On the phenomenon of the combination type resonance in non-linear two-degree-of-freedom systems

The phenomena of the almost-periodic response, called combination type resonance, is analyzed in two-degree-of-freedom dissipative, non-linear systems subject to sinusoidal excitation. The problem is solved theoretically by means of the modified “small parameter technique” as well as by means of the Ritz-Galerkin method. The results are then compared and checked against an analog computer analysis. Conclusions are drawn about a range of applicability of the two methods. The theoretical investigations are completed by solving the stability problem for both methods.

The results indicate that the phenomenon of the combination resonance might occur in wide regions of initial conditions, since one of the phase angle proves to be arbitrary.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

Solutions of some classes of second order non-linear ordinary differential equations

A second order non-linear ordinary differential equation satisfied by a homogeneous function of u and v where u is a solution of the linear equation ÿ + p(t)ÿ + r(t)y = 0 and v = ωu, ω being an arbitrary function of t, is obtained. Defining ω suitably in two specific cases, solutions are obtained for a non-linear equation of the form ÿ + p(t)ÿ + q(t)y = μÿ²y^{−1 +} f(t)yⁿ where μ ≠ 1, n≠ 1. Applying our results, some classes of equations of the above type possessing solutions involving two or one or no arbitrary constants are derived. Some illustrative examples are also discussed.     

Non-linear oscillations with multiple forcing terms

The problem of determining the transient response of a non-linear oscillator of the form ü + u = εƒ(u,u) + E(t) is studied by the method of multiple time scales, using the symbolic computation system MACSYMA. when the excitation E(t) consists of a finite number of harmonic forcing terms. Here ε is a small parameter and ƒ(u,u) is a non-linear function of its arguments. In particular, the Van der Pol and Duffing oscillators are studied in detail. It is found that when the forcing frequencies are not close to each other or close to the primary resonance of the system, then the response of the system is analogous to the behavior when only one forcing term is present. However, when the forcing frequencies are close to each other or close to the primary resonance, then the behavior is quite different, exhibiting certain oscillations not observed in the case of one forcing term.     

A time-reversal method for an acoustical pulse propagating in randomly layered media

In the recent years a considerable amount of mathematical work has been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33 (4), 519–625 (1991)] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the context of ultrasounds, time-reversal mirrors have been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCI-Paris). We refer to: [M. Fink, “Time reversal mirrors”, J. Phys. D: Appl. Phys. 26, 1333–1350 (1993)]. Our goal is to present a mathematical analysis of a time-reversal method for analyzing reflected signals in the model described in [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33(4), 519–625 (1991)]. We restrict our analysis to the one-dimensional case, the three-dimensional layered case being the content of a forthcoming paper. It is noticeable that we do not introduce new mathematics in the problem but simply put together an already existing mathematical theory and a new device, the time-reversal mirror.     

Elastic wave scattering by a three-dimensional inhomogeneity in an elastic half space

In this paper we will consider scattering of elastic waves in a half space. The half space is an isotropic, linear and homogeneous medium except for a finite inhomogeneity. The T-matrix method (also called the “extended boundary condition method” or “null field approach”) is extended to derive expressions for the elastic field inside the half space and the surface field on the interface. The assumptions on the source that excites the half space are fairly weak. In the numerical applications found in this paper we assume a Rayleigh surface wave to be the incoming field, and we only compute the surface displacements. We make illustrations on some simple types of scatterers (spheres and spheroids; the latter ones can be arbitrarily oriented).     

Interaction and “apparent” reconnection of 3D vortex tubes via direct numerical simulations

We present evidence of: “binding” of anti-parallel vortex tube segments; strong noncircular core development; evolution of new secondary finger-like vortex structures: and finally “apparent” vortex reconnections due to entanglement. The latter three processes are not present in Biot-Savart filament simulations.     

Mathematical modelling of transformation plasticity in steels II: Coupling with strain hardening phenomena

The models for the plastic behaviour of steels during phase transformations proposed in Part I and in a previous paper ( et al. [1986b]) for the case of ideal-plastic phases are extended to include strain-hardening effects (isotropic or kinematic hardening). An expression for the transformation plastic strain rate is obtained by modifying the treatment of Part I in a suitable manner. The classical plastic strain rate is also studied in a similar way. Complementary evolution equations for the hardening parameters are finally given, taking into account the possible “recovery” of strain hardening during transformations (i.e., the fact that the newly formed phase can “forget,” partially or totally, the previous hardening).     

Autoparametric quasiperiodic excitation

The dynamics of an autonomous conservative three degree of freedom system which exhibits autoparametric quasiperiodic excitation is investigated. The system is a generalization of a classical system known as the “particle in the plane”. The system exhibits a motion, the z=0 mode, whose stability is governed by a linear second order ODE with quasiperiodic coefficients. The behavior of the latter ODE is studied by using three different methods: numerical integration, harmonic balance and perturbation methods.     

Counter current gas-liquid vertical flow, model for flow pattern and pressure drop

Counter-current flow pattern transition and pressure drop are modeled. Emphasis is placed on the understanding of the transition mechanisms from a mechanistic point of view.

Unlike the case of co-current flow, in counter-current flow, the situations of “no solution” as well as “multiple solutions” for the flow pattern and pressure drop exist. These possibilities are discussed and criteria for the actual flow pattern that will take place are suggested.

Some of the results are supported by data (from the literature), others are somewhat tentative suggesting future experimental verification is needed.     

A survey of experimental heat transfer data for nucleate pool boiling of liquid metals and a new correlation

The experimental data for heat transfer during nucleate pool boiling of saturated liquid metals on plain surfaces are surveyed and a new correlation is presented. The correlation is h = Cq^0.7p_r^m, where C and m are, respectively, 13.7 and 0.22 p_r < 0.001 and 6.9 and 0.12 for p_r > 0.001 (h is in W/m² K and q in W/m²). This correlation has been verified with data for K, Na, Cs, Li, and Hg from 17 sources over the reduced pressure (p_r) range of 4.3 × 10⁻⁶ to 1.8 × 10⁻². The correlation of Subbotin et al. was found unsatisfactory, but a modified correlation was developed that also gives good agreement with most of the data.     

Endochronic analysis for finite elasto-plastic deformation and application to metal tube under torsion and metal rectangular block under biaxial compression

The ordinary differential constitutive equations of endochronic theory are extended to simulate elasto-plastic deformation in the range of finite strain using the concept of corotational rate. Different corotational stress rates (Jaumann, Cotter-Rivlin, Truesdell, Dienes and Mandel) are incorporated into the theory. In addition, a new formulation of the plastic spin, which can be used in the Mandel stress rate, is derived. Theoretical simulations of the axial effects for various materials subjected to simple and pure torsional loading cases are discussed in this study. It is shown that the endochronic theory incorporated with the Mandel stress rate yields the most satisfactory result, as indicated from comparison with the experimental data found in literature.

Finally, theoretical investigation of the deformation subjected to finite proportional and non-proportional biaxial compression is presented. The true relationship between stress and strain can be converted to a nominal stress-strain relationship for biaxial loading through the explicit transformation equations derived in this paper. Experimental data tested by Khan and Wang [1990] (“An Experimental study of Large finite Plastic Deformation in Annealed 1100 Aluminum During Proportional and Non-proportional Biaxial Compression” Int. J. Plasticity, 6, 485) are suitably described by the theory demonstrated from a comparison with the theoretical prediction according to rigid-plastic and elastic-plastic models employed by Huang and Khan [1991]. “An Analysis of Finite Elastic-Plastic Deformation under Biaxial Compression”, Int. J. Plasticity, 7, 219).     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Exact analysis of Burgers equation on semiline with flux condition at the origin

The initial boundary value problem for the Burgers equation in the domain x 0, t > 0 with flux boundary condition at x = 0 has been solved exactly. The behaviour of the solution as t tends to infinity is studied and the “asymptotic profile at infinity” is obtained. In addition, the uniqueness of the solution of the initial boundary value problem is proved and its inviscid limit as → 0 is obtained.     

A geometric approach for establishing dynamic buckling loads of autonomous potential N-degree-of-freedom systems

Non-linear dynamic buckling of autonomous non-dissipative N-degree-of-freedom systems whose static instability is governed either by a limit point or by an unstable symmetric bifurcation is thoroughly discussed using energy and geometric considerations. Characteristic distances associated with the geometry of the zero level total potential energy “hypersurface” in connection with total energy-balance equation lead to dynamic (global) instability criteria. These criteria allow the determination of “exact” dynamic buckling loads without solving the non-linear initial-value problem. The reliability and efficiency of the proposed geometric approach is demonstrated via several dynamic buckling analyses of 3-degree-of-freedom systems which subsequently are compared with corresponding numerical analyses based on the Verner–Runge–Kutta scheme.     

The cumulative interaction of finite amplitude waves in strings

A “two time scale” asymptotic expansion procedure describing the modulation of a propagating simple wave governed by a system of non-linear partial differential equations is applied to the deflection waves of non-linear elastic strings. Rapid deflection signals propagating into a general slowly varying disturbance are modulated. In addition, they themselves affect the equations for that disturbance. The two effects are separated naturally when, to prevent the cumulative growth inherent in most “high frequency” procedures, an averaging technique is introduced. The interaction of two deflection waves is given as a specific example.     

Subharmonic generation in physical systems: An interaction-box approach

A so-called “interaction-box” formalism, which has recently been introduced to describe hysteresis in dynamical systems in the case of higher harmonic generation, is further discussed and generalized to describe the phenomenon of subharmonic generation. In this case, the increase in the periodicity of the response is reflected in the formation of multiple loops in the Effect (output) vs. Cause (input) diagrams. Conversely, we show how this type of response represents a sort of “signature” of the system, and can thus be employed to draw general conclusions about the features of the latter. A specific example of a nonlinear system is chosen to illustrate the approach, namely a vibrating cantilever beam with a breathing crack. Effect vs. Cause curves are calculated for this system in the presence of higher harmonics and subharmonics.