Diagonal Stiffness Research Articles

Dynamic contact and stiffness characteristic analysis of angular contact ball bearings under combined external loads

A nonlinear dynamic model of angular contact ball bearing under combined external loads is established based on the nonlinear elastic Hertz contact theory and raceless control theory. The established dynamic model is solved by employing variable step size Newton Raphson iteration method and the validation of model is verified by comparing the proposed results with the corresponding results come from the existing literatures. The solution of the presented model has high accuracy compared with the existing experimental results. In addition, the computational efficiency of the proposed model is improved significantly by introducing iteration step size adjustment factor. According to the proposed model, the dynamic contact and stiffness characteristics of angular contact ball are studied systematically by investigating the effects of combined external load working conditions on the contact parameters including contact angle and contact force and the stiffness parameters including diagonal stiffness and off-diagonal stiffness. This research can provide the theoretical basis and technique guidance for the designation and manufacture of angular contact ball bearing under combined external loads.

Cross-coupling stiffness for natural goal-directed robot motion

The capability of humans to use their natural dynamics for generating explosive motions in a highly-coordinated sequence is a feat that has yet to be reached in robotics. With the introduction of intrinsically elastic joints, great progress towards this goal has been made. However, there are still some challenges associated with this type of actuation, which limits its application. Generating goal-directed sequences has proven difficult as optimal control solutions tend to result in uncoordinated swing-up motions. This can be explained when viewing the structure of the stiffness matrix: If the elastic elements are placed in series with the motor, a diagonal stiffness matrix is generated. This in turn leads to a multitude of frequencies at which the system can oscillate. By adding off-diagonal elements, a dominant primary resonance behaviour can be achieved. Leveraging this cross-coupling stiffness, we show that robots can produce natural goal-directed oscillatory and explosive movements that closely resemble human throwing.

Control of magnetically levitated rotors using stabilizing effects of gyroscopes

Stable control of magnetically levitated rotors with a high gyroscopic effect is an important subject in the field of rotating machinery. This paper describes a method to address this problem using a cross-coupled control approach. In the low and middle speed range a constant diagonal controller and a speed dependent cross-coupled controller is used. In the high speed range the controller takes advantage of the stabilizing effect of gyroscopes. Hence, no positive diagonal stiffness is required in the high speed range in the tilting control path. The damping of the nutation and precession mode is provided only by the cross-coupled control paths. Thus, no diagonal controller is required for the stabilization of the tilting modes. This circumstance reduces the gain and the bandwidth of the control structure, and therefore decreases the impact of sensor noise and the possibility of destabilizing high frequency flexible modes. In this study the stabilizing effect of magnetically stabilized gyroscopes is derived analytically. Furthermore, the damping effects of the cross-coupled controllers are explained, using the eigenvalues of a simplified system. Simulation results show the output sensitivity of the control structures. Based on this knowledge, the optimal switching point between the control structures can be derived. Finally, experimental results on a turbo-molecular pump validate the effectiveness of the proposed control method.

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in ℝd. As a generalisation of the G. Szegö’s family in 1D (1939), the first family of multivariate GHPs (resp. GHFs) are orthogonal with respect to the weight function |x|2μe-|x|2(resp. |x|2μ) in ℝd. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product [u,v]Hs(ℝd)= ((-Δ)s/2u, (-Δ)s/2v)ℝdassociated with the IFL of orders> 0. As an immediate consequence, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient in solving PDEs with the fractional Schrödinger operator: (-Δ)s+ |x|2μwiths∈ (0,1] andμ> −1/2 in ℝdWe construct the second family of multivariate nontensorial Müntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrödinger eigenvalue problems.

Sensitivity Study of the Influence of Blade Sectional Stiffness Parameters on the Aeroelastic Response of Wind Turbines

With the development of wind turbines as a result of large-scale and offshore trends, the wind turbine size is becoming increasingly larger. The passive control technique is used to alleviate the increasing loads on the blade for the sake of improving the durability of the wind turbine. The ply design of shells considering the coupling effect of bending and torsion is one of the passive control techniques. The bending torsion coupling stiffness is one of the parameters of the blade section stiffness matrix. In order to fully understand the influence of each blade stiffness parameter on the aeroelastic responses of wind turbines and to consider the influence of structural characteristics on the aeroelastic responses in blade design, the influences and sensitivity of each stiffness parameter in the 6 × 6 stiffness matrix of the blade sections on the aeroelastic responses of the wind turbines are systematically studied under steady wind condition. The aerodynamic forces in the aeroelastic model are calculated by an AeroDyn module based on blade element momentum theory, and the structural dynamic responses of the blade are calculated using generalized Timoshenko beam theory and geometric exact beam theory. The NREL baseline 5 MW wind turbine and blade properties are used in this study, where the diagonal stiffness parameters and non-diagonal stiffness parameters of the matrixes of each blade section are scaled according to certain principles. The results show that the axial stiffness, the flap-wise stiffness, and the torsional stiffness in the diagonal are sensitive to the root loads and tip displacement of the blade. The flap-wise bending torsion coupling stiffness, the flap-wise shear-torsion coupling stiffness, and the edge-wise shear-torsion coupling stiffness in the non-diagonal are also sensitive to the aeroelastic responses. For completeness, the effects of other stiffness parameters on the aeroelastic responses are also analyzed and discussed.

Effect of angular misalignment of inner ring on the contact characteristics and stiffness coefficients of duplex angular contact ball bearings

Ring misalignment is unavoidable in most applications of bearing and significantly affects bearing performance. However, it has not been thoroughly studied for duplex angular contact ball bearings. To overcome the deficiency, this paper presents the analysis for duplex angular contact ball bearings by considering inner ring angular misalignment. A highly modular five degree-of-freedom model of duplex angular contact ball bearings is proposed based on vector-and-matrix method for back-to-back, face-to-face and tandem configurations. Five-dimensional stiffness matrix of bearing is also presented. On the basis, influences of ring misalignment on load distribution, maximum and average contact loads of bearing with back-to-back and face-to-face configurations are analyzed. Finally, translation and tilting stiffness coefficients of bearing are further discussed under misalignment condition. The results show that, ring misalignment considerable changes the contact characteristic of ball-raceway and causes uneven load distribution. Ring misalignment affects the position of maximum contact load and significantly impacts the diagonal stiffness of bearing. Contact characteristics and stiffness are also dependent on the bearing configuration. The results suggest the importance of bearing alignment and demonstrate the necessity of misalignment effect analysis.

Ultrasounds could be considered as a future tool for probing growing bone properties

Juvenile bone growth is well described (physiological and anatomical) but there are still lacks of knowledge on intrinsic material properties. Our group has already published, on different samples, several studies on the assessment of intrinsic material properties of juvenile bone compared to material properties of adult bone. The purpose of this study was finally to combine different experimental modalities available (ultrasonic measurement, micro-Computed Tomography analysis, mechanical compression tests and biochemical measurements) applied on small cubic bone samples in order to gain insight into the multiparametric evaluation of bone quality. Differences were found between juvenile and adult groups in term of architectural parameters (Porosity Separation), Tissue Mineral Density (TMD), diagonal stiffness coefficients (C33, C44, C55, C66) and ratio between immature and mature cross-links (CX). Diagonal stiffness coefficients are more representative of the microstructural and biochemical parameters of child bone than of adult bone. We also found that compression modulus E was highly correlated with several microstructure parameters and CX in children group while it was not at all correlated in the adult group. Similar results were found for the CX which was linked to several microstructure parameters (TMD and E) only in the juvenile group. To our knowledge, this is the first time that, on a same sample, ultrasonic measurements have been combined with the assessment of mechanical and biochemical properties. It appears that ultrasonic measurements can provide relevant indicators of child bone quality (microstructural and biochemical parameters) which is promising for clinical application since, B-mode ultrasound is the preferred first-line modality over other more constraining imaging modalities (radiation, parent–child accessibility and access to the patient's bed) for pediatric patients.

Multi-objective optimization of parallel manipulators using a game algorithm

• A multi-objective optimization method using game algorithm for parallel manipulators is proposed. • A new stiffness index considering the coupling effect of the non-diagonal elements is proposed. • A new index which defines the difference between the total and principal diagonal stiffness indices is proposed. • A four-dimensional intuitive image of slice distribution of the index is presented. In this paper, a multi-objective optimization game algorithm (MOOGA) for parallel manipulators (PMs) is proposed. The proposed algorithm considers the volume of the regular cylindrical workspace, motion/force transmission performance, and stiffness performance as objective functions. First, the distributions of the objective functions in the complete parameter space are calculated and sorted by importance. Second, game weighting factors and lower bound values are assigned to different objective functions according to the engineering requirements. Finally, after multiple rounds of gaming according to the weighting factors and lower bound values, the objective functions reach an optimal balance point and obtain a balance intersection subspace. In addition, a new comprehensive stiffness index (CSI) is proposed, that takes the coupling of the non-diagonal elements into consideration. This index decouples the linear and angular stiffness and has definite physical dimensions as well as a clear physical meaning. A Lagrangian function is used to obtain the maximum and minimum stiffnesses at a given position along with their corresponding directions. To compare the difference between the CSI and the principal diagonal stiffness index (PDSI), a divergence index κ is proposed. 2UPR–RPU and 2UPR–2RPU PMs are employed as examples to implement the proposed algorithm, where U, P and R denote a universal joint, prismatic pair and revolute pair, respectively. The corresponding slice distributions of the local CSI and κ in the regular cylindrical workspace are presented. Additionally, the distributions of the extreme linear stiffness indices and their corresponding directions are presented. The results show that the CSI is decreased by 99% relative to the PDSI. The numerical results demonstrate the effectiveness of the algorithm proposed in this paper.

A Novel Geometric Design Method of Elastic Structure for 6-Axis Force/Torque Sensor

We present a novel design method of a 6-axis Force/Torque (F/T) sensor with identical stiffness in all directions. In contrast to common F/T sensor designs, the proposed method is an analytical approach to the realization of a desired diagonal stiffness matrix which implies the complete decoupling of the stiffnesses in all directions. The first step of the design is to group the column vectors of a given rank 6 diagonal stiffness matrix into in-plane type and out-of-plane type stiffnesses. Each type of stiffnesses is then synthesized by means of three line vectors and designed as the corresponding serial-chain for a limb of an F/T sensor using only circular hinges. The F/T sensor can be formed by connecting the designed two serial-chains in series. However, for a practical design with minimum structural error and better strength, each of two type stiffnesses is further divided into a set of n separate serial-chains. Finally, the n-sets of serial-chains each consisting of six circular hinges that correspond in-plane and out-of-plane type stiffnesses are connected to the moving platform in parallel to complete the design. The FEM analysis and experiments are conducted to verify the proposed design method.

Seismic behaviour of reinforced concrete frame with infill panels

Infill panels are usually used in structures; therefore, it is very important to study their behaviour and their influence on bare frames. Several models had been proposed to understand the effect of the infill panels on structure. The aim of this work is to identify the effect of the presence of infill panels on bare frames. The diagonal stiffness and shear compression failure of the structure, the nonlinear behaviour of the infilled frame under axial loading and the relation between the strength, ductility and stress-strain of different previous experiments will be presented. Experimental research on the interaction between the steel frames and infill panels has been done to determine the behaviour of steel frames with different types of infill walls. Many frames with composite panels subjected to cyclic-loading were realized to determine the damping ratio, the static and dynamic response, strength and stiffness degradation of infill elements. The objective of this paper is to represent and analyze the hysteretic behaviour, failure modes, and the variation of different parameters of the infilled bare frame like stiffness, strength, ductility and loading displacement curves.

Seismic behaviour of reinforced concrete frame with infill panels

Infill panels are usually used in structures; therefore, it is very important to study their behaviour and their influence on bare frames. Several models had been proposed to understand the effect of the infill panels on structure. The aim of this work is to identify the effect of the presence of infill panels on bare frames. The diagonal stiffness and shear compression failure of the structure, the nonlinear behaviour of the infilled frame under axial loading and the relation between the strength, ductility and stress-strain of different previous experiments will be presented. Experimental research on the interaction between the steel frames and infill panels has been done to determine the behaviour of steel frames with different types of infill walls. Many frames with composite panels subjected to cyclic-loading were realized to determine the damping ratio, the static and dynamic response, strength and stiffness degradation of infill elements. The objective of this paper is to represent and analyze the hysteretic behaviour, failure modes, and the variation of different parameters of the infilled bare frame like stiffness, strength, ductility and loading displacement curves.

A Beam Finite Element Based on the Explicit Finite Element Method

The present article presents a new three-dimensional beam finite element, the element EB6, based on explicit theory. To the element is assigned a set of modes. A diagonal Modal elemental stiffness matrix is formed which through a series of matrix transformations relates to the local and global coordinate systems. An initial load due to temperature is derived, as well as the elemental mass matrix and the geometric stiffness. All quantities are explicitly defined. The element is used to study a number of beam, arc, and frame structures. Computational aspects of the element are also provided.

Ultrasonic assessment of diagonal stiffness coefficients in children cortical bone

"Ultrasonic assessment of diagonal stiffness coefficients in children cortical bone." Computer Methods in Biomechanics and Biomedical Engineering, 18(sup1), pp. 1978–1979

Stiffness synthesis of 3-DOF planar 3RPR parallel mechanisms

SUMMARYForce control is important in robotics research for safe operation in the interaction between a manipulator and a human operator. The elasticity center is a very important characteristic for controlling the force of a manipulator, because a force acting at the elasticity center results in a pure displacement of the end-effector in the same direction as the force. Similarly, a torque acting at the elasticity center results in a pure rotation of the end-effector in the same direction as the torque. A stiffness synthesis strategy is proposed for a desired elasticity center for three-degree-of-freedom (DOF) planar parallel mechanisms (PPM) consisting of three revolute-prismatic-revolute (3RPR) links. Based on stiffness analysis, the elasticity center is derived to have a diagonal stiffness matrix in an arbitrary configuration. The stiffness synthesis is defined to determine the configuration when the elasticity center and the diagonal matrix are given. The seven nonlinear system equations are solved based on one reference input. The existence and the solvability of the nonlinear system equations were analyzed using reduced Gröbner bases. A numerical example is presented to validate the method.

6축 병렬형 순응기구를 이용한 위치/힘 동시제어

In this paper, the kinestatic control algorithm using a six-axis compliant device is presented. Unlike the traditional control methods using a force/torque sensor with very limited compliance, this method employs a compliant device to provide sufficient compliance between an industrial robot and a rigid environment. This kinestatic control method is used to simply control the position of an industrial robot with twists of compensation, which can be decomposed into twists of compliance and twists of freedom. A simple design method of a six-axis parallel-type compliant device with a diagonal stiffness matrix is presented. A compliant device prototype and kinestatic control hardware system and programming were developed. The effectiveness of the kinestatic control algorithm was verified through two kinds of kinestatic control experiments.

Closure to “Discussion on ‘Optimum placement and characteristics of velocity-dependent dampers under seismic excitation’ by SA Mousavi and AK Ghorbani-Tanha” by Izuru Takewaki

The authors would like to thank the discusser for his considerations and comments. The discusser believes that some of the derived formulations need to be referred to his previously published works and also some related studies have not been cited.Up to knowledge of the authors, the idea of drift-based equations of motion presented in the paper is new. While mathematical aspects of the idea are simple, its specifi c form would greatly accelerate optimization process due to the fact that stiffness and damping matrices, which are supposed to be optimized, are diagonal in the drift-based form. The technique is not limited to damper placement and can also be used in the other cases. The authors would like to note thatEqs. (18) and (21) are well known basic formulations which step by step procedure of their derivation is presented in the paper. Equation (18) is simply obtained by taking the Fourier transform of the drift-based equation of motion (Eq. (13)). Zhang and Soong (1992), Cimellaro (2007), Cimellaro and Retamales (2007) and many others have also presented this formulation without mentioning any reference. Moreover, Eq. (21) is a famous relation in random vibration theory which says that the mean square value of a random process equals the area under spectral density graph of that process (Newland, 1993). These well known relations are so familiar to the audiences which do not need any specifi c reference at all. Due to space limitation, only strictly related papers have been cited and other studies discussing other aspects of passive dampers were excluded, as stated in the “Introduction” section. Some of the stated references by the discusser are essentially similar in terms of formulations and optimality criteria, such as Takewaki (2000), Takewaki (1999), and Takewaki and Uetani (1999).As stated in the paper's “Introduction”, adopted optimization index was a combination between those proposed by Zhang and Soong (1992) and Takewaki (1997). However, derived formulations were different and well suited for practical applications. In the paper, considering higher modes effects, optimization procedure selected to be based on target modal damping ratios rather than target total damping coeffi cient which was the case in earlier studies (Takewaki 1997, Takewaki 1999, Takewaki and Uetani 1999, Takewaki 2000, Lopez-Garcia 2001, etc.). As shown in the “Example” section, this new philosophy leads to a lower value of total damping coeffi cient. In our paper, higher mode effects can be easily taken into account by defi ning discrete modal-based transfer matrices (Eq. 22). The discusser mentioned that Eq. (22) of the paper is similar to the formulations in Takewaki (2001). After fi nding and reviewing the paper, the authors found no relation similar to Eq. (22). However, similar concept has been presented schematically for power spectral density (PSD) of the critical excitation. As suggested by Takewaki (2001), the critical excitation has a step-wise Dirac-based PSD function. In other words, in our study, discrete transfer function is achieved based on concept of the shape functions (Eq. (23)), while Takewaki (2001) has conceptually derived discrete PSD function using Dirac delta function with no specifi c equation.As a conclusion, this paper was aimed to address optimal placement of velocity-dependent dampers using a simple yet accurate procedure. The authors believe that the procedure is well suited for professional engineers. The main new contributions of the paper can be summarized as follows: (1) Introducing drift-based equation of motion which has diagonal stiffness and damping matrices.(2) Defi ning optimization procedure based on target modal damping ratios.(3) Considering higher mode effects by defi ning discrete modal-based transfer functions.

An Approximate Method of Determining the Equivalent Diagonal Strut Stiffness of the Completely Infilled Masonry in Plane Reinforced Concrete Frames

Based on some hypotheses and experiment results, the formulas which determine the diagonal strut stiffness are proposed. Inheriting from the previous research, this paper presents a method to determine the diagonal strut stiffness based on the concept of beam on an elastic foundation

A model for computing the dual stiffness matrix of the human knee joint

This paper discusses an application of dual algebra to the analysis of human knee joint stiffness matrix. According to the proposed methodology, the general stiffness matrix characterizing the elasticity properties of the knee ligaments is reduced to a dual diagonal stiffness matrix. For this purpose, the similarity transform was extended to the field of dual numbers. The theoretical framework proposed seems particularly useful when comparing stiffness matrices obtained from different experimental setup and within different Cartesian coordinate systems.

Improving eigenpairs of automated multilevel substructuring with subspace iterations

This paper improves the eigenpair approximations obtained from the automated multilevel substructuring (AMLS) method by subspace iterations. Two variants of AMLS hybrid Subspace Iteration Method (AMLS-SIMa and AMLS-SIMb) are proposed. AMLS-SIMa is a derivative of the basic subspace iteration by utilizing the AMLS approximations as initial vectors. AMLS-SIMb further takes advantage of the AMLS transformed block diagonal stiffness matrix to avoid factorization of the original stiffness matrix. Numerical experiments show that: (a) the error of AMLS approximate eigenpairs can be significantly reduced with just a few iteration steps; (b) AMLS-SIMb is more efficient than AMLS-SIMa with less execution time.

Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.