this method is investigated through extensive parametrical study using different types of earthquake excitations,and some modification is made to the method. Numerical calculation results indicate that the modified AMS method is effective for estimating the maximum displacements relative to the ground and the maximum interstorey drifts of MDOF structures equipped with AVS systems.

Variable Stiffness Identification and Configuration Optimization of Industrial Robots for Machining Tasks

Industrial robots are increasingly being used in machining tasks because of their high flexibility and intelligence.However,the low structural stiffness of a robot significantly affects its positional accuracy and the machining quality of its operation equipment.Studying robot stiffness characteristics and optimization methods is an effective method of improving the stiffness performance of a robot.Accordingly,aiming at the poor accuracy of stiffness modeling caused by approximating the stiffness of each joint as a constant,a variable stiffness identification method is proposed based on space gridding.Subsequently,a task-oriented axial stiffness evaluation index is proposed to quantitatively assess the stiffness performance in the machining direction.In addition,by analyzing the redundant kinematic characteristics of the robot machining system,a configuration optimization method is further developed to maximize the index.For numerous points or trajectory-processing tasks,a configuration smoothing strategy is proposed to rapidly acquire optimized configurations.Finally,experiments on a KR500 robot were conducted to verify the feasibility and validity of the proposed stiffness identification and configuration optimization methods.

In-Plane Impact Dynamics Analysis of Re-Entrant Honeycomb with Variable Cross-Section

Due to the unique deformation characteristics of auxetic materials(Poisson’s ratioμ<0),they have better shock resistance and energy absorption properties than traditional materials.Inspired by the concept of variable crosssection design,a new auxetic re-entrant honeycomb structure is designed in this study.The detailed design method of re-entrant honeycomb with variable cross-section(VCRH)is provided,and five VCRH structures with the same relative density and different cross-section change rates are proposed.The in-plane impact resistance and energy absorption abilities of VCRH under constant velocity are investigated by ABAQUS/EXPLICIT.The results show that the introduction of variable cross-section design can effectively improve the impact resistance and energy absorption abilities of auxetic re-entrant honeycombs.The VCRH structure has better Young’s modulus,plateau stress,and specific energy absorption(SEA)than traditional re-entrant honeycomb(RH).The influence of microstructure parameters(such as cross-section change rateα)on the dynamic impact performance of VCRH is also studied.Results show that,with the increase in impact velocity andα,the plateau stress and SEA of VCRH increase.A positive correlation is also found between the energy absorption efficiency,impact load uniformity andαunder both medium and high impact speeds.These results can provide a reference for designing improved auxetic re-entrant honeycomb structures.

IGA Based Bi-Layer Fiber Angle Optimization Method for Variable Stiffness Composites

This paper presents a topology optimization method for variable stiffness composite panels with varying fiber orientation and curvilinear fiber path.Non-uniform rational B-Splines(NURBS)based Isogeometric analysis(IGA)is utilized for the numerical computation of the general minimum compliance problem.The sensitivity analysis of the structure compliance function for the density and bi-layer orientation is conducted.The bi-layer fiber paths in the design domain are generated using streamline method and updated by divided pieces reselection method after the optimization process.Several common examples are tested to demonstrate the effectiveness of the method.The results show that the proposed method can generate more manufacturable fiber paths than some typical topology optimization methods.

An XBi-CFAO Method for the Optimization of Multi-Layered Variable Stiffness Composites Using Isogeometric Analysis

This paper presents an effective fiber angle optimization method for two and multi-layered variable stiffness composites.A gradient-based fiber angle optimization method is developed based on isogeometric analysis(IGA).Firstly,the element densities and fiber angles for two and multi-layered composites are synchronously optimized using an extended Bi-layered continuous fiber angle optimization method(XBi-CFAO).The densities and fiber angles in the base layer are attached to the control points.The structure response and sensitivity analysis are accomplished using the non-uniform rational B-spline(NURBS)based IGA.By the benefit of the B-spline space,this method is free from checkerboards,and no additional filtering is needed to smooth the sensitivity numbers.Then the curved fiber paths are generated using the streamline method and the discontinuous fiber paths are smoothed using a partitioned selection process.The proposed method in the paper can alleviate the phenomenon of fiber discontinuity,enhance information retention for the optimized fiber angles of the singular points and save calculating resources effectively.

Equivalent bending stiffness of simply supported preflex beam bridge with variable cross-section

In order to understand mechanical characters and find out a calculating method for preflex beams used in particular bridge engineering projects, two types of simply supported preflex beams with variable cross-section, preflex beam with alterative web depth and preflex beam with alterative steel flange thickness, are discussed on how to achieve the equivalent moment of inertia and Young’s modulus. Additionally, methods of calculating the equivalent bending stiffness and post-cracking deflection are proposed. Results of the experiments on 6 beams agree well with the theoretical analysis, which proves the correctness of the proposed formulas.

The theoretical analysis of dynamic response on cantilever beam of variable stiffness

The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable coefficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last,the study obtains the solution of dynamic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied,the dynamic characteristics are analyzed under several cases. The results demonstrate that the natural angular frequency reduces as the flexural rigidity reduces. When the rigidity of clamped end is higher than that of free end,low-level mode contributes the larger displacement response to the total response. On the contrary,the contribution of low-level mode is lesser than that of high-level mode.

Vibration Absorption Efficiency and Higher Branches Elimination of Variable-Stiffness Nonlinear Energy Sink

Vibration absorption efficiency of a variable⁃stiffness nonlinear energy sink(NES)was investigated when the main oscillator was subjected to harmonic and impulse excitations.The slow flow equations of the coupled system were derived by using the complexification⁃averaging method,and the nonlinear equations which describe the steady⁃state response were obtained.As the harmonic excitation force increased,the system which comprises constant⁃stiffness NES generated higher branch responses,greatly reducing the vibration absorption efficiency.The influence of nonlinear stiffness on the responses of the system was investigated.Results show that,with the increase of harmonic exciting force,a reduction of NES stiffness can eliminate the higher branch responses and even the frequency band of strongly modulated responses.The vibration absorption efficiency of variable⁃stiffness NES attached to the linear oscillator for different amplitudes of impulse excitation was investigated.Results show that the proper reduction of nonlinear stiffness under increasing impulse excitation can greatly increase the vibration absorption efficiency of NES,and the variable⁃stiffness design can effectively mitigate the negative influences of the increase of the excitation amplitude on the efficiency of constant⁃stiffness NES.