Energy Methods in Time-Varying System Stability and Instability Analyses

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The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated. Based on this verification error, it is shown that the radial basis function of neural network has the average error of 4. Further analysis of pull-in instability of beam under different input conditions has been investigated and comparison results of modeling with numerical considerations shows a good agreement, which also proves the feasibility and effectiveness of the adopted approach. The results reveal significant influences of size effect and geometric parameters on the static pull-in instability voltage of MEMS.

Microelectromechanical systems MEMS are widely being used in today's technology. So investigating the problems referring to MEMS owns a great importance. One of the significant fields of study is the stability analysis of the parametrically excited systems. Parametrically excited microelectromechanical devices are ever increasingly being used in radio, computer, and laser engineering [ 1 ].

Parametric excitation occurs in a wide range of mechanics, due to time-dependent excitations, especially periodic ones; some examples are columns made of nonlinear elastic material, beams with a harmonically variable length, parametrically excited pendulums, and so forth.

1 Introduction

Investigating stability analysis on parametrically excited MEM systems is of great importance. In Gasparini et al. Applying voltage difference between an electrode and ground causes the electrode to deflect towards the ground. At a critical voltage, which is known as pull-in voltage, the electrode becomes unstable and pulls in onto the substrate.

The pull-in behavior of MEMS actuators has been studied for over two decades without considering the casimir force [ 3 — 5 ]. Osterberg et al. Sadeghian et al. A comprehensive literature review on investigating MEMS actuators can be found in [ 6 ]. Further information about modeling pull-in instability of MEMS has been presented in [ 7 , 8 ]. The classical continuum mechanics theories are not capable of prediction and explanation of the size-dependent behaviors which occur in micron- and sub-micron-scale structures. However, some nonclassical continuum theories such as higher-order gradient theories and the couple stress theory have been developed such that they are acceptably able to interpret the size dependencies.

In s some researchers such as Koiter [ 9 ], Mindlin and Tiersten [ 10 ], and Toupin [ 11 ] introduced the couple stress elasticity theory as a nonclassic theory capable of predicting the size effects with appearance of two higher-order material constants in the corresponding constitutive equations. In this theory, beside the classical stress components acting on elements of materials, the couple stress components, as higher-order stresses, are also available which tend to rotate the elements. Utilizing the couple stress theory, some researchers investigated the size effects in some problems [ 12 ].

Employing the equilibrium equation of moments of couples beside the classical equilibrium equations of forces and moments of forces, a modified couple stress theory was introduced by Yang et al. Recently, size-dependent nonlinear Euler-Bernoulli and Timoshenko beams modeled on the basis of the modified couple stress theory have been developed by Xia et al. Rong et al. Their method is Rayleigh-Ritz method and assumes one deflection shape function. They derive the two governing equations by enforcing the pull-in conditions that the first and second order derivatives of the system energy functional are zero.

In their model, the pull-in voltage and displacement are coupled in the two governing equations. This paper investigates the pull-in instability of microbeams with a curved ground electrode under action of electric field force within the framework of von-Karman nonlinearity and the Euler-Bernoulli beam theory. The static pull-in voltage instability of clamped-clamped and cantilever microbeam is obtained by using MAPLE commercial software.

The effects of geometric parameters such as beam lengths, width, thickness, gaps, and size effect are discussed in detail through a numerical study. The objective of this paper is to establish neural network model for estimation of the pull-in instability voltage of cantilever beams.

More specifically, radial basis function RBF is used to construct the pull-in instability voltage. Effective parameters influencing pull-in voltage and their levels for training were selected through preliminary calculations carried out on instability pull-in voltage of microbeam. Networks trained by the same numerical data are then verified by some numerical calculations different from those used in training phase, and the best model was selected based on the criterion of having the least average values of verification errors. To the best of authors' knowledge, no previous studies which cover all these issues are available.

To the authors' best knowledge, no previous studies which cover all these issues are available.

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In the modified couple stress theory, the strain energy density u - for a linear elastic isotropic material in infinitesimal deformation is written as [ 17 ]. Assuming the above displacement field, after deformation, the cross-sections remain plane and always perpendicular to the center line, without any change in their shapes. It is noted that parameter z represents the distance of a point on the section with respect to the axis parallel to y -direction passing through the centroid. In this section, the governing equation and corresponding classical and nonclassical boundary conditions of a nonlinear microbeam modeled on the basis of the couple stress theory are derived.

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The coordinate system and loading of an Euler-Bernoulli beam have been depicted in Figure 1. In this figure, F x , t and G x , t refer to the intensity of the transverse distributed force and the axial body force, respectively, both as force per unit length. By assuming small slopes in the beam after deformation, the axial strain, that is, the ratio of the elongation of a material line element initially in the axial direction to its initial length, can be approximately expressed by the von-Karman strain as. It is noted that finite deflection w is permissible and only it is needed that the slopes be very small.

Hereafter, we use 8 for the axial strain, instead of the infinitesimal definition presented in 3. Substitution of 7 and 8 into 3 — 5 yields the nonzero components. Also, combination of 6 and 7 gives [ 19 ]. Substitution of 9 into 5 yields the following expression for the only nonzero component of the symmetric curvature tensor:. It is assumed that the components of strains, rotations, and their gradients are sufficiently small. By neglecting Poisson's effect, substitution of 8 into 2 gives the following expressions for the main components of the symmetric part of the stress tensor in terms of the kinematic parameters:.

In order to write the nonzero components of the deviatoric part of the couple stress tensor in terms of the kinematic parameters, one can substitute 10 into 4 to get. The work done by the external loads acting on the beam is also expressed as. The resultant axial and transverse forces are work conjugate to u and w , respectively.

Now, the Hamilton principle can be applied to determine the governing equation:. By applying 13a , 13b , 13c , and 14 , the governing equilibrium microbeam is derived as. The cross-sectional area and length of beam are A and L , respectively. F x , t is the electrostatic force per unit length of the beam. The electrostatic force enhanced with first order fringing correction can be presented in the following equation [ 20 ]:.

For clamped-clamped beam, the boundary conditions at the ends are. Table 1 shows the geometrical parameters and material properties of microbeam. Hence, 15 is reduced to. A uniform microbeam has a rectangular cross-section with height h and width B , subjected to a given electrostatic force per unit length. Let us consider the following dimensionless parameters:. The normalized nonlinear governing equation of motion of the beam can be written as [ 21 ]. A neural network is a massive parallel system comprised of highly interconnected, interacting processing elements or nodes. Neural networks process through the interactions of a large number of simple processing elements or nodes, also known as neurons.

Knowledge is not stored within individual processing elements, rather represented by the strengths of the connections between elements. Each piece of knowledge is a pattern of activity spread among many processing elements, and each processing element can be involved in the partial representation of many pieces of information. In recent years, neural networks have become a very useful tool in the modeling of complicated systems because they have an excellent ability to learn and to generalize interpolate the complicated relationships between input and output variables [ 22 ].

Also, the ANNs behave as model-free estimators; that is, they can capture and model complex input-output relations without the help of a mathematical model [ 23 ]. In other words, training neural networks, for example, eliminates the need for explicit mathematical modeling or similar system analysis. In this research radial basis function RBF neural network has been used for modeling the pull-in instability voltage of microcantilever beams.

The radial basis network has some additional advantages such as rapid learning and low error. In particular, most RBFNs involve fixed basis functions with linearly unknown parameters in the output layer. In practice, the number of parameters in RBFN starts becoming unmanageably large only when the number of input features increases beyond 10 or 20, which is not the case in our study. Hence, the use of RBFN was practically possible in this research.

The construction of a radial basis function RBF neural network in its most basic form involves three entirely different layers. The input layer is made up of source nodes sensory units. The second layer is a single hidden layer of high enough dimension, which serves a different purpose in a feedforward network. The output layer supplies the response of the network to the activation patterns applied to the input layer. The input units are fully connected though unit-weighed links to the hidden neurons, and the hidden neurons are fully connected by weighed links to the output neurons.

Each hidden neuron receives input vector X and compares it with the position of the center of Gaussian activation function with regard to distance. Finally, the output of the j th-hidden neuron can be written as. The structure of a radial basis neuron in the hidden layer can be seen in Figure 3. Output neurons have linear activation functions and form a weighted linear combination of the outputs from the hidden layer:. Basically, the RBFN has the properties of rapid learning, easy convergence, and low error, generally possessing the following characteristics.

RBFN is being used for an increasing number of applications, proportioning a very helpful modeling tool [ 25 ]. When the applied voltage between the two electrodes increases beyond a critical value, the electric field force cannot be balanced by the elastic restoring force of the movable electrode and the system collapses onto the ground electrode. The voltage and deflection at this state are known as the pull-in voltage and pull-in deflection, which are of utmost importance in the design of MEMS devices.

The pull-in voltage of cantilever and fixed-fixed beams is an important variable for analysis and design of microswitches and other microdevices. Typically, the pull-in voltage is a function of geometry variable such as length, width, and thickness of the beam and the gap between the beam and ground plane. To study the instability of the nanoactuator, 22 is solved numerically and simulated.

For a small gap length of 0. As shown in Figure 6 , the difference in the pull-in voltage is even larger when a gap length of 4. It is evident that pull-in voltage of fixed-fixed beam is larger than fixed-free beam. We observe that, for all cases, the pull-in voltages obtained with linear model are with significant error larger than 5. When the gap increases, the error in pull-in voltage with linear model increases significantly. Furthermore, contrary to the case of cantilever beams, the thickness has a significant effect on the error in pull-in voltages.

The thinner the beam, the larger the error. Another observation is that the length of the beam has little effect on the error in pull-in voltage. This observation is also different from the case of cantilever beams. From the results, it is clear the linear model is generally not valid for the fixed-fixed beams case, except when the gap is very small, such as the 0. These figures represent that the size effect increases the pull-in voltage of the nanoactuators.

Comparison of linear and nonlinear geometry model results for a fixed-fixed beam with a gap of 0.

Comparison of linear and nonlinear geometry model results for a fixed-fixed beam with a gap of 4. Comparison of linear and nonlinear geometry model results for a fixed-free beam with a gap of 0. Comparison of linear and nonlinear geometry model results for a fixed-free beam with a gap of 4.

Gap versus pull-in voltage for fixed-fixed beams with a thickness of 0. Observe the large difference in pull-in voltage obtained from linear and nonlinear geometry model of beam. Pull-in voltage versus size effect for fixed-fixed beam with gap of 2. Pull-in voltage versus size effect for cantilever beam with gap of 2.

Modeling of pull-in instability of microbeam with RBF neural network is composed of two stages: training and testing of the networks with numerical data. A total of such data sets were used, of which were selected randomly and used for training purposes whilst the remaining 10 data sets were presented to the trained networks as new application data for verification testing purposes. Thus, the networks were evaluated using data that had not been used for training. These normalized data were used as the inputs and output to train the ANN.

Figure 16 shows the general network topology for modeling the process. Table 2 shows 10 numerical data sets, which have been used for verifying or testing network capabilities in modeling the process. Therefore, the general network structure is supposed to be 4-n-1, which implies 4 neurons in the input layer, n neurons in the hidden layer, and 1 neuron in the output layer.

Then, by varying the number of hidden neurons, different network configurations are trained, and their performances are checked. It has to be larger than the distance between adjacent input vectors, so as to get good generalization, but smaller than the distance across the whole input space. Therefore, in order to have a network model with good generalization capabilities, the spread factor should be selected between 0. For training the RBF network, at first, a guess is made for the value of spread factor in the obtained interval. Also, the number of radial basis neurons is originally set as 1.

We would now like to understand the non-linear outcome of the instability, and its non-axisymmetric development, which was ignored in the analytic analysis. To do this, we have integrated the hydrodynamic equations in a series of numerical experiments. The numerical model retains the basic assumptions used in the linear analysis. The parent warp mode is axisymmetric. The equations of motion are those appropriate to the local model see equation 1. The disc is Keplerian, so The fluid is assumed strictly isothermal. We do not require that the system remain axisymmetric.

These perturbations around the equilibrium seed the parametric instability; we would otherwise need to wait for the instability to grow from machine roundoff error. The variables lie on a staggered mesh, so that scalar quantities are zone-centred, while vector quantities are centred on zone faces. It conserves mass and linear momentum to machine roundoff error. Our implementation has also been tested on a number of standard problems. It can, for example, reproduce standard linear results such as sound wave propagation, and standard non-linear results such as the Sod shock tube.

One relevant test of our shearing box implementation is uniform epicyclic motion. This test is non-trivial: for a naive implementation of the Coriolis and tidal forces the amplitude of the epicycle will grow or decay. This is because the time-step determined from the Courant condition depends on epicyclic phase. A second, relevant test is the evolution of a linear amplitude bending wave.

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This test was performed in axisymmetry. We introduced a bending wave in the initial conditions with wavelength L x. The numerical model has seven important parameters: L x , L y , L z , n x , n y , n z and S. Before studying the effect of these parameters we will consider a single, fiducial run in detail. The reflecting vertical boundaries are therefore three scaleheights away from the mid-plane. The velocity arising from the parent bending wave has been removed.

From Fig. As a result shocks are produced through which part of the energy is dissipated. The remainder of the dissipation is via numerical averaging at the grid scale. The epicyclic energy is reduced by a factor of 5. Evolution, in the fiducial model, of NA 1,2,3 , which measure lowest-order non-axisymmetric structure see the text for a definition.

The ordinate units are arbitrary. At early times in Fig. This is caused by the limited accuracy of the time integration of epicyclic motions. As a result epicyclic energy varies with epicyclic phase. The oscillation is stable, however, the epicyclic energy neither grows nor decays from cycle to cycle.

We have confirmed that the amplitude of this oscillation decreases as zone size and time-step decrease. Several factors contribute to lower the growth rate below the analytic value. We shall discuss these below. Black is highest density, blue lowest. Density variations are visible because compressive waves are strongly excited. The extended, sharp features are shocks. Colour image of density on slices through the fiducial model. Black is highest density, while blue is lowest. We have explored the effect of each of the main model parameters on the development of the parametric instability and its non-linear outcome.

A full list of relevant models, with model parameters, is given in Table 1. Run 1 is the fiducial model. One concern in formulating a numerical model such as this is whether an artificial aspect of the boundary conditions controls the outcome. In this respect, the azimuthal and radial boundary conditions are somewhat less worrisome than the vertical boundary conditions.


The former restrict the scale of structure that can develop to the size of the box, while the latter introduces a reflection of outgoing waves that would otherwise dissipate in the disc atmosphere. Although there could be a sharp boundary between a hot disc atmosphere and the body of the disc that would also produce reflections, albeit from a free rather than fixed boundary. We have tested for variation in linear growth rate with the vertical size of box: compare, for example, runs 2 and 3. Thus for values of L z close to the fiducial value, 6 H , we observe no change in the numerically measured growth rate.

For the non-linear development, we observe a trend in that the decay of epicyclic energy is slower in the larger boxes.

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The small size of this effect suggests that the boundary condition is not controlling the outcome. The growth rates measured in the periodic vertical runs are not distinguishably different from those measured in reflecting boundary condition runs at similar resolution. The decay of epicyclic energy also follows a similar trajectory under the two different boundary conditions.

This again suggests that the boundary conditions are not controlling the outcome. One might be concerned that the limited azimuthal size of the box is somehow limiting the linear development of the instability or the non-linear outcome. This concern turns out to be justified over a limited range in L y for both the linear and non-linear development of the instability. The numerical resolution n y L y is constant between these runs. This evidence suggests that there is a non-axisymmetric counterpart of the parametric instability for, and only for, sufficiently low m L y.

In retrospect this result is easy to understand. The radial wavenumber of the disturbance would then evolve according to The amplitudes of these shearing waves will grow so long as the radial wavenumber does not change so much over a growth time that the resonance between the daughter modes and the parent mode is detuned. The non-linear development of the instability is not strongly altered by variation of L y.

There is a tendency for E epi to decay slightly more slowly in models with larger L y. Presumably, this is because large-scale, slowly dissipating motions are available in the large L y box, and these motions act as a reservoir for the epicyclic energy. Runs 16—21, 1, 22 and 23 test the effect of numerical resolution. The measured linear growth rate is slightly dependent on resolution in the meridional plane.

3 Lyapunov Stability Analysis

The non-linear decay rate of the warp also has a trend with resolution in that high resolution models decay more slowly. To test this, we measured a numerical growth rate for a series of values of S. In each case, we used the run with the highest available numerical resolution in the meridional plane we will be concerned only with the development of axisymmetric instabilities here. The results are listed in Table 2. We believe that this is primarily a resolution effect, since the growth rate grows monotonically with increasing resolution. This is observed in the numerical models listed in Table 2.

We have shown that warps in unmagnetized, non-self-gravitating, inviscid Keplerian discs are subject to an instability that leads, in the non-linear regime, to damping of the warp. The damping is non-linear in that the growth rate of the instability is proportional to the amplitude of the warp. How might this process influence the development of warps in astrophysical discs? A full answer to this question awaits an understanding of other damping mechanisms that might act on a warp.

If the disc is already turbulent, the parametric instability may not occur. In the Appendix, we estimate the influence of an isotropic effective turbulent viscosity on our analysis. Assuming that the instability develops into the non-linear regime, we may hazard a guess at the consequences for the warp, using a simple model for the interaction of in-plane and out-of-plane motions in the warp.

We then apply the result to the warped, masing disc in NGC It is evident that the in-plane and out-of-plane motions behave as coupled harmonic oscillators. At very small a r , we expect n 2 because the growth rate of the parametric instability is proportional to a r. At larger amplitudes f models the effect of stationary, fully developed turbulence.

The decay time of the epicyclic energy shown in Fig. Let us apply these results to the masing disc in NGC Miyoshi et al. It is therefore difficult to see how any local mechanism can limit the warp if the growth rate is as large as the estimate of Maloney et al. Part of this discrepancy might be explained by uncertainties in the parameters of the model by Maloney et al. It is also possible that non-linear effects reduce the growth rate of the Pringle instability.

There is some evidence for this in that the disc in NGC is not as strongly twisted as one would expect from linear theory [see Herrnstein for a warp model that fits the observations]. We thank the Isaac Newton Institute for their support during the programme on Dynamics of Astrophysical Discs, where this work was initiated. We also thank John Papaloizou and Caroline Terquem for their comments.

The effect of a small viscosity on the mode-coupling analysis of Section 5 can also be estimated. Provided that bulk viscosity effects may be neglected, the eigenfunctions of wave modes in an isothermal disc Section 3 can still be obtained in terms of Hermite polynomials when viscosity is included.

Generally speaking, d n increases with increasing n and k. We therefore consider the first parametric resonance, which occurs between modes 1 and 2 i. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation.

Close mobile search navigation Article Navigation. Volume Article Contents. Appendix A: Effect of viscosity on the instability. Linear and non-linear theory of a parametric instability of hydrodynamic warps in Keplerian discs Charles F. Oxford Academic. Google Scholar. Jeremy Goodman. Gordon I. Cite Citation. Permissions Icon Permissions. Abstract We consider the stability of warping modes in Keplerian discs. These perturbations obey. One can then obtain a single second-order equation for the pressure perturbation,.

The eigenfunctions are Hermite polynomials 1 e. For future reference we also note the corresponding Lagrangian displacement,. This happens because of the coincidence of the natural frequencies of horizontal and vertical oscillations. The form of the dispersion relation for small kH is. The physical velocity field, with arbitrary normalization, is. The basic state is now regarded as infinite in both radial and vertical extent, and has uniform density and pressure. Note that one cannot find normal modes because the basic flow is itself time-dependent. Here one considers a perturbation in the form of a Fourier mode, the wavevector of which evolves in time according to the strain field.

The amplitudes of the perturbations obey. Recall that k z is time-dependent. These equations can be reduced to a single Floquet equation a linear equation with periodic coefficients of the form. To this order the equation for the evolution of the perturbation is the Mathieu equation, which has the normal form. admin