Coupled Bending Research Articles

Mass Minimization of Axially Functionally Graded Euler–Bernoulli Beams with Coupled Bending and Axial Vibrations

Mass Minimization of Axially Functionally Graded Euler–Bernoulli Beams with Coupled Bending and Axial Vibrations

Design of Soft Pneumatic Actuator with Two Oblique Chambers for Coupled Bending and Twisting Movements

Soft pneumatic network (Pneu-net) actuators are frequently used to achieve sophisticated movements, but they face challenges in producing both bending and twisting motions concurrently. In this paper, we present a new Pneu-net twisting and bending actuator (PTBA) design that enables them to perform complex motions. We achieved this by adjusting the chamber angle, ranging from 15 to 75 degrees, to optimize the bending and twisting movements through finite element analysis and experimental verification. We also investigated the variation trends in bending and twisting motions and determined the actuator’s workspace and maximum grasping force for a variety of objects with different shapes, materials, and sizes. Our findings suggest that PTBA is a promising candidate for advanced applications requiring intricate and bioinspired movements. This new design method offers a path toward achieving these goals.

Coupled Bending and Axial Vibrations of Axially Functionally Graded Euler–Bernoulli Beams

Coupled Bending and Axial Vibrations of Axially Functionally Graded Euler–Bernoulli Beams

Analytical Investigation into the Rotational Performance of Glulam Bolted Beam-Column Connections under Coupled Bending Moment and Shear Force

Considering the glulam beam-column connection form and the number of bolts, monotonic loading test and finite element analysis was carried out on 9 connection specimens in 3 groups to study the rotational performance and failure mode of the connection. The test results revealed that compared with U-shaped connectors, T-shaped connectors can effectively improve the ductility of connections, and the increase in the number of bolts can reduce the initial stiffness and ductility of connections. By theoretical analysis, formulas for calculating the initial stiffness and ultimate moment of connections were deduced. Subsequently, the moment-rotation theoretical model of connections was established based on the formulas, which were validated according to the test data and simulation results. The proposed model can not only improve the current theoretical system of heavy-duty glulam beam-column structure but also provide a theoretical basis for calculating the mechanical properties of the glulam beam-column connection.

Mechanical Properties of Coupled Bending and Contact to Hollow Cylindrical Roller Contacts

A numerical analysis model of the hollow cylindrical roller is established by coupling the bending of the curved beam and elastic half-space contact theory. Slicing technology is applied to solve the bending deformation of every circle slice. Discrete fast Fourier transform (DC-FFT) and a conjugate gradient method are employed to solve the contact deformation and contact pressure, considering the bending effect in both radial and axial directions. The influence of hollowness on deformation, contact pressure, stiffness, and strain energy for a straight roller contacting a flat plane is studied using the numerical model. The results reveal that with the increase of hollowness, the contact pressure decreases at the end, the edge effect weakens apparently, the contact width and contact deformation increase slightly in the middle part, and the stiffness decreases. The bending deformation and strain energy increase sharply as the hollowness reaches 60%. What's more, the profile of the contact region gradually changes from "outer convex" to "inner concave" at the end, which is also validated by photographing the shapes of the static contact region under various hollowness conditions using an optical interference test.

Coupled bending torsional vibrations of viscoelastic rotors with fractional damper

The behavior of a Jeffcott rotor with lateral-torsional coupling is investigated in the presence of internal and external damping and eccentricity. The governing equations are derived based on the Lagrange method. Also, the Laplace method and linearization is used to solve the governing equations for free vibrations analysis. For a rotor with unbalance, the instability occurs when the real part of eigenvalues has positive values, and at the same time, it is the intersection point between the lines of natural frequencies. The instability speed increases with increasing the external damping, yet dependent on the internal damping and unbalance. Also, it is demonstrated that the rotor critical speed is affected by the order of fractional derivatives.

An Analysis of Piezomagnetic-Piezoelectric Semiconductor Unimorphs in Coupled Bending and Extension under a Transverse Magnetic Field

An Analysis of Piezomagnetic-Piezoelectric Semiconductor Unimorphs in Coupled Bending and Extension under a Transverse Magnetic Field

In vivo primary and coupled segmental motions of the healthy female head-neck complex during dynamic head axial rotation

While previous studies have greatly improved our knowledge on the motion capability of the cervical spine, few reported on the kinematics of the entire head-neck complex (C0-T1) during dynamic activities of the head in the upright posture. This study investigated in vivo kinematics of the entire head-neck complex (C0-T1) of eight female asymptomatic subjects during dynamic left–right head axial rotation using a dual fluoroscopic imaging system and 3D-to-2D registration techniques. During one-sided head rotation (i.e., left or right head rotation), the primary rotation of the overall head-neck complex (C0-T1) reached 55.5 ± 10.8°, the upper cervical spine region (C0-2) had a primary axial rotation of 39.7 ± 9.6° (71.3 ± 8.5% of the overall C0-T1 axial rotation), and the lower cervical spine region (C2-T1) had a primary rotation of 10.0 ± 3.7° (18.6 ± 7.2% of the overall C0-T1 axial rotation). Coupled bending rotations occurred in the upper and lower cervical spine regions in similar magnitude but opposite directions (upper: contralateral bending of 18.2 ± 5.9° versus lower: ipsilateral bending of 21.4 ± 5.1°), resulting in a compensatory cervical lateral curvature that balances the head to rotate horizontally. Furthermore, upper cervical segments (C0-1 or C1-2) provided main mobility in different rotational degrees of freedom needed for head axial rotations. Additionally, we quantitatively described both coupled segmental motions (flexion–extension and lateral bending) by correlation with the overall primary axial rotation of the head-neck complex. This investigation offers comprehensive baseline data regarding primary and coupled motions of craniocervical segments during head axial rotation.

Coupled bending and torsional beam vibrations excited by a moving load

AbstractFor a simply supported Bernoulli‐Euler beam under a moving load with constant velocity the bending and torsion vibrations are analysed. An analytical solution of the decoupled partial differential equations is derived based on the eigenmode expansion considering the boundary and initial conditions. For the pure bending and the pure torsion the analysis is performed and the beam vibrations are calculated. For various longitudinal speeds the results are plotted. The coupled bending and torsion deformations are computed using a modal transformation. The modal decoupling yields two separated ordinary differential equations in time which are computed using the derived closed form solutions. The solutions of the coupled equations are plotted and discussed. A comparison with a numerical calculation of a finite element model which shows good agreement.

Coupled Bending Inwards Motion and Control Strategy Analysis of a Cable-Driven Underactuated Finger

Motion strategy analysis at the pre-bending stage is a fundamental component of underactuated finger grasp research. This study presents the pre-bending motion strategy and corresponding analysis of cable driving forces a 3-DOF underactuated finger comprising cable truss units. The structure and four motion strategies at the pre-bending stage are illustrated. The equivalent joint-driven and quasi-static motion models are established in the case where one or two cable driving forces drive the finger. Quasi-static motion model is established, and four typical quasi-static coupled motion space and distributions of corresponding cable driving forces are analysed. In accordance with the kinetic energy theorem and Lagrange’s dynamical method, a sampled and general dynamic model is established. Through matrix transformation, a hypothetical “PPA” type dynamic model is obtained. On the basis of the “PPA” dynamic model, three grasping control strategies at pre-bending stage are carried out. Valid simulations and experiments are conducted to verify the accuracy of the coupled bend inwards motion and control strategy.

Coupled bending and torsional vibration characteristics analysis of inhomogeneous wind turbine tower with variable cross section under elastic constraint

The separation of cross section’ mass center from its centroid will lead to coupled bending and torsional vibration of the wind turbine tower (Referred to as tower). Three governing differential equations of vibration of the tower with inhomogeneous material and variable cross section are formulated based on D'Alembert's principle and Euler-Bernoulli beam theory. Considering the installation and operation factors of the wind turbine, the dimensionless natural frequencies and mode shapes of the tower under multiple constraint conditions are obtained by the differential quadrature method. Compared with those available in the literature, numerical results demonstrate the accuracy and validity of the proposed method. A series of parametric study is accomplished to reveal the influence of the foundation stiffness, wall thickness, eccentricity, and the ratio of the inner diameter of the top and bottom cross section of tower on the vibration characteristics of tower. The frequencies veering for the mode of tower is discussed. These numerical results have guiding significance for the design of tower.

Coupled bending torsional vibrations of non-ideal energy source rotors under non-stationary operating conditions

In the automotive industry, high-speed elecrical engines will be used in a very large range of speeds, leading to non-stationary operating conditions. Thus more critical speeds may be crossed a very large number of times during the whole life-time of the engine. Therefore, estimating accurately the non-stationary loads and deformations during these transient regimes is of first importance for a correct design. In this paper, we propose a novel dynamic model for unbalanced high-speed rotors with less restrictive assumptions. The finite element model accounts for flexion, torsion and traction-compression leading to six degrees of freedom on each node. The non-ideal energy source is considered and the rotor is running under non-stationary operating conditions and crossing supercritical speeds. The angular displacement is defined in such a way that it combines simultaneously the intrinsic nominal rotation and the torsional deformation. The comparison of the proposed model with other models under different assumptions on the energy source and on the bending-torsion coupling shows that our model offers more accurate prediction for both lateral and torsional vibrations when crossing critical speeds. The ability of the proposed model to account for the mutual influence between lateral and torsional behavior is highlighted and is exhibited through time-frequency analyses.

Coupled Bending–Bending–Axial–Torsional Vibrations of Rotating Blades

In this paper, the coupled bending–bending–axial–torsional free vibrations of rotating blades are investigated based on the Euler–Bernoulli beam model. The coupled partial differential equations governing flapwise, edgewise, axial and torsional motions are derived by the Hamilton’s principle, wherein three types of velocity-dependent terms, namely static centrifugal terms, dynamic centrifugal terms and gyroscopic coupling terms, are focused. The ordinary differential equations are acquired by the Galerkin truncation, and the natural frequencies in all directions and complex mode shapes of the rotating blades are analyzed in detail. It is revealed that the three types of velocity-dependent terms have different effects on the natural frequencies. The natural frequencies are noticeably dependent on the rotating speed and preset angle, except for the axial vibration, which is almost immune to the preset angle. The complex modal motions are displayed by a series of positions of the central line and free-end cross section for different time instants, showing the coupled vibrations among different directions.

Coupled bending and torsional vibrations of non-uniform thin-walled beams by the transfer differential transform method and experiments

This paper describes a new methodology capable of analyzing coupled bending and torsional vibrations in non-uniform thin-walled beams. The transfer matrix method (TMM) and the differential transform method (DTM) are combined to create the proposed transfer differential transform method (TDTM) for solving equations coupling bending and torsional vibrations in non-uniform beams. Compared with the finite element method (FEM), the TDTM utilizes a type of changeable mode shape functions so that the number of meshed elements can be reduced greatly when the beam's geometric size is uniform. The equations of motion of the non-uniform thin-walled beams are established using Hamilton's principle. The method considers both the warping and rotary effects of the beam section. The natural frequencies and mode shapes of the bending and torsional components are obtained using the TDTM. The accuracy of solutions can be controlled by the mesh density and the series expansion order of the mode shape whose ranges are also discussed. For illustrative purposes, the natural frequencies and the frequency response curves of a uniform beam and a non-uniform beam are studied respectively and are validated by experiments. The effects of warping and the distance between the centroid of a given section and the corresponding shear center on the vibration properties of the beam are also investigated.

Coupled bending–torsional frequency response of beams with attachments: exact solutions including warping effects

This paper deals with the coupled bending–torsional vibrations of beams carrying an arbitrary number of viscoelastic dampers and attached masses. Exact closed analytical expressions are derived for the frequency response under harmonically varying, arbitrarily placed polynomial loads, making use of coupled bending–torsion theory including warping effects and taking advantage of generalized functions to model response discontinuities at the application points of dampers/masses. In this context, the exact dynamic Green’s functions of the beam are also obtained. The frequency response solutions are the basis to derive the exact dynamic stiffness matrix and load vector of a two-node coupled bending–torsional beam finite element with warping effects, which may include any number of dampers/masses. Remarkably, the size of the dynamic stiffness matrix and load vector is $$8\times 8$$ and $$8\times 1$$ , respectively, regardless of the number of dampers/masses and loads along the beam finite element.

Coupled bending and torsional free vibrations of beams with in-span supports and attached masses

Abstract This paper proposes an exact approach to coupled bending and torsional free vibration analysis of beams with monosymmetric cross section, featuring an arbitrary number of in-span elastic supports and attached masses. The proposed method relies on the elementary coupled bending-torsion theory and makes use of the theory of generalized functions to handle the discontinuities of the response variables. Based on a simple procedure, exact natural frequencies and closed-form eigenfunctions are calculated from a characteristic equation built as determinant of a 6 × 6 matrix, for any number of supports/masses. Likewise, the exact dynamic stiffness matrix of the beam is obtained in a closed form, with size 6 × 6 regardless of the number of supports/masses, to be assembled for frame analysis. In addition, the orthogonality condition among the eigenfunctions is derived. Two numerical examples show the advantages of the proposed exact method.

Coupled bending and torsional vibration of a rotor system with nonlinear friction

Unacceptable vibrations induced by the nonlinear friction in a rotor system seriously affect the health and reliability of the rotating machinery. To find out the basic excitation mechanism and characteristics of the vibrations, a coupled bending and torsional nonlinear dynamic model of rotor system with nonlinear friction is presented. The dynamic friction characteristic is described with a Stribeck curve, which generates nonlinear friction related to relative velocity. The motion equations of unbalance rotor system are established by the Lagrangian approach. Through numerical calculation, the coupled vibration characteristics of a rotor system under nonlinear friction are well investigated. The influence of main system parameters on the behaviors of the system is discussed. The bifurcation diagrams, waterfall plots, the times series, orbit trails, phase plane portraits and Poincare maps are obtained to analyze dynamic characteristics of the rotor system and the results reveal multiform complex nonlinear dynamic responses of rotor system under rubbing. These analysis results of the present paper can effectively provide a theoretical reference for structural design of rotor systems and be used to diagnose selfexcited vibration faults in this kind of rotor systems. The present research could contribute to further understanding on the self-excited vibration and the bending and torsional coupling vibration of the rotor systems with Stribeck friction model.

Fiber Bragg Grating Sensing-Based Online Torque Detection on Coupled Bending and Torsional Vibration of Rotating Shaft

This paper has presented a novel approach based on fiber Bragg grating (FBG) sensing to analyze and decouple the coupled bending and torsional vibration of a rotating shaft. The theoretical FBG-based strain detection models under pure torsional vibration and coupled bending and torsional vibration have been derived. Two FBG sensors have been attached on the opposite sides of a rotating shaft with an angle of 45° with respect to the axial direction to support the use of the proposed decoupling method and determine the decoupled bending and torsion signals. A new supporting device has been designed to protect the fiber optical rotary joint from damage under large load impact, and ensure the stable signal transmit. The calibrated sensitivity of the pasted FBGs for measuring torque is 7.02 pm/ $\text{N}\cdot \text{m}$ . Dynamic experiments on a rotor platform under different rotating speeds have been performed to effectively decouple the bending strain and torsional strain in real time. The decoupled torque measurements have shown a close agreement with the data detected from a commercial tacho-torquemeter, which demonstrated the effectiveness of the proposed decoupling method.

Wave vibration control of planar frame structures based on the advanced Timoshenko bending theory

Coupled bending and axial vibrations in a planar multi-story frame structure are controlled from a wave vibration standpoint, in which vibrations are described as waves that propagate along uniform waveguides, and are reflected and transmitted upon structural discontinuities. The bending vibrations are modeled using a Timoshenko model, which takes into account the effects of rotary inertia and shear distortion. The axial vibrations are modeled using elementary theory, as it is typically valid for frequencies up to twice the cutoff frequency of Timoshenko bending waves. Regardless of the complexity of a structure, when it is modeled from a wave point of view, it consists of only two basic types of structural components, namely, structural elements and joints. In this paper, both structural element and structural joint controllers are designed based on various control strategies such as optimal damping and optimal energy absorbing. Numerical examples are presented. Results are compared to those obtained based on the classical Euler–Bernoulli model that are available in the literature. As was expected, good agreement between the results of the models at low frequencies was obtained.

A Study on Coupled Bending and Torsional Vibrations of Wind Turbine Blades

A parametric study is developed to investigate the effect of geometry, material stiffness and the rotational motion on the coupled flapwise bending and torsional vibration modes of a wind turbine blade. The assumed modes method is used to discretize the derived kinetic and potential energy terms. Lagrange’s equations are used to derive the modal equations from the discretized terms, which are solved for the vibration frequencies. The parametric study utilizes dimensional analysis techniques to study the collective influence of the investigated parameters by combining them into few non-dimensional parameters, thus providing deeper insight to the physics of the dynamic response. Results would be useful in providing rules and guidelines to be used in blade design.