A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

This paper presents equivalent circuits representing the partial differential equations of the theory of elasticity for bodies of arbitrary shapes. Transient, steady-state, or sinusoidally oscillating elastic-field phenomena may now be studied, within any desired degree of accuracy, either by a “network analyzer,” or by numerical- and analytical-circuit methods. Such problems are the propagation of elastic waves, determination of the natural frequencies of vibration of elastic bodies, or of stresses and strains in steady-stressed states. The elastic body may be non-homogeneous, may have arbitrary shape and arbitrary boundary conditions, it may rotate at a uniform angular velocity and may, for representation, be divided into blocks of uneven length in different directions. The circuits are developed to handle both two- and three-dimensional phenomena. They are expressed in all types of orthogonal curvilinear reference frames in order to simplify the boundary relations and to allow the solution of three-dimensional problems with axial and other symmetry by the use of only a two-dimensional network. Detailed circuits are given for the important cases of axial symmetry, cylindrical co-ordinates (two-dimensional) and rectangular co-ordinates (two- and three-dimensional). Nonlinear stress-strain relations in the plastic range may be handled by a step-by-step variation of the circuit constants. Nonisotropic bodies and nonorthogonal reference frames, however, require an extension of the circuits given. The circuits for steady-state stress and small oscillation phenomena require only inductances and capacitors, while the circuits for transients require also standard (not ideal) transformers. A companion paper deals in detail with numerical and experimental methods to solve the equivalent circuits.

The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler–Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.

A commonly used idealization when describing separation of a chemical bond between molecules is that of an energy well which prescribes the dependence of energy of interaction between the molecules in terms of a reaction coordinate. The energy difference between the peak to be overcome and the root of the well is the so-called activation energy, and the overall shape of the well dictates the kinetics of separation through a constitutive assumption concerning transport. An assumption tacit in this description is that the state of the bond evolves with only a single degree of freedom—the reaction coordinate—as the system explores its energy environment under random thermal excitation. In this discussion we will consider several bonds described by one and the same energy profile. The cases differ in that the energy profile varies along a line extending from the root of the well in the first case, along any radial line in a plane extending from the root of the well in a second case, and along any radial line in space extending from the root of the well in a third case. To focus the discussion we determine the statistical rate of escape of states from the well in each case, requiring that the profile of the well is the same in all three cases. It is found that the rates of escape each depend exponentially on the depth of the well but that the coefficients of the exponential vary with depth of the well differently in the three cases considered.

The traveling charge (TC) concept is theoretically capable of producing higher muzzle velocities without a large increase in maximum operating pressure, compared with the conventional charge. This work presents experimental and numerical studies on a 35 mm test gun system using liquid fuels as traveling charge. Eight firings with 2 different configurations of booster charge and traveling charge are performed in this paper. The firing experimental results indicate that the liquid traveling charge configuration performs better, in terms of increased muzzle velocity, than a conventional propellant charge by approximately 94 m/s, corresponding to about 8% velocity increase. A mathematical model for the two-phase flows in the 35 mm test gun system using liquid fuels as traveling charge is established and simulated by using the two-phase flow method and computational fluid dynamics technology. The mathematical model for the two-phase gas-dynamical processes consists of a system of first-order, nonlinear coupled partial differential equations. An adaptive grid generation algorithm is developed to account for the expansion of the computational domain due to the motion of the system’s payload in the tube. The numerical code is well validated by comparing its predictions with the experimental results. The calculated pressure-time profiles and projectile muzzle velocity are in good agreement with the experimental data. The numerical results show that the mathematical model developed gives the correct trend and can provide useful calculated parameters for the structural design of liquid traveling charge.

This study extends a recently developed cellular automata (CA) modeling approach ( Leamy, 2008, “Application of Cellular Automata Modeling to Seismic Elastodynamics,” Int. J. Solids Struct., 45(17), pp. 4835–4849 ) to arbitrary two-dimensional geometries via the development of a rule set governing triangular automata (cells). As in the previous rectangular CA method, each cell represents a state machine, which updates in a stepped manner using a local “bottom-up” rule set and state input from neighboring cells. Notably, the approach avoids the need to develop and solve partial differential equations and the complexity therein. The elastodynamic responses of several general geometries and loading cases (interior, Neumann, and Dirichlet) are computed with the method and then compared with results generated using the earlier rectangular CA and finite element approaches. Favorable results are reported in all cases with numerical experiments indicating that the extended CA method avoids, importantly, spurious oscillations at the front of sharp wave fronts.

The vibrations and stability are investigated for an axially moving rectangular antisymmetric cross-ply composite plate supported on simple supports. The partial differential equations governing the in-plane and out-of-plane displacements are derived by the balance of linear momentum. The natural frequencies for the in-plane and out-of-plane vibrations are calculated by both the Galerkin method and differential quadrature method. It can be found that natural frequencies of the in-plane vibrations are much higher than those in the out-of-plane case, which makes considering out-of-plane vibrations only is reasonable. The instability caused by divergence and flutter is discussed by studying the complex natural frequencies for constant axial moving velocity. For the axially accelerating composite plate, the principal parametric and combination resonances are investigated by the method of multiple scales. The instability regions are discussed in the excitation frequency and excitation amplitude plane. Finally, the axial velocity at which the instability region reaches minimum is detected.

In this paper, the nonsimilarity boundary-layer flows of second-order fluid over a flat sheet with arbitrary stretching velocity are studied. The boundary-layer equations describing the steady laminar flow of an incompressible viscoelastic fluid past a semi-infinite stretching flat sheet are transformed into a partial differential equation with variable coefficients. An analytic technique for highly nonlinear problems, namely, the homotopy analysis method, is applied to give convergent analytical approximations, which agree well with the numerical results given by the Keller box method. Furthermore, the effects of physical parameters on some important physical quantities, such as the local skin-friction coefficient and the boundary-layer thickness, are investigated in detail. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of nonlinear partial differential equations with variable coefficients in physics.

Dynamic systems with lumped parameters, which experience random temporal variations, are considered. The variations “smear” the boundary between the system’s states, which are dynamically stable and unstable in the classical sense. The system’s response within such a “twilight zone” of marginal instability is found to be of an intermittent nature, with alternating periods of zero (or almost-zero) response and rare short outbreaks. As long as it may be impractical to preclude completely such outbreaks for a designed system, subject to highly uncertain dynamic loads, the corresponding system’s response should be analyzed. Results of such analyses are presented for cases of slow and rapid (broadband) parameter variations in Papers I and II, respectively. The former case has been studied in Paper I ( 2008, “Marginal Instability and Intermittency in Stochastic Systems—Part I: Systems With Slow Random Variations of Parameters,” ASME J. Appl. Mech., 75(4), pp. 041002 ) for a linear model of the system using a parabolic approximation for the variations in the vicinity of their peaks (so-called Slepian model) together with Krylov–Bogoliubov averaging for the transient response. This resulted in a solution for the probability density function (PDF) of the response, which was of an intermittent nature indeed due to the specific algorithm of its generation. In the present paper (Paper II), rapid broadband parameter variations are considered, which can be described by the theory of Markov processes. The system is assumed to operate beyond its stochastic instability threshold—although only slightly—and its nonlinear model is used accordingly. The analysis is based on the solution of the Fokker–Planck–Kolmogorov partial differential equation for the relevant stationary PDF of the response. Several such PDFs are analyzed; they are found to have integrable singularities at the origin, indicating an intermittent nature of the response. Asymptotic analysis is performed for the first-passage problem for such response processes with highly singular PDFs, resulting in explicit formulas for an expected time interval between outbreaks in the intermittent response.

In 1970, F. Y. M. Wan derived a single, complex-valued ordinary differential equation for an elastically isotropic right circular conical shell ( “On the Equations of the Linear Theory of Elastic Conical Shells,” Studies Appl. Math., 49, pp. 69–83 ). The unknown was the n th Fourier component of a complex combination of the midsurface normal displacement and its static-geometric dual, a stress function. However, an attempt to formally replace the Fourier index n by a partial derivative in the circumferential angle θ results in a partial differential equation, which is eighth order in θ . The present paper takes as unknowns the traces of the bending strain and stress resultant tensors, respectively, and derives static-geometric dual partial differential equations of fourth order in both the axial and circumferential variables. Because of the explicit appearance of Poisson ratios of bending and stretching, these two equations cannot be combined into a single complex-valued equation. Reduced equations for beamlike (axisymmetric and lateral) deformations are also derived.

The nonlinear static response and vibration behavior of cross-ply laminated cylindrical shell panels subjected to axial compression combined with other secondary loading are examined. The shell theory adopted in the present case is based on a higher-order shallow shell theory, includes geometric imperfection and von Kármán-type geometric nonlinearity. The solutions to the governing nonlinear partial differential equations are sought using the multiterm Galerkin technique. The nonlinear equilibrium paths through limit points and bifurcation points are traced using the Newton–Raphson method coupled with the Riks approach. The free vibration frequencies of post-buckled cylindrical panels about the static equilibrium state are reported by solving the associated linear eigenvalue problem. Results are presented for simply supported cross-ply laminated cylindrical shell panels, which illustrates the influence of initial geometric imperfection, temperature field, lateral pressure loads, and mechanical edge loads on the static response and vibration behavior of the shell panel.

In this paper, a 2D wavelet-based spectral finite element (WSFE) is developed for a anisotropic laminated composite plate to study wave propagation. Spectral element model captures the exact inertial distribution as the governing partial differential equations (PDEs) are solved exactly in the transformed frequency-wave-number domain. Thus, the method results in large computational savings compared to conventional finite element (FE) modeling, particularly for wave propagation analysis. In this approach, first, Daubechies scaling function approximation is used in both time and one spatial dimensions to reduce the coupled PDEs to a set of ordinary differential equations (ODEs). Similar to the conventional fast Fourier transform (FFT) based spectral finite element (FSFE), the frequency-dependent wave characteristics can also be extracted directly from the present formulation. However, most importantly, the use of localized basis functions in the present 2D WSFE method circumvents several limitations of the corresponding 2D FSFE technique. Here, the formulated element is used to study wave propagation in laminated composite plates with different ply orientations, both in time and frequency domains.

A dynamic model of a diamond mesh cod-end subject to harmonic forcing is developed. The partial differential equations governing the displacements of the cod-end and the tension in the twine are first derived and then analyzed using the harmonic balance method by substituting a harmonic series for the dependent variables and the forcing term. A closed-form solution is derived for the case of rigid-body motion, where there is no deformation of the cod-end geometry, along with the conditions for the forcing under which this motion occurs. A pressure loading, which varies linearly over a portion of the cod-end and varies harmonically with time, is then introduced as a first representation of the loading on the cod-end that results from the pressure and acceleration forces on the catch due to surge motion of the towing vessel. The resulting sets of equations for the static and the first and second harmonic terms are solved numerically in a sequential manner, and the results presented for a number of cases. These results show that, due to the nonlinearity of the system, the oscillatory motion of the cod-end is asymmetric, and that the deformation of the net and the amplitude of oscillation increases as the region over which the forcing is applied increases. The model is the basis for a more complete coupled catch/cod-end model.

In this paper, consideration is given to the dynamic response of a rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end. Starting with the basic geometrical relations and energy formulation for a rotating Timoshenko beam constrained at the hub in a centrifugal force field, a system of coupled partial differential equations are derived for the combined axial, lateral and twisting motions which includes the transverse shear, rotary inertia, and Coriolis effects, as well. In the mathematical formulation, the torsion of the thin airfoil also considers a very general case of shear center not being coincident with the CG (center of gravity) of the cross section, which allows the equations to be used also for analyzing eccentric tip-rub loading of the blade. Equations are presented in terms of axial load along the longitudinal direction of the beam which enables us to solve the dynamic pulse buckling due to the tip being loaded in the longitudinal as well as transverse directions of the beam column. The Rayleigh–Ritz method is used to convert the set of four coupled-partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices. Natural frequencies are computed for beams with varying “slenderness ratio” and “aspect ratio” as well as “twist angles.” Dynamical equations account for the full coupling effect of the transverse flexural motion of the beam with the torsional and axial motions due to pretwist in the airfoil. Some transient dynamic responses of a rotating beam repeatedly rubbing against the outer casing is shown for a typical airfoil with and without a pretwist.

Finding the thermoelastic damping in a vibrating body, for the most general case, involves the simultaneous solving of the three equations for displacements and one equation for temperature (called the heat equation). Since these are a set of coupled nonlinear partial differential equations there is considerable difficulty in solving them, especially for finite geometries. This paper presents a single degree of freedom (SDOF) model that explores the possibility of estimating thermoelastic damping in a body, vibrating in a particular mode, using only its geometry and material properties, without solving the heat equation. In doing so, the model incorporates the notion of “modal temperatures,” akin to modal displacements and modal frequencies. The procedure for deriving the equations that determine the thermoelastic damping for an arbitrary system, based on the model, is presented. The procedure is implemented for the specific case of a rectangular cantilever beam vibrating in its first mode and the resulting equations solved to obtain the damping behavior. The damping characteristics obtained for the rectangular cantilever beam, using the model, is compared with results previously published in the literature. The results show good qualitative agreement with Zener’s well known approximation. The good qualitative agreement between the predictions of the model and Zener’s approximation suggests that the model captures the essence of thermoelastic damping in vibrating bodies. The ability of this model to provide a good qualitative picture of thermoelastic damping suggests that other forms of dissipation might also be amenable for description using such simple models.

The generation of a random walk path under the action of an external potential field has been of interest for decades. The motivation derives largely from the prospect of incorporating the nonlocal excluded volume effect through such a potential in characterizing the statistical behavior of a long flexible polymer molecule. In working toward a continuum mean-field model, a central feature is a partial differential equation incorporating the influence of the potential and governing the generating function for the dependence of end to end separation distance of the molecule on its pathlength. The purpose here is to describe an approach in which the differential equation is recast as a global minimization of a functional. The variational approach is illustrated by an application to familiar configurations, the first of which is a molecule attached at one end to a noninteracting plane barrier in the presence of a uniform potential field. As a second illustration, the generating function is sought for a free molecule for the case in which conformations must be consistent with the excluded volume condition. This is accomplished by adapting a local form of the Flory approach to the phenomenon and extracting estimates of the expected end to end separation distance, the entropy and other statistical features of behavior. By means of the variational principle, the problem is recast into a form that admits a direct, noniterative analysis of conformations within the context of the self-consistent field theory.

The work presented introduces correlation moment analysis. This technique can be employed to explore the growth of determinism from stochastic initial conditions in physical systems described by non-linear partial differential equations (PDEs) and is also applicable to wholly deterministic situations. Correlation moment analysis allows the analytic determination of the time dependence of the spatial moments of the solutions of certain types of non-linear partial differential equations. These moments provide measures of the growth of processes defined by the PDE, furthermore the results are obtained without requiring explicit solution of the PDE. The development is presented via case studies of the linear diffusion equation and the non-linear Kortweg de-Vries equation which indicate strategies for exploiting the various properties of correlation moments developed in the text. In addition, a variety of results have been developed which show how various classes of terms in PDEs affect the structure of a sequence of correlation moment equations. This allows results to be obtained about the behavior of the PDE solution, in particular how the presence of certain types of terms affects integral measures of the solution. It is also demonstrated that correlation moments provide a very simple, natural approach to determining certain subsets of conserved quantities associated with the PDEs.

Heat transfer in a two-dimensional moving packed bed consisting of pellets surrounded by a gaseous atmosphere is numerically investigated. The governing equations are formulated based on the volume averaging method. A two-equation model, representing the solid and gas phases separately, and a one-equation model, representing both the solid and gas phases, are considered. The models take the form of partial differential equations with a set of boundary conditions, some of which were determined experimentally, and design parameters in addition to the operating conditions. We examine and discuss the parameters in order to reduce temperature differences from pellet to pellet. The calculation results show that by adopting a constant temperature along the preheater outer wall and decreasing the velocity of the pellets in the preheater, the difference in temperature from pellet to pellet is reduced from ∼ 120 ° C to ∼ 55 ° C , and the thermal efficiency of the preheater is tremendously improved.

A statically admissible solution for a perfectly plastic material in plane stress is presented for the mode I crack problem. The yield condition employed is an alternative type first proposed by von Mises in order to approximate his original yield condition for plane stress while eliminating most of the elliptic region as pertaining to partial differential equations. This yield condition is composed of two intersecting parabolas rather than a single ellipse in the principal stress space. The attributes of this particular solution of the mode I problem over that previously obtained are that it contains neither stress discontinuities nor compressive stresses anywhere in the field.

An acceptable variant of the Koiter–Morley equations for an elastically isotropic circular cylindrical shell is replaced by a constant coefficient fourth-order partial differential equation for a complex-valued displacement-stress function. An approximate formal solution for the associated “free-space” Green’s function (i.e., the Green’s function for a closed, infinite shell) is derived using an inner and outer expansion. The point wise error in this solution is shown rigorously to be of relative order ( h ∕ a ) ( 1 + h ∕ a ∣ x ∣ ) , where h is the constant thickness of the shell, a is the radius of the mid surface, and a x is distance along a generator of the mid surface.