M. Amin Changizi

A New Approach for the Non-linear Analysis of the Deflection of Beams Using Lie Symmetry Groups

The linear formulation does not yield acceptable results when is used to the beams that experience large deflections. Further, the linear models could accommodate large deflections such as those encountered in some machinery where bending does not exceed three times the thickness of the beam. However, the defection of beams subjected to arbitrary loads that yield non-linear deflection has been solved so far only for two loading conditions; point moment and point force. The present work presents a general method based on Lie symmetry groups that yields an exact solution to the general problem involving any arbitrary loading.

A Closed Form Solution for Non-linear Deflection of Non-straight Ludwick Type Beams Using Lie Symmetry Groups

Micro Electro Mechanical Systems (MEMS) have found a large range of applications over the recent years. One of the prodigious application of micro-cantilever beams that is in use is represented by AFM probes (Atomic Force Microscopy). The AFM principle is based on the real-time measurement of the deflection of a micro-beam while following a surface profile. Hence, the prior knowledge of the deflection of beams has been of great interest to designers. Although both analytical and numerical solutions have been found for specific type of loads, there is no general solution specifically formulated for micro-cantilever beams that are not geometrically perfectly straight. Hence, the problem has not been specifically considered so far. The current work presents an analytical method based on Lie symmetry groups. The presented method produces an exact analytical solution for the deflection of Ludwick type beams subjected to any point load for non-straight beams. The Lie symmetry method is used to reduce the order of the Ordinary Differential Equation (ODE) and formulate an analytical solution of the deflection function. The result is compared with an analytical solution for a particular case that is available in the open literature. It was found that the two results coincide.

Viscosity Measurement for Picoliter Liquid Volumes Using Twin Micro-Cantilever Beams

In this paper a novel method to measure viscosity of a liquid with a volume of fraction of pico-litre is presented. The measurement of viscosity keeps into account the effect of evaporation which can result in the total evaporation of the liquid in a short time. At the present time there are several standard methods to measure viscosity of a fluid. All methods require larger volume of liquid and it takes rather long time to measure viscosity. No effect of evaporation is considered in these methods given the large volume of fluid that is used. In the proposed method viscosity of a droplet is measured by considering that in ambient condition such small droplet evaporates in less than a second. The principle of measurement includes two parallel micro-cantilever beams for which their individual dynamic response could be measured. Micro-cantilever beams are being used in MEMS mostly as inertial sensors. In this research the small volume of liquid is assumed to be positioned between the two parallel micro-cantilever beams. The two micro-beams are assumed to be respond to the dynamic excitation with a non-linear response which will be proved below that is related to the viscosity of the fluid. The dynamic system was modelled as a discreet three degree of freedom massdamping-spring. The nonlinear differential equations governing the performance of the above mentioned system is further solved and the results are analysed. The initial deflection of micro-cantilever beams due to capillary force yields to the dynamic behaviour of micro-cantilever beams expressed in three differential equations that are nonlinear and stiff. The used algorithm to solve the coupled differential equations is called LSODE and in usually capable to perform the computation solution for the stiff system of nonlinear initial value differential equations. The time response and phase diagram of the deflection of the beams for the massspring-damper system are numerically derived and the results are compared for two circumstances: the liquid would not evaporate and the liquid evaporates. Conclusion with regards to the equipment capable to perform such measurements are drawn.

Effect of asymmetry in the restoring force of the “click” mechanism in insect flight

The aim of this paper is to examine the effect of asymmetry in the force-deflection characteristics of an insect flight mechanism on its nonlinear dynamics. An improved simplified model for insect flight mechanism is suggested and numerical methods are used to study its dynamics. The range at which the mechanism may operate is identified. The asymmetry can lead to differences in the velocity in the upward and downward movements which can be beneficial for the insect flight.

Lie group analysis of non-linear dynamic of micro structures under electrostatic field

This paper presents an analytical solution of nonlinear differential equation of micro-structures subjected to electro- static fields. The constitutive equation of such a model is a second order differential equation (ODE). The problem is solved when the assumption of linear deflection is considered. However, deflection of micro cantilevers in practical applications is non- linear. Moreover, the constitutive ODE is stiff and various numerical algorithms used to solve it yield non-consistent numerical solutions. A deduction order method - Lie group symmetry is employed to reduce the order of the ODE. Although the resulting first order ODE has no symmetry that would guarantee an explicit close form solution, it enables an analytical formulation for the no-damping assumption only. The restoring force term in the first order ODE reveals the pull-in voltage as expressed in classical MEMS textbooks. It is shown that the numerical solution for the second order ODE and the reduced first order ODE are same. Finding any symmetry other than translation, scaling or rotation will enable the reduction of the first order ODE and thus, the formulation of an analytical solution to this highly non-linear problem.

MEMS Wind Speed Sensor: Large Deflection of Curved Micro-Cantilever Beam Under Uniform Horizontal Force

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force.Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides.We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid.The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs.Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple.Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.Copyright

A Complete Parametric Study of Pull-In Voltage by Nonlinear Differential Equation

Micro-cantilever beams are interested structures in MEMS because of their fabrication is very easy and its versatility. The importance of micro-cantilevers beam in MEMS has driven various investigations like static and dynamic performances under different loading such as potential fields. In this research the non-linear differential equation which models dynamics of a micro-cantilever beams vibration subjected to electrostatic field has been studied. The model which has one degree of freedom is used to calculate the pull-in voltage. This model adopted based on different method of calculating stiffness of micro-cantilever beam. The nonlinear ordinary differential equation which used to model the dynamics of the cantilever subjected to electric field close to snap on is highly stiff. Investigation on solving of nonlinear stiff ordinary equation showed that only Lsode algorithm yield to correct solution to the problem. Lsode is equipped with a robust adaptive time step selection mechanism that enables solutions to very stiff problems, as the one under discussion. The best match in the resonant frequency for equivalent stiffness based on four different models was considered. The stiffness model suitable for the best match in deflection is proved to be different from the model that yields. Pull-in voltage under electric field was studied. Pull-in voltage has been investigated from the analytical and numerical perspective. A complete parametric study of structural damping effect on large deflection of micro-cantilever beam was studied was done numerically in this work. Different kind of impulse voltages were considered and effect of them on pulling voltage numerically was studied. A cumbersome mathematical method, Lie symmetry, was used to drive a closed from of time response to step voltage for undamped system and pull in voltage of such system was calculated. Finally, a closed form driven from the nonlinear ODE for calculating pulling voltage was presented.Copyright