Design of Helical Springs for Minimum Weight, Volume, and Length

1959 ◽  
Vol 81 (1) ◽  
pp. 37-40 ◽  
Author(s):  
R. T. Hinkle ◽  
I. E. Morse
Keyword(s):  
Infinite Number ◽  
Minimum Weight ◽  
Allowable Stress ◽  
Number Of Solutions ◽  
Helical Springs ◽  
Additional Requirement ◽  
Outside Diameter ◽  
Selection Of

In the design of helical springs where the load, deflection, allowable stress, and material are specified, there are an infinite number of solutions. In this paper, equations and graphs are presented for the selection of a spring index that will result in a spring of minimum weight, volume, or length. If, an addition to these requirements, the inside or outside diameter of the spring is fixed, there is only one solution. Equations and graphs are included for the selection of the spring index which will satisfy this additional requirement.

Free-surface flows with two stagnation points

1996 ◽  
Vol 324 ◽  
pp. 393-406 ◽  
Author(s):  
J.-M. Vanden-Broeck ◽  
F. Dias
Keyword(s):  
Free Surface ◽  
Infinite Number ◽  
Wave Breaking ◽  
Free Surface Flows ◽  
Number Of Solutions ◽  
Surface Flows ◽  
Stagnation Points ◽  
Surface Profiles ◽  
Countably Infinite

Symmetric suction flows are computed. The flows are free-surface flows with two stagnation points. The configuration is related to the modelling of wave breaking at the bow of a ship. It is shown that there is a countably infinite number of solutions and that the free-surface profiles are characterized by waves.

A Proposed Method to Group the Solutions From Dimensional Synthesis: Planar Triads for Six Precision Position

1992 ◽  
Author(s):  
Xiancheng Lu ◽  
Chuen-Sen Lin
Keyword(s):  
Infinite Number ◽  
Numerical Solutions ◽  
Angular Displacement ◽  
Dimensional Synthesis ◽  
Number Of Solutions ◽  
Jacobian Matrices ◽  
Design Task ◽  
Numerical Examples ◽  
Computer Automation ◽  
Constraint Sets

Abstract In this paper, a method has been proposed to group into six sets the infinite number of solutions from dimensional synthesis of planar triads for six precision positions. The proposed method reveals the relationships between the different configurations of the compatibility linkage and the sets of numerical solutions from dimensional synthesis. By checking the determinant signs and the contunities of values of the sub-Jacobian matrices and their derivatives with respect to the independent angular displacement for all constraint sets in the compatibility linkage, it enables the computer to identify and group the synthesized solutions. Numerical examples have been given to verify the applicability of this method. Six sets of the partial triad Burmester curves have been plotted based on grouped solutions. Suitable solutions can be easily found from the partial triad Burmester curves and utilized for the prescribed design task. This method provides a useful tool to group the dimensional synthesis solutions and enhances the computer automation in the design of linkage mechanisms.

scholarly journals Problems of Menotactic Orientation According to the Polarized Light of the Sky

1975 ◽  
Vol 30 (1-2) ◽  
pp. 88-90 ◽  
Author(s):  
Kuno Kirschfeld ◽  
M. Lindauer ◽  
H. Martin
Keyword(s):  
Infinite Number ◽  
Polarized Light ◽  
The Sun ◽  
Number Of Solutions ◽  
Special Cases ◽  
Linearly Polarized ◽  
Vector Direction

Abstract It is shown that the knowledge of the E-vector direction of the linearly polarized light at any point of the sky alone is insufficient for the determination of the position of the sun. If the E-vector direction of a second point is not known the knowledge of at least one other parameter is necessary. This parameter might be the height of the sun over the horizon. With the knowledge of the height the infinite number of solutions for the sun’s position becomes reduced to two, or in special cases to one. These cases are derived.

scholarly journals Inductive analysis of the deflection of a beam truss allowing kinematic variation

2018 ◽  
Vol 239 ◽  
pp. 01012
Author(s):  
Mikhail Kirsanov ◽  
Evgeny Komerzan ◽  
Olesya Sviridenko
Keyword(s):  
Optimal Design ◽  
External Load ◽  
Analytical Calculation ◽  
System Of Equations ◽  
Number Of Solutions ◽  
Recurrence Equations ◽  
General Terms ◽  
Cutting Out ◽  
Selection Of ◽  
Inductive Analysis

A scheme of a statically determinate planar truss is proposed and an analytical calculation of its deflection and displacement of the mobile support are obtained. The forces in the rods from the external load, uniformly distributed over the nodes of the lower or upper belt, are determined by the method of cutting out nodes using the computer mathematic system Maple. In the generalization of a number of solutions of trusses with a different number of panels to the general case, the general terms of the sequence of coefficients in the formulas are found from solutions of linear homogeneous recurrence equations. To compose and solve these equations, Maple operators were used. In the process of calculation it was revealed that for even numbers of panels in half the span, the determinant of the system of equations degenerates. This corresponds to the kinematic degeneracy of the structure. The corresponding scheme of possible speeds of the truss is given. The displacement was determined by the Maxwell-Mohr’s formula. The graphs of the obtained dependences have appreciable jumps, which in principle can be used in the selection of optimal design sizes.

scholarly journals THERMODYNAMICS OF QUASI-PARTICLES

2007 ◽  
Vol 16 (09) ◽  
pp. 3024-3027
Author(s):  
FERNANDO G. GARDIM ◽  
FERNANDO M. STEFFENS
Keyword(s):  
Infinite Number ◽  
Temperature Dependent ◽  
Number Of Solutions

We present in this work a generalization of the solution of Gorenstein and Yang to the inconsistency problem of thermodynamics for systems with a temperature dependent Hamiltonian. We show that there are, in principle, an infinite number of solutions.

WORD EQUATIONS WITH ONE UNKNOWN

2011 ◽  
Vol 22 (02) ◽  
pp. 345-375 ◽  
Author(s):  
MARKKU LAINE ◽  
WOJCIECH PLANDOWSKI
Keyword(s):  
Infinite Number ◽  
Linear Time ◽  
Time Algorithm ◽  
Solution Set ◽  
Infinitely Many Solutions ◽  
Linear Time Algorithm ◽  
Number Of Solutions ◽  
Word Equation ◽  
Word Equations

We consider properties of the solution set of a word equation with one unknown. We prove that the solution set of a word equation possessing infinite number of solutions is of the form (pq)*p where pq is primitive. Next, we prove that a word equation with at most four occurrences of the unknown possesses either infinitely many solutions or at most two solutions. We show that there are equations with at most four occurrences of the unknown possessing exactly two solutions. Finally, we prove that a word equation with at most 2k occurrences of the unknown possesses either infinitely many solutions or at most 8 log k + O(1) solutions. Hence, if we consider a class εk of equations with at most 2k occurrences of the unknown, then each equation in this class possesses either infinitely many solutions or O( log k) number of solutions. Our considerations allow to construct the first alphabet independent linear time algorithm for computing the solution set of an equation in a nontrivial class of equations.

Parametrization of Reduced Order MIMO Tracking Prefilters With Optimality Considerations

2002 ◽  
Vol 124 (2) ◽  
pp. 307-312
Author(s):  
Matt Bement ◽  
Suhada Jayasuriya
Keyword(s):  
Infinite Number ◽  
Internal Model ◽  
Linear Equations ◽  
Optimal Solution ◽  
Optimization Procedure ◽  
Number Of Solutions ◽  
Existence Of A Solution ◽  
Nonminimum Phase Systems ◽  
All Solutions ◽  
Asymptotic Tracking

A primary disadvantage of using an internal model to achieve multivariable tracking is the high order of the internal model. In situations where it is known that each output is to track only its associated reference input, the internal model formulation results in an overdesign of sorts. A method is presented through which a prefilter may be constructed to achieve asymptotic tracking of only the required reference inputs. It is shown that obtaining the prefilter requires the solution of a polynomial matrix equation. Conditions for existence of a solution to this equation, as well as an algorithm for its construction, are presented. Since existence of a solution implies an infinite number of solutions, the algorithm provides a means of parametrizing all solutions of a given order. Unlike prefilter techniques such as plant inversion, the method presented may be applied to nonminimum phase systems and results in proper, physically realizable systems. Since an infinite number of solutions exist, criteria for defining and obtaining the optimal solution are presented. In fact, it is shown that obtaining the optimal prefilter reduces to solving a set of linear equations. A multivariable system is used to demonstrate the effectiveness of the optimization procedure. In addition, the tracking is shown to be robust with respect to certain structured plant perturbations.

Minimum Torque Placement of a Kinematic Structure

2001 ◽  
Author(s):  
Wei Yu ◽  
Jingzhou Yang ◽  
Karim Abdel-Malek
Keyword(s):  
Design Process ◽  
Infinite Number ◽  
Significant Contribution ◽  
Optimization Methods ◽  
Working Environment ◽  
Kinematic Structure ◽  
Ergonomic Design ◽  
Number Of Solutions ◽  
The Given ◽  
Minimum Torque

Abstract A general formulation for calculating where a kinematic structure must be positioned (and oriented) is presented. This study is applicable to both humans; to place a human in a working environment while minimizing stress on extremity joints, or for robot manipulators. In recent years, there has been focused interest on ergonomics and ergonomic design with emphasis on the disposition of a worker while performing tasks at prolonged periods of time and where repetitive motions are exerted. To a great extent, the effect of stress on a joint is major factor leading to potential injuries. We believe there is a unique opportunity for the mechanical engineering community to make a significant contribution to this field. Furthermore, because the ergonomic design process encompasses many parameters, it is evident that it must be formulated using optimization methods where the best possible solution is calculated from an infinite number of solutions. This study presents a rigorous formulation for placement of a worker based on minimizing the torque (as a cost function) induced at a joint, whereby satisfying constraints imposed by the given task. The reverse of this problem is the calculation of the coordinates of a number of target points in the reachable space of a fixed kinematic structure, which is also addressed. Both problems are mathematically formulated and numerically solved. Examples are illustrated.

scholarly journals Some remarks on the paper by B. G. Marsden “An attempt to reconcile the dynamical and radar determinations of the Astronomical Unit”

1965 ◽  
Vol 21 ◽  
pp. 237-239
Author(s):  
Eugene Rabe
Keyword(s):  
Least Squares ◽  
Infinite Series ◽  
Infinite Number ◽  
Least Squares Method ◽  
Fundamental Principle ◽  
Astronomical Unit ◽  
Number Of Solutions ◽  
Least Squares Solution ◽  
Minimum Value

While Marsden's solution C leaves residuals with the relatively small [vv] of 13.73, it should be realized that this representation of the observations of Eros does not satisfy the fundamental principle of the least squares method, in so far as the associated value of [vv] is not a minimum with respect to small arbitrary deviations from solution C. As a matter of fact, there is an infinite number of “solutions” with [vv] between the 13.73 of Marsden's solution C and the 8.66 of his solution A, each of these being associated with a certain arbitrarily prescribed value of the mass of Mars and with a related mass of Earth + Moon. Of this infinite series of solutions, only solution A is a least squares solution in the true sense, with a minimum value of [vv]. This can be seen and verified as follows.

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