The Lure of the Mean Axes

2006 ◽  
Vol 74 (3) ◽  
pp. 497-504 ◽  
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.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

A Benchmark Study on Crushing and Cutting of Plated Structures

1997 ◽  
Vol 41 (02) ◽  
pp. 147-160
Author(s):  
Jeom Kee Paik ◽  
Tomasz Wierzbicki
Keyword(s):  
Experimental Data ◽  
Closed Form ◽  
Dynamic Effects ◽  
Crushing Strength ◽  
Closed Form Solutions ◽  
Cutting Resistance ◽  
Benchmark Study ◽  
Stiffened Structures ◽  
The Mean

A benchmark study on several closed-form solutions for the mean crushing strength and the cutting resistance of plated structures during collision or grounding is carried out by comparing theoretical solutions with experimental data. Based on expressions which have been derived for unstiffened structures, an extension of the methods is proposed for longitudinally and/or transversely stiffened structures. Dynamic effects on the crushing and cutting response are discussed, and applicability of the quasistatic formulations to analyze the crushing and cutting damage of the structure in the dynamic situations is investigated.

Closed-form solutions of linear stationary state equations

1967 ◽  
Vol 55 (7) ◽  
pp. 1244-1245 ◽  
Author(s):  
M.L. Liou ◽  
W.R. Broyles
Keyword(s):  
Closed Form ◽  
Stationary State ◽  
State Equations ◽  
Closed Form Solutions

Assessment of Aided-INS Performance

2011 ◽  
Vol 65 (1) ◽  
pp. 169-185 ◽  
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.

Active-Passive Hybrid Constrained Layer for Structural Damping Augmentation

2000 ◽  
Vol 122 (3) ◽  
pp. 254-262 ◽  
Author(s):  
Yanning Liu ◽  
K. W. Wang
Keyword(s):  
Closed Form ◽  
Viscoelastic Material ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Generic Model ◽  
Structural Damping ◽  
Active Materials ◽  
Closed Form Solutions ◽  
Fail Safe ◽  
Active Constrained Layer

A new surface-damping concept with an active-passive hybrid constraining layer (HCL) is proposed to improve the damping performance of traditional active constrained layer (ACL) systems. Instead of using a pure piezoelectric constraining layer, passive and active materials are used together to constrain the viscoelastic material layer. A generic model of the HCL treatment is presented. Nondimensional equations of motion and boundary and connecting conditions are derived. The closed-form solutions to the equations are developed and analyzed. Tabletop tests are also performed to verify the feasibility of the new damping concept. It is shown that by properly selecting a passive constraining material and assigning appropriate lengths for the active and passive constraining parts, HCL can outperform a system with a pure active PZT coversheet, both in terms of its fail-safe ability and closed-loop damping performance. [S0739-3717(00)01503-8]

Analysis of transient amplification for a torsional system passing through resonance

2014 ◽  
Vol 229 (13) ◽  
pp. 2341-2354 ◽  
Author(s):  
Laihang Li ◽  
Rajendra Singh
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Classical Problem ◽  
Peak Frequency ◽  
Empirical Formulas ◽  
Closed Form Solutions ◽  
Analytical Approximations ◽  
Numerical Predictions ◽  
Prior Literature ◽  
Run Up

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.

On Wave Propagation in Thermoplastic Media

1966 ◽  
Vol 33 (3) ◽  
pp. 514-520 ◽  
Author(s):  
A. D. Fine ◽  
H. Kraus
Keyword(s):  
Wave Propagation ◽  
Closed Form ◽  
Wave Front ◽  
Dynamic Behavior ◽  
Static Analysis ◽  
Equations Of Motion ◽  
Normal Velocity ◽  
Wave Fronts ◽  
Closed Form Solutions ◽  
Infinite Region

The dynamic behavior of a medium, according to the uncoupled thermoplastic theory, is presented and is compared to the behavior that would be obtained from an uncoupled quasi-static analysis. Since the inertia terms are retained in the equations of motion, wave fronts (or surfaces of discontinuity) are produced in the medium. The normal velocity of the wave front separating the elastic and plastic regions is determined. General closed-form solutions of the displacement (according to both the dynamic and the quasi-static approaches) are obtained; their unique forms are found for the semi-infinite region, and an illustrative numerical example is then presented.

Efficient Coupling of Absolute Nodal Coordinate Formulation Flexible Bodies With an Existing Multibody Dynamics Code

2012 ◽  
Author(s):  
Jeff Liu ◽  
Abdel-Nasser A. Mohamed
Keyword(s):  
Multibody Dynamics ◽  
Absolute Nodal Coordinate Formulation ◽  
Sparse Matrix ◽  
Equations Of Motion ◽  
Matrix Solution ◽  
State Equations ◽  
Combined System ◽  
Flexible Bodies ◽  
Absolute Nodal Coordinate ◽  
Efficient Coupling

A couple of issues are identified in the process to embed absolute nodal coordinate formulation (ANCF) flexible bodies in an existing multibody dynamics code. (1) The generalized coordinates of ANCF must be solved together with those of the rest of the mechanism in a combined system of the equations of motion. (2) The various constraints, joints, and forces elements supported in the multibody dynamics code must be extended to the ANCF flexible bodies without major code restructuring. This paper describes two novel techniques that were devised to solve these issues. The first is the idea of interface triad. We will demonstrate how to construct the interface triad such that all exiting constraints, joints, and forces elements are automatically supported. The second idea is to represent the equations of motion of the ANCF body as a user-defined subroutine element representing a set of implicit general state equations subroutine (GSESUB). By treating each ANCF body modularly as a user-defined subroutine, not only all existing integration options of its host solver, e.g., HHT or DAE index-1, 2, and 3, etc., are automatically supported, but also the existing features such as parallel computing and sparse matrix solution of the existing multibody dynamics software are supported with minimum programming. Numerical examples are presented to demonstrate the efficiency and the success of these two techniques.

Variation Analysis of Position, Velocity, and Acceleration of Two-Dimensional Mechanisms by the Direct Linearization Method

2009 ◽  
Author(s):  
Robert C. Leishman ◽  
Kenneth W. Chase
Keyword(s):  
Closed Form ◽  
Process Variations ◽  
Linearization Method ◽  
Analysis Tool ◽  
Large Set ◽  
Computationally Efficient ◽  
Closed Form Solutions ◽  
Direct Linearization ◽  
The Mean ◽  
Kinematic Performance

Velocity and acceleration analysis is an important tool for predicting the motion of mechanisms. The results, however, may be inaccurate when applied to manufactured products, due to the process variations which occur in production. Small changes in dimensions can accumulate and propagate in an assembly, which may cause significant variation in critical kinematic performance parameters. A new statistical analysis tool is presented for predicting the effects of variation on mechanism kinematic performance. It is based on the Direct Linearization Method developed for static assemblies. The solution is closed form, and may be applied to 2-D, open or closed, multi-loop mechanisms, employing common kinematic joints. It is also shown how form, orientation, and position variations may be included in the analysis to analyze variations that occur in kinematic joints. Closed form solutions eliminate the need of generating a large set of random assemblies, and analyzing them one-by one, to determine the expected range of critical variables. Only two assemblies are analyzed to characterize the entire population. The first determines the performance of the mean, or average assembly, and the second estimates the range of variation about the mean. The system is computationally efficient and well suited for design iteration.

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