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

Volume 5: 35th Design Automation Conference, Parts A and B ◽

10.1115/detc2009-86236 ◽

2009 ◽

Cited By ~ 1

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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Direct Linearization Method Kinematic Variation Analysis

Journal of Mechanical Design ◽

10.1115/1.4001531 ◽

2010 ◽

Vol 132 (7) ◽

Cited By ~ 5

Author(s):

Robert C. Leishman ◽

Kenneth W. Chase

Keyword(s):

Process Variations ◽

Linearization Method ◽

Variation Analysis ◽

Computationally Efficient ◽

Design Iteration ◽

Direct Linearization ◽

Manufactured Products ◽

Acceleration Analysis ◽

The Mean ◽

Mechanism Kinematic

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 that occur in production. Small changes in mechanism dimensions can accumulate and propagate, causing a significant variation in the performance of the mechanism. A new application of statistical analysis is presented for predicting the effects of variation on mechanism kinematic performance. The new method is an extension of the direct linearization method developed for static assemblies. This method provides a solution that is a closed form. It may be applied to two-dimensional mechanisms to predict variation in velocity and acceleration due to dimensional variations. It is also shown how form, orientation, and position variations may be included in the analysis to analyze variations that occur within the joints. Only two assemblies are analyzed to characterize the distribution: The first determines the mean, and the second estimates the variance. The system is computationally efficient and well suited for design iteration.

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A Benchmark Study on Crushing and Cutting of Plated Structures

Journal of Ship Research ◽

10.5957/jsr.1997.41.2.147 ◽

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.

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The Combinatorics of Neurite Self-Avoidance

Neural Computation ◽

10.1162/neco_a_00186 ◽

2011 ◽

Vol 23 (11) ◽

pp. 2746-2769 ◽

Cited By ~ 6

Author(s):

Elizabeth M. Forbes ◽

Jonathan J. Hunt ◽

Geoffrey J. Goodhill

Keyword(s):

Closed Form ◽

Neural Development ◽

Stochastic Simulations ◽

Large Set ◽

Unique Combination ◽

Closed Form Solutions ◽

Specific Parameter ◽

Key Variables

During neural development in Drosophila, the ability of neurite branches to recognize whether they are from the same or different neurons depends crucially on the molecule Dscam1. In particular, this recognition depends on the stochastic acquisition of a unique combination of Dscam1 isoforms out of a large set of possible isoforms. To properly interpret these findings, it is crucial to understand the combinatorics involved, which has previously been attempted only using stochastic simulations for some specific parameter combinations. Here we present closed-form solutions for the general case. These reveal the relationships among the key variables and how these constrain possible biological scenarios.

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The Impact of Arbitrary Oriented Ellipsoidal Shell With a Barrier: Analytical Study

Journal of Applied Mechanics ◽

10.1115/1.4029780 ◽

2015 ◽

Vol 82 (4) ◽

Cited By ~ 1

Author(s):

Shahab Mansoor-Baghaei ◽

Ali M. Sadegh

Keyword(s):

Closed Form ◽

Numerical Solutions ◽

Closed Form Solution ◽

Linearization Method ◽

Form Solution ◽

Ellipsoidal Shell ◽

Explicit Finite Element ◽

Closed Form Solutions ◽

Transmitted Force ◽

The Impact

In this paper, a closed form solution of an arbitrary oriented hollow elastic ellipsoidal shell impacting with an elastic flat barrier is presented. It is assumed that the shell is thin under the low speed impact. Due to the arbitrary orientation of the shell, while the pre-impact having a linear speed, the postimpact involves rotational and translational speed. Analytical solution for this problem is based on Hertzian theory (Johnson, W., 1972, Impact Strength of Materials, University of Manchester Institute of Science and Technology, Edward Arnold Publication, London) and the Vella’s analysis (Vella et al., 2012, “Indentation of Ellipsoidal and Cylindrical Elastic Shells,” Phys. Rev. Lett., 109, p. 144302) in conjunction with Newtonian method. Due to the nonlinearity and complexity of the impact equation, classical numerical solutions cannot be employed. Therefore, a linearization method is proposed and a closed form solution for this problem is accomplished. The closed form solution facilitates a parametric study of this type of problems. The closed form solution was validated by an explicit finite element method (FEM). Good agreement between the closed form solution and the FE results is observed. Based on the analytical method the maximum total deformation of the shell, the maximum transmitted force, the duration of the contact, and the rotation of the shell after the impact were determined. Finally, it was concluded that the closed form solutions were trustworthy and appropriate to investigate the impact of inclined elastic ellipsoidal shells with an elastic barrier.

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Alternative closed-form solutions for the mean rate of exceedance of structural limit states

Earthquake Engineering & Structural Dynamics ◽

10.1002/eqe.2300 ◽

2013 ◽

Vol 42 (12) ◽

pp. 1827-1845 ◽

Cited By ~ 7

Author(s):

Xavier Romão ◽

Raimundo Delgado ◽

Aníbal Costa

Keyword(s):

Closed Form ◽

Limit States ◽

Closed Form Solutions ◽

The Mean

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Modeling Variation Propagation Through a Manufacturing Process

Volume 3a: 8th Design for Manufacturing Conference ◽

10.1115/detc2003/dfm-48149 ◽

2003 ◽

Author(s):

Daniel Kern ◽

Anna Thornton

Keyword(s):

Standard Deviation ◽

Closed Form ◽

Manufacturing Process ◽

Dimensional Change ◽

New Method ◽

Closed Form Solutions ◽

Effective Manner ◽

Variation Propagation ◽

The Mean ◽

High Level

Companies are under increased pressure to manufacture products that have a high level of quality. Manufacturing products with a high level of quality in a cost-effective manner requires the products to be designed so that they can be manufactured with an acceptable level of variation. Creating a new design that can be produced with the necessary level of variation has historically been a very challenging problem. A new method for calculating the effect a manufacturing process has on the mean and standard deviation of a distribution is presented. This new method is founded on the concept of characterizing a manufacturing process with two math functions called DeltaP and SigmaP. DeltaP and SigmaP represent the theory of Process Imparted Dimensional Change and Process Imparted Variation. Using these functions, closed-form solutions for the mean and standard deviation of a distribution exiting a manufacturing process can be calculated. The authors present the background of the theory as well as the derivation of the closed form solutions for the output mean and standard deviation from a generic manufacturing process. The derivation is followed by a simple example to demonstrate the method.

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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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