Translational Stiffness Research Articles

Size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams with elastic boundary conditions and thermal effects based on two‐phase local/nonlocal elastic model

AbstractWe present predictive models of the size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams under thermal effects based on the two‐phase local/nonlocal elastic model. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. In contrast to most literature in this field, both the bending and thermal deformations of the beams are simultaneously considered to be two‐phase local/nonlocal of two phases, that is, the thermal effect is taken as equivalent to a size‐dependent thermal load. By using the fully equivalent differential form of the local/nonlocal equation with a set of constitutive boundary conditions, the problem is solved numerically with the aid of the generalized differential quadrature method (GDQM). Through conducting validation study, several parametric studies are given for examining the effects of the slope of beams’ thickness variation, nonlocal parameter, and elastically supported conditions on the buckling loads of non‐uniform beams. The results show that constrained stiffness has a drastic influence on the critical bucking loads of the beams. Furthermore, the consideration of the two‐phase thermal load will further reduce the actual buckling load of the beams.

Origami-Inspired Collapsible Structures for Small Rotorcraft Collision Resilience and Landing

Abstract This article reports on the design, fabrication, and testing of flexible, collapsible structures inspired by origami and kirigami that can be used to protect small rotorcraft from collision impacts and to serve as landing gear. Twenty structures created from variations of Pako Pako, Magic Ball, and Herringbone fold patterns were laser cut from cardstock and evaluated experimentally. Some of the Pako Pako structures were designed to have graduated mechanical stiffness. An empirical procedure based on acceleration measurements has been developed that enables quantitative evaluation of each structure based on the characterization parameters of peak acceleration, peak velocity, translational kinetic energy, impact duration, mass, and mechanical stiffness. Structures were mounted onto the front of a radio-controlled (RC) car and driven on a linear test track into a rigid wall. Accelerations measured during the collisions were numerically integrated to determine velocities over the impact and collapse/compression of the structures. Flight tests conducted with small RC quadcopters demonstrated that the collapsible structures could successfully be used to mitigate collisions, enough to enable the quadcopters to continue flying after a direct impact and to land indoors or on outdoor terrain with varying slopes and surfaces. Based on trade-off comparisons between the evaluative metrics of the cases studied, conical Pako Pako structures with uniform stiffness are shown to be the most effective for collision resilience and landing.

Determination of the mechanical properties of the sacroiliac joint of a dog by imaging measurement methods and model update in a multi‐body model

AbstractThe sacroiliac joint (SIJ) is the articulated connection between the sacrum and the ilium and is a tight, combined, and only slightly mobile joint. In many dogs, the joint becomes stiff with age. As a result, the animal experiences pain while moving. Too little information is known about the degree of stiffening and the physiological range of motion for therapy. Consequently, recording the range of motion of different individuals (healthy and diseased) is a step toward a better understanding of the SIJ syndrome. For the basic investigation of the mechanical properties of the SIJ, post‐mortem examinations are performed on different individuals. To determine the mobility of the joint, the sacrum is fixed and a movement is forced on the ilium using actuators and recording the required forces. In addition, the spatial movement of the ilium and sacrum is measured. The kinematic and dynamic variables can be used to determine the translational and rotational stiffness of the SIJ. As it is not possible to apply directed loads to the ilium, the measurement data is used to determine the elastic parameters of the joint ligaments using heuristic methods. The multi‐body model includes the ilium, the ligaments, and the actuators of the test rig with their spatial orientation.

Semi-analytical formulation to predict the vibroacoustic response of a fluid-loaded plate with ABH stiffeners

Stiffened structures are widely used in aeronautics, marine and rail industries. When stiffeners are integrated into host structures, so-called Bloch-Floquet waves are generated due to interactions between the host’s flexural waves and the stiffeners’ flexural and torsional waves. It is reported in the literature that these waves are often the source of undesirable noise and vibrations when the stiffened structure is excited by a force. To mitigate unwanted noise and vibrations from the stiffened structures, this study proposes to replace common rectangular stiffeners with acoustic black hole (ABH) stiffeners. To do this, a semi-analytical model is initially developed in the wavenumber domain to predict the forced vibroacoustic response of a 2D fluid-loaded infinite plate with stiffeners on one side. In the proposed model, the stiffeners are characterised by their translational and rotational dynamic stiffnesses which can be estimated by a finite element method (FEM). These dynamic stiffnesses are then coupled with the analytical formulation of the fluid-loaded plate to obtain the expressions of the spectral displacement and radiated pressure. Comparisons of the results in terms of the plate’s mean quadratic velocity and radiated sound power for the rectangular and ABH stiffeners show that by using the ABH stiffeners instead of the conventional stiffeners, one can significantly reduce the vibroacoustic response of light/heavy fluid-loaded plates.

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

Theoretical Evaluation of Lubrication Performance of Thrust-Type Foil Bearings in Liquid Nitrogen

The development of reusable liquid rocket turbopumps has gradually highlighted the disadvantages of rolling bearings, particularly the contradiction between long service life and high rotational speed. It is critical to explore a feasible bearing scheme offering a long wear life and high stability to replace the existing rolling bearings. In this study, liquid nitrogen is adopted to simulate the ultra-low temperature environment of liquid rocket turbopumps, and theoretical evaluations of the lubrication performance of thrust-type foil bearings in liquid nitrogen are conducted. A link-spring model for the bump foil structure and a thin-plate finite element model for the top foil structure are established. The static and dynamic characteristics of the bearings are analyzed using methods including the finite difference method, the Newton–Raphson iteration method, and the finite element method. Detailed analysis includes the effects of factors such as rotational speed, fluid film thickness, thrust disk tilt angle, and the friction coefficient of the bump foil interface on the static and dynamic characteristics of thrust-type foil bearings. The research results indicate that thrust-type foil bearings have a good load-carrying capacity and low frictional power consumption. The adaptive deformation of the foil structure increases the fluid film thickness, preventing dry friction due to direct contact between the rotor journal and the bearing surface. When faced with thrust disk tilt, the direct translational stiffness and damping coefficient of the bearing do not undergo significant changes, ensuring system stability. Based on the results of this study, the exceptional performance characteristics of thrust-type foil bearings make them a promising alternative to rolling bearings for the development of reusable liquid rocket turbopumps.

Non-linear modeling and parameter identification of a bolted joint using substructure synthesis theory

Engineering applications require accurate modeling and parameter identification to anticipate bolted joint behavior under different loading circumstances. Bolted joints are extensively used in civil and mechanical structures. In this research study, cubic stiffness non-linearity is examined and a comprehensive model, which can exhibit the non-linearity of joint behavior, is developed. This is achieved by modeling the bolted joint which is connected at one end of a cantilever beam. Mathematical model is established using concept of sub structure synthesis theory by considering non-linear translational stiffness and non-linear rotational stiffness as a boundary condition. Non-dimensional natural frequencies are estimated by assuming stiffness parameters and joint parameter estimation for translational spring having nonlinear stiffness is calculated using second order curve fitting technique. Furthermore, estimated and exact value of cubic stiffness coefficient are compared. The experimental results shows that the percentage error of estimation for all the modes are less than 8%. The experimentally determined natural frequencies are further contrasted with an analytical results to validate a precision of generated model. Present work contributes in improving the bolted joint analysis by providing a solid non-linear modeling and parameter identification methodology. This methodology enhances design practices and ensures the durability of engineering structures and mechanical systems by improving bolted joint comprehension.

Statics and stability of diagonalless beams of particular kind – preliminary studies

Attractive large-scale glazing is currently an architectural trend. However, achieving adequate stiffness for larger glazing spans requires the use of complex cross-sections, generally aluminium sections of considerable height. Members with openwork webs are sometimes used in order to achieve increased load-bearing capacity and stiffness with reduced weight. The disadvantage is that this solution takes up a lot of space inside the building. A recently patented diagonalless member attempts to solve the above-mentioned problems. The member is fully demountable and allows glass units to be installed in the space between the chords. It consists of two chords spaced apart by metal sleeves with bolts passed through them. In this study, preliminary qualitative experimental tests were carried out to determine the behaviour of the member under load and to identify zones sensitive to local deformation. On this basis, numerical models (bar and 3D solid models, including contact interactions) were created and tested. Subsequently, the optimum sleeve spacing was determined, the effect of rotational and translational stiffness reduction at the nodes was investigated, and stress concentration zones and forms of stability loss were identified. A new form of local loss of stability of the chord facewall was identified, the so-called sliding push effect of the chord walls on the sleeve (within the larger openings). This is a completely different type of chord facewall failure from that found in known tubular welded joints. The research programme focused on identifying the phenomena occurring in the new member in order to provide a basis for further, more advanced analyses.

Dynamic property evaluation of pipeline clamp based on analyzing the propagation characteristics of flexural wave in pipeline

The external clamp of hydraulic pipeline plays an important role in strengthening pipeline stiffness and improving vibration attenuation ability. In this study, wave propagation method is proposed to evaluate the translational and rotational stiffness and their loss factors of the clamp by analyzing the flexural wave propagation characteristics of pipeline. In order to validate the measured dynamic properties, experimental modal analysis was performed on the two-end elastically-supported pipeline. By modelling the pipeline with translational and rotational edge restraints, the analytical modal frequency parameters, wave numbers, and waveform coefficients were obtained. By comparing of the first-order flexural mode frequencies between experiment and analysis, the dynamic stiffness predicted by the wave propagation method was validated. The effects of the magnitude of torque, direction, clamping torque cycle, and clamp size on the complex dynamic properties of pipeline clamp were investigated. The proposed wave propagation method allows non-destructive continuous monitoring of a clamp stiffness, provides additional information about the vibration attenuation feature. It has advantages in clamp applications and is an effective methodology for measuring the dynamic properties of viscoelastic materials.

Natural Frequency Maximization of Beam with an Internal Sliding Joint of Translational Restraint

The natural frequency variation of a uniform Euler–Bernoulli beam with an internal sliding joint of the translational restraint is studied intensively such that the optimal design of the connecting point is pursued for maximization of a natural frequency of the system. First, the compatibility conditions at the linkage are presented in detail, and the dynamic characteristic equation of the beam system is accordingly derived. Next, the influences of both the translational restraint stiffness and the position of the sliding joint upon the natural frequency and mode shape are described comprehensively. Then, the optimal position and the corresponding minimum translational stiffness of the internal joint are explored exclusively for maximizing a natural frequency of the beam structure. In this context, the essential requirements for the optimization of the internal sliding joint are presented, respectively, in two different ways to reveal their mechanical implications. It turns out that the optimal joint location can be determined readily from the corresponding mode shape of the equivalent continuous beam. Afterward, numerical results are provided with two classical beams of a single internal sliding joint to demonstrate the variations of their dynamic characteristics as well as to offer significant physical insight into the vibratory behaviors of the beam system with maximization of a natural frequency of interest.

Numerical-Computational Model for Dynamic Nonlinear Analysis of Frames With Semi-Rigid Connection Considering the Damping Effect

Objective: the scope of this work is to present a numerical-computational model for transient dynamic nonlinear analysis of frames with geometric nonlinearity and semi-rigid connection. Methodology: the equation of motion, which describes the structural dynamical system, is solved by the a-Generalized direct integration method associated with the standard Newton-Raphson method. The structures are discretized through the co-rotational formulation of the Finite Element Method considering the Euler-Bernoulli beam theory. The semi-rigid connection of structural members (beam-column and beam-support) is simulated by a connection element with zero length, which is described in terms of axial, translational and rotational stiffness. Results and conclusion: From the numerical results of three structural systems (bi-fixed beam, L-shaped frame, and single-story frame) obtained with the free Scilab program, it is concluded that the definition of the type of connection is an important factor to be considered in the analysis of frames subjected to dynamic loads. Furthermore, structural damping, which is a measure of energy dissipation, drives the structure from a vibrating state to a resting state. Research implications: Structural Engineering has been designing systems that cannot be analyzed and dimensioned without dynamic effects being considered. lack of knowledge of the levels and characteristics of the dynamic response can lead to system failure during the application of repetitive loading due to accumulation of structural damage. In this sense, the numerical model can represent a valuable engineering tool when it comes to the dynamic analysis of plane metallic structures with geometric nonlinearity.

An Examination of the Stiffness Terms Needed to Model the Dynamics of an Eddy Current Based Maglev Vehicle

This paper re-examines the basis for each eddy current stiffness term computed from prior published steady-state eddy current models. The paper corrects prior analysis work by confirming, through the use of 2-D and 3-D dynamic finite element analysis modelling, that when a magnetic source is moving over an infinite-wide and infinite-long conductive sheet guideway the steady-state lateral and translational stiffness terms will be zero and only the vertical coupled stiffness terms need to be modelled. Using these observations, a much simplified 6 degrees-of-freedom (DoF) linearized eddy current dynamic force model can be used to compute the steady-state force changes in eddy current based maglev vehicles when operating over a wide uniform conductive track.

Biomechanical comparison of open Bankart repair vs. conjoint tendon transfer in a 10% anterior glenoid bone loss shoulder instability model

The treatment of shoulder instability in patients with subcritical glenoid bone loss poses a difficult problem for surgeons as new evidence supports a higher failure rate when a standard arthroscopic Bankart repair is used. The purpose of this study was to compare a conjoint tendon transfer (soft-tissue Bristow) to an open Bankart repair in a cadaveric instability model of 10% glenoid bone loss. Eight cadaveric shoulders were tested using a custom testing system that allows for a 6-degree-of-freedom positioning of the glenohumeral joint. The rotator cuff muscles were loaded to simulate physiologic muscle conditions. Four conditions were tested: (1) intact, (2) Bankart lesion with 10% bone loss, (3) conjoint tendon transfer, and (4) open Bankart repair. Range of motion, glenohumeral kinematics, and anterior-inferior translation at 60° of external rotation with 20 N, 30 N, and 40 N were measured in the scapular and coronal planes. Glenohumeral joint translational stiffness was calculated as the linear fit of the translational force-displacement curve. Force to anterior-inferior dislocation was also measured in the coronal plane. Repeated measures analysis of variance with a Bonferroni correction was used for statistical analysis. A Bankart lesion with 10% bone loss increased the range of motion in both the scapular (P=.001) and coronal planes (P=.001). The conjoint tendon transfer had a minimal effect on the range of motion (vs. intact P=.019, .002), but the Bankart repair decreased the range of motion to intact (P=.9, .4). There was a significant decrease in glenohumeral joint translational stiffness for the Bankart lesion compared with intact in the coronal plane (P=.021). The conjoint tendon transfer significantly increased stiffness in the scapular plane (P=.034), and the Bankart repair increased stiffness in the coronal plane (P=.037) compared with the Bankart lesion. The conjoint tendon transfer shifted the humeral head posteriorly at 60° and 90° of external rotation in the scapular plane. The Bankart repair shifted the head posteriorly in maximum external rotation in the coronal plane. There was no significant difference in force to dislocation between the Bankart repair (75.8±6.6 N) and the conjoint tendon transfer (66.5±4.4 N) (P=.151). In the setting of subcritical bone loss, both the open Bankart repair and conjoint tendon transfer are biomechanically viable options for the treatment of anterior shoulder instability; further studies are needed to extrapolate these data to the clinical setting.

Effects of boundary conditions on phase shift of a dual-tube conveying fluid in the Coriolis mass flowmeter

This paper investigates the effects of boundary conditions on the performance of a Coriolis mass flowmeter with double straight tubes conveying fluid. The non-self-adjoint governing equations of motion are derived in dimensionless form, and the Galerkin method is used to discretize them into a set of ordinary differential equations. The free vibration analysis of the structure, including extraction of the natural frequencies and the symmetric and antisymmetric mode shapes of double tubes, is presented. In addition, the effects of the stiffness of the elastic supports on the phase shift induced between specific points of the tubes are investigated. The greater the phase shift between two special points, the easier it will be to measure by pickoffs; consequently, and the Coriolis flowmeter will be more accurate. The results revealed that the phase shift approaches zero for small translational stiffness but approaches a limited non-zero value when the rotational stiffness approaches zero. The maximum phase shift is obtained for a specific translational and rotational stiffness. The effects of the springs’ stiffness on the stability of the tubes are also investigated.

Design and optimization of a decoupled RP flexure joint for an adjustable-motion-direction compliant mechanism

Callus deformations, including amplitudes and directions, greatly impact bone healing. Many existing devices can hardly adjust the callus-deformed directions since their unexpected deformations are not constrained properly. This study presents a way to adjust the callus-deformed direction by suppressing unexpected callus deformations. This is realized by designing a decoupled revolute-prismatic(RP) flexure joint and integrating it into an adjustable-motion-direction compliant mechanism (AMD-CM). This decoupled RP flexure joint is designed by serially combining a 3-leaf cross flexure joint and a parallelogram flexure joint according to their eigenstructures. The decoupled property eliminates the rotation axis misalignment and suppresses unexpected callus rotational deformations. First, eigencompliance analysis of decoupled RP flexure joints is performed and shows that the rotation axis misalignment is eliminated. Second, the AMD-CM’s pseudo-rigid-body model is developed, and the constant translational stiffness condition is found. Then, optimization is performed to obtain large compliance ratios and constant translational stiffness. Finite element analysis shows that by using the optimized decoupled RP flexure joint, the callus-deformed direction can be adjusted. The unexpected callus rotational deformation is also reduced greatly compared to previously proposed devices. Finally, experiments show that the optimized decoupled RP flexure joint can realize an adjustable callus-deformed direction.

Measurement and identification of translational static stiffness in workspace of a machine tool

This work presents a measurement for the identification of translational quasi-static stiffness of machine tool using Stiffness Workspace System (SWS). The novelty of this work is a significant modification of the SWS. The changes in measurement procedure and data analysis as well as new technical solutions are described in detail. A methodology for generalised translational static stiffness determination is presented. The major purpose of this paper is to provide the information about quasi-static stiffness values that characterise a typical machine tool. The measurement procedure is implemented in a case study on a 5-axis machining centre. Measurement results were used to determine the generalised translational stiffness indicators of points in the workspace. Then, the static stiffness distribution on the XY plane over machining space was estimated. The obtained results confirm the importance of analyses of static stiffness distribution as they point out areas where some possible changes occur. In the conclusion of the paper the utility value of the information of translational static stiffness is emphasised.

Robot stiffness theory reconsideration based on Schur complement eigenvalues: Extension to GSP dynamic stiffness evaluation

The known robot stiffness evaluation algorithm based on the Schur complements’ eigenvalues (SCE) is refined and its application is extended to the dynamic stiffness problem. For the refinement, the Schur complements synthesis is reduced to standard transformations of the stiffness matrix into the triangular block matrices. In these terms, the Schur complements of the robot dynamic stiffness matrix are formulated for steady-state forced sinusoidal vibrations of a damping-free system. The interrelationships between the forced-frequency-dependent SCE and natural frequencies are established: the frequencies resulting in the SCE zero values are proven to be identical to the natural frequencies, and vice versa. The SCE of the dynamic stiffness matrix defines the range and extremes of the dynamic stiffness values that allow the building of the translational and rotational stiffness ellipsoids and formulation of the stiffness-related robot performance indices. In four application examples, the SCE-based dynamic stiffness values of the Gough-Stewart platform (GSP) on weightless supports are defined. The definitions allow the formulation of the dynamic stiffness-related performance indices which are used for quantitative evaluation and comparison of the GSP designs.

Model-based analysis of temperature-dependent dynamics in CFRP spindle unit

The application of CFRP to machine tool structures can potentially improve their dynamic characteristics and energy efficiency owing to its highly specific material properties. The different thermal characteristics of CFRP and steel in spindle units likely induce unique thermal deformation, which affects the bearing dynamics. This research proposes a modeling method of CFRP spindle unit dynamics via generalized multipoint receptance coupling with contact characteristics at the spindle-holder interface. The temperature-dependent mechanical properties are analyzed by parameter identification from the model. The translational stiffness and damping of bearings in the CFRP spindle increase with temperature increase at high rotational speed.

Robotics Benchmark on Transfer Learning: a Human-Robot Collaboration Use Case

In the context of human-robot collaboration (HRC), the model of the robots needs to be adapted to describe new tasks in new environments and under new operating conditions. A long-standing challenge in HRC is to transfer the acquired robot's skills and adapt models from a limited amount of data and/or with limited computational resources. To facilitate the research addressing this issue, this paper proposes a transfer learning benchmark in a HRC setting using data acquired from a 7-DOF Franka Emika Panda robot. The goal is to estimate a dynamical model mapping set-point pose, measured pose, and velocity of the end-effector into the external interaction wrench. This type of problem may arise in real applications to design virtual sensors for forces/torques. In the proposed benchmark, the model can be estimated based on a long dataset acquired under a nominal operating condition of the robot, along with 5 shorter trajectories (to be used for model adaptation) gathered for 5 different values of translational stiffness. Performance is measured on 5 test trajectories where the same translational stiffness values of the transfer experiments are applied. Baseline results are presented using a transfer learning approach tailored to dynamical systems recently proposed in the literature.

Orthotropic plates with dynamic vertical seismic load modeled as multi line

Calculation plate floor concrete, using a static load which is a gravity load consisting of a live load and a dead load, with various of boundary conditions, floor slabs are orthotropic plate, and rarely account for dynamic loads due to vertical seismic loads, with other boundary conditions, such as Clamped, simply supported, ES (Elastic Support), ER (Elastic Restraint), and ESR (Elastic Support and Restraint). Analytical solution based on the Modified Bolotin Method to analyze floor slab under Vertical Peak Ground Acceleration (PGAv), the natural frequency solution based on auxiliary Levy’s type problems. Dynamic vertical seismic loads using multiline, first line at 0 < t < 0.5 is linear equation, second line at 0.05 < t < 0.15 is quadratic equation, third line 0.15 > t < 0.6 is sextic equation, last line, t > 0.6 is linear equation, vertical seismic load with two conditions far fault and near fault, multi-line equation are depending on (PGAv/g). A numerical example is given, for various boundary conditions, and far fault, translational stiffness (kx, ky) and rotational stiffness (cx ,cy), from the results of plate calculations due to dynamic vertical seismic loads with 5 types of edge support, ES (elastic support) is the best result.