Characteristics Of Parametric Resonance Research Articles

Hyperelastic Microcantilever AFM: Efficient Detection Mechanism Based on Principal Parametric Resonance

The impetus of writing this paper is to propose an efficient detection mechanism to scan the surface profile of a micro-sample using cantilever-based atomic force microscopy (AFM), operating in non-contact mode. In order to implement this scheme, the principal parametric resonance characteristics of the resonator are employed, benefiting from the bifurcation-based sensing mechanism. It is assumed that the microcantilever is made from a hyperelastic material, providing large deformation under small excitation amplitude. A nonlinear strain energy function is proposed to capture the elastic energy stored in the flexible component of the device. The tip–sample interaction is modeled based on the van der Waals non-contact force. The nonlinear equation governing the AFM’s dynamics is established using the extended Hamilton’s principle, obeying the Euler–Bernoulli beam theory. As a result, the vibration behavior of the system is introduced by a nonlinear equation having a time-dependent boundary condition. To capture the steady-state numerical response of the system, a developed Galerkin method is utilized to discretize the partial differential equation to a set of nonlinear ordinary differential equations (ODE) that are solved by the combination of shooting and arc-length continuation method. The output reveals that while the resonator is set to be operating near twice the fundamental natural frequency, the response amplitude undergoes a significant drop to the trivial stable branch as the sample’s profile experiences depression in the order of the picometer. According to the performed sensitivity analysis, the proposed working principle based on principal parametric resonance is recommended to design AFMs with ultra-high detection resolution for surface profile scanning.

A broadband magnetically coupled bistable energy harvester via parametric excitation

Owing to the excitation level of ambient vibrations, the wideband performance of multi-stable energy harvesters and parametrically excited energy harvesters is restricted. The bistable periodic interwell motion and parametric resonance initial threshold amplitude primarily depend on the base excitation level. In this paper, a wideband two-element piezoelectric energy harvester with both bistability and parametric resonance characteristics is presented (i.e. designed, theoretically tested, fabricated and experimentally tested) to tackle the issue of reducing the potential barrier and triggering the parametric threshold amplitude. This device is the first to utilise a directly excited element as an external oscillation source to trigger the large amplitude oscillation of a parametrically excited element through magnetic coupling effects; the periodic interwell motion revealed in the parametrically excited device infers its bistability feature; and a continuously effective operational bandwidth (10–17.3 Hz) is achieved for both elements, which infers the high-energy orbit motions. Theoretical modelling of the proposed device and the expression of the magnetic coupling effects are presented to predict the dynamics of the system, as well as verifying the experimental results. The effects of the base excitation level and the gap distance between magnets are also analysed to shed light on the flexibilities and limitations of the device.

Opening the air‐chamber of an oscillating water column spar buoy wave energy converter to avoid parametric resonance

The oscillating-water-column (OWC) spar-buoy is a type of wave energy converter that may exhibit undesirable large roll and pitch amplitudes caused by a dynamic instability induced by parametric resonance. The occurrence of this phenomenon not only reduces the power extraction but significantly increases the structural loads on the buoy, the turbine rotor and on the mooring system. The paper compares the parametric resonance behaviour of two configurations of an OWC spar-buoy using experimental data obtained in a wave flume at a scale of 1:100. The configurations investigated were: (1) closed and (2) fully open-air chamber. The experimental tests covered a wide range of regular and irregular waves, as well as in free decay experiments. Results showed that opening the air chamber reduces the coupling between the buoy and the OWC within, thus shifting the damped natural heave frequency of the system in comparison with the closed chamber configuration. This effect changes parametric resonance characteristics of the two configurations due to the coupling between roll/pitch and heave modes. Moreover, for specific wave frequencies, the occurrence of parametric resonance observed when the chamber is closed do not occur while the air chamber is fully open. These results suggest the possibility of controlling a pressure relief valve installed on top of the device to reduce parametric resonance whenever this dynamic instability is detected.

Mechanotunable optical filters based on stretchable silicon nanowire arrays

Abstract Nano-structural optical filters embedded in elastomers having high mechanical tunability provide the geometric degree of freedom for selective light manipulation. The active control of spectral information in typical structural optical filters is highly limited due to the substrate rigidity. Herein, we present mechanochromic transmissive optical filters by employing flexible and stretchable polymer-embedded silicon nanostructures. Si-based nanowire arrays (Si-NWAs) have been introduced to exhibit parametric resonance characteristics by controlling the period and/or diameter. Furthermore, the spectral shift phenomenon by increased diffraction efficiency was observed after the application of a uniaxial tensile force, which depends on the period of Si-NWAs with a large index contrast between the silicon nanowire and elastomer. The strain-sensitive properties of tunable Si-NWAs filters induced by light diffraction were calculated by simulation based on wave optics. The spectral tunability and light filtering features were simply demonstrated by stretching the Si-NWAs’ optical filters. Our proposed structure provides potential opportunities for a wide variety of applications, including dynamic color display, visual strain sensor and anti-counterfeiting.

On the static and dynamic stability of spherical sandwich shell panels with viscoelastic material core and laminated composite face sheets under uniaxial and biaxial harmonic excitations

The buckling and parametric resonance characteristics of laminated composite spherical sandwich shell panels with viscoelastic material (VEM) core are investigated in the present analysis considering full geometric nonlinearity in the Green–Lagrange sense. The study includes the longitudinal strain and normal strain in the transverse direction along with transverse shear deformation of the VEM core. The core displacements are considered to be varying linearly along the thickness and those of the face sheets follow first-order shear deformation theory. An eight-noded sandwich shell finite element of the serendipity family is adopted to discretize the sandwich shell panel domain. The finite element-based equation of motion is derived using Hamilton’s principle in the form of the Mathieu–Hill-type equation. The dynamic instability regions are obtained by applying Hsu’s criteria-based Saito–Otomi conditions to the transformed equation motion. An in-house finite element-based code is developed in the MATLAB platform to solve the stability problem and to establish the stability regions. A parametric study is carried out to investigate the influence of different system parameters on the critical buckling load and the parametric resonance of the sandwich shell panels. It is noted that an increase in core and constraining layer thicknesses increases the critical buckling load of the sandwich shell panels. The stability boundaries are observed to shift toward a higher-excitation-frequency region in the stability diagram with an increase in constraining layer thickness and a decrease in aspect ratio.

Parametric Resonance Characteristics of Woven Fiber Metal Laminated Plates

The present investigation deals with the assessment of parametric resonance behavior of new aircraft material, i.e., woven fiber metal laminated (FML) plates subjected to in-plane static and harmonic loading using finite element (FE) technique and Bolotin’s method. In this analysis, a four-node isoparametric element with five degrees of freedom per node is adopted. Based on the first-order Reissner–Mindlin theory, the parametric instability of FML plate subjected to in-plane harmonic loading is examined. A MATLAB code is developed for the parametric study on the dynamic stability of FML plates. The reliability of present formulation is checked by comparing numerical results obtained from present FE analysis with the published researches in the field. The influences of several factors, viz. static load factor, aspect ratio, length-to-thickness ratio, number of layers, ply orientation and boundary conditions on the dynamic instability regions are discussed. Significant variations of these factors on dynamic instability zones of FML plates are observed. The instability zones can be used as guidelines for the prediction of the dynamic behavior of FML plates.

A physically meaningful new approach to parametric excitation and attenuation of oscillations in nonlinear systems

Parametric excitation of a nonlinear physical pendulum by modulation of its moment of inertia is analyzed in terms of physics as an example of the suggested approach. The modulation is provided by a redistribution of auxiliary masses. The system is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of nonlinear properties of the physical system in restricting the resonant swinging is emphasized. Phase locking between the drive and oscillations of the pendulum and the phenomenon of parametric autoresonance are investigated. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. The feedback providing active optimal control of amplification and attenuation of oscillations is analyzed. An effective method of suppressing undesirable rotary oscillations of suspended constructions is suggested.

Free Vibration and Dynamic Stability of Functionally Graded Material Plates on Elastic Foundation

The study of parametric resonance characteristics of functionally-graded material (FGM) plates on elastic foundation is proposed under biaxial in plane periodic load. Finite element method in conjunction with Hamilton’s principle is utilised to establish the governing equations in a discrete form, Floquet’s theory was applied to determine the instability regions of FGM plate resting on elastic foundation. The effects of power law index, temperature rise, and foundation coefficients on the natural frequencies and dynamic stability of the plate have been examined in detail through parametric studies. The first two natural frequencies decrease with increase in temperature and power law index values, on the contrary, these two frequencies increase with increase in the foundation constants. Increase in power law index enhances the instability of the FGM plate. Increased foundation stiffness enhances the stability of the plate. Influence of shear layer constant is more dominant compared to the Winkler foundation constant. Defence Science Journal, Vol. 65, No. 3, May 2015, pp.245-251, DOI: http://dx.doi.org/10.14429/dsj.65.8621

Parametric Instability of Beams with Transverse Cracks Subjected to Harmonic In-Plane Loading

Cracks in structural members lead to local changes in their stiffness and consequently their static, dynamic and stability behavior is altered. The influence of cracks on dynamic characteristics like free vibration, buckling and parametric resonance characteristics of a cracked beam with a transverse crack using finite element method (FEM) is investigated in the present work. Modal testing of beams with transverse open crack is conducted using FFT analyzer to verify the frequencies of vibration of beams. The crack is assumed to be open type and the analysis is linear based on small deformation theory neglecting damping. The loading on the beam is considered as axial with a simple harmonic fluctuation with respect to time. A two-noded Timoshenko beam element with provision of crack is used in this study. The equation of motion represents a system of second-order differential equations with periodic coefficients of the Mathieu–Hill type. The development of the regions of instability arises from Floquet's theory and the periodic solution is obtained by Bolotin's approach using FEM. It is observed that the frequencies of vibration and buckling load of the beam are influenced significantly by location and depth of cracks. It is observed that, for a given location of crack, the onset of instability occurs earlier with increase in depth of crack. As the location of crack moves from the fixed end to the free end the excitation frequency increases. The instability occurs later and the width of the instability regions reduces. When the damage is near to the free end, the instability region almost coincides with the instability region for the undamaged beam. This means that the damage near the fixed end is more severe on the dynamic instability behavior than that of the crack located at other positions. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance and safety.

Pendulum with a square-wave modulated length

Parametric excitation of a rigid planar pendulum caused by a square-wave modulation of its length is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of non-linear properties of the pendulum in restricting the resonant swinging is emphasized. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. Stationary oscillations at these boundaries and at the threshold conditions are investigated. The feedback providing active optimal control of pumping and damping is analyzed. Phase locking between the drive and the pendulum at large amplitudes and the phenomenon of parametric autoresonance are discussed.

Parametric Resonance Characteristics of Laminated Composite Curved Shell Panels in a Hygrothermal Environment

The present study deals with the parametric resonance behaviour of laminated composite curved shell panels in a hygrothermal environment using Bolotin’s approach. A simple laminated model is developed using first order shear deformation theory (FSDT) for the vibration and dynamic stability analysis of laminated composite shells subjected to hygrothermal conditions. A computer program based on the finite element method (FEM) in a MATLAB environment is developed to perform all necessary computations. Quantitative results are presented to show the effects of curvature, ply-orientations, degree of orthotropy and geometry of laminates on the parametric instability of composite curved shell panels for different temperature and moisture concentrations. The excitation frequencies of laminated composite panels decrease with the increase of temperature and moisture due to reduction of stiffness for all laminates.

Parametric resonance in a linear oscillator at square-wave modulation

The phenomenon of parametric resonance in a linear torsion spring oscillator caused by a square-wave modulation of its moment of inertia is explained and investigated both analytically and with the help of a computer simulation. Characteristics of parametric resonance and regeneration are found and discussed in detail. Ranges of frequencies within which parametric excitation is possible are determined. Stationary oscillations at the boundaries of these ranges and at the threshold conditions are investigated.

Parametric excitation of a linear oscillator

The phenomenon of parametric resonance is explained and investigated both analytically and with the help of a computer simulation. Parametric excitation is studied for the example of the rotary oscillations of a simple linear system—mechanical torsion spring pendulum excited by periodic variations of its moment of inertia. Conditions and characteristics of parametric resonance and regeneration are found and discussed in detail. Ranges of frequencies within which parametric excitation is possible are determined. Stationary oscillations at the boundaries of these ranges are investigated. The simulation experiments aid greatly an understanding of basic principles and peculiarities of parametric excitation and complement the analytical study of the subject in a manner that is mutually reinforcing.

Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor

Parametric resonance has been well established in many areas of science, including the stability of ships, the forced motion of a swing and Faraday surface wave patterns on water. We have previously investigated a linear parametrically driven torsional oscillator and along with other groups have mentioned applications including mass sensing, parametric amplification, and others. Here, we thoroughly investigate the design of a highly sensitive mass sensor. The device we use to carry out this study is an in-plane parametrically resonant oscillator. We show that in this configuration, the nonlinearities (electrostatic and mechanical) have a large impact on the dynamic response of the structure. This result is not unique to this oscillator—many MEMS oscillators display nonlinearities of equal importance (including the very common parallel plate actuator). We report the effects of nonlinearity on the behavior of parametric resonance of a micro-machined oscillator. A nonlinear Mathieu equation is used to model this problem. Analytical results show that nonlinearity significantly changes the stability characteristics of parametric resonance. Experimental frequency response around the first parametric resonance is well validated by theoretical analysis. Unlike parametric resonance in the linear case, the jumps (very critical for mass sensor application) from large response to zero happen at additional frequencies other than at the boundary of instability area. The instability area of the first parametric resonance is experimentally mapped. Some important parameters, such as damping co-efficient, cubic stiffness and linear electrostatic stiffness are extracted from the nonlinear response of parametric resonance and agree very well with normal methods.

Parametric Resonance Characteristics of Laminated Composite Doubly Curved Shells Subjected to Non-Uniform Loading

The parametric resonance characteristics of laminated composite doubly curved panels subjected to various in-plane static and periodic compressive edge loadings, including partial and concentrated edge loading are studied using finite element analysis. The first order shear deformation theory is used to model the doubly curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to the Sander’s first approximation for doubly curved laminated shells, which can be reduced to Love’s and Donnell’s theories by means of tracers. The effects of number of layers, static load factor, side to thickness ratio, shallowness ratio, boundary conditions, degree of orthotropy, ply orientations and various load parameters on the principal instability regions of doubly curved panels are studied in detail using Bolotin’s method. Quantitative results are presented to show the effects of shell geometry, lamination details and load parameters on the stability boundaries. Results of plates and cylindrical shells are also presented as special cases and are compared with those available in the literature.

Parametric Resonance Characteristics of Laminated Composite Doubly Curved Shells Subjected to Non-Uniform Loading

The parametric resonance characteristics of laminated composite doubly curved panels subjected to various in-plane static and periodic compressive edge loadings, including partial and concentrated edge loading are studied using finite element analysis. The first order shear deformation theory is used to model the doubly curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to the Sander’s first approximation for doubly curved laminated shells, which can be reduced to Love’s and Donnell’s theories by means of tracers. The effects of number of layers, static load factor, side to thickness ratio, shallowness ratio, boundary conditions, degree of orthotropy, ply orientations and various load parameters on the principal instability regions of doubly curved panels are studied in detail using Bolotin’s method. Quantitative results are presented to show the effects of shell geometry, lamination details and load parameters on the stability boundaries. Results of plates and cylindrical shells are also presented as special cases and are compared with those available in the literature.

Parametric resonance

Computer simulation experiments are especially good tools for helping students understand basic principles of physics. In this a, I have developed a package of simulation programs called Physics of Oscillations. One of its programs is suited for examining the phenomenon of paramagnetic resonance in a linear system. The program simulates the parametric excitation of the rotary oscillations of a mechanical torsion-spring pendulum whose moment of inertia is subject to periodic variations. I discuss the conditions and characteristics of parametric resonance, including paramagnetic regeneration. Instructors and their students can also use the Physics of Oscillations package to explore other problems, such as ranges of frequencies within which parametric excitation is possible and stationary oscillations on the boundaries of these ranges. The simulation experiments complement the analytical study of the subject in a manner that is mutually reinforcing.