Force-response Curves Research Articles

In-plane and out-of-plane nonlinear size-dependent dynamics of microplates

The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Karman plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.

Nonlinear dynamics of microplates

In this paper, the nonlinear dynamics of a microplate is investigated based on the modified couple stress theory. The von Kármán plate theory is employed to model the system by retaining in-plane displacements and inertia. The equations of motion are derived via an energy method based on the Lagrange equations, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. These equations are recast into a set of first-order nonlinear ordinary differential equations and the resulting equations are solved by means of the pseudo-arclength continuation technique. The nonlinear dynamics is examined through plotting the frequency-response and force-response curves of the system. The influence of system parameters on the resonant responses is highlighted. The differences in the response amplitude of the system modelled based on the modified couple stress theory and the classical one are discussed.

Atomic force microscopy of red-light photoreceptors using peakforce quantitative nanomechanical property mapping.

Atomic force microscopy (AFM) uses a pyramidal tip attached to a cantilever to probe the force response of a surface. The deflections of the tip can be measured to ~10 pN by a laser and sectored detector, which can be converted to image topography. Amplitude modulation or "tapping mode" AFM involves the probe making intermittent contact with the surface while oscillating at its resonant frequency to produce an image. Used in conjunction with a fluid cell, tapping-mode AFM enables the imaging of biological macromolecules such as proteins in physiologically relevant conditions. Tapping-mode AFM requires manual tuning of the probe and frequent adjustments of a multitude of scanning parameters which can be challenging for inexperienced users. To obtain high-quality images, these adjustments are the most time consuming. PeakForce Quantitative Nanomechanical Property Mapping (PF-QNM) produces an image by measuring a force response curve for every point of contact with the sample. With ScanAsyst software, PF-QNM can be automated. This software adjusts the set-point, drive frequency, scan rate, gains, and other important scanning parameters automatically for a given sample. Not only does this process protect both fragile probes and samples, it significantly reduces the time required to obtain high resolution images. PF-QNM is compatible for AFM imaging in fluid; therefore, it has extensive application for imaging biologically relevant materials. The method presented in this paper describes the application of PF-QNM to obtain images of a bacterial red-light photoreceptor, RpBphP3 (P3), from photosynthetic R. palustris in its light-adapted state. Using this method, individual protein dimers of P3 and aggregates of dimers have been observed on a mica surface in the presence of an imaging buffer. With appropriate adjustments to surface and/or solution concentration, this method may be generally applied to other biologically relevant macromolecules and soft materials.

Frequency response analysis of nonlinear tapping-contact mode atomic force microscopy

The nonlinear vibrations of the tapping-mode atomic force microscopy probe are investigated due to both nonlinearity in tip–sample contact force and curvature of the microcantilever probe. The nonlinear equations of motion for vibrations of the probe are obtained using Hamilton’s principle. In this work, the contact force is considered to be more dominant while previous works only consider Van der Waals force. The nonlinear contact force is expanded using a Taylor series to provide a polynomial with coefficients that are functions of the tip–sample distance. The outcome of this work allows the proper distance to be chosen before scanning to avoid instability in the response. Instability regions must be avoided for accurate imaging. The results show that initial tip–sample distance has a major effect on the stability of the frequency response and force response curves. For analytical investigation, the mode shapes of the atomic force microscopy probe are derived based on the presence of the nonlinear contact force as a boundary condition at the free end of the probe. The frequency response curve is obtained using the method of multiple scales. The results show that the effects of the nonlinearities due to probe geometry and contact force can be minimized. Minimizing the effects of nonlinearities allows for less cumbersome and calculation intensive software packages for atomic force microscopies. This research shows that one possible method of decreasing the nonlinearity effect is increasing the excitation force; however, this can increase the contact region and is not the best solution for canceling the nonlinearity effect. The superior method which is the major contribution of this paper is to find the optimal initial tip–sample distance and excitation force that minimize the nonlinearity effect. It is shown that at a certain tip–sample distance the quadratic and cubic nonlinearities cancel each other and the system responds linearly.

Analysis of vibration suppression of master structure in nonlinear systems using nonlinear delayed absorber

In this paper, a nonlinear delayed absorber is proposed by a delayed feedback loop and is utilized to absorb or suppress the vibration of a two-degree-of-freedom nonlinear system when the primary resonance and the 1:1 internal resonance occur simultaneously. To explain analytically mechanism of performance of the absorber, an integral equation method is provided to obtain the second order approximation and the amplitude equations. As a result, the feedback gain and time delay which make the amplitude of the main system equal to zero can be derived analytically. Optimal working conditions of the system are extracted from the time delay-response curves, the force-response curves and the frequency-response curves. In the illustrative example, the appropriate choice of feedback gains and time delays can reduce the amplitude of the main system by more than $$98\,\%$$ in comparison with the amplitude with no time delay. All analytical predictions are in excellent agreement with the numerical simulations.

Parametric resonances in the two-to-one resonant beams on elastic foundation

The nonlinear vibration of the inextensional beam on the elastic foundation under parametric resonance and two-to-one internal resonance is investigated. Considering the inextensional condition and the second-order moment of the subgrade reaction, the extended Hamilton principle is applied to derive the motion equation of the beam on elastic foundation. Then the multimodal discretization and the method of multiple scales are used to obtain the modulation equations. The nonlinear response is examined by means of the frequency- and force–response curves. It is shown that the two-mode solution is born when the single-mode solution undergoes the pitchfork bifurcation. The shooting method and numerical simulations are applied to investigate the dynamic solutions. Particular attention is placed on the effects of the cut-off frequency on the nonlinear response.

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.

Analysis of subcombination internal resonances in a non-linear cantilever beam of varying orientation with tip mass

In this study, sub-combination internal resonance of a uniform cantilever beam of varying orientation with a tip mass under vertical base excitation was investigated. The Euler–Bernoulli theory for the slender beam was used to derive the governing non-linear partial differential equation. The governing equation, which retains the cubic non-linearities of geometric and inertial type, was discretised using Galerkin's method. The resulting second-order temporal differential equation was then reduced by the method of multiple scales to a set of first order six non-linear ordinary-differential equations, governing the amplitudes and phases of the three interacting modes. Both frequency–response and force–response curves were plotted for the case Ω≈ω4=1/2(ω2+ω5). Two possible responses occurred: single-mode and three-mode responses. The single-mode periodic response was observed to undergo supercritical and subcritical pitchfork bifurcations, which caused three-mode interactions. In the event of three-mode responses, there are conditions for which the low-frequency mode becomes effective over the response, resulting in high-amplitude oscillations.

Nonlinear resonant behavior of microbeams over the buckled state

The present study investigates the nonlinear resonant behavior of a microbeam over its buckled (non-trivial) configuration. The system is assumed to be subjected to an axial load along with a distributed transverse harmonic load. The axial load is increased leading the system to lose the stability via a pitchfork bifurcation; the postbuckling configuration is obtained and the nonlinear resonant response of the system over the buckled state is examined. More specifically, the nonlinear equation of motion is obtained employing Hamilton’s principle along with the modified couple stress theory. The continuous system is truncated into a system with finite degrees of freedom; the Galerkin scheme is employed to discretize the nonlinear partial differential equation of motion into a set of ordinary differential equations. This set of equations is solved numerically employing the pseudo-arclength continuation technique; first a nonlinear static analysis is performed upon this set of equations so as to obtain the onset of buckling (supercritical pitchfork bifurcation) and the buckled configuration of the microbeam. The frequency-response and force-response curves of the system are then constructed over the buckled configurations. A comparison is made between the frequency-response curves obtained by means of the modified couple stress and the classical theories. The effect of different system parameters on the frequency-response and force-response curves is also examined.

Linear and nonlinear dynamics of suspended cable considering bending stiffness

This paper reports a systematic investigation on the linear and nonlinear dynamics of a suspended cable, taking bending stiffness into account. Firstly, the linear dynamics features, for example, eigen frequencies and modes for in-plane and out-of-plane motions, are formulated. Secondly, parametrical studies are conducted to explore the effect of bending stiffness on the natural frequencies and mode shapes of the symmetrical/antisymmetrical in-plane and out-of-plane modes. Then, the three-to-one internal resonance between the first- and third-order in-plane symmetrical modes is analyzed by applying directly the method of multiple scales dealing with the nonlinear partial differential equation and boundary conditions. Finally, the frequency-response curves and force-response curves are obtained through solving the modulation equations using the Newton–Raphson method and the pseudo-arclength scheme. The results show that the bending stiffness plays a considerable role in changing the natural frequencies and mode shapes, shifting the conditions for the occurring of nonlinear interaction, saddle-node bifurcation and Hopf bifurcation of suspended cables.

Nonlinear behaviour of electrically actuated MEMS resonators

The present study investigates the nonlinear size-dependent behaviour of an electrically actuated MEMS resonator based on the modified couple stress theory; the microbeam is excited by an AC voltage which is superimposed on a DC voltage. A high-dimensional reduced order model of the continuous system is obtained by applying the Galerkin scheme to the nonlinear partial differential equation of motion. The pseudo-arclength continuation technique is employed to examine the nonlinear static and dynamic behaviour of the system. Specifically, the nonlinear static behaviour of the system is investigated when the microbeam is excited by the electrostatic excitation (DC voltage); this analysis yields the static deflected configuration of the system and the value of the DC voltage corresponding to the static pull-in instability. The size-dependent dynamic behaviour of the system is examined under primary and superharmonic excitations; the frequency- and force-response curves of the system as well as time histories and phase-plane portraits are constructed. Moreover, the effect of taking into account the length-scale parameter on the static and dynamic behaviour of the system is examined by comparing the results obtained by means of the classical and modified couple stress theories.

Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams

The geometrically nonlinear size-dependent behaviour of a Timoshenko microbeam is examined numerically, taking into account the coupled longitudinal–transverse displacements as well as the rotation. The strain energy of a Timoshenko microbeam is obtained based on the modified couples stress theory. Hamilton’s principle is then employed to derive the nonlinear partial differential equations of motion for the longitudinal, transverse, and rotational motions. The Galerkin scheme is applied to these nonlinear partial differential equations, resulting in a set of nonlinear ordinary differential equations with coupled terms. The nonlinear resonant response of the system is examined by solving the discretized equations of motion via the pseudo-arclength continuation technique and constructing the frequency–response and force–response curves. In particular, the effect of the length scale parameter is investigated by comparing the results obtained using the modified couple and classical theories. The frequency–response curves of the present model are compared to those of the one in which the longitudinal displacement is neglected so as to highlight the importance of taking into account the longitudinal displacement. The effect of other system parameters on the frequency–response and force–response curves is also investigated.

Coupled nonlinear size-dependent behaviour of microbeams

The nonlinear resonant behaviour of a microbeam, subject to a distributed harmonic excitation force, is investigated numerically taking into account the longitudinal as well as the transverse displacement. Hamilton’s principle is employed to derive the coupled longitudinal-transverse nonlinear partial differential equations of motion based on the modified couple stress theory. The discretized form of the equations of motion is obtained by applying the Galerkin technique. The pseudo-arclength continuation technique is then employed to solve the discretized equations of motion numerically. Different types of bifurcations as well as the stability of solution branches are determined. The numerical results are presented in the form of frequency-response and force-response curves for different sets of parameters. The effect of taking into account the longitudinal displacement is highlighted.

Vibration suppression of a symmetrically cantilever composite beam using internal resonance under chordwise base excitation

Large flexible structures can be modeled as flexible beams with an appendage. Passive, semi-active or active vibration absorber can be used as an appendage in such structures. In this paper, the performance of an oscillator consisting of mass and spring at the tip of a symmetrically cantilever composite beam is investigated under chordwise base excitation. The oscillator is located at the tip of the beam in order to relate first-order uncoupled modes of the composite beam through internal resonance. Natural frequencies which are dependent on the oscillator parameters have been selected such that two-to-one and one-to-one internal resonance can be produced. The motion of the combined system is described by a set of four non-linear coupled partial differential equations, corresponding to the four describing variables of motion. The equations of motion are then solved using the method of multiple-scales in the occurrence of two-to-one and one-to-one internal resonances. Saturation phenomenon is detected in the force modulation response at the one-to-one internal resonance. Reduction of amplitudes is shown in both frequency and force–response curves under primary resonance.

The core domain as the force sensor of the yeast mechanosensitive TRP channel

Stretch-activated conductances are commonly encountered in careful electric recordings. Those of known proteins (TRP, MscL, MscS, K2p, Kv, etc.) all share a core, which houses the ion pathway and the gate, but no recognizable force-sensing domain. Like animal TRPs, the yeast TRPY1 is polymodal, activated by stretch force, Ca2+, etc. To test whether its S5–S6 core senses the stretch force, we tried to uncouple it from the peripheral domains by strategic peptide insertions to block the covalent core–periphery interactions. Insertion of long unstructured peptides should distort, if not disrupt, protein structures that transmit force. Such insertions between S6 and the C-terminal tail largely removed Ca2+ activation, showing their effectiveness. However, such insertions as well as those between S5 and the N-terminal region, which includes S1–S4, did not significantly alter mechanosensitivity. Even insertions at both locations flanking the S5–S6 core did not much alter mechanosensitivity. Tryptophan scanning mutations in S5 were also constructed to perturb possible noncovalent core–periphery contacts. The testable tryptophan mutations also have little or no effects on mechanosensitivity. Boltzmann fits of the wild-type force–response curves agree with a structural homology model for a stretch-induced core expansion of ∼2 nm2 upon opening. We hypothesize that membrane tension pulls on S5–S6, expanding the core and opening the TRPY1 gate. The core being the major force sensor offers the simplest, though not the only, explanation of why so many channels of disparate designs are mechanically sensitive. Compared with the bacterial MscL, TRPY1 is much less sensitive to force, befitting a polymodal channel that relies on multiple stimuli.

Lumbar Vertebral Growth Is Governed by “Chondral Growth Force Response Curve” Rather Than “Hueter-Volkmann Law”

Vertebral defects were created in a validated 3D finite element model (FEM) to simulate destructive tubercular lesions of increasing severity. Forces in various parts of the spine were then calculated and correlated to deformity progression and growth modulation (GM) changes. To understand the biomechanical basis of GM, which governs spinal growth and the progression of kyphosis in posttubercular kyphotic (PTK) deformities. Hueter-Volkmann Law (HVL), chondral growth force response curve (CGFRC), and regional growth acceleratory phenomenon have all been proposed to explain the modulation of growth in limbs but have not been tested in vertebral end plates (VEP). We have previously documented various GM changes in posttubercular kyphotic. By simulating the kyphotic collapse in a validated FEM, the mechanical basis of GM can be established. Sixty-three children with tuberculosis treated conservatively formed the clinical material. The progress of deformity and GM changes in the fusion mass and the kyphotic curve was documented. Defects simulating lesions of four levels of severity (types A, B, C, and D) were created in a validated 3D FEM and subjected to load till restabilization occurred. The stresses at the end plates, discs, facet joints, and the points of contact were calculated. Regional growth acceleratory phenomenon and favorable growth changes were found in type A collapse where the facets were intact. With increasing destruction, the forces in the facet capsules increased beyond 30 MPa predicting facet dislocations in types B, C, and D collapse. As the contact stress on the VEP increased to 16.6 MPa (type B) and 40 MPa (type C), this was associated with growth suppression. Type D collapse involved facet dislocation at multiple levels leading to "buckling collapse". Acceleratory growth was found both in tension and compression phases proving that VEP growth followed principles of CGFRC rather than HVL. This is the first study in the current literature to demonstrate that spinal growth follows CGFRC rather than HVL. This observation opens a potential window of opportunity to treat spinal deformities by mechanical GM.

1:2 and 1:3 internal resonance active absorber for non-linear vibrating system

Vibration and dynamic chaos should be controlled in either structures or machines. An active vibration absorber for suppressing the vibration of the non-linear plant when subjected to external and parametric excitations is studied in the presence of one-to-two and one-to-three internal resonance. The main attention is focused on the study of the active control and stability of two systems, which can be used to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to determine four first-order non-linear ordinary differential equations that govern the modulation of the amplitudes and phases in the presence of internal resonance of the two systems with quadratic and cubic order of control. These equations are used to determine the steady state solutions and their stability. The stability study of non-linear periodic solution for two cases (1:2 and 1:3 internal resonance) and the stability of the obtained numerical solution are investigated using frequency, force–response curves and phase-plane method. Also, effects of some parameters on the steady state solution of the vibrating system are investigated and reported in this paper. Variation of some parameters leads to the bending of the frequency, force–response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.

Non-Linear Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports

In this study, nonlinear transverse vibrations of a tensioned Euler-Bernoulli beam resting on multiple supports are investigated. The immovable end conditions due to simple supports cause stretching of neutral axis and introduce cubic nonlinearity to the equations of motion. Forcing and damping effects are included in the analysis. The general arbitrary number of support case is investigated and 3, 4, and 5 support cases analyzed in detail. A perturbation technique (the method of multiple scales) is applied to the equations of motion to obtain approximate analytical solutions. 3:1 internal resonance case is also considered. Natural frequencies and mode shapes for the linear problem are found for the tensioned beam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different forcing and damping cases. Frequency-response and force-response curves are drawn. Different internal resonance cases between modes are investigated.

Effects of Additional Mass on Modal Experimental Analysis of 3-Dimension and 5-Direction Braided Composites

The work of vibration test has significant meanings for the researches and applications of 3-dimension and 5-direction braided composites. This article discusses the effects of added mass with different weight on the modal test of 3-dimension and 5-direction braided composites. The comparison of the modal parameters of 3-dimension and 5-direction braided composites tested by different weight of mass reveals that the additional mass is a mostly influence factor for vibration property of 3-dimension and 5-direction braided composites. The results of frequency response and force response curves show that smaller mass accelerometer is more effective for a wider range of frequencies around the resonance frequency, a higher natural frequency and a larger peak in these points. Force-response curves show that force response amplitude increases with the increase of additional mass weight, and the larger additional mass, the shorter time taken for reaching stationary state. The errors of natural frequency and damping ratio increase when the weight of additional mass increases. With the increase of modal orders, relative errors of modal characteristics have slighter decreasing degrees. The results derived from this article will provide a useful reference for precise modal analysis of 3-dimension and 5-direction braided composites.

Focus on nanoscience and nanotechnology

Focus on nanoscience and nanotechnology