Dynamic Stiffening Effect Research Articles

Dynamical stiffening of dsDNA confined and stretched in a nanochannel

The internal dynamics of double-stranded DNA (dsDNA) confined within channels of diameters ranging from the persistence length to twice that length were investigated using fluorescence microscopy. By analysing spatiotemporal intensity fluctuations, we derived the intermediate dynamic structure factor. The structure factor consistently exhibited behaviour characteristic of Rouse dynamics, regardless of the channel diameter. Notably, as the channel diameter decreased, the DNA molecules became increasingly elongated along the channel, leading to shorter relaxation times. These findings indicate a dynamic stiffening effect, where the effective spring constant of a stretched polymer chain increases. We propose that this effect has significant implications for controlling DNA motion in biological and biotechnological applications.

Dynamics analysis of rotating soft core sandwich beams using the absolute nodal coordinates formulation with zigzag theory

With the widespread use of rotating sandwich structures in engineering applications, it is significant to understand their vibration characteristics and dynamic behavior. Since the absolute nodal coordinates formulation (ANCF) is one of the most effective methods for solving rotational dynamics, it is feasible to use the ANCF with zigzag theory (ANCF-ZZ) to analyze the dynamics of these structures. Therefore, this study builds upon existing research to further refine the theory of ANCF-ZZ and proposes diverse models for determining the element complexity required in modeling rotating sandwich structures. In order to better solve the dynamics of rotating sandwich beams, the ANCF-ZZ is described in a floating coordinate system by coordinate transformation. Subsequently, the dimensionless frequency and dynamic responses of rotating sandwich beams are analyzed, and the influence of core-to-face elastic modulus ratio and thickness ratio on them are investigated. The results show that the segmental description of the cross-section based on the zigzag theory is more influential than the warping description by higher-order interpolating polynomials. Consequently, reducing the order of original high-order ANCF-ZZ beam elements can improve computational efficiency without affecting accuracy. In addition, the ANCF-ZZ captures the dynamic stiffening effect and frequency veering phenomena, which are essential characteristics in rotational frequency analysis. Compared with commercial finite element software, the ANCF-ZZ not only can accurately analyze the dynamic stiffening effect of rotating sandwich beams but also can obtain the deformation response at non-constant rotational speeds and higher rotational speeds using a few elements, which is valuable for engineering applications.

Viscoelastic stiffening of gelatin hydrogels for dynamic culture of pancreatic cancer spheroids

The tumor microenvironment (TME) in pancreatic adenocarcinoma (PDAC) is a complex milieu of cellular and non-cellular components. Pancreatic cancer cells (PCC) and cancer-associated fibroblasts (CAF) are two major cell types in PDAC TME, whereas the non-cellular components are enriched with extracellular matrices (ECM) that contribute to high stiffness and fast stress-relaxation. Previous studies have suggested that higher matrix rigidity promoted aggressive phenotypes of tumors, including PDAC. However, the effects of dynamic viscoelastic matrix properties on cancer cell fate remain largely unexplored. The focus of this work was to understand the effects of such dynamic matrix properties on PDAC cell behaviors, particularly in the context of PCC/CAF co-culture. To this end, we engineered gelatin-norbornene (GelNB) based hydrogels with a built-in mechanism for simultaneously increasing matrix elastic modulus and viscoelasticity. Two GelNB-based macromers, namely GelNB-hydroxyphenylacetic acid (GelNB-HPA) and GelNB-boronic acid (GelNB-BA), were modularly mixed and crosslinked with 4-arm poly(ethylene glycol)-thiol (PEG4SH) to form elastic hydrogels. Treating the hybrid hydrogels with tyrosinase not only increased the elastic moduli of the gels (due to HPA dimerization) but also concurrently produced 1,2-diols that formed reversible boronic acid-diol bonding with the BA groups on GelNB-BA. We employed patient-derived CAF and a PCC cell line COLO-357 to demonstrate the effect of increasing matrix stiffness and viscoelasticity on CAF and PCC cell fate. Our results indicated that in the stiffened environment, PCC underwent epithelial-mesenchymal transition. In the co-culture PCC and CAF spheroid, CAF enhanced PCC spreading and stimulated collagen 1 production. Through mRNA-sequencing, we further showed that stiffened matrices, regardless of the degree of stress-relaxation, heightened the malignant phenotype of PDAC cells. Statement of significanceThe pancreatic cancer microenvironment is a complex milieu composed of various cell types and extracellular matrices. It has been suggested that stiffer matrices could promote aggressive behavior in pancreatic cancer, but the effect of dynamic stiffening and matrix stress-relaxation on cancer cell fate remains largely undefined. This study aimed to explore the impact of dynamic changes in matrix viscoelasticity on pancreatic ductal adenocarcinoma (PDAC) cell behavior by developing a hydrogel system capable of simultaneously increasing stiffness and stress-relaxation on demand. This is achieved by crosslinking two gelatin-based macromers through orthogonal thiol-norbornene photochemistry and post-gelation stiffening with mushroom tyrosinase. The results revealed that higher matrix stiffness, regardless of the degree of stress relaxation, exacerbated the malignant characteristics of PDAC cells.

Influence of geometric nonlinearity on static and dynamic response of flexible beam

The flexible beam theory is widely applied in engineering, due to that many engineering structures can be regarded as flexible beam. For the flexible beam under large rotation, researches have shown that consideration of geometric nonlinearity is necessary: beam deflection will be erroneously larger or diverge without consideration of geometric nonlinearity. The reason lies on the dynamic stiffening effect caused by geometric nonlinearity. The dynamic stiffening effect of flexible beam under large rotation is well studied already, however, most studies only focus on flexible beam with constant cross section and rarely considers flexible beam without rotation. In view of this, a flexible beam with variable and twisted cross section is adopted to analyze the influence of geometric nonlinearity, and the response of flexible beam without rotation or static response is also analyzed. The results of dynamic beam deflection with and without consideration of geometric nonlinearity is compared, the results of static beam deflection under different load is also compared. On the basis of result comparison, the influence of geometric nonlinearity on static and dynamic response of flexible beam is then explained.

Size-Dependent Rigid–Flexible Coupling Dynamics of Functionally Graded Rotating Moderately Thick Microplates

Micro air vehicles, which are typical small-sized rotating-motion systems, have seen major advancements in recent years. To provide some theoretical basis for developing and producing micro air vehicles, this study establishes a novel rigid–flexible coupling dynamic model for functionally graded (FG) moderately thick rectangular microplates attached to a central rotating rigid hub based on the modified couple stress theory and first-order shear deformation theory. The proposed model incorporates nonlinear coupling term of in-plane deformation to reflect the dynamic stiffening effect caused by rotational motion. Material characteristics of the FG microplate have a linear power-law distribution along the thickness axis. Further, the discrete form dimensionless coupling dynamic equations and their numerical solutions are obtained by combining the Euler–Lagrange equation and the Chebyshev–Ritz method. Convergence and comparative studies are carried out to demonstrate the accuracy and validity of the proposed model. Thereafter, the influence of material length scale parameter, rotational speed, gradient index, and aspect ratio on the frequency of the microplates is investigated. Numerical results reveal that couple stress and dynamic stiffening effects both enhance the rigidity of the microplates, whereas the gradient index decreases the rigidity. Nonlinear coupling term which leads to significant differences in frequency value and trace line can’t be ignored for rotative structure. In-plane motion and its coupling terms play a significant function for the moderately thick or thick microplates. The increase of rotational speed and gradient index will reduce the size dependency of the microplate. Furthermore, the frequency trajectory steering and corresponding mode transition phenomenon are graphically represented.

Study on the Rotation Effect on the Modal Performance of Wind Turbine Blades

With the large-scale development of wind turbines, large flexible blades bear heavier loads. In the actual rotating work of blades, the coupling of structural deformation and motion produces a dynamic stiffening effect and spin softening effect, which affects the dynamic characteristics of blades. In this study, the finite element method is used to model the NREL 5MW blade, and the dynamic stiffening and spin softening effects are investigated using the modal analysis. The influence of rotating effects on the blade’s natural frequency is revealed. It is concluded that the effect of dynamic stiffening is more significant than that of spin softening, and the comprehensive result of the two effects is not simply the superposition of them but presents obvious nonlinearity.

Dynamic modeling and analysis of center rigid body Timoshenko beam model based on unconstrained mode

The transverse deformation of the beam will lead to the longitudinal shortening deformation, and this transverse-longitudinal deformation coupling will bring the dynamic stiffening effect term on the generalized rigidity of the beam model. For the rotating beam structure, the centrifugal force will cause axial tension, with coupling axial and transverse deformation of the beam and bring additional geometric stiffness, which is more obvious for the thick short beam. The central rigid body-Timoshenko beam model with a large-range-motion center was investigated. Firstly, the dynamic model with centrifugal forces was established by means of the Timoshenko beam theory and the Hamilton principle. Secondly, the unconstrained mode concept was introduced, and the unconstrained mode shape functions and natural frequencies were solved with the Frobenius method. Finally, numerical simulations were carried out to explore the difference of generalized stiffness between the unconstrained mode and the constrained mode at different constant speeds, and the effects of centrifugal forces on the model under unconstrained mode condition were discussed.

First-order approximate rigid-flexible coupled dynamics analysis of a simple aero-engine blade model with dynamic stiffening effect

First-order approximate rigid-flexible coupled dynamics analysis of a simple aero-engine blade model with dynamic stiffening effect

The influence of rotation on natural frequencies of wind turbine blades with pre-bend

The natural frequencies of wind turbine blades in the rotating state differ significantly from those in free vibration conditions. In this paper, an established model blade analysis with the multi-body method and modal parameter identification were used to investigate the influence of rotation on natural frequencies of wind turbine blades. For the blades with rated powers of 100 kW, 1.5 MW, and 2.3 MW, the frequency growth with the increase in rotating speed, as well as the influence of gravity and pre-bend on frequencies and intensity of vibrations, was investigated. Additionally, the Campbell diagrams were analyzed. The results reveal that the combined effect of centrifugal force and pre-bend can induce the vibration of flap-edge coupling, and the intensity of the coupled vibration enhances with the increase in rotating speed. Gravity has little effect on the natural frequencies of rotating blades, but it enhances the edgewise vibration. Blades with stronger flexibility have a greater dynamic stiffening effect mainly in the flapwise direction. As rotating speed increases, the first-order flapwise (1st flap) natural frequency increases significantly, while the growth rate of higher-order frequencies decreases successively. When the 100 kW, 1.5 MW, and 2.3 MW wind turbines reached their rated speeds, the 1st flap natural frequencies increased by 19.41%, 20%, and 15.91%, respectively. Additionally, in the Campbell diagram, the corresponding rotating speed at the intersection of the 1st flap frequency and 3P (3P is three times the rotation frequency of the wind rotor) increases when rotation is considered.

Vibration analysis of a free moving thin plate with fully covered active constrained layer damping treatment

A comprehensive dynamic model for a free moving thin plate with fully covered active constrained layer damping (ACLD) treatment is developed. The discrete equations of motion of the moving ACLD plate are derived by using Ritz method and Lagrange’s equations. Free vibration analysis of the composite plate undergoing large rotational or translational motions are performed by solving the complex modal eigenvalue problem of the system in open-loop and closed-loop cases. Interesting frequency curve veering, mode shape switching as well as damping ratio jumping phenomena are observed and discussed for the plate rotating around two different axes. It is demonstrated that the flexible plate undergoing rigid motions can not only generate dynamic stiffening effect but also dynamic softening effect. Effects of parameters such as angular velocity, acceleration, and control gain on the modal and damping characteristics of the ACLD plate are also investigated.

Modal Analysis of a Compressor Disk

The compressor disk is one of the key parts of the aeroengine. Its structure and strength play an important role in the reliability of the compressor. Most of the damage are caused by resonance stress. At the same time, the dynamic design and analysis of the key components are required because of the rotating state. When the compressor disk is in normal operation, the disk rotates around the rotating shaft in a large scale. Centrifugal inertia force will affect the deformation of the structure, and induce a certain change of natural frequency when compared to the static condition, which has increased complexity to modelling and numerical calculation. In this paper, the effect of dynamic stiffening on the vibration characteristics of disc is studied. In order to derive the equations of motion, Hamilton’s principle is used, and the boundary conditions are obtained at the same time. We studied the natural frequency of the isotropic plate and laminate plate by Ritz method. Using the popular finite element analysis software ANSYS, we carry on simulation computation of vibration modes on a certain type of compressor in the working process. Besides, by investigation of the influence of structure parameter and rotating speed on the forward travelling wave and back travelling wave, we studied the critical speed of the compressor disk, which is essential in the rotator.

A regularized approach for frictional impact dynamics of flexible multi-link manipulator arms considering the dynamic stiffening effect

The present study offers a regularized approach for multi-link flexible manipulator arms with frictional impacts. The complex risks of global dynamics simulation, which involve nonlinear frictional impact, stick–slip, and foreshortening deformation, as well as multi-scale numerical problems, were implemented. The system is described as an assembly of $n$ flexible links connected by $n$ rotary joints. The stretching, bending, and the torsional deformations of the flexible links were considered in addition to the flexibility and mass of the joint. The introduction of a contact force potential energy approach transformed the non-differentiable functions of the normal and tangential frictions into differentiable ones, thereby generating Lagrange equations for the general recursive formulation of the systems. A numerical simulation for the double pendulum and spatial manipulator arms collision with targets was generated, thereby allowing the calculation of the frequent switching between the stick/sliding and forward/backward sliding. Several normal contact and friction models were adopted, and their corresponding results were analyzed. The generated ordinary differential equations of the proposed smoothed algorithm were solved using explicit solvers to verify any improvements in the global computational efficiency of the frictional collision dynamics for the flexible manipulator arms.

Dynamic modeling and analysis of a rotating flexible beam with smart ACLD treatment

Abstract Dynamics of a rotating flexible beam with fully covered active constrained lamped damping (ACLD) treatment is investigated. The ACLD beam consists of three sub-layer beams: a piezo-constraining layer, a visco-elastic layer, and a base beam layer. By using the method of assumed modes to describe the deformations of the three sub-layer beams, with the longitudinal shrinking caused by transverse deformation included and all the high-order terms retained, the high-order approximate coupling (HOAC) dynamic equations of a hub-ACLD beam system are derived. The dynamic stiffening effect is considered in this new dynamic model and the model can be used to study the dynamics of a hub-ACLD system undergoing large overall rotating motions. An active/passive hybrid vibration control strategy is adopted to control the vibration of the rotating beam. Simulation results indicate that the proposed method has better parametric adaptability and numerical stability than the other available in the literature. Performances of the system controlled by ACLD technology are superior to those controlled by pure active control technology. By solving the characteristic complex eigenvalue problems of the system numerically, vibration frequencies and damping ratios of the system are obtained and verified. The results of this work can be helpful to the design of smart composite structures for vibration suppression and control in rotating rigid-flexible coupling structures such as robotic arms.

An element-free Galerkin approach for rigid–flexible coupling dynamics in 2D state

Based on an improved element-free Galerkin (EFG) method and the geometric nonlinear elastic theory, an element-free numerical approach for rigid–flexible coupling dynamics of a rotating hub–beam system is implemented. Using the Hamilton principle, the coupled nonlinear integral-differential governing equations are derived, in which the dynamic stiffening effect is expressed by the longitudinal shrinking of the beam induced by the transverse displacement. By the EFG method where the global interpolating moving least squares (IMLS) and generalized moving least squares (IGMLS) were used for discretizing the longitudinal and transverse deformation variables, respectively, the spatially element-free discretized dynamic equations of the system are obtained and the Newmark scheme was selected as the time integration method for the numerical computation. The superiority of the global interpolating property makes the ease for imposing both displacement and derivative boundary conditions. The transverse bending vibration and structural dynamics of the flexible beam in non-inertial system are analyzed in the numerical simulations and influence parameters of the EFG method are also fully discussed. Simulation results verify the feasibility of the improved EFG approach for rigid–flexible coupling dynamics.

Three-dimensional vibration of rotating functionally graded beams

The three-dimensional free vibration and time response of rotating functionally graded (FG) cantilevered beams are studied. Material properties of functionally graded beams are assumed to change gradually through both the width and the thickness in power-law form. The second-kind Lagrange’s equations are used in conjunction with the Ritz method to derive the comprehensive coupling dynamic equations for the axial, chordwise, and flapwise motions. The trial functions of deformations are taken as the products of the Chebyshev polynomials and the corresponding boundary functions. Nonlinear coupling deformations are considered to capture the dynamic stiffening effect due to the rotating motion. The influences of the material gradient index and rotational speed on modal characteristics are investigated by the state space method. The eigenvalue loci veering phenomena with modal conversions are exhibited. The time responses indicate that the deformations of rotating functionally graded beams are greatly affected by the material gradient index. It is shown that for large deformation problems, using Chebyshev polynomials is more efficient in computing precision and robustness than using other polynomials.

Influences of physical and structural parameters on vibration modes for large-scale rotating wind turbine blades

For large-scale offshore wind turbine blades, ANSYS and UG were respectively used to complete the solid modeling and the calculation of vibration modes, and the influences of the rotating speed on each order vibration mode and their main reasons were analyzed. Furthermore, the effects of material categories, the deviations of physical parameters in the process of material manufacture and structural parameters on the natural frequencies of rotating blades were respectively compared. Numerical results show that the dynamic stiffening effect of rotating blades is obvious, and the stress stiffness and the geometric stiffness play a dominant role on the natural frequencies from the first to the sixth order and from the seventh to the tenth order respectively. The influences of material categories on natural frequencies of the blades are significantly higher than those of physical parameter deviations and chord length changes. The effects of the equal amplitude increase of elasticity modulus or the chord length and the equal amplitude decrease of density on the tenth order frequency for the torsional vibration are less than those of them on the first nine orders' frequencies for the shimmy and flapping vibration, making that the increase amplitude of the blade natural frequencies first increases and then decreases with the increase of the order number. In addition, the results in this work can provide technical references for the optimization design and the further analysis of vibration characteristics of wind turbine blades.

Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system

The aim of this paper is to compare the accuracy of the absolute nodal coordinate formulation and the floating frame of reference formulation for the rigid-flexible coupling dynamics of a three-dimensional Euler–Bernoulli beam by numerical and experimental validation. In the absolute nodal coordinate formulation, based on geometrically exact beam theory and considering the torsion effect, the material curvature of the beam is derived, and then variational equations of motion of a three-dimensional beam are obtained, which consist of three position coordinates, two slope coordinates, and one rotational coordinate. In the floating frame of reference formulation, the displacement of an arbitrary point on the beam is described by the rigid-body motion and a small superimposed deformation displacement. Based on linear elastic theory, the quadratic terms of the axial strain are neglected, and the curvatures are simplified to the first order. Considering both the linear damping and the quadratic air resistance damping, the equations of motion of the multibody system composed of air-bearing test bed and a cantilevered three-dimensional beam are derived based on the principle of virtual work. In order to verify the results of the computer simulation, two experiments are carried out: an experiment of hub–beam system with large deformation and a dynamic stiffening experiment. The comparison of the simulation and experiment results shows that in case of large deformation, the frequency result obtained by the floating frame of reference formulation is lower than that obtained by the experiment. On the contrary, the result obtained by the absolute nodal coordinate formulation agrees well with that obtained by the experiment. It is also shown that the floating frame of reference formulation based on linear elastic theory cannot reveal the dynamic stiffening effect. Finally, the applicability of the floating frame of reference formulation is clarified.

Free vibration analysis of rotating functionally graded rectangular plates

A dynamic model of a functionally graded rectangular plate undergoing large overall motions is presented in this paper. The material properties of the FGM plate are assumed to vary continuously in its thickness direction according to a power-law distribution. The dynamic model with the dynamic stiffening effect included is used to study the free vibration characteristics of various kinds of rotating cantilever rectangular FGM plates. Simulations results from the present dynamic equations are verified to have higher accuracy than those from the previous method in literature, which induces different frequency variation phenomena for the same case. Phenomena of frequency loci veering rather than crossing can be observed in either rotating homogeneous plates or rotating FGM plates. Complicated frequency loci veering and associated mode shift phenomena occur in a rotating hub-FGM plate system, where a frequency locus may veer more than once due to different types of modal coupling between the bending and torsional vibrations of the FGM plate. The effects of dimensionless parameters such as the hub radius ratio, the aspect ratio, and the volume fraction exponent on the variations of the natural frequencies of rotating FGM plates are also investigated.

Modeling and control of lateral vibration of an axially translating flexible link

Manipulators used for the transportation of large panel-shape payloads often adopt long and slender links (or forks) with translational joins to carry the payloads. As the size of the payload increases, the length of the links also increases to hold the payload securely. The increased length of the link inevitably amplifies the effect of the flexure in the link. Intuitively, the translational motion of the link in its longitudinal direction should have no effect on the lateral vibration of the link because of the orthogonality between the direction of the translational motion and the lateral vibration. If, however, the link was flexible and translated horizontally (perpendicular to the gravitational field) the asymmetric deflection of the link caused by gravity would break the orthogonality between the two directions, and the longitudinal motion of the link would excite lateral motion in the link. In this paper, the lateral oscillatory motion of the flexible link in a large-scale solar cell panel handling robot is investigated where the links carry the panel in its longitudinal direction. The Newtonian approach in conjunction with the assumed modes method is used for derivation of the equation of motion for the flexible forks where non-zero control force is applied at the base of the link. The analysis illustrates the effect of longitudinal motion on the lateral vibration and dynamic stiffening effect (variation of the natural frequency) of the link due to the translational velocity. Lateral vibration behavior is simulated using the derived equations of the motion. A robust vibration control scheme, the input shaping filter technique, is implemented on the model and the effectiveness of the scheme is verified numerically.

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

Abstract A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.