scholarly journals I. Normal Frequencies of a One‐Dimensional Crystal. II. An Approximation to the Lattice Frequency Distribution in Isotropic Solids

1951 ◽  
Vol 19 (11) ◽  
pp. 1375-1379 ◽  
Author(s):  
J. O. Halford
Keyword(s):  
Frequency Distribution ◽  
Lattice Frequency ◽  
One Dimensional ◽  
Isotropic Solids ◽  
Dimensional Crystal

Using Tolerance-Maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-Hole Assemblies

2007 ◽  
Vol 7 (4) ◽  
pp. 347-359 ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Statistical Method ◽  
Frequency Distribution ◽  
Probabilistic Representation ◽  
Worst Case ◽  
Frequency Distributions ◽  
One Dimensional ◽  
Material Condition ◽  
Geometric Tolerances ◽  
Case Basis

A new mathematical model for representing the geometric variations of lines is extended to include probabilistic representations of one-dimensional (1D) clearance, which arise from positional variations of the axis of a hole, the size of the hole, and a pin-hole assembly. The model is compatible with the ASME/ ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242), a hypothetical volume of points that models the 3D variations in location and orientation for a segment of a line (the axis), which can arise from tolerances on size, position, orientation, and form. Here, it is extended to model the increases in yield that occur when maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis; the statistical method includes the specification of both size and position tolerances on a feature. The frequency distribution of 1D clearance is decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is built from geometric bias, and it is presumed that manufacturing bias is uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled pin and hole. A comparison is made between the effects of choosing the optional material condition MMC and not choosing it with the tolerances that determine the allowable variations in position.

Surface states of a deformed one-dimensional crystal

Physica ◽  
1966 ◽  
Vol 32 (7) ◽  
pp. 1274-1282 ◽  
Author(s):  
Maria Stȩślicka ◽  
K.F. Wojciechowski
Keyword(s):  
Surface States ◽  
One Dimensional ◽  
Dimensional Crystal

A One Dimensional Crystal With Nearest Neighbors Coupled Through Their Velocities

1981 ◽  
Vol 103 (3) ◽  
pp. 293-296 ◽  
Author(s):  
J. N. Boyd ◽  
P. N. Raychowdhury
Keyword(s):  
Natural Frequencies ◽  
Nearest Neighbors ◽  
Projection Operators ◽  
Circular Array ◽  
One Dimensional ◽  
Neighboring Point ◽  
Large Numbers ◽  
Kinetic And Potential Energies ◽  
Dimensional Crystal ◽  
Number Of Particles

We consider a circular array of point masses connected by springs of non-negligible mass. In the Lagrangian for the harmonic motions of this system, the movements of neighboring point masses are coupled through both the kinetic and potential energies. By use of transformations derived from the theory of projection operators, we simplify the Lagrangian and obtain the natural frequencies of the motions of the system as functions of the number of particles present. We note that for large numbers of particles, the results for our circular array will yield the frequencies of a one dimensional crystal.

A new solvable one-dimensional crystal model

1983 ◽  
Vol 16 (18) ◽  
pp. L771-L775 ◽  
Author(s):  
L Wille ◽  
H Verschelde ◽  
P Phariseau
Keyword(s):  
One Dimensional ◽  
Crystal Model ◽  
Dimensional Crystal
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