An Inverse Force Analysis of a Tetrahedral Three-Spring System

1995 ◽  
Vol 117 (2A) ◽  
pp. 286-291 ◽  
Author(s):  
P. Dietmaier
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Force Analysis ◽  
Equilibrium Configurations ◽  
Spring System ◽  
The Common ◽  
The Stability ◽  
Inverse Force

A tetrahedral three-spring system under a single load has been analyzed and a closed-form solution for the equilibrium positions is given. Each of the three springs is attached at one end to a fixed pivot in space while the other three ends are linked by a common pivot. The springs are assumed to behave in a linearly elastic way. The aim of the paper at hand was to find out what the maximum number of equilibrium positions of such a system might be, and how to compute all possible equilibrium configurations if a given force is applied to the common pivot. First a symmetric and unloaded system was studied. For such a system it was shown that there may exist a maximum of 22 equilibrium configurations which may all be real. Second the general, loaded system was analyzed, revealing again a maximum of 22 real equilibrium configurations. Finally, the stability of this three-spring system was investigated. A numerical example illustrates the theoretical findings.

A new closed-form solution to inverse static force analysis of a spatial three-spring system

2014 ◽  
Vol 229 (14) ◽  
pp. 2599-2610 ◽  
Author(s):  
Ying Zhang ◽  
Qizheng Liao ◽  
Hai-Jun Su ◽  
Shimin Wei
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Polynomial Equation ◽  
Form Solution ◽  
Common Point ◽  
Static Force ◽  
Force Analysis ◽  
Intrinsic Geometry ◽  
Equilibrium Configurations ◽  
Spring System

In this paper, a new closed-form solution to the inverse static force analysis of a spatial three-spring system is presented. The system is formed by three springs each of which connects the ground at one end and joins a common point at the other. When a known force is applied to the common point of the system, the goal of inverse static analysis is to determine all the equilibrium configurations. A system of three polynomial equations in three variables is derived based on the geometric constraint and static force balancing. A 20 by 20 Dixon resultant matrices firstly derived from these three polynomials and then reduced to an 18 by 18 matrix. A 46th-degree univariate polynomial equation is yielded from the above 18 by 18 matrix. By further analysis, we found that 24 roots were degenerated and only the remaining 22 roots are the ones for the three-spring system. The result agrees with previous results. At last, two numerical examples are given to verify the elimination procedure. The presented algebraic elimination solution reveals some intrinsic geometry nature of this challenging problem.

An Inverse Force Analysis of a Planar Two-Spring System

1994 ◽  
Author(s):  
Thomas M. Pigoski ◽  
Joseph Duffy
Keyword(s):  
The Other ◽  
Spring Stiffness ◽  
Force Analysis ◽  
Degree Polynomial ◽  
Real Solutions ◽  
Variable Elimination ◽  
Spring System ◽  
The Common ◽  
Resultant Length ◽  
Inverse Force

Abstract A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

An Inverse Force Analysis of a Planar Two-Spring System

1995 ◽  
Vol 117 (4) ◽  
pp. 548-553 ◽  
Author(s):  
T. M. Pigoski ◽  
J. Duffy
Keyword(s):  
The Other ◽  
Spring Stiffness ◽  
Force Analysis ◽  
Degree Polynomial ◽  
Real Solutions ◽  
Variable Elimination ◽  
Spring System ◽  
The Common ◽  
Resultant Length ◽  
Inverse Force

A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

Scattering by hard and soft parallel half-planes

1981 ◽  
Vol 59 (12) ◽  
pp. 1879-1885 ◽  
Author(s):  
R. A. Hurd ◽  
E. Lüneburg
Keyword(s):  
Plane Wave ◽  
Closed Form ◽  
Half Plane ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Normal Derivative ◽  
Form Solution ◽  
The Other ◽  
Total Field

We consider the diffraction of a scalar plane wave by two parallel half-planes. On one half-plane the total field vanishes whilst on the other its normal derivative is zero. This is a new canonical diffraction problem and we give an exact closed-form solution to it. The problem has applications to the design of acoustic barriers.

DIVISIBLE LOAD DISTRIBUTION IN A NETWORK OF PROCESSORS

2008 ◽  
Vol 09 (01n02) ◽  
pp. 31-51 ◽  
Author(s):  
SAMEER BATAINEH
Keyword(s):  
Closed Form ◽  
Load Distribution ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Optimum Number ◽  
Sequencing Problem ◽  
Divisible Load ◽  
Finish Time ◽  
Optimum Sequence

The paper presents a closed form solution for an optimum scheduling of a divisible job on an optimum number of processor arranged in an optimum sequence in a multilevel tree networks. The solution has been derived for a single divisible job where there is no dependency among subtasks and the root processor can either perform communication and computation at the same time. The solution is carried out through three basic theorems. One of the theorems selects the optimum number of available processors that must participate in executing a divisible job. The other solves the sequencing problem in load distribution by which we are able to find the optimum sequence for load distribution in a generalized form. Having the optimum number of processors and their sequencing for load distribution, we have developed a closed form solution that determines the optimum share of each processor in the sequence such that the finish time is minimized. Any alteration of the number of processors, their sequences, or their shares that are determined by the three theorems will increase the finish time.

Bi-BCM: A Closed-Form Solution for Fixed-Guided Beams in Compliant Mechanisms

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen
Keyword(s):  
Closed Form ◽  
Compliant Mechanisms ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Boundary Line ◽  
Final Solution ◽  
Second Mode ◽  
Design Calculations ◽  
Constraint Model

A fixed-guided beam, with one end is fixed while the other is guided in that the angle of that end does not change, is one of the most commonly used flexible segments in compliant mechanisms such as bistable mechanisms, compliant parallelogram mechanisms, compound compliant parallelogram mechanisms, and thermomechanical in-plane microactuators. In this paper, we split a fixed-guided beam into two elements, formulate each element using the beam constraint model (BCM) equations, and then assemble the two elements' equations to obtain the final solution for the load–deflection relations. Interestingly, the resulting load–deflection solution (referred to as Bi-BCM) is closed-form, in which the tip loads are expressed as functions of the tip deflections. The maximum allowable axial force of Bi-BCM is the quadruple of that of BCM. Bi-BCM also extends the capability of BCM for predicting the second mode bending of fixed-guided beams. Besides, the boundary line between the first and the second modes bending of fixed-guided beams can be easily obtained using a closed-form equation. Bi-BCM can be immediately used for quick design calculations of compliant mechanisms utilizing fixed-guided beams as their flexible segments (generally no iteration is required). Different examples are analyzed to illustrate the usage of Bi-BCM, and the results show the effectiveness of the closed-form solution.

Spontaneous potential log response expressed as convolution

Geophysics ◽  
1982 ◽  
Vol 47 (9) ◽  
pp. 1335-1337
Author(s):  
E. A. Nosal
Keyword(s):  
Closed Form ◽  
Signal Analysis ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Individual Contribution ◽  
Spontaneous Potential ◽  
The Earth ◽  
The Individual ◽  
Special Case

A special case of spontaneous potential (SP) logging, which has a closed‐form solution, will be expressed as a convolutional operation. Such a formal demonstration serves two purposes. First, it separates the individual contribution of the tool from that of the earth. Second, it places this logging device within the mathematical context of signal analysis. The special case for which a closed‐form solution is known is that where all resistivities are equal. Fourier analysis applied to this solution leads to a product of two functions, of which one is identified as the contribution of the earth and the other of the tool.

On the Stability of Waves in a Thin Orthotropic Spinning Disk

1982 ◽  
Vol 49 (3) ◽  
pp. 570-572 ◽  
Author(s):  
J. L. Nowinski
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elementary Functions ◽  
Wave Stability ◽  
Spinning Disk ◽  
Perturbation Parameters ◽  
The Stability ◽  
Temporal Perturbation ◽  
Stability Of Waves

A system of two ordinary coupled differential equations with periodic coefficients of the Mathieu type for two temporal perturbation parameters is derived. A closed-form solution of the system in terms of elementary functions is found and discussed. A condition for the wave stability involving the coefficients of anisotropy is established. Illustration involves a specific range of these coefficients.

Diffraction by an infinite set of soft/hard parallel half-planes: the Riemann approach

1982 ◽  
Vol 60 (8) ◽  
pp. 1125-1138 ◽  
Author(s):  
E. Lüneburg
Keyword(s):  
Plane Wave ◽  
Boundary Value Problem ◽  
Closed Form ◽  
Closed Form Solution ◽  
Normal Derivative ◽  
Form Solution ◽  
The Other ◽  
Bragg Angle ◽  
Total Field ◽  
Infinite Set

We consider the diffraction of a plane wave by an infinite set of parallel equidistant half-planes. On each plate the total field vanishes on one side and the normal derivative vanishes on the other side. A closed-form solution for Bragg angle incidence is obtained by reducing the boundary value problem to a solvable Riemann problem.

Sign in / Sign up

Export Citation Format

Share Document