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%.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Research of Elastic and Elastic-Plastic Deformation on Bending Problems of Variable Stiffness Beams

A novel variable stiffness model was proposed for analyzing elastic-plastic bending problems with arbitrary variable stiffness in detail.First,it was assumed that the material of a rectangular beam is an ideal isotropic elastic-plastic material,whose elastic modulus,yield strength,and section height are functions of the axial coordinates of the beam respectively.Considering the effect of shear on the deformation of the beam,the elastic and elastic-plastic bending problems of the axially variable stiffness beam were studied.Then,the analytical solutions of the elastic and elastic-plastic deformation of the beam were derived when the cross-section height and the elastic modulus of the material were varied by special function along the length of the beam respectively.The elastic and elastic-plastic analysis of the variable stiffness beam was carried out using Differential Quadrature Method(DQM)when the bending stiffness varied arbitrarily.The influence of the axial variation of the bending stiffness on the elastic and elastic-plastic deformation of the beam was analyzed by numerical simulation,DQM,and finite element method(FEM).Simulation results verified the practicability of the proposed mechanical model,and the comparison between the results of the solutions of DQM and FEM showed that DQM is accurate and effective in elastic and elastic-plastic analysis of variable stiffness beams.

Variable stiffness control algorithm for high-rising buildings of closely spaced frequencies based on wavelet transforms

To establish the algorithm of SAT-TMD system with the wavelet transform(WT),the modal mass participation ratio is proposed to distinguish if the high-rising structure has the characteristic of closely distributed frequencies.A time varying analytical model of high-rising structure such as TV-tower with the SAT-TMD is developed.The proposed new idea is to use WT to identify the dominant frequency of structural response in a segment time,and track its variation as a function of time to retune the SAT-TMD.The effectiveness of SAT-TMD is investigated and it is more robust to change in building stiffness and damping than that of the TMD with a fixed frequency corresponding to a fixed mode frequency of the building.It is proved that SAT-TMD is particularly effective in reducing the response even when the building stiffness is changed by ±15%;whereas the TMD loses its effectiveness under such building stiffness variations.

An EMG-Based Biomimetic Variable Stiffness Modulation Strategy for Bilateral Motor Skills Relearning of Upper Limb Elbow Joint Rehabilitation

Bilateral rehabilitation systems with bilateral or unilateral assistive robots have been developed for hemiplegia patients to recover their one-side paralysis.However,the compliant robotic assistance to promote bilateral inter-limb coordination remains a challenge that should be addressed.In this paper,a biomimetic variable stiffness modulation strategy for the Variable Stiffness Actuator(VSA)integrated robotic is proposed to improve bilateral limb coordination and promote bilateral motor skills relearning.An Electromyography(EMG)-driven synergy reference stiffness estimation model of the upper limb elbow joint is developed to reproduce the muscle synergy effect on the affected side limb by independent real-time stiffness control.Additionally,the bilateral impedance control is incorporated for realizing compliant patient-robot interaction.Preliminary experiments were carried out to evaluate the tracking performance and investigate the multiple task intensities’influence on bilateral motor skills relearning.Experimental results evidence the proposed method could enable bilateral motor task skills relearning with wide-range task intensities and further promote bilateral inter-limb coordination.

A practical PID variable stiffness control and its enhancement for compliant force-tracking interactions with unknown environments

Compliant interaction control is a key technology for robots performing contact-rich manipulation tasks.The design of the compliant controller needs to consider the robot hardware because complex control algorithms may not be compatible with the hardware performance,especially for some industrial robots with low bandwidth sensors.This paper focuses on effective and easy-to-use compliant control algorithms for position/velocity-controlled robots.Inspired by human arm stiffness adaptation behavior,a novel variable target stiffness(NVTS)admittance control strategy is proposed for adaptive force tracking,in which a proportional integral derivative(PID)variable stiffness law is designed to update the stiffness coefficient of the admittance function by the force and position feedback.Meanwhile,its stability and force-tracking capability are theoretically proven.In addition,an impact compensator(Impc)is integrated into the NVTS controller to enhance its disturbance-suppression capability when the robot is subjected to strong vibration disturbances in complicated surface polishing tasks.The proposed controllers are validated through four groups of experimental tests using different robots and the corresponding results demonstrate that they have high-accuracy tracking capability and strong adaptability in unknown environments.

The Novel Variable Stiffness Composite Systems with Characteristics of Repeatable High Load Bearing and Response Rate

On the base of controllable variable stiffness property,variable stiffness composites were the main components of functional materials in aerospace.However,the relatively low mechanical strength,stiffness range,and response rate restricted the application of variable stiffness composite.In this work,the novel variable stiffness composite system with characteristics of repeatable high load bearing and response rate was successfully prepared via the double-layer anisotropic structure to solve the bottlenecks of variable stiffness composites.The novel variable stiffness composite systems were composed of variable stiffness layer of polycaprolactone(PCL)and the driven layer of silicone elastomer with alcohol,which continuously changed Young’s modulus from 0.1 to 7.263 MPa(72.63 times variation)in 200 s and maintained maximum weight of 11.52 times its own weight(8.5 g).Attributed to the relatively high variable stiffness range and load bearing value of variable stiffness composite system,the repeatable response process led to the efficient high load driven as“muscle”and diversified precise grab of objects with different shapes as“gripper”,owning widespread application prospects in the field of bionics.

Optimal Variable Stiffness Control and Its Applications in Bionic Robotic Joints:A Review

Variable Stiffness Actuation(VSA)is an efficient,safe,and robust actuation technology for bionic robotic joints that have emerged in recent decades.By introducing a variable stiffness elastomer in the actuation system,the mechanical-electric energy conversion between the motor and the load could be adjusted on-demand,thereby improving the performance of the actuator,such as the peak power reduction,energy saving,bionic actuation,etc.At present,the VSA technology has achieved fruitful research results in designing the actuator mechanism and the stiffness adjustment servo,which has been widely applied in articulated robots,exoskeletons,prostheses,etc.However,how to optimally control the stiffness of VSAs in different application scenarios for better actuator performance is still challenging,where there is still a lack of unified cognition and viewpoints.Therefore,from the perspective of optimal VSA performance,this paper first introduces some typical structural design and servo control techniques of common VSAs and then explains the methods and applications of the Optimal Variable Stiffness Control(OVSC)approaches by theoretically introducing different types of OVSC mathematical models and summarizing OVSC methods with varying optimization goals and application scenarios or cases.In addition,the current research challenges of OVSC methods and possible innovative insights are also presented and discussed in-depth to facilitate the future development of VSA control.

Vibrations of tandem cold rolling mill: coupled excitation of rolling force and variable stiffness of fault-free back-up roll bearing

Vibration issues of a five-stand tandem cold rolling mill were found in the steel production practice,and the experimental observation and numerical analysis indicated that the vibrations were related to the back-up roll bearing.The results were validated by replacing the back-up roll bearing with the new bearing resulting in 30%decline in vibration amplitude.Models describing the four-row cylindrical roller bearing and the vertical system of the cold rolling mill including the bearing were established.Moreover,the mechanisms of periodic excitation and amplified vibrations of fault-free bearing were explained theoretically,along with the analysis of bifurcation behaviors of the motion states of the roller bearing and rolling mill system.It is found that the energy transmitted between vibrations with different frequencies if multiple excitation frequencies in the rolling mill system were close.

Structural design and stiffness matching control of bionic variable stiffness joint for human–robot collaboration

The physical compliance of interaction is an important requirement for safe and efficient collaboration between robots and humans,and the realization of human–robot compliance requires robot joints with variable stiffness similar to those of human joints.In this study,based on the tissue structure and driving principle of the human arm muscle ligament,a robot joint with variable stiffness is designed,consisting of an elastic belt and serial elastic actuator in parallel.The variable stiffness of the joint is realized by adjusting the tension length of the elastic belt.Surface electromyography(sEMG)signals of the human arm are used as the characterization quantity of joint stiffness to establish the pseudostiffness model of the elbow joint.The stiffness of the robot joints is adjusted in real-time to match the human arm stiffness based on the changes in sEMG signals of the human arm during operation.Real-time compliant interaction of human–robot collaboration is realized based on an end stiffness matching strategy.Additionally,to verify the effectiveness of the human joint stiffness matching-based compliance control strategy,a human–robot cooperative lifting experiment was designed.The bionic variable stiffness joint shows good stiffness adjustment,and the human–robot joint stiffness matching strategy based on human sEMG signals can improve the effectiveness and comfort of human–robot collaboration.

A Variable Stiffness Soft Gripper Using Granular Jamming and Biologically Inspired Pneumatic Muscles

As the domains, in which robots operate change the objects a robot may be required to grasp and manipulate, are likely to vary sig- nificantly and often. Furthermore there is increasing likelihood that in the future robots will work collaboratively alongside people. There has therefore been interest in the development of biologically inspired robot designs which take inspiration from nature. This paper pre- sents the design and testing of a variable stiffness, three fingered soft gripper, which uses pneumatic muscles to actuate the fingers and granular jamming to vary their stiffness. This gripper is able to adjust its stiffness depending upon how fragile/deformable the object being grasped is. It is also lightweight and low inertia, making it better suited to operation near people. Each finger is formed from a cylindrical rubber bladder filled with a granular material. It is shown how decreasing the pressure inside the finger increases the jamming effect and raises finger stiffness. The paper shows experimentally how the finger stiffness can be increased from 21 N·m^-1 to 71 N·m^-1. The paper also describes the kinematics of the fingers and demonstrates how they can be position-controlled at a range of different stiffness values.

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.