Parametric instability in a taut string with a periodically moving boundary, which is governed by a one-dimensional wave equation with a periodically varying domain, is investigated. Parametric instability usually occurs when coefficients in governing differential equations of a system periodically vary, and the system is said to be parametrically excited. Since the governing partial differential equation of the string with a periodically moving boundary can be transformed to one with a fixed domain and periodically varying coefficients, the string is parametrically excited and instability caused by the periodically moving boundary is classified as parametric instability. The free linear vibration of a taut string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The exact response of the linear model is obtained using the wave or d'Alembert solution. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-i (i≥1) parametric instability regions, where period-1 parametric instability regions are analytically obtained using the wave solution and the fixed point theory, and period-i (i>1) parametric instability regions are numerically calculated using bifurcation diagrams. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. Three corresponding nonlinear models that consider coupled transverse and longitudinal vibrations of the string, only the transverse vibration, and coupled transverse and axial vibrations are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using three numerical methods: Galerkin's method, the explicit finite difference method, and the implicit finite difference method; advantages and disadvantages of each method are discussed. Numerical results for the linear model can be verified using the exact wave solution, and those for the nonlinear models are compared with each other since there are no exact solutions for them. It is shown that for parameters in the parametric instability regions of the linear model, the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models. The mechanism of the parametric instability is explained in the linear model and through axial strains in the third nonlinear model.

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

Analyzing the dynamic behavior of microelectrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect, the residual stress, the tensile stress, and the nonlinear electrostatic force. In this study, this problem is resolved by modeling the electrostatic system using a continuous model and solving the resulting governing equation of motion using a hybrid scheme comprising the differential transformation method and the finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two fixed-fixed microbeams when actuated by a dc voltage. It is shown that the numerical results for the pull-in voltage deviate by no more than 1.74% from those presented in the literature. The hybrid scheme is then applied to examine the nonlinear behavior of one clamped microbeam actuated by a combined dc/ac scheme. The beam displacement is analyzed as a function of both the magnitude and the frequency of the ac voltage. Finally, the actuating conditions, which ensure the stability of the microbeam, are identified by reference to phase portraits and Poincaré maps. Overall, the results presented in this study show that the hybrid differential transformation and finite difference method provides a suitable means of analyzing a wide variety of common electrostatically actuated microstructures.

The fully implicit finite-difference method is used to solve the continuity, momentum, and energy equations for flow within a gas pipeline. This methodology (1) incorporates the convective inertia term in the conservation of momentum equation, (2) treats the compressibility factor as a function of temperature and pressure, and (3) considers the friction factor as a function of the Reynolds number and pipe roughness. The fully implicit method representation of the equations offers the advantage of guaranteed stability for a large time step, which is very useful for gas pipeline industry. The results show that the effect of treating the gas in a nonisothermal manner is extremely necessary for pipeline flow calculation accuracies, especially for rapid transient process. It also indicates that the convective inertia term plays an important role in the gas flow analysis and cannot be neglected from the calculation.

Inaccuracies in the modeling assumptions about the distributional characteristics of the monitored signatures have been shown to cause frequent false positives in vehicle monitoring systems for high-risk aerospace applications. To enable the development of robust fault detection methods, this work explores the deterministic as well as variational characteristics of failure signatures. Specifically, we explore the combined impact of crack damage and manufacturing variation on the vibrational characteristics of beams. The transverse vibration and associated eigenfrequencies of the beams are considered. Two different approaches are used to model beam vibrations with and without crack damage. The first approach uses a finite difference approach to enable the inclusion of both cracks and manufacturing variation. The crack model used with both approaches is based on a localized decrease in the Young’s modulus. The second approach uses Myklestad’s method to evaluate the effects of cracks and manufacturing variation. Using both beam models, Monte Carlo simulations are used to explore the impacts of manufacturing variation on damaged and undamaged beams. Derivations are presented for both models. Conclusions are presented on the choice of modeling techniques to define crack damage, and its impact on the monitored signal, followed by conclusions about the distributional characteristics of the monitored signatures when exposed to random manufacturing variations.

In this work, the stability of a two-phase, natural circulation circuit is analyzed, using a specially developed model. This thermohydraulic model results in a set of coupled, nonlinear, first-order partial differential equations, which are solved by means of the up-wind finite difference method, using combinations of explicit and implicit methods for the numerical integration of the different balance equations. An adaptive nodalization scheme is implemented, minimizing the error of the propagation of small perturbations through the discretized volumes and especially the ones having two-phase flow regime. A linearization method is implemented by means of numerical perturbations. Frequency domain calculations are carried out, allowing a rapid visualization of the stability of the linearized system. Two cases are analyzed: a test case, where the code is compared in a wide range of qualities with an analytical model, and an application case, where the model is used to analyze the stability of an integral reactor cooled by natural circulation. The CAREM prototype is taken as a reference. In both cases, the numerical diffusion and integration errors are analyzed in the stability limit prediction by means of a convergence analysis using different nodalization and numerical integration criteria.

Previously proposed methods for subgrid-scale (SGS) stress modeling were re-investigated and extended to SGS heat-flux modeling, and various anisotropic and isotropic eddy viscosity/diffusivity models were obtained. On the assumption that they are used in a finite-difference (FD) simulation, the models were constructed in such a way that they are insensitive to numerical parameters on which calculated flows are strongly dependent in the conventional Smagorinsky model. The models obtained, as well as those previously proposed, were evaluated a priori in a stably stratified open channel flow, which is considered to be a challenging application of large eddy simulation and suitable for testing both SGS stress and heat-flux models. The most important feature of the models proposed is that they are insensitive to the discretized test filtering parameter required in the dynamic procedure of Germano et al. (1991, Phys. Fluids, 3, pp. 1760–1765) in FD simulation. We also found in SGS heat-flux modeling that the effect of the grid (resolved)-scale (GS) velocity gradient plays an important role in the estimation of the streamwise heat flux, and an isotropic eddy diffusivity model with the effect of the GS velocity is proposed.

The dynamic rupture process of a thin power-law type non-Newtonian liquid film on a cylinder has been analyzed by investigating the stability to finite amplitude disturbances. The dynamics of the liquid film is formulated using the balance equations including a body force term due to van der Waals attractions. The governing equation for the film thickness was solved by finite difference method as part of an initial value problem for spatial periodic boundary conditions. A decrease in the cylinder radius will induce a stronger lateral capillary force and thus will accelerate the rupture process. The influence of the power-law exponent on rupture is discussed.

A hybrid spectral/finite difference method is developed in this paper for the analysis of three-dimensional unsteady viscous flows between concentric cylinders subjected to fully developed laminar flow and executing transverse oscillations. This method uses a partial spectral collocation approach, based on spectral expansions of the flow parameters in the transverse coordinates and time, in conjunction with a finite difference discretization of the axial derivatives. The finite difference discretization uses central differencing for the diffusion derivatives and a mixed central-upwind differencing for the convective derivatives, in terms of the local mesh Reynolds number. This mixed scheme can be used with coarser as well as finer axial mesh spacings, enhancing the computational efficiency. The hybrid spectral/finite difference method efficiently reduces the problem to a block-tridiagonal matrix inversion, avoiding the numerical difficulties otherwise encountered in a complete three-dimensional spectral-collocation approach. This method is used to compute the unsteady fluid-dynamic forces, the real and imaginary parts of which are related, respectively, to the added-mass and viscous-damping coefficients. A parametric investigation is conducted to determine the influence of the Reynolds and oscillatory Reynolds (or Stokes) numbers on the axial variation of the real and imaginary components of the unsteady forces. A semi-analytical method is also developed for the validation of the hybrid spectral method, in the absence of previous accurate solutions or experimental results for this problem. Good agreement is found between these very different methods, within the applicability domain of the semi-analytical method.

This paper presents a numerical investigation of fluid flow, frequency response in the fully developed region, and convective heat transfer in the entrance region of the crescent-shaped lumen catheter. The catheter is commonly used in the biomedical field to clinically diagnose heart disease and also to treat vessel blockage in surgery. The catheter is subjected to a constant wall temperature. The solution to discretization of the momentum and energy equations is obtained by using the numerically generated boundary fitted coordinate system. According to this method, the complex domain in the physical plane is transformed into a regular square domain in the computational plane. The control volume-based finite difference method is then used to discretize the transformed governing equations. Results for the thermal entry region flow, frequency response, and heat transfer are presented in graphical form. The representative curves illustrating variations of the flow rate, frequency response, damping coefficient, bulk temperature, and the Nusselt numbers with pertinent parameters in the entire thermal entry region are plotted. The optimized catheter design for diagnostic use in the medical industry is also presented graphically.

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

A new boundary equation method is presented for analyzing plates of arbitrary geometry. The plates may have holes and may be subjected to any type of boundary conditions. The boundary value problem for the plate is formulated in terms of two differential and two integral coupled boundary equations which are solved numerically by discretizing the boundary. The differential equations are solved using the finite difference method while the integral equations are solved using the boundary element method. The main advantages of this new method are that the kernels of the boundary integral equations are simple and do not have hyper-singularities. Moreover, the same set of equations is employed for all types of boundary conditions. Furthermore, the use of intrinsic coordinates facilitates the modeling of plates with curvilinear boundaries. The numerical results demonstrate the accuracy and the efficiency of the method.

On the basis of the Donnell-type shell equations with the effect of nonlinear prebuckling deformations taken into consideration, a theoretical analysis is performed on the buckling of clamped truncated conical shells under two loads combined out of uniform pressure, axial load, and uniform heating. The problem is solved by a finite difference method. It is found that the interaction curves of buckling loads are changed remarkably by the difference in the shape of conical shells. This is due to the large nonlinear prebuckling deformation and the difference in the buckling mode between two cases of single load.

A plane-stress thermoelastic problem in a multiply connected region of variable thickness is formulated in terms of a stress function by deriving new Michell integral conditions necessary for the assurance of single-valuedness of rotation and displacements. The system of fundamental equations is solved by means of a finite difference method and numerical calculations are carried out for the cases of a rectangular plate of variable thickness with a rectangular hole.

Linear spatial stability of the nonparallel developing flow in a rigid circular pipe has been studied at several axial locations for nonaxisymmetric disturbances. The main flow velocity profile is obtained by Hornbeck’s finite-difference method assuming uniform flow at entry to the pipe. The method of multiple scales is used to account for all the nonparallel effects. It is found that the nonparallel developing flow is most unstable to nonaxisymmetric disturbances with azimuthal wave number n equal to unity. Axisymmetric disturbances are, however, more unstable than nonaxisymmetric disturbances with n ≥ 2 except in the near-entry region. The results show that the parallel flow theory overpredicts the critical Reynolds number by as much as 136.5 percent in the near entry region for the n = 1 disturbance. The present results compare well with the available experimental data.

A method based on the variational principles in conjunction with the finite difference technique is applied to determine the dynamic characteristic of eccentrically stiffened plates. The inplane deformations in both directions of the plate have been considered and the inplane inertia has been included into the analysis. The strain and kinetic energy for the plate and the stiffener are expressed in terms of discrete displacement components using the finite difference method. The energy functional is minimized with respect to discretized displacement components and the natural frequencies and corresponding mode shapes are obtained from the solution of a linear eigenvalue problem. The effect of inplane deformations of the plate and the stiffener and also the effect of inplane inertia on free vibration characteristics of uniaxial and cross-stiffened plates have been examined.

A theoretical investigation into the linear, spatial instability of the developing flow in a rigid circular pipe, incorporating the effects of nonparallelism of the main flow, has been made at several axial locations. The velocity profile in the developing flow region is obtained by a finite-difference method assuming uniform flow at the entry to the pipe. For the stability analysis, the continuity and momentum equations have been integrated separately using fourth-order Runge-Kutta integration scheme and applying selectively the Gram-Schmidt orthonormalization procedure to circumvent the parasitic error-growth problem. It is found that the critical frequency, obtained from different growth rates, decreases first sharply and then gradually with increasing X , where X = x/aR = X/R; x being the streamwise distance measured from the pipe inlet, a being the radius of the pipe, and R the Reynolds number based on a and average velocity of flow. However, the critical Reynolds number versus X curves pass through a minima. The minimum critical Reynolds number corresponding to gψ( X , O), the growth rate of stream function at the pipe axis, to g E ( X ), the growth rate of energy density, and to the parallel flow theory are 9700 at X = 0.00325, 11,000 at X = 0.0035, and 11,700 at X = 0.0035, respectively. It is found that the actual developing flow remains unstable over a larger inlet length of the pipe than its parallel-flow approximate. The first instability of the flow on the basis of gψ( X , O), g E ( X ) and the parallel flow theory, is found to occur in the range 30 ≤ X ≤ 36, 35 ≤ X ≤ 43, and 36 ≤ X ≤ 45, respectively. The critical Reynolds numbers obtained on the basis of gψ( X , O) are closest to the experimental values.

On the basis of the dynamic version of the nonlinear von Karman equations, a theoretical analysis is performed on the elastic instability of a uniformly heated, thin, annular plate which has suffered a finite axisymmetric deformation due to lateral pressure. The linear free vibration problems around the finite axisymmetric deformation of the plate are solved by a finite-difference method. By examining the frequency spectrum with various asymmetric modes, the critical temperature rise under which the axisymmetric deformation becomes unstable due to the bifurcation buckling is determined, which is found to jump up to 7.2 times within a range of very small lateral pressure.

The elastic instability of a thin clamped annular plate which has suffered a finite axisymmetric deformation due to simultaneous loading of uniform compression and lateral pressure is studied by examining the asymmetric small free vibration in the neighborhood of the nonlinear axisymmetric equilibrium state. The problem is solved by applying a finite-difference method to the dynamic version of the nonlinear von Karman plate theory. The numerical results indicate that there are the ranges of the magnitude of combined loads under which the axisymmetric deformation of the plate becomes unstable.