scholarly journals A Semihydrostatic Theory of Gravity-Dominated Compressible Flow

2014 ◽  
Vol 71 (12) ◽  
pp. 4621-4638 ◽  
Author(s):  
Thomas Dubos ◽  
Fabrice Voitus
Keyword(s):  
Acoustic Waves ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Potential Temperature ◽  
Normal Modes ◽  
Additional Term ◽  
Internal Gravity Waves ◽  
Least Action Principle ◽  
Vertical Coordinate ◽  
Hydrostatic Balance

Abstract From Hamilton’s least-action principle, compressible equations of motion with density diagnosed from potential temperature through hydrostatic balance are derived. Slaving density to potential temperature suppresses the degrees of freedom supporting the propagation of acoustic waves and results in a soundproof system. The linear normal modes and dispersion relationship for an isothermal state of rest on f and β planes are accurate from hydrostatic to nonhydrostatic scales, except for deep internal gravity waves. Specifically, the Lamb wave and long Rossby waves are not distorted, unlike with anelastic or pseudoincompressible systems. Compared to similar equations derived by A. Arakawa and C. S. Konor, the semihydrostatic system derived here possesses an additional term in the horizontal momentum budget. This term is an apparent force resulting from the vertical coordinate not being the actual height of an air parcel but its hydrostatic height (the hypothetical height it would have after the atmospheric column it belongs to has reached hydrostatic balance through adiabatic vertical displacements of air parcels). The Lagrange multiplier λ introduced in Hamilton’s principle to slave density to potential temperature is identified as the nonhydrostatic vertical displacement (i.e., the difference between the actual and hydrostatic heights of an air parcel). The expression of nonhydrostatic pressure and apparent force from λ allow the derivation of a well-defined linear symmetric positive definite problem for λ. As with hydrostatic equations, vertical velocity is diagnosed through Richardson’s equation. The semihydrostatic system has therefore precisely the same degrees of freedom as the hydrostatic primitive equations, while retaining much of the accuracy of the fully compressible Euler equations.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

scholarly journals The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Minimum Problem ◽  
Infinite Class ◽  
Coupled Equations ◽  
Geometrical Problem ◽  
Maximum Minimum

A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.

Gravity waves as a mechanism of coupling oceanic and atmospheric acoustic waveguides to seismic sources

2020 ◽  
Author(s):  
Oleg Godin
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Seismic Waves ◽  
Wind Waves ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Body Waves ◽  
Surface Scattering ◽  
Seismic Sources ◽  
T Waves

<p>Direct excitation of acoustic normal modes in horizontally stratified oceanic waveguides is negligible even for shallow earthquakes because of the disparity between velocities of seismic waves and the sound speed in the water column. T-phases, which propagate at the speed of sound in water, are often reported to originate in the open ocean in the vicinity of the epicenter of an underwater earthquake, even in the absence of prominent bathymetric features or significant seafloor roughness. This paper aims to evaluate the contribution of scattering by hydrodynamic waves into generation of abyssal T-waves. Ocean is modeled as a range-independent waveguide with superimposed volume inhomogeneities due to internal gravity waves and surface roughness due to wind waves and sea swell. Guided acoustic waves are excited by volume and surface scattering of ballistic body waves. The surface scattering mechanism is shown to explain key observational features of abyssal T-waves, including their ubiquity, low-frequency cutoff, presence on seafloor sensors, and weak dependence on the earthquake focus depth. On the other hand, volume scattering due to internal gravity waves proves to be ineffective in coupling the seismic sources to T-waves. The theory is extended to explore a possible role that scattering by gravity waves may play in excitation of infrasonic normal modes of tropospheric and stratospheric waveguides by underwater earthquakes. Model predictions are compared to observations [L. G. Evers, D. Brown, K. D. Heaney, J. D. Assink, P. S. M. Smets, and M. Snellen (2014), Geophys. Res. Lett., <strong>41</strong>, 1644–1650] of infrasonic signals generated by the 2004 Macquarie Ridge earthquake.</p>

Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation

2003 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Beam Model ◽  
Periodic Excitation ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to periodic excitation. The approach is an extension of the nonlinear normal mode formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree of freedom, whose response is known. A reduced order model for the forced system is then determined by the usual nonlinear normal mode procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple two-degree-off-reedom mass-spring system with cubic nonlinearities, and for a discretized beam model with 12 degrees of freedom. The results show that this method provides very accurate responses over a range of frequencies near resonances.

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

2002 ◽  
Vol 124 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Eric Pesheck ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Coupling ◽  
Rotating Beam ◽  
Internal Resonances ◽  
Modal Expansion ◽  
Nonlinear Normal

A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses.

scholarly journals Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

2004 ◽  
Vol 10 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Mathias Legrand ◽  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotating Shaft ◽  
Nonlinear Mode ◽  
Large Amplitude Motions ◽  
Nonlinear Normal ◽  
The Individual

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig–Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig–Bampton model.

Computing Nonlinear Normal Modes Using Numerical Continuation and Force Appropriation

2012 ◽  
Author(s):  
Robert J. Kuether ◽  
Matthew S. Allen
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Continuation ◽  
Gradient Based ◽  
Nonlinear Normal ◽  
Definition Of ◽  
Nonlinear Equations Of Motion ◽  
Mass Spring

Many structures can behave nonlinearly, exhibiting behavior that is not captured by linear vibration theory such as localization and frequency-energy dependence. The nonlinear normal mode (NNM) concept, developed over the last few decades, can be quite helpful in characterizing a structure’s nonlinear response. In the definition of interest, an NNM is a periodic solution to the conservative nonlinear equations of motion. Several approaches have been suggested for computing NNMs and some have been quite successful even for systems with hundreds of degrees of freedom. However, existing methods are still too expensive to employ on realistic nonlinear finite element models, especially when the Jacobian of the equations of motion is not available analytically. This work presents a new approach for numerically calculating nonlinear normal modes by combining force appropriation, numerical integration and continuation techniques. This method does not require gradients, is found to compute the NNMs accurately up to moderate response amplitudes, and could be readily extended to experimentally characterize nonlinear structures. The method is demonstrated on a nonlinear mass-spring-damper system, computing its NNMs up to a 35% shift in frequency. The results are compared with those from a gradient based algorithm and the relative merits of each method are discussed.

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

2018 ◽  
Vol 13 (4) ◽  
Author(s):  
S. Mahmoudkhani
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Oscillatory Systems ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

scholarly journals Design of a Dynamical Core Based on the Nonhydrostatic “Unified System” of Equations*

2014 ◽  
Vol 142 (1) ◽  
pp. 364-385 ◽  
Author(s):  
Celal S. Konor
Keyword(s):  
Acoustic Waves ◽  
Potential Temperature ◽  
Time Derivative ◽  
System Of Equations ◽  
Dynamical Core ◽  
Vertical Coordinate ◽  
Discrete Equations ◽  
Wide Range ◽  
Supplementary Material ◽  
Vertically Propagating

Abstract This paper presents the design of a dry dynamical core based on the nonhydrostatic “unified system” of equations. The unified system filters vertically propagating acoustic waves. The dynamical core predicts the potential temperature and horizontal momentum. It uses the predicted potential temperature to determine the quasi-hydrostatic components of the Exner pressure and density. The continuity equation is diagnostic (and used to determine the vertical mass flux) because the time derivative of the quasi-hydrostatic density is obtained from the predicted potential temperature. The nonhydrostatic component of the Exner pressure is obtained from an elliptic equation. The main focus of this paper is on the integration procedure used with this unique dynamical core. In the implementation described in this paper, height is used as the vertical coordinate, and the equations are vertically discretized on a Lorenz-type grid. Cartesian horizontal coordinates are used along with an Arakawa C grid. A detailed description of the discrete equations is presented in the supplementary material, along with the rationale behind the decisions made during the discretization process. Only a short description is given here. To demonstrate that the model is capable of simulating a wide range of dynamical scales, the results from cyclone- and cloud-scale simulations are presented. The solutions obtained for the selected cloud-scale simulations are compared to those from a fully compressible, anelastic, and pseudo-incompressible models that (as far as possible) uses the same schemes used in the unified dynamical core. The results show that the unified dynamical core performs reasonably well in all these experiments.

The Transfer Matrix Method for the Vibration of Compressed Helical Springs

1982 ◽  
Vol 24 (4) ◽  
pp. 163-171 ◽  
Author(s):  
D. Pearson
Keyword(s):  
Transfer Matrix ◽  
Axial Force ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Helix Angle ◽  
Sinusoidal Vibration ◽  
Helical Springs ◽  
Round Wire

The partial differential equations of motion are obtained for a helical spring subject to a static axial force, the typical element of the spring having six degrees of freedom. The wire cross-section can be any doubly symmetrical shape. The overall transfer matrix is calculated and its application is discussed for obtaining the response to forced sinusoidal vibration. Natural frequencies are found from the transfer matrix by iteration. Comparisons are made with published experiments on the natural frequencies of helical springs, made from round wire, with and without a static axial force. Comparison is also made with published theory for the static buckling of helical springs. Information is given on the effect on the natural frequencies of the static axial force, helix angle, number of active turns, ratio of helix to wire diameter, Poisson's ratio, shear coefficient, and the end conditions. The calculation of the normal modes is discussed.

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