Analytical Stiffness Model Research Articles

Analysis and experiment of a novel compact magnetic spring with high linear negative stiffness

Improving the isolation performance of vibration isolators by introducing magnetic negative stiffness mechanisms (NSMs) has attracted extensive attention in recent years. Nevertheless, common magnetic NSMs are always accompanied by nonlinear stiffness, resulting in degradation of isolation performance and system instability when subjected to excitations with large amplitude. To improve the stiffness linearity and working range of magnetic NSMs in a limited space, a novel compact magnetic spring with high linear negative stiffness (CMS-HLNS) is proposed in this paper. The CMS-HLNS consists of two sets of permanent magnets (PMs) arranged in an alternate manner, which has both high linearity and high negative stiffness properties. A simplified analytical stiffness model of the CMS-HLNS is deduced based on the magnetic charge method. The effects of design parameters on the stiffness characteristics of the CMS-HLNS are investigated. Finally, experimental prototypes are constructed to verify the feasibility of the analytical model and the superior vibration isolation performance when introducing the CMS-HLNS. The experimental results demonstrate that the CMS-HLNS can achieve linear negative stiffness in a large working range under the constraint of space and is suitable for (ultra-) low-frequency isolation under excitations with large amplitude. It’s believed that the magnetic spring proposed in this paper can be applied as an innovative technology in engineering practice, and its good linearity and compact structure make it extremely adaptable.

Dynamic behaviors and contact characteristics of ball bearings in a multi-supported rotor system under the effects of 3D clearance fit

Clearance fit is a common assembly method for bearings in rotor systems, which leads to a certain clearance between the outer ring of the bearing and the bearing pedestal, that is, the fit clearance. The nonlinear factors caused by the fit clearance may affect the mechanical properties of the bearing and rotor system. Considering the fillet and six-degree-of-freedom (6-DOF) motion of the outer ring, a three-dimensional (3D) clearance fit model is proposed. In addition, considering the interference fit between the inner ring and spindle, an analytical stiffness model of ball bearings is developed. By coupling the clearance fit model and the analytical bearing stiffness model with a finite element (FE) model of a spindle, a dynamic model of a multi-supported rotor system is established and verified by the experiment, which fully considers the nonlinear factors of ball-raceway contact and outer ring-bearing pedestal contact. The effects of fit clearance, radial force, and axial force on the vibration behaviors and contact characteristics of the ball bearing in the system are analyzed. The results indicate that the larger fit clearance of the unilateral bearing may lead to misalignment of the bearing and reduce the bearing force. Under radial force, the bearing clearance and fit clearance are eliminated, which improves the stability of the system. At this time, further application of axial force causes the load-bearing area to become larger and weakens the variable compliance (VC) characteristics of the bearing. This investigation can provide technical support for the assembly and design of the bearing-rotor system.

Design of a redundant actuated 4-PPR planar 3-DOF compliant nanopositioning stage

The compliant planar positioning stage actuated by voice coil motors can obtain nanoscale motion accuracy within a millimeter motion range and has received increasing attention in precision engineering. However, there is always a contradiction between the actuation force and the output stiffness in the design of the stage. This study presents the design of a novel redundant actuated planar 3-DOF-compliant nanopositioning stage. A redundant actuated configuration can increase the output stiffness of the stage and decrease the peak actuating force. The analytical stiffness model, kinetostatic model, and dynamic model of the stage are established, which illustrate the high force transmission efficiency and dynamic decoupling feature of the stage. The geometric parameters were optimized based on the established models, and model validation and performance evaluation of the stage were conducted via finite element analysis. Finally, a prototype positioning stage is fabricated. Experimental results show that the proposed stage can achieve a motion range of up to ±2.5 mm × ±2.5 mm × ±2.5° with a resolution of 80 nm and 0.28″, and the tracking error ratio for a composite Lissajous trajectory is within 0.102%.

Analytical bending stiffness model of composite shaft with breathing fatigue crack

In this paper, an improved stiffness model of the breathing crack of the composite shaft tube was established based on the Layer-wise theory, which considered the influence of orientation angle and stacking sequence of the cracked layers. Then, the model was compared with the horizontal neutral axis model, Mayes' model and analytical model. The results showed that the improved model can more effectively reflect the influence of the laminate parameters on the stiffness. Based on the model, the bending stiffness characteristics of the cracked composite shaft were studied, and the effects of laminate parameters on the stiffness were obtained.

Mechanical design and determination of bandwidth for a two-axis inertial reference unit

The inertial reference unit (IRU) serves as a master reference for acquisition, tracking and pointing (ATP) system due to its ability to maintain stability with respect to inertial frame. However, the bandwidth of IRU system is usually limited by the dynamics of mounted inertial sensors and structural resonances. In this paper, magnetohydrodynamics (MHD) angular rate sensor (ARS) with extremely low noise at high frequencies (>3Hz) was combined with a conventional gyro serving as inertial sensing unit of the developed IRU. This sensing unit was verified experimentally to exhibit a nearly perfect transfer function with “unity” gain and “zero” phase error. A central hinge that allows only rotation about the actuating axes was used as the supporting structure. A structural parameter design approach based on analytical stiffness model was developed for the central hinge. The relative errors between theoretical stiffness values, finite element analysis (FEA) simulations and experimental data were found to be within ±13%. A double-T network notch filter with optimal parameters was implemented to suppress the low-frequency mechanical resonance. With the notch filter in the forward path, the open-loop dynamic model of the IRU was measured with the results indicating a completely damped resonant peak and an identification error less than ±0.5 dB in magnitude and ±5° in phase. The developed IRU was ultimately certified to achieve more than ±6 mrad as a drive range and more than 120 Hz as a closed-loop bandwidth under a digital controller designed with frequency shaping technique.

Parametric sensitivity research of interference-fit bolted single-lap laminates joint based on an improved analytical stiffness model

Because of the excellent static and fatigue performance, the interference-fit bolted structure has a wide application prospect in the joint field. In this paper, an improved spring-mass stiffness analytical prediction model is established for the interference-fit bolted single-lap laminated composite structure. The influences of interference-fit percentage, bolt preload, secondary bending and interface frictions are considered in the model. Combined with experimental research, the value of secondary bending moment coefficient ε is studied, and the correctness of the analytical model is verified. Based on the improved stiffness model, parametric research and regression analysis on the interference-fit percentage, preload, friction, laminate width and material properties are carried out and show that the overall structure stiffness is obviously affected by ε value, laminate width and laminates properties. The stiffness decreases with the increase of ε and increases with the increase of laminate width. And as the key factors, the interference-fit percentage mainly affects the joint local friction and bolt shear stiffness, the preload and friction coefficient mainly affect the local friction, and the laminates sizes and properties directly affect the overall structural stiffness.

Optimizing the Rigid or Compliant Behavior of a Novel Parallel-Actuated Architecture for Exoskeleton Robot Applications.

The purpose of this work is to optimize the rigid or compliant behavior of a new type of parallel-actuated robot architecture developed for exoskeleton robot applications. This is done in an effort to provide those that utilize the architecture with the means to maximize, minimize, or simply adjust its stiffness property so as to optimize it for particular tasks, such as augmented lifting or impact absorption. This research even provides the means to produce non-homogeneous stiffness properties for applications that may require non-homogeneous dynamic behavior. In this work, the new architecture is demonstrated in the form of a shoulder exoskeleton. An analytical stiffness model for the shoulder exoskeleton is created and validated experimentally. The model is then used, along with a method of bounded nonlinear multi-objective optimization to configure the parallel substructures for desired rigidity, compliance or nonhomogeneous stiffness behavior. The stiffness model and its optimization can be applied beyond the shoulder to any embodiment of the new parallel architecture, including hip, wrist and ankle robot applications. In order to exemplify this, we present the rigidity optimization for a theoretical hip exoskeleton.

Design and experimental study of a passive power-source-free stiffness-self-adjustable mechanism

Passive variable stiffness joints have unique advantages over active variable stiffness joints and are currently eliciting increased attention. Existing passive variable stiffness joints rely mainly on sensors and special control algorithms, resulting in a bandwidth-limited response speed of the joint. We propose a new passive power-source-free stiffness-self-adjustable mechanism that can be used as the elbow joint of a robot arm. The new mechanism does not require special stiffness regulating motors or sensors and can realize large-range self-adaptive adjustment of stiffness in a purely mechanical manner. The variable stiffness mechanism can automatically adjust joint stiffness in accordance with the magnitude of the payload, and this adjustment is a successful imitation of the stiffness adjustment characteristics of the human elbow. The response speed is high because sensors and control algorithms are not needed. The variable stiffness principle is explained, and the design of the variable stiffness mechanism is analyzed. A prototype is fabricated, and the associated hardware is set up to validate the analytical stiffness model and design experimentally.

The influence of 3D microstructural features on the elastic behaviour of tow-based discontinuous composites

Tow-based discontinuous composites consist of carbon-fibre tows dispersed in a polymeric matrix, resulting in a complex microstructure with inherent 3D features such as tow waviness, non-uniform tow thickness, and fibre content variations; however, most models for these materials are based on 2D formulations, or in very computationally-expensive FE models. This work proposes a computationally-efficient microstructure generator able to recreate the 3D features of tow-based discontinuous composites, coupled with an analytical stiffness model able to quantify the effect of tow waviness on the modulus of these materials, for the first time in the literature. The features of the microstructures generated are validated against 3D measurements obtained through micro-CT scans and optical micrographs. The results of the stiffness model are successfully validated against experimental results from the literature, showing that tow waviness can trigger a knock-down of over 20% on the stiffness of tow-based discontinuous composites. It is also shown that current testing standards can lead to uncertainties over 20% on the measurement of the Young’s modulus of tow-based discontinuous composites, and alternatives to define suitable experimental campaigns are proposed.

Influence of joints flexibility on overall stiffness of a 3-PRUP compliant parallel manipulator

The main aim of this paper is the influence assessment of the one of the most significant non-geometrical errors called compliance errors on accuracy of a 3-PRUP parallel robot. In this paper, a detailed and comprehensive study is performed to evaluate the effects of the joints and bodies flexibility on position accuracy of the robot. For this purpose, a continuous modeling method based on the energy method and Castigliano's 2nd theorem is presented. The capabilities of this method significantly reduce the limiting assumptions to obtain a highly accurate analytical stiffness model. Using the capabilities of this method, the compliance errors modeling of flexible bodies with complex geometry will be possible as well as the bending and torsional moments, shear forces and weight of all robot compliant modules can be considered as continuous loads. Using the concept of Wrench Compliant Module Jacobian, WCMJ, matrix and physical/structural properties matrix of each module, the compliance matrix of each flexible module is independently obtained. Using a computational algorithm, a comprehensive study is performed to obtain the contribution of the joints flexibility on the robot accuracy throughout the workspace. Finally, to evaluate the compliance errors boundaries throughout the workspace, the concept of General Compliance Error Range, GCER, is presented. The GCER represents a range between maximum and minimum accuracy of robot relative to the compliance errors.

Deterministic Design for a Compliant Parallel Universal Joint With Constant Rotational Stiffness

Compliant universal joints have been widely employed in high-precision fields due to plenty of good performance. However, the stiffness characteristics, as the most important consideration for compliant mechanisms, are rarely involved. In this paper, a deterministic design for a constraint-based compliant parallel universal joint with constant rotational stiffness is presented. First, a constant stiffness realization principle is proposed by combination of the freedom and constraint topology (FACT) method and beam constraint model (BCM) to establish a mapping relationship between stiffness characteristics and topology configurations. A parallel universal joint topology is generated by the constant stiffness realization principle. Then, the analytical stiffness model of the universal joint with some permissible approximations is formulated based on the BCM, and geometrical prerequisites are derived to achieve the desired constant rotational stiffness. After that, finite element analysis (FEA), experimental testing, and detailed stiffness analysis are carried out. It turns out that the rotational stiffness of the universal joint can keep constant with arbitrary azimuth angles even if the rotational angle reaches up to ±5 deg. Meanwhile, the acceptable relative errors of rotational stiffness are within 0.53% compared with the FEA results and 2.6% compared with the experimental results, which indicates the accuracy of the theoretical stiffness model and further implies the feasibility of constant stiffness realization principle on guiding the universal joint design.

Analytical stiffness model of a fluid-filled U-shaped bellows based three-parameter fluid damper for micro-vibration suppression

Bellows are widely employed to construct three-parameter fluid damper in application to on-orbit micro-vibration isolation thanks to its ability to provide accurate stiffness and deformation without dry friction. An analytical stiffness model is established to provide a designing method for a U-shaped bellows based three-parameter fluid damper, which considers coupling between bellows and fluid to account for fluid–structure interaction (FSI). Firstly, the stiffness and volume deformation for a U-shaped bellow under composite axial load and internal pressure are obtained based on the analytical model of a U-shaped bellow. Then, the FSI model of a three-parameter damper is established to obtain the output force and internal pressure under harmonic displacement input. The stiffness model employs expressions of volume deformation, the pressure equation and the continuity equation in the orifice to derive the coupling equation, and the force and pressure are obtained by solving the equation with the Laplace transform method. The stiffness model of the bellow is verified by a finite element model. The stiffness model of the U-shaped bellows based three-parameter damper is validated with a two-way FSI analysis carried out in ANSYS WORKBENCH. The results demonstrate that analytical prediction is in good agreement with numerical simulation with greatly enhanced efficiency.

Development of a novel 3-DOF suspension mechanism for multi-function stylus profiling systems

This paper proposes a novel 3-DOF suspension mechanism for multi-function stylus profiling systems. Incorporating an electromagnetic force actuator, the 3-DOF suspension mechanism provides a controlled loading force. For reasons of the thermal and mechanical stability, a triangular flexure structure is utilized to support the stylus. The stiffness matrix method is used to establish the analytical stiffness model of the 3-DOF suspension mechanism. Considering the 3-DOF suspension mechanism as a 3-DOF lumped-mass-spring system, the dynamic model is established. Finite element analysis (FEA) is used to validate the established static and dynamic models of the 3-DOF suspension mechanism. A prototype is fabricated and experimental tests are carried out to characterize the mechanism’s performance. The results show that the 3-DOF suspension mechanism provides a controlled force in a range of up to 10 mN and has a working range in excess of 10 μm with a first natural frequency of 342 Hz in Z axis, indicating good capability for multi-function measurements at the micro/nano scale.

Hybrid-model-based stiffness analysis of a three-revolute-prismatic-spherical parallel kinematic machine

A three-revolute-prismatic-spherical parallel kinematic machine is proposed as an alternative solution for high-speed machining tool due to its high rigidity and high dynamics. Considering the parallel kinematic machine module as a typical compliant parallel mechanism, whose three limb assemblages have bending, extending and torsional deflections, this article proposes a hybrid modeling methodology to establish an analytical stiffness model for the three-revolute-prismatic-spherical device. The developed analytical model is further used to evaluate the stiffness mapping of the three-revolute-prismatic-spherical module over a given work plane which is then validated by experimental tests. The simulations and experiments indicate that the present hybrid methodology can predict the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness in a quick and accurate manner. The solution for eigenvalue problem of the stiffness matrix leads to the stiffness characteristics of the parallel module including eigenstiffnesses and the corresponding eigenscrews as well as the equivalent screw spring constants. Based on the eigenscrew decomposition, the parallel kinematic machine is physically interpreted as a rigid platform suspending by six screw springs. The minimum, maximum and average of the screw spring constants are chosen as indices to assess the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness performance. The distributions of the proposed indices throughout the workspace reveal a strong dependency on the mechanism’s configurations. At the final stage, the effects of some design parameters on system stiffness characteristics are investigated with the purpose of providing useful information for the conceptual design and performance improvement of the parallel kinematic machine.

Semi-Analytical Approach for Stiffness Estimation of 3-DOF PKM

Due to their advantageous of high stiffness, high speed, large load carrying capacity and complicated surface processing ability, PKMs (Parallel Kinematic Manipulators) have been applied to machine tools. This paper mainly addresses the issue of stiffness formulation of a three-prismatic- revolute-spherical PKM (3-PRS PKM). The manipulators consist of three kinematic limbs of identical topology structure, and each limb is composed of an actuated prismatic-revolute-spherical. In order to build up the stiffness model, kinematics, Jacobian and finite element analysis are also performed as the basis. Main works in this paper can be outlined as follows. By use of approaches of vector, inverse position analysis of 3-PRS PKM is conducted. When the independent position and orientation parameters of the end-effectors are provided, the translational distances of active prismatic joints can be determined. Then with the aid of the wrench and reciprocal screw theory, the overall Jacobian of this manipulator is formulated quickly, and which is a six by six dimensional matrix and can reflect all information about actuation and constraint singularity. After for- mulating the position analysis and Jacobian matrix, the next step is stiffness analysis. Analytical stiffness model, a function of Jacobian matrix and components stiffness matrix, is obtained first using the principle of virtual work. Stiffness model is also a six by six dimensional matrix and can provide the information of actuation and constraint stiffness simultaneously. For the complex geometry shape of some components, it is impossible to know their stiffness distributions with the varying configuration. Therefore, ANSYS technology has to be applied to compute the stiffness coefficients of these components at different configurations. Then, the computed data are used to obtain the stiffness distribution by use of the numerical fitting method. Up to now, the semi-ana- lytical stiffness model of the manipulator is completely formulated and can be applied to estimate the system stiffness of 3-PRS PKM. The model enables the stiffness of a 3-PRS PKM to be quickly estimated. Provided with the geometry parameters and load situation on tool tip, the stiffness of 3-PRS PKM system is estimated based on the stiffness matrix about tool tip which is obtained by transforming the point from the center of circle composed of three S joints to the tool tip. Then, the stiffness of system along x, y and z directions can be solved. In order to testify the correctness, the corresponding stiffness is also obtained by use of FEA software. The stress distribution and fre- quency of system are also gained by solving the FEA model.

Anisotropic force ellipsoid based multi-axis motion optimization of machine tools

The existing research of the motion optimization of multi-axis machine tools is mainly based on geometric and kinematic constraints, which aim at obtaining minimum-time trajectories and finding obstacle-free paths. In motion optimization, the stiffness characteristics of the whole machining system, including machine tool and cutter, are not considered. The paper presents a new method to establish a general stiffness model of multi-axis machining system. An analytical stiffness model is established by Jacobi and point transformation matrix method. Based on the stiffness model, feed-direction stiffness index is calculated by the intersection of force ellipsoid and the cutting feed direction at the cutter tip. The stiffness index can help analyze the stiffness performance of the whole machining system in the available workspace. Based on the analysis of the stiffness performance, multi-axis motion optimization along tool paths is accomplished by mixed programming using Matlab and Visual C++. The effectiveness of the motion optimization method is verified by the experimental research about the machining performance of a 7-axis 5-linkage machine tool. The proposed research showed that machining stability and production efficiency can be improved by multi-axis motion optimization based on the anisotropic force ellipsoid of the whole machining system.