Analysis of transient amplification for a torsional system passing through resonance

The classical problem of vibration amplification of a linear torsional oscillator excited by an instantaneous sinusoidal torque is re-examined with focus on the development of new analytical solutions of the transient envelopes. First, a new analytical method in the instantaneous frequency (or speed) domain is proposed to directly find the closed-form solutions of transient displacement, velocity, and acceleration envelopes for passage through resonance during the run-up or run-down process. The proposed closed-form solutions are then successfully verified by comparing them with numerical predictions and limited analytical solutions as available in prior literature. Second, improved analytical approximations of maximum amplification and corresponding peak frequency are found, which are also verified by comparing them with prior analytical or empirical formulas. In addition, applicability of the proposed analytical solution is clarified, and their error bounds are identified. Finally, the utility of analytical solutions and approximations is demonstrated by application to the start-up process of a multi-degree-of-freedom vehicle driveline system.

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Assessment of Aided-INS Performance

Journal of Navigation ◽

10.1017/s0373463311000609 ◽

2011 ◽

Vol 65 (1) ◽

pp. 169-185 ◽

Cited By ~ 2

Author(s):

Itzik Klein ◽

Sagi Filin ◽

Tomer Toledo ◽

Ilan Rusnak

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Navigation Systems ◽

Inertial Navigation Systems ◽

Closed Form Solutions ◽

Land Vehicles ◽

Error Covariance ◽

Model Dependency ◽

Error State ◽

Insight Into

Aided Inertial Navigation Systems (INS) systems are commonly implemented in land vehicles for a variety of applications. Several methods have been reported in the literature for evaluating aided INS performance. Yet, the INS error-state-model dependency on time and trajectory implies that no closed-form solutions exist for such evaluation. In this paper, we derive analytical solutions to evaluate the fusion performance. We show that the derived analytical solutions manage to predict the error covariance behavior of the full aided INS error model. These solutions bring insight into the effect of the various parameters involved in the fusion of the INS and an aiding sensor.

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Analytical Solutions for Flow of a Dusty Fluid Between Two Porous Flat Plates

Journal of Fluids Engineering ◽

10.1115/1.2910280 ◽

1994 ◽

Vol 116 (2) ◽

pp. 354-356 ◽

Cited By ~ 2

Author(s):

Ali J. Chamkha

Keyword(s):

Exact Solutions ◽

Closed Form ◽

Skin Friction ◽

Analytical Solutions ◽

Velocity Profiles ◽

Flat Plates ◽

Friction Coefficients ◽

Dusty Fluid ◽

Closed Form Solutions

Equations governing flow of a dusty fluid between two porous flat plates with suction and injection are developed and closed-form solutions for the velocity profiles, displacement thicknesses, and skin friction coefficients for both phases are obtained. Graphical results of the exact solutions are presented and discussed.

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Analytical Approximations in Modeling Contacting Rough Surfaces

Journal of Tribology ◽

10.1115/1.2833926 ◽

1999 ◽

Vol 121 (2) ◽

pp. 234-239 ◽

Cited By ~ 74

Author(s):

Andreas A. Polycarpou ◽

Izhak Etsion

Keyword(s):

Closed Form ◽

Gaussian Distribution ◽

Rough Surfaces ◽

Critical Examination ◽

Height Distribution ◽

Closed Form Solutions ◽

Real Area Of Contact ◽

Analytical Approximations ◽

Classic Paper ◽

Real Area

A critical examination of the analytical solution presented in the classic paper of Greenwood and Williamson (1966), (GW) on the statistical modeling of nominally flat contacting rough surfaces is undertaken in this study. It is found that using the GW simple exponential distribution to approximate the usually Gaussian height distribution of the asperities is inadequate for most practical cases. Some other exponential type approximations are suggested, which approximate the Gaussian distribution more accurately, and still enable closed form solutions for the real area of contact, the contact load, and the number of contacting asperities. The best-modified exponential approximation is then used in the case of elastic-plastic contacts of Chang et al. (1987) (CEB model), to obtain closed-form solutions, which favorably compare with the numerical results using the Gaussian distribution.

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Analytical and Semi-Analytical Solutions for the Self and Mutual Inductances of Concentric Coplanar Ordinary and Bitter Disk Coils

10.20944/preprints202103.0153.v1 ◽

2021 ◽

Author(s):

Slobodan Babic

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Simple Form ◽

Elliptic Integral ◽

Mutual Inductance ◽

The Self ◽

Computational Time ◽

Closed Form Solutions

In this paper, closed and semi-closed form solutions are presented for the self - and mutual inductance of ordinary and Better disk coils which lie concentrically in a plane. The solutions are given as the combination of the elliptic integral of the first kind and a simple integral or only as the second elliptic integral of the second kind. All formulas or obtained in remarkably simple form and give extremely accurate results with significantly neglectable computational time. All cases either regular or singular (disks in contact or overlap) are covered. The formulas for the mutual inductances can be directly used to calculating the self-inductance of the ordinary disk coil or the Bitter disk coil, respectively. Many presented examples show the excellent numerical agreement with previous publish methods.

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The Lure of the Mean Axes

Journal of Applied Mechanics ◽

10.1115/1.2338060 ◽

2006 ◽

Vol 74 (3) ◽

pp. 497-504 ◽

Cited By ~ 19

Author(s):

Leonard Meirovitch ◽

Ilhan Tuzcu

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Numerical Solutions ◽

Equations Of Motion ◽

Space Structures ◽

Aerodynamic Forces ◽

State Equations ◽

Flexible Bodies ◽

Closed Form Solutions ◽

The Mean

A variety of aerospace structures, such as missiles, spacecraft, aircraft, and helicopters, can be modeled as unrestrained flexible bodies. The state equations of motion of such systems tend to be quite involved. Because some of these formulations were carried out decades ago when computers were inadequate, the emphasis was on analytical solutions. This, in turn, prompted some investigators to simplify the formulations beyond all reasons, a practice continuing to this date. In particular, the concept of mean axes has often been used without regard to the negative implications. The allure of the mean axes lies in the fact that in some cases they can help decouple the system inertially. Whereas in the case of some space structures this may mean complete decoupling, in the case of missiles, aircraft, and helicopters the systems remain coupled through the aerodynamic forces. In fact, in the latter case the use of mean axes only complicates matters. With the development of powerful computers and software capable of producing numerical solutions to very complex problems, such as MATLAB and MATHEMATICA, there is no compelling reason to insist on closed-form solutions, particularly when undue simplifications can lead to erroneous results.

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Analytical Estimation for Static Deformation of Wire Ropes With Fibrous Core

Journal of Applied Mechanics ◽

10.1115/1.2897617 ◽

1990 ◽

Vol 57 (4) ◽

pp. 1000-1003 ◽

Cited By ~ 5

Author(s):

K. Kumar ◽

J. E. Cochran

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Close Agreement ◽

Helix Angle ◽

Closed Form Solutions ◽

Wire Ropes ◽

Axial Forces ◽

End Conditions ◽

Analytical Estimation ◽

Rigidity Modulus

This papers develops closed-form solutions for the extension of twisted wire ropes with fibrous cores which are subjected to axial forces as well as axial moments. The analytical results are compared with the corresponding numerical results obtained by Costello and Phillips. A close agreement between the two establishes validity of the analytical solutions. Finally, an expression for the effective rigidity modulus of wire ropes with fibrous core is obtained in terms of the helix angle and the number of helical wires in the rope for each of the two end conditions.

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On the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2021/6624287 ◽

2021 ◽

Vol 2021 ◽

pp. 1-19

Author(s):

I. L. Ferreira ◽

A. Garcia ◽

A. L. S. Moreira

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Single Phase ◽

Moving Boundary ◽

Error Function ◽

Solid State Diffusion ◽

Two Phase ◽

Closed Form Solutions ◽

State Diffusion ◽

Cylinders And Spheres

Analytical solutions for the transient single-phase and two-phase inward solid-state diffusion and solidification applied to the radial cylindrical and spherical geometries are proposed. Both solutions are developed from the differential equation that treats these phenomena in Cartesian coordinates, which are modified by geometric correlations and suitable changes of variables to achieve closed-form solutions. The modified differential equations are solved by using two well-known closed-form solutions based on the error function, and then equations are obtained to analyze the diffusion interface position as a function of time and position in cylinders and spheres. These exact correlations are inserted into the closed-form solutions for the slab and are used to update the roots for each radial position of the moving boundary interface. The predictions provided by the proposed analytical solutions are validated against the results of a numerical approach. Finally, a comparative study of diffusion in slabs, cylinders, and spheres is also presented for single-phase and two-phase solid-state diffusion and solidification, which shows the importance of the effects imposed by the radial cylindrical and spherical curvatures with respect to the Cartesian coordinate system in the process kinetics. The analytical model is proved to be general, as far as, a semi-infinite solution for diffusion problems with phase change exists in the form of the error function, which enables exact closed-form analytical solutions to be derived, by updating the root at each radial position the moving boundary interface.

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