Designing extensometric resistors for measuring mechanical quantities requires solving multiparametric problems that account for the geometric form and ...
The maximum pressure at the base x3 = 0 is calculated from the formula
o'83(0, (-)O, 0 ) = 0.230887. o'33 This equation determines the maximum pressure of the plate on the elastic halfspace. LITERATURE CITED I. 2. 3.
I. N. Vekua, "Theory of thin sloping shells of variable thickness," Tr. Tbilis. Mat. Inst., 30, 5-103 (1965). I. N. Vekua, Some General Methods of Constructing Various Versions of Shell Theory [in Russian], Nauka, Moscow (1982). N. N. Lebedev, Special Functions and Their Applications [in Russian], Gostekhteorizdat, Moscow (1953).
SELECTING OPTIMAL GEOMETRIC DIMENSIONS OF ELASTIC ELEMENTS FOR EXTENSOMETRIC RESISTORS G. F. Stolyar and A. S. Skaliukh
UDC 531.787.087.92
Designing extensometric resistors for measuring mechanical quantities requires solving multiparametric problems that account for the geometric form and structure of elastic elements and their loading characteristics. Practically, the geometric dimensions or the physical parameters of elastic extensometers must be determined under the deformation conditions expected for the extensometric resistor. Selecting optimal geometric dimensions is done primarily with a mathematical model through numerical checking over a wide range of variation in the parameters. Preparation of experimental samples of extensometers is costly, bur theoretical calculations allow us to select optimal dimensions and greatly reduce material expenditures on these samples. We will present an algorithm for numerically solving the related integral problem and we will apply one of the Laplace transformation methods [7]: to finding the optimal geometric dimensions of a solid, circular, elastic element, with extensometric resistors glued on its lateral surface, that is acted on by a nonstationary impulsive force under extreme deformation conditions. We will consider the following model of an extensometer: a solid, circular cylinder with an extensometric resistor mounted on its lateral surface (Fig. i). The cylinder is made of an isotropic elastic material with the following elastic parameters: a shear modulus G, a Poisson coefficient v, and a volume of V = {0 ~ r ~ . ~ b ; 0 ~ 27; I Z I ~ h}. The extensometric resistor has a size that is much less than that of the elastic element, but, to preserve symmetry, we will assume that the extensometric resistor is glued along the entire circumference of the element's lateral surface. It is assumed that the extensometric resistor along with the glue comprise an elastic, homogeneous, hollow cylinder with a height of 2h~ and a thickness of 6. This simplification leads to slight overestimation of the maximum deformation of the extensometric resistor, but, because the thickness 6 is small and the Young modulus El is two orders of magnitude less than the shear modulus of the cylinder G, we can conclude that the obtained estimates differ little from the true estimates. The extensometer is subjected to a nonstationary impulsive force at its end section. It is necessary to find the levels of vertical displacement at the edge of the extensometric resistor and the increase in the middle Circumference; These quantities play the greatest role in design of extensometers. One should note that these quantities cannot exceed critical values beyond which the extensometric resistor would be destroyed. Mathematically, this means that we must find the solution to the equation of motion of an elastic isotropic medium for the axially symmetric case (U~ = 0, 8/8 9 = 0) [7]. Novocherkassk. Translated from Prikladnaya Mekhanika, Vol. 24, No. 12, pp. 24-30, December, 1988. Original article submitted September 29, 1986. 1160
0038-5298/88/2412-1160512.50
9 1989 Plenum Publishing Corporation
zr r
Fig. 1
t,
1-9~
UN-e-+FF\-; -
1 1--~
+~-+
O2U~ Oraz
2(l--v) I--v
dz \ Or
+
1 - - 2v
+ aoU~ +
a2U, Oz2
p O~U, . O at 2 '
p O
a:U, Ot2 '
and the equation of motion for the extensometric resistor [lJ is the following, where it is assumed that only axial deformation occurs (as is known, the extensometric resistor has an effect only along the axis OZ)
OW r
l
~h
~2Vr
(2)
with the boundary conditions
(r~,=P~(Z,O r=b, cr,==O (rn=O 1
r=b,
]Z ~
