Uniqueness in the inversion of inaccurate gross Earth data

1970 ◽  
Vol 266 (1173) ◽  
pp. 123-192 ◽  
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Variance Matrix ◽  
The Earth ◽  
Mass Moment ◽  
Frequency Independent ◽  
Finite Set ◽  
Mass Moment Of Inertia ◽  
Limits Of Error ◽  
Local Average

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.

Backus-Gilbert style inversions for mantle anisotropy using normal mode data 

2021 ◽  
Author(s):  
Federica Restelli ◽  
Paula Koelemeijer ◽  
Christophe Zaroli
Keyword(s):  
Normal Mode ◽  
Ad Hoc ◽  
Seismic Anisotropy ◽  
Normal Modes ◽  
Least Square ◽  
Mantle Flow ◽  
Tomographic Images ◽  
The Earth ◽  
Initial Work ◽  
Mantle Anisotropy

<p>Seismic tomography is essential for imaging the Earth’s interior in order to better understand the dynamic processes at work. However, robust physical interpretation of tomographic images remain difficult as the inverse problem is under-determined, model amplitudes are biased and uncertainties are usually not quantified.</p><p>Commonly-used techniques, such as damped least-square inversions, break the non-uniqueness of the model solution by adding a subjective, ad hoc, regularization, which can lead to biased amplitudes and potential physical misinterpretations. The SOLA method (Zaroli, 2016; Zaroli et al., 2017), based on a Backus-Gilbert approach, removes the non-uniquess by averaging, rather than introducing a subjective regularization. The method explicitly constrains the amplitudes to be unbiased and the computation of the model resolution and uncertainty is inherent and efficient. Instead of aiming to minimize the data fit, the SOLA approach aims to minimize the size of the averaging volume <!-- Think it is clear enough without the extra sentence – as averaging is mentioned before -->and the associated uncertainties.</p><p>We aim to build a new tomographic model of the Earth’s mantle using the SOLA method. We focus our observations on normal mode data, the standing waves of the Earth observed after very large earthquakes, which are not affected by an uneven data distribution. As normal modes are sensitive to multiple seismic parameters, we treat the sensitivity to different parameters as so called “3D noise” within the SOLA framework. We are specifically interested in constraining seismic anisotropy, which provides more direct information on mantle flow.</p><p>Here, we report on some forward modelling results, fundamental to understanding normal mode sensitivity to seismic anisotropy at different depths and identifying which modes to focus on during inversions. We also show our initial work towards building a new tomography model, including the calculation of 3D noise and target kernels.</p><p> </p>

Vibration Frequencies for a Uniform Beam With One End Elastically Supported and Carrying a Mass at the Other End

1975 ◽  
Vol 42 (4) ◽  
pp. 878-880 ◽  
Author(s):  
D. A. Grant
Keyword(s):  
Normal Modes ◽  
Frequency Equation ◽  
The Other ◽  
Concentrated Mass ◽  
Mass Moment ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Mass Moment Of Inertia ◽  
Elastically Supported ◽  
Range Of Values

In this paper the author obtains the frequency equation for the normal modes of vibration of uniform beams with linear translational and rotational springs at one end and having a concentrated mass at the other free end. The eigenfrequencies for the fundamental mode are given for a wide range of values of mass ratio, mass moment of inertia ratios, and stiffness ratios.

The influence of rotation on the free oscillations of the Earth

1975 ◽  
Vol 279 (1292) ◽  
pp. 583-624 ◽  
Keyword(s):  
Rigid Body ◽  
Normal Mode ◽  
Body Force ◽  
Normal Modes ◽  
Earth Model ◽  
Static Response ◽  
The Earth ◽  
Rigid Body Modes ◽  
Effects Of Rotation ◽  
Zero Frequency

We pursue an abstract investigation of the theory of the infinitesimal free elasticgravitational oscillations of a fairly general rotating Earth model. By considering in some detail the transition to the non-rotating case, we are able to delineate certain of the intrinsic effects of rotation on the normal mode eigensolutions, and to show how profoundly rotation alters the fundamental mathematical and physical properties of these eigensolutions. In particular, we show that the displacement eigenfunctions of a rotating Earth model are not mutually orthogonal, and that the corresponding normal modes of oscillation cannot in general be represented by pure standing waves. We consider the excitation of the normal modes of oscillation of a rotating Earth model by a transient imposed body force distribution, and we show that the complex dynamical amplitude of each normal mode may, in many geophysical applications, be determined separately, in spite of the lack of orthogonality among the displacement eigenfunctions. The calculation of the associated static response after the decay of the normal modes of oscillation is, on the other hand, complicated considerably by the absence of orthogonality. We specifically examine the influence of rotation on the zero-frequency rigid body translational and rotational modes of any non-rotating Earth model, and show how to account for the corresponding rigid body modes of any rotating Earth model in excitation calculations.

Modeling and validation of crankshaft speed fluctuations of a single-cylinder four-stroke diesel engine

2021 ◽  
pp. 095440702110262
Author(s):  
Mustafa Babagiray ◽  
Hamit Solmaz ◽  
Duygu İpci ◽  
Fatih Aksoy
Keyword(s):  
Diesel Engine ◽  
Dynamic Model ◽  
Moment Of Inertia ◽  
Mass Motion ◽  
Cylinder Pressure ◽  
Motion Equations ◽  
Single Cylinder ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Speed Fluctuations

In this study, a dynamic model of a single-cylinder four-stroke diesel engine has been created, and the crankshaft speed fluctuations have been simulated and validated. The dynamic model of the engine consists of the motion equations of the piston, conrod, and crankshaft. Conrod motion was modeled by two translational and one angular motion equations, by considering the kinetic energy resulted from the mass moment of inertia and conrod mass. Motion equations involve in-cylinder gas pressure forces, hydrodynamic and dry friction, mass inertia moments of moving parts, starter moment, and external load moment. The In-cylinder pressure profile used in the model was obtained experimentally to increase the accuracy of the model. Pressure profiles were expressed mathematically using the Fourier series. The motion equations were solved by using the Taylor series method. The solution of the mathematical model was performed by coding in the MATLAB interface. Cyclic speed fluctuations obtained from the model were compared with experimental results and found compitable. A validated model was used to analyze the effects of in-cylinder pressure, mass moment of inertia of crankshaft and connecting rod, friction, and piston mass. In experiments for 1500, 1800, 2400, and 2700 rpm engine speeds, crankshaft speed fluctuations were observed as 12.84%, 8.04%, 5.02%, and 4.44%, respectively. In simulations performed for the same speeds, crankshaft speed fluctuations were calculated as 10.45%, 7.56%, 4.49%, and 3.65%. Besides, it was observed that the speed fluctuations decreased as the average crankshaft speed value increased. In the simulation for 157.07, 188.49, 219.91, 251.32, and 282.74 rad/s crankshaft speeds, crankshaft speed fluctuations occurred at rates of 10.45%, 7.56%, 5.84%, 4.49%, and 3.65%, respectively. The effective engine power was achieved as 5.25 kW at an average crankshaft angular speed of 219.91 rad/s. The power of friction loss in the engine was determined as 0.68 kW.

Acoustic Green's Function Approximations

1998 ◽  
Vol 06 (04) ◽  
pp. 435-452 ◽  
Author(s):  
Robert P. Gilbert ◽  
Zhongyan Lin ◽  
Klaus Hackl
Keyword(s):  
Inverse Problem ◽  
Normal Mode ◽  
Eigenvalue Problems ◽  
Normal Modes ◽  
Modal Parameters ◽  
Mindlin Plate ◽  
Ocean Bottom ◽  
Hankel Transformation ◽  
Poroelastic Materials ◽  
Function Approximations

Normal-mode expansions for Green's functions are derived for ocean–bottom systems. The bottom is modeled by Kirchhoff and Reissner–Mindlin plate theories for elastic and poroelastic materials. The resulting eigenvalue problems for the modal parameters are investigated. Normal modes are calculated by Hankel transformation of the underlying equations. Finally, the relation to the inverse problem is outlined.

Experimental Study for Extraction of Normal Modes

1995 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Frequency Response ◽  
Normal Mode ◽  
Normal Modes ◽  
Measurement Data ◽  
Mode Shapes ◽  
Complex Frequency ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Incomplete System ◽  
Coupled Structures

Abstract A frequency-domain technique to extract the normal mode from the measurement data for highly coupled structures is developed. The relation between the complex frequency response functions and the normal frequency response functions is derived. An algorithm is developed to calculate the normal modes from the complex frequency response functions. In this algorithm, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. In addition, the developed technique is independent of the damping types. It is only dependent on the model of analysis. Two experimental examples are employed to illustrate the applicability of the technique. The effects due to different measurement locations are addressed. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled incomplete system.

Optimal Performance of the TLCD in Structural Pitching Vibration Control

2002 ◽  
Vol 8 (5) ◽  
pp. 619-642 ◽  
Author(s):  
S. D. Xue ◽  
J. M. Ko ◽  
Y. L. Xu
Keyword(s):  
Numerical Optimization ◽  
Sensitivity Study ◽  
Optimization Approach ◽  
Optimum Control ◽  
Head Loss Coefficient ◽  
Tuning Ratio ◽  
Mass Moment ◽  
Practical Design ◽  
Mass Moment Of Inertia ◽  
Structural Uncertainties

A detailed optimal parametric study is performed for a tuned liquid column damper (TLCD) in suppressing the pitching vibration of structures. Due to the difficulty of finding analytical solutions for the damped structure, a numerical optimization approach is proposed and applied to the system to find the optimum TLCD parameters. The variations of the optimum control parameter with system parameters are determined and discussed. Using various numerical searching data, a set of practical design formulas for the optimum tuning ratio and optimum head loss coefficient of the TLCD are then derived through regression analysis. The comparison between practical design formula and numerical optimization shows a very close agreement between the two results. The practical design formulas provide a convenient tool for designers. In order to account for the possible effects of structural uncertainties, a parametric sensitivity study on the de-tuning of optimum damper parameters is also carried out. It is found that the detuning effect is more severe for low damped structure with lower ratios of mass moment of inertia, especially for the detuning of tuning ratio.

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