Approximate Response of Beam-Type Resonant Biosensors

2018 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Exponential Function ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Impact Force ◽  
Equations Of Motion ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
The Impact

In this paper, the effect of absorption of antigens to the functionalized surface of a biosensor is modeled using a single degree-of-freedom mass-spring-damper system. The change in the mass of the system due to absorption is modeled with an exponential function. The governing equations of motion is derived considering the change in the mass of the system as well as the impact force due to absorption. It has been demonstrated that this equation is a linear second-order ordinary differential equation with time-varying coefficients. The solution of this differential equation is approximated by expanding the exponential function with a Taylor series and applying the method of multiple scales. The advantage of using the method of multiple scales to derive an approximate solution is in the insight it provides on the effect of each parameter on the response of the system. The free vibration response of the biosensor is derived using the approximate solution under different conditions, namely, with and without viscous damping, with and without considering the impact force, and for different binding rates.

Modeling and Dynamic Analysis of a Slider-Crank Mechanism With Flexible Coupler and Joint

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz Haque
Keyword(s):  
Dynamic Analysis ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Joint Stiffness ◽  
Mechanism Model ◽  
Double Frequency ◽  
Varying Coefficients ◽  
Matrix Expansion ◽  
Time Varying Coefficients ◽  
Crank Mechanism

Abstract Modeling and dynamic analysis of a slider-crank mechanism with flexible joint and coupler is presented. The equations of motion of the mechanism model are formulated using a virtual work multibody formalism and cast in terms of a minimum set of generalized coordinates through a Jacobian matrix expansion. Numerical results show the influence of time-varying coefficients on the mechanism dynamic behavior due to a repeated task. The results illustrate that the joint motion and coupler deformation are highly coupled. The joint response is dominated by double frequency of input, however, the coupler deformation is influenced by the same frequency as that of excitation. Increase in joint stiffness tends to decrease the variations in coupler deformation.

scholarly journals An Evolutionary Analytic Method of Multi-DOF Nonlinear Coupling Dynamic Model for Controllable Close-Chain Linkage Mechanism System

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Ru-Gui Wang ◽  
Gan-Wei Cai ◽  
Xiao-Rong Zhou
Keyword(s):  
Approximate Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Responses ◽  
Analytic Method ◽  
Nonlinear Coupling ◽  
Linkage Mechanism ◽  
First Order ◽  
Coupling System ◽  
Coupling Dynamic

The 2-DOF controllable close-chain linkage mechanism is investigated in this paper. Based on the characteristics of the multi-DOF nonlinear coupling dynamic equation of the system established by the finite element method, an analytic method of multiple-scales Newmark is presented after thinking about the method of perturbation and the method of numerical analysis. Firstly, the first-order approximate solution of the dynamic responses of the system at the time of t is calculated by the multiple scales method. Then, taken the first-order approximate solution as the initialization of the generalized coordinate of the system, the stable dynamic response of the system is obtained by the implicit Newmark method. The simulation and experimental results are given in the end. The studies indicate that the method of multiple-scales Newmark is correct and practicable to study the dynamic characteristics of such kind of multi-DOF nonlinear coupling system.

SOLITON-LIKE PHONON LOCALIZED MODES IN DIATOMIC NONLINEAR LATTICE CHAIN

1991 ◽  
Vol 05 (13) ◽  
pp. 2237-2252 ◽  
Author(s):  
ZHU-PEI SHI ◽  
GUOXING HUANG ◽  
RUIBAO TAO
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Localized Modes ◽  
Wave Approximation ◽  
Long Wave ◽  
Model Hamiltonian ◽  
Nonlinear Lattice ◽  
Gap Solitons ◽  
Long Wave Approximation

We have studied the localization of multivibrational states in diatomic nonlinear lattice chain. A simple model Hamiltonian presented here describes a system containing two kinds of phonons. The equations of motion for these boson operators are two partial differential equations with nonlinear coupling in long-wave approximation. With the help of the method of multiple scales, these equations are reduced to the nonlinear Schrödinger equation. It is shown that soliton-like phonon localized modes, multi-phonon localized modes can exist. The possibility of observing the gap solitons (phonon localized modes in the frequency gap) in diatomic nonlinear lattice chain is predicted.

Effects of Flexural Stiffness, Axial Velocity and Internal Support Locations on Translating Beam’s Natural Frequencies

2014 ◽  
Vol 532 ◽  
pp. 316-319 ◽  
Author(s):  
Ferid Köstekci
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Flexural Rigidity ◽  
Equations Of Motion ◽  
Flexural Stiffness ◽  
Transverse Vibrations ◽  
Internal Support ◽  
Simple Support ◽  
Moving Speed

The aim of this paper is to examine the natural frequencies of beams for different flexural stiffness, internal simple support locations and axial moving speed. In the present investigation, the linear transverse vibrations of an axially translating beam are considered based on Euler-Bernoulli model. The beam is passing through two frictionless guides and has an internal simple support between the guides. The governing differential equations of motion are derived using Hamiltons Principle for two regions of the beam. The method of multiple scales is employed to obtain approximate analytical solution. Some numerical calculations are conducted to present the effects of flexural rigidity, mean translating speed and different internal support locations on natural frequencies.

Nonlinear Free Vibrations of a Timoshenko Beam Using Multiple Scales Method

Volume 2 ◽  
2004 ◽  
Author(s):  
Asghar Ramezani ◽  
Mehrdaad Ghorashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Free Vibrations ◽  
Partial Differential ◽  
Cantilevered Beam

In this paper, the large amplitude free vibration of a cantilever Timoshenko beam is considered. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the beam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the thick-cantilevered beam.

Superharmonic Resonance of Cross-Ply Laminates by the Method of Multiple Scales

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Hadj Youzera ◽  
Sid Ahmed Meftah ◽  
El Mostafa Daya
Keyword(s):  
Composite Materials ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Transversely Isotropic ◽  
Superharmonic Resonance ◽  
Plastic Composite ◽  
Isotropic Composite ◽  
Forced Vibration Analysis

General differential equations of motion in nonlinear forced vibration analysis of multilayered composite beams are derived by using the higher-order shear deformation theories (HSDT's). Viscoelastic properties of fiber-reinforced plastic composite materials are considered according to the Kelvin–Voigt viscoelastic model for transversely isotropic composite materials. The method of multiple scales is employed to perform analytical frequency amplitude relationships for superharmonic resonance. Parametric study is conducted by considering various geometrical and material parameters, employing HSDT's and first-order deformation theory (FSDT).

scholarly journals Nonlinear tuning of microresonators for dynamic range enhancement

2015 ◽  
Vol 471 (2179) ◽  
pp. 20140969 ◽  
Author(s):  
M. Saghafi ◽  
H. Dankowicz ◽  
W. Lacarbonara
Keyword(s):  
Nonlinear Response ◽  
Dynamic Range ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Path Following ◽  
Critical Amplitude ◽  
Inertia Effects ◽  
Response Characteristics ◽  
The Galerkin Method

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

Nonlinear Marine Structures With Random Excitation

1988 ◽  
Vol 110 (3) ◽  
pp. 246-253 ◽  
Author(s):  
E. R. Jefferys
Keyword(s):  
Narrow Band ◽  
Equations Of Motion ◽  
Random Excitation ◽  
Wave Excitation ◽  
Offshore Structure ◽  
Chaotic Motions ◽  
Varying Coefficients ◽  
Excitation Time ◽  
Vessel Response ◽  
Time Varying Coefficients

Various important types of offshore structure contain significant nonlinearities or time-varying coefficients in their equations of motion. Well-known examples include tension leg platforms, free-hanging risers, single-buoy moorings, ships moored against fenders and vessels constrained by stiffening moorings. When subject to sinusoidal wave excitation, time domain mathematical models of these structures can display large subharmonic or chaotic motions. This paper shows that such behavior is often an artifact of the regularity of the excitation and is usually unlikely to present a significant problem in a random sea. Narrow-band vessel response can, however, generate near-harmonic motions to create conditions in which these instabilities may become important.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

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