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# Department of Biomedical Engineering and Mechanics

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A collaboration between School of Biomedical Engineering and Sciences and the Department of Engineering Science and Mechanics to form the Department of Biomedical Engineering and Mechanics.

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- Nonlinear analysis of the forced response of structural elementsNayfeh, Ali H.; Mook, Dean T.; Sridhar, Seshadri (Acoustical Society of America, 1974)
Show more A general procedure is presented for the nonlinear analysis of the forced response of structural elements to harmonic excitations. Internal resonances (i.e., modal interactions) are taken into account. All excitations are considered, with special consideration given to resonant excitations. The general procedure is applied to clamped-hinged beams. The results reveal that exciting a higher mode may lead to a larger response in a lower interacting mode, contrary to the results of linear analyses.Show more - Sound waves in two-dimensional ducts with sinusoidal wallsNayfeh, Ali H. (Acoustical Society of America, 1974)
Show more The method of multiple scales is used to analyze the wave propagation in two-dimensional hard-walled ducts with sinusoidal walls. For traveling waves,resonance occurs whenever the wall wavenumber is equal to the difference of the wave numbers of any two duct acoustic modes. The results show that neither of these resonating modes could occur without strongly generating the other.Show more - Nonlinear acoustic propagation in two-dimensional ductsNayfeh, Ali H.; Tsai, Ming-Shing (Acoustical Society of America, 1974)
Show more The method of multiple scales is used to obtain a second-order uniformly valid expansion for the nonlinear acoustic wave propagation in a two-dimensional duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear effects tend to flatten and broaden the absorption versus frequency curve, in qualitative agreement with the experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.Show more - Parallel_plate waveguide with sinusoidally perturbed boundariesNayfeh, Ali H.; Asfar, Omar R. (American Institute of Physics, 1974)
Show more The method of multiple scales is used to obtain a uniformly valid asymptotic expansion for the propagation of TM modes on a parallel_plate waveguide with perfectly conducting boundary surfaces that are sinusoidally perturbed in the direction of propagation. The analysis shows that resonance occurs whenever the wave number of the wall distortion function is equal to the difference between the wave numbers of two propagating modes. It is further shown that the generated mode is the same order of magnitude as the excited mode due to resonance and that energy is continuously exchanged between the two modes as they propagate down the guide.Show more - Algebraically growing waves in ducts with sheared mean flowNayfeh, Ali H.; Telionis, Demetri P. (Acoustical Society of America, 1974)
Show more Standing or traveling waves which vary algebraically with the axial distance in uniform ducts with sheared mean velocity profiles are investigated. The results show that such waves are not possible for ducts with uniform cross sections and fully developed mean flows.Show more - Asymptotic Solutions Of Second-Order Linear Equations with 3 Transition PointsNayfeh, Ali H. (AIP Publishing, 1974-12-01)
Show more A uniformly valid asymptotic expansion is obtained for the regular solution of a class of second_order linear differential equations with three transition points_a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.Show more - Nonlinear propagation of a wave packet in a hard-walled circular ductNayfeh, Ali H. (Acoustical Society of America, 1975)
Show more The method of multiple scales is used to derive a nonlinear Schridinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cutoff frequencies.Show more - Nonparallel Stability of Boundary-Layer FlowsSaric, W. S.; Nayfeh, Ali H. (AIP Publishing, 1975)
Show more The spatial stability of two‐dimensional incompressible boundary‐layer flows is analyzed using the method of multiple scales. The analysis takes into account the streamwise variations of the mean flow, the disturbance amplitude, and the wavenumber. The theory is applied to the Blasius and the Falkner–Skan flows. For the Blasius flow, the nonparallel analytical results are in good agreement with the experimental data. The results show that the nonparallel effects increase as the pressure gradient decreases.Show more - Numerical-perturbation technique for the transverse vibrations of highly prestressed platesNayfeh, Ali H.; Kamat, Manohar P. (Acoustical Society of America, 1975)
Show more Under the usual assumptions of small strains with moderately large rotations, the problem of the transverse vibrations of highly prestressed nonuniform annular plates is reduced to the solution of the differential equation governing the transverse vibration of the corresponding prestressed membrane subject to modified boundary conditions that account for the effects of bending. The method of composite expansions is used to determine these modified boundary conditions. The agreement of the present solution or results with known exact solutions for simple geometries demonstrates the efficiency of this method when compared with other well-known numerical techniques.Show more - Acoustic waves in ducts with sinusoidally perturbed walls and mean flowNayfeh, Ali H. (Acoustical Society of America, 1975)
Show more An analysis is presented of the propagation of acoustic waves in a hard-walled duct with sinusoidally perturbed walls and carrying mean flow. The results show that resonance occurs whenever the wavenumber of the wall undulations is approximately equal to the difference between the wavenumber of any two propagating modes. It is shown that neither of the resonating modes could exist in the duct without strongly exciting the other resonating mode.Show more - Nonlinear resonances in a class of multi-degree-of-freedom systemsSridhar, Seshadri; Nayfeh, Ali H.; Mook, Dean T. (Acoustical Society of America, 1975)
Show more An analysis is presented of the superharmonic, subharmonic, and combination resonances in a multi-degree-of-freedom system which has cubic nonlinearity and modal viscous damping and is subject to harmonic excitation. It is shown that in the absence of internal resonances, the steady-state response contains only the modes which are directly excited. It is shown that in the presence of internal resonances, modes other than those that are directly excited can appear in the response. The strong influence of internal resonances is exhibited in numerical examples involving hinged-clamped beams. It is shown that when a multimode solution exists the lowest mode can dominate the response, even when it is not directly excited.Show more - Population difference of two_level atomic system due to a running pulsed fieldNayfeh, Munir H.; Nayfeh, Ali H. (American Institute of Physics, 1975)
Show more The method of multiple scales is used to derive an expression for the population difference in an absorbing atomic system due to a running pulsed field. The main contribution to this expression comes from a quasi_steady_state part which has the same functional form as the hole produced by a continuous running field, except that the saturation parameter contains the time dependence of the field. The expression includes also an oscillatory term and a quasisteady term, which decay with a rate that is equal to the inverse of the lifetime of the levels.Show more - Simulation of multicorrelated random processes using the FFT algorithmWittig, Larry E.; Sinha, A. Krishna (Acoustical Society of America, 1975)
Show more A technique for the digital simulation of multicorrelated Gaussian random processes is described. This technique is based upon generating discrete frequency functions which correspond to the Fourier transform of the desired random processes, and then using the fast Fourier transform (FFT) algorithm to obtain the actual random processes. The main advantage of this method of simulation over other methods is computation time; it appears to be more than an order of magnitude faster than present methods of simulation. One of the main uses of multicorrelated simulated random processes is in solving nonlinear random vibration problems by numerical methods of simulation. One of the main uses of multicorrelated simulated random integration of the governing differential equations. The response of a nonlinear string to a distributed noise input is presented as an example. nonlinear string to a distributed noise input is presented as an example.Show more - Finite-amplitude plane waves in ducts with varying propertiesNayfeh, Ali H. (Acoustical Society of America, 1975)
Show more The method of multiple scales is used to determine a first-order uniform expansion for finite-amplitude plane waves of continuous waveforms propagating in a duct having a slowly varying cross section and filled with an inhomogeneous fluid. Losses due to the acoustic boundary layer or a slight wall admittance can be accounted for by decomposing the continuous waveform into its Fourier components and correcting the amplitude and phase of each component independently. Losses at shocks can be accounted for by using weak-shock theory. The results show that the shock formation distance tends to be shortened in a converging section and tends to be lengthened in a diverging section of the duct.Show more - Nonlinear modulation of TM waves in a circular waveguideAsfar, Omar R.; Nayfeh, Ali H. (American Institute of Physics, 1976)
Show more The method of multiple scales is used to derive a nonlinear Schrödinger equation for the temporal and spatial amplitude and phase modulations of TM waves in a perfectly conducting guide containing a nonlinear isotropic medium. This equation is used to show that monochromatic waves are stable if the mechanism producing the nonlinearity is an electric or magnetic polarization and unstable if the nonlinearity is due to electrostriction or magnetostriction. It is also used to determine the amplitude dependence of the cutoff frequencies.Show more - Compressible Boundary Layers Over Wavy WallsLekoudis, S. G.; Nayfeh, Ali H.; Saric, W. S. (AIP Publishing, 1976)
Show more An analysis is presented of compressible viscous flows past wavy walls without restricting the mean flow to be linear in the disturbance layer. Linearization of the compressible disturbance equations results in a system of six first_order differential equations for the perturbation quantities. The method of orthonormalization is used to control the error growth in the numerical solution of these equations. The present results agree more closely with experimental data than the results obtained by using Lighthill's theory, which restricts the mean flow to be linear in the disturbance layer.Show more - Influence of anisotropic liners on the attenuation of sound in circular ductsSun, John; Nayfeh, Ali H. (Acoustical Society of America, 1976)
Show more An analysis is presented of the sound propagation and attenuation in a circular duct carrying a uniform mean flow and lined with an anisotropicporous material backed by cellular cavities. A combination of a fourth-order Runge-Kutta routine and a Newton-Raphson procedure is used to determine the effects of the liner properties, the flowMach number, and the sound frequency on the attenuation of spinning and nonspinning modes. The results show that low-frequency noise is better attenuated by anisotropic liners. The optimum liner is the one whose axial resistivity increases with increasing frequency.Show more - Critical angle for reflection at a liquid-solid interface in single crystalsHenneke, Edmund G. II; Jones, Gerald L. (Acoustical Society of America, 1976)
Show more Recent investigations have utilized the measurement of the critical angle for reflection from a liquid-solid interface for determination of the elastic constants of the solid. For anisotropic media, this technique is appropriate only for certain special cases of the incident plane and reflecting surface. We discuss here the general condition for the critical angel in anisotropic media and show that for some planes in quartz, major errors may arise if one employs the usual statement of Snell's law for definition of the critical angle.Show more - Optical resonance of a two_level atomic systemNayfeh, Munir H.; Nayfeh, Ali H. (American Institute of Physics, 1976)
Show more The method of multiple scales is used to derive a solution of the damped optical Bloch equations of a two_level atomic system due to a strong pulsed field. The time dependence of the oscillations of the atomic inversion influenced by detuning and power broadening is found. The population inversion consists, in general, of three terms: a quasisteady term, quasisteady term that decays with time, and an oscillatory term that also decays with time. In the limit of constant fields, the solution of Torrey for damped systems and that of Rabi for undamped systems are recovered. For an adiabatic switching of the field, the solution for undamped systems reduces to that of Crisp in the adiabatic following limit. An equation describing the field envelope is derived for an arbitrary amount of detuning. At exact resonance, this equation reduces to a pendulum equation, in agreement with previous analyses.Show more - Closed 1st-Order And 2nd-Order Moment Equations For Stochastic Nonlinear Problems with Applications To Model Hydrodynamic And Vlasov-Plasma TurbulenceBesieris, Ioannis M.; Stasiak, W. B. (AIP Publishing, 1976-09-01)
Show more Working along the lines of a procedure outlined by Keller, a technique is developed for deriving closed first_ and second_order moment equations for a general class of stochastic nonlinear equations by performing a renormalization at the level of the second moment. The work of Weinstock, as reformulated recently by Balescu and Misguich, is extended in order to obtain two equivalent representations for the second moment using an exact, nonperturbative, statistical approach. These general results, when specialized to the weak_coupling limit, lead to a complete set of closed equations for the first two moments within the framework of an approximation corresponding to Kraichnan's direct_interaction approximation. Additional restrictions result in a self_consistent set of equations for the first two moments in the stochastic quasilinear approximation. Finally, the technique is illustrated by considering its application to two specific physical problems: (1) modelhydrodynamicturbulence and (2) Vlasov_plasma turbulence in the presence of an external stochastic electric field.Show more