Then, for the sinusoidal voltage frequency response curves are illustrated to analysis the effects of hardening and softening behaviors of the system and the property of jumping phenomenon. The renormalization group method is one of the most powerful techniques for studying the effective behavior of a complex system in the space of scales . The basic object of interest is a dynamical system for the effective model in which the time parameter is replaced by scale. Therefore this dynamical system describes how the effective model changes as the scale changes.

What is the multiple scales method

Many ideas have been proposed, among which we mention the linked atom methods, hybrid orbitals, and the pseudo-bond approach. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next. For type A problems, we need to decide where fine scale models should be used and where macro-scale models are sufficient. This requires developing new style of error indicators to guide the refinement algorithms. Several proposals have been made regarding general methodologies for designing multiscale algorithms.

The other extreme is to work with a microscale model, such as the first principle of quantum mechanics. As was declared by Dirac back in 1929 , the right physical principle for most of what we are interested in is already provided by the principles of quantum mechanics, there is no need to look further. We simply have to input the atomic numbers of all the participating atoms, then we have a complete model which is sufficient for chemistry, much of physics, material science, biology, etc. Dirac also recognized the daunting mathematical difficulties with such an approach — after all, we are dealing with a quantum many-body problem. With each additional particle, the dimensionality of the problem is increased by three.

A Novel Nonlinear Elasticity Approach for Analysis of Nonlinear and Hyperelastic Structures

Analytical and numerical approaches are used to obtain the dynamic solution of the device under different prestretch and electrical conditions. The effect of the prestretch on the first transverse and radial natural frequencies is first calculated analytically and validated by a finite element software. One-to-one, two-to-one and three-to-one internal resonances are possible as the prestretch applied to the membrane is varied. Using a Galerkin procedure and a multiple scale technique, the frequency response curves are obtained for different applied voltages. The results are validated using a Runge–Kutta time discretization technique. Finally, the effect of the applied voltage is analyzed for the frequency response curves of the radial and transverse displacements.

The coastal transition zone is an underexplored frontier in hydrology … –

The coastal transition zone is an underexplored frontier in hydrology ….

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In this paper, the nonlinear size-dependent vibration of a dielectric elastomer microbeam resonator was analyzed. The constitutive model of the system was considered based on a hyperelastic Cosserat continuum model. The equation of transverse motion was derived and discretized through the use of Hamilton’s principle and the Galerkin method, respectively. The resulting ordinary differential equation was solved numerically in time domain via the Runge-Kutta method. The influence of the system’s parameters -e.g., the length-scale parameter, polarization voltage, amplitude of AC voltage, and Lame’s modulus on the dynamic response of the system was analyzed.

Mathematics > Analysis of PDEs

In particular, we focus on many-body resonant and non-resonant continuum states observed in unstable nuclei. The main ideas behind this procedure are quite general and can be carried over to general linear or nonlinear models. The procedure allows one to eliminate a subset of degrees of freedom, and obtain a generalized Langevin type of equation for the remaining degrees of freedom. However, in the general case, the generalized Langevin equation can be quite complicated and one needs to resort to additional approximations in order to make it tractable. The basic idea is to use microscale simulations on patches (which are local spatial-temporal domains) to mimic the macroscale behavior of a system through interpolation in space and extrapolation in time. The incomplete macroscale model represents the knowledge we have about the possible form of the effective macroscale model.

By applying a high voltage to the two electrodes, the dimension of the membrane decreases in the direction of thickness while increases in the direction of the surface area. DEs have attracted attention due to the unique features like high flexibility, low cost, large deformation, and high stretch-ability . W. E , Principles of multiscale modeling, Cambridge University Press, Cambridge. The spherical balloon is a key configuration of DEs in biomedical and industrial applications such as pumps (Ho et al., 2017), tactile devices (Matysek et al., 2009), and Soft Composites (Hosoya et al., 2015). The renormalization group method has found applications in a variety of problems ranging from quantum field theory, to statistical physics, dynamical systems, polymer physics, etc.

What is the multiple scales method

Then, the obtained equations are solved using a meshless solution method named the semi-analytical polynomial method. The stress and deformation results of the structure under uniform and non-uniform transverse external loadings are obtained for different types of boundary conditions, loading, and material properties. Consequently, this research can be widely used as a suitable essential reference for researchers studying the mechanical behavior of nonlinear elastic structures. Averaging methods were developed originally for the analysis of ordinary differential equations with multiple time scales. The main idea is to obtain effective equations for the slow variables over long time scales by averaging over the fast oscillations of the fast variables .

Internal resonance and nonlinear dynamics of a dielectric elastomer circular membrane

The dynamic stretches therefore can be extremely large, and are limited only by the locking stretch of the elastomer. Because extremely large stretches are possible without snap-through instability, resonances in the frequency response transition from softening nonlinearity at moderate stretches to hardening nonlinearity at large stretches. Two types of atypical frequency response that depend on the amplitude of voltage fluctuations that dynamically excite the membrane are reported here for the first time. First, for moderate-amplitude voltage fluctuations, three steady-state vibrations are possible within a frequency band near resonance. Second, for large-amplitude voltage fluctuations, a frequency band occurs near resonance where only one steady-state vibration is possible, and this vibration corresponds to extreme dynamic stretches. Because of these unique steady-state solutions near resonance, new atypical jump-phenomenon may occur during frequency sweeps through resonance.

  • As was declared by Dirac back in 1929 , the right physical principle for most of what we are interested in is already provided by the principles of quantum mechanics, there is no need to look further.
  • Homogenization methods can be applied to many other problems of this type, in which a heterogeneous behavior is approximated at the large scale by a slowly varying or homogeneous behavior.
  • The other extreme is to work with a microscale model, such as the first principle of quantum mechanics.
  • Dielectric elastomers are smart soft systems that belong to electroactive polymers.
  • In the multiscale approach, one uses a variety of models at different levels of resolution and complexity to study one system.
  • The outcome of the numerical results indicates that both the chaos and quasiperiodicity arise in the micro reasoner.
  • In the language used below, the quasicontinuum method can be thought of as an example of domain decomposition methods.

By the claim of the task, you will not find the region of divergent resonance with this linear approach. Connect and share knowledge within a single location that is structured and easy to search. This term is O and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution. For this reason, the Taylor expansion is employed to construct the general form, i.e. A classical example in which matched asymptotics has been used is Prandtl’s boundary layer theory in fluid mechanics.

Chapter 6 The method of multiple scales

It would be worthwhile to develop software packages with formula manipulation such that these calculations can be performed with the aid of a computer. The method of two scales is illustrated in which the examples of the weakly nonlinear spring, the perihelium precession, and the linear oscillator with weak damping, are examined. The chapter presents the application of the method to the Mathieu-equation. A generalization of the method is discussed, such that it may also be applied to partial differential equations of wave type. The weakly nonlinear free oscillations are also presented in the chapter. In concurrent multiscale modeling, the quantities needed in the macroscale model are computedon-the-fly from the microscale models as the computation proceeds.

Chen and Wang have investigated dynamic performance of a dielectric elastomer balloon subject to a combination of pressure and periodic voltage. They obtained the governing equation for the balloon according to the law of conservation of the energy and simulated the problem using the parametric excitation numerically . Dynamic electromechanical instability of a dielectric elastomer balloon has been studied by Chen et al. They numerically presented time history responses and the phase plane to analysis effects of different factors on the dynamic electromechanical instability of a dielectric elastomer balloon (Chen et al., 2015). Yong et al. have carried out the dynamic analysis of a thick-walled dielectric elastomer spherical shell using the hyper-elastic neo-Hookean model and spherical capacitor assumptions (Yong et al., 2011). Zhang and Chen have studied the electromechanical performance of a viscoelastic dielectric elastomer balloon through thermodynamic assumptions and the hyper-elastic neo-Hookean model .

The governing equation of motion is formulated by means of Hamilton’s principle and then truncated into a reduced-order model through Galerkin’s technique. Approximate analytical solution in the primary resonant case is obtained using multiple time scales method. The stabilities of steady-state responses in the vicinity of the equilibrium states and critical buckling voltage are analyzed. Moreover, the bifurcation phenomenon has been studied according to the different values of some control parameters such as the applied DC voltage, the frequency detuning parameter, and the amplitude of the excitation force. The results have been compared with those of some previous studies and can provide a better understanding of the design of dielectric elastomer resonators, which are designed in the form of micro-beam structure.

The method is sometimes attributed to Poincare, although Poincare credits the basic idea to the astronomer Lindstedt . Later Krylov and Bogoliubov and Kevorkian and Cole introduced the two-scale expansion, which is now the more standard approach. The formula for $x_1(\tau,T)$ will have terms from homogeneous solution like $C\cos(\tau+D)$ also. A person using the map can use a pair of dividers to measure a distance by comparing it to the linear scale.

What is the multiple scales method

To ensure that there are no secular terms in \(Y_1\ ,\) the resonant terms on the right hand side of are forced to be zero, i.e. The use of the symbol \(E\) is deliberate since the Duffing oscillator is a Hamiltonian system with total energy \(E\) given by . However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999).

Method of multiple scales on Mathieu’s equation

These slowly varying quantities are typically the Goldstone modes of the system. For example, the densities of conserved quantities such as mass, momentum and energy densities are Goldstone modes. The equilibrium states of macroscopically homogeneous systems are parametrized by the values of these quantities.

Instability has been recognized as one of the major issues restricting the full potential applications of dielectric elastomer -based devices. As observed in the experiments, the intrinsic material viscosity of elastomers varies with deformation, which becomes more manifest particularly for DE actuators undergoing large deformation. A highly customized user-element subroutine in Abaqus is developed for the FE implementation. The accuracy and robustness of the FE framework are validated by comparison with existing experimental data and analytical studies. There is some literature on the dynamic performance of dielectric elastomer balloons under different electromechanical loading conditions.

Dynamics of a thick-walled dielectric elastomer spherical shell

The proposed investigation can find its potential use in the design and analysis of interconnected DE actuators subjected to a dynamic electromechanical actuation. In the multiscale approach, one uses a variety of models at different levels of resolution and complexity to study one system. The multi-scale analysis different models are linked together either analytically or numerically. For example, one may study the mechanical behavior of solids using both the atomistic and continuum models at the same time, with the constitutive relations needed in the continuum model computed from the atomistic model.

Multiscale analysis

But we have additional degree of freedom when we use the method of two timing/ multiple scales due to separate dependence on $\tau,T,etc$. This additional freedom is used to eliminate the secular terms that pop up when dealing with multiple time scales in a differential equation. This way the $\tau cos\tau$ and $\tau sin \tau$ term disappear and we get dependence of A and B in terms of slower time scales T($\epsilon t$), etc. You can check examples of it in strogatz’s nonlinear dynamics book in the section on weakly nonlinear oscillators or search two timing in internet. Dielectric elastomers are smart soft systems that belong to electroactive polymers. Microbeam resonators have been introduced as a new application of the dielectric elastomers.

Averaging methods

In the language used below, the quasicontinuum method can be thought of as an example of domain decomposition methods. Note that the frequency one components of the homogeneous/complementary solution were left out, as they would only replicate some variant of the base solution. As one of them is positive, this gives an exponentially growing term in the solution, leading to divergence as per the claim.