Dokukova N.A., Kaftaikina E.N.
Belarusian state university, Belarus
Dynamics problem research of the longitudinal-radial
oscillations of the ring
The problem of the dynamics of longitudinal-radial oscillations of the
ring is researched. Complex stress-strain state of the annular member ds will consist of the twisting of the cross
section by the angle x4(j, t), longitudinal extensions u(j, t), the
bending that occurs in the plane of the ring with an offset cross-section by
the x3(j, t)angle its center of gravity of the
plane from the ring and tension-compression in the plane of the center line in
the direction of the polar radius v(j, t). Under the ring meant inextensible
circular beam of constant cross section with a radius R. There are
the bending, torque, lateral and longitudinal forces in the cross section of
the circular timber (Figure 1) [1]. During the
motion position of the cross-section of the ring is characterized by the offset x3
of the gravity center of the ring plane and by the
rotation angle of the section x4
(Figure 2).
|
Figure 1 - The beam segment with length ds |
Complex
dynamic equilibrium of the element ds of
the ring with all the force factors, shown in Figure 1, and leads to a system
of four partial differential equations, the first two of which are different
from those already known [2] a missed inertial term in the second equation. |
, (1)
, (2)
, (3)
. (4)
Figure 2 - Cross-section
of the ring and scheme of the ring deformation
The two
last equations (3) and (4) describe the longitudinal vibrations and transverse
shear in the plane of the ring. General equations system of the stress-strain
state dynamics of the ring is a system of mutually dependent partial
differential equations, free oscillations integration of the first of them
presents certain difficulties. For future studies will use the standard methods
of solution, developed in [3], the latter system of the longitudinally radial
oscillations of the ring (3), (4) with a special non-uniform right side
interpreted distributed along the longitudinal axis by the internal pressure:
, 
, (5)
where 
(ñ-1), a - acceleration value (m/s2), acting on the outer ring as the load in
the form of a standing wave, referred to the
mass 
of the considered infinitesimal element ds, n 
N , g - frequency (rad / s).
Using the initial conditions at the following form:
, 
, (6)
homogeneous
boundary conditions and periodic
. (7)
Here 
is a constant initial
elongation (m). Then the complete solution of the
inhomogeneous differential equation will be the following:
. (8)
The
second equation integration of the system (1)-(4) is allowed to determine the
unknown displacement v(j,t) in the radial direction with
homogeneous initial conditions
, 
, (9)
and
boundary conditions
, (10)
, 
, (11)
. (12)
The
changing of the ring radius caused by the deformation in the radial and
longitudinal directions, can be determined on the basis of formulas (8) and
(12)
(13)
Setting specific parameters r =0.015 m, R =0.15 m, m0=6.9 kg/m, F
- cross-sectional area of the ring (m2), r=7860 kg/m3,
E=2.1·1011 N/m2, G=8.0·1010
N/m2, J1 = 2·10-5 m4, Jê = 7.95·10-8 m4,
we can visually interpret longitudinal radial strain
state of the ring (Figures 3,5-7).
Figure 3 shows the longitudinal-radial strain state ring for n = 1, corresponding to the
first form of the oscillations (Fig. 3 a). This view is consistent with the
results obtained in [2]. In addition, when the speed of the load close to the
velocity of the traveling wave in the material become apparent deflection of
the elastic line study rods beam constructions, disks [2] (Figure 4).
|
a -view
of the oscillations in the plane of the ring at t = |
b – general
view of natural and forced vibrations in the plane of the ring at t=0.8 s D1R=0.025cosj×(2cos(34459×t)-1)-0.1497× ×10-5cosj×(5cos(34459×t)-34459 sin(5t)) |
|
c - the
spatial view at t = 0.8 s Figure 3 - Longitudinal radial strain state ring when n = 1, q1=0.025m, a =120m/s2,
g=5rad/s |
|
In our
example, the speed of the load is 
, the speed of the traveling wave in the
steel is 5 m / s. Obviously, the deflection of the material is clearly visible
(Figure 3 b, c). This is consistent with experimental studies of the critical
speed of rolling tire vehicle. [2] If the rolling velocity less than critical,
the strain is localized in the area of tire contact with the road, if it’s above
than the critical speed, then waves are formed on the side surface (Figure 4).
Figure 4 - Experimental study of the critical velocity of rolling car
tire [2]
|
a- view of the oscillations in
the plane of the ring at t = |
b - general view of
natural and forced vibrations in the plane of the ring at t = 0.2 s D1R=0.008cos2j(2cos(68918×t)-1)- -0.7798×10-6×cos2j(5cos(68918t)-68918 sin(5t)) |
||
|
c- the spatial view
for t = 0.2s Figure
5 - Longitudinal radial strain state ring when n = 2, q1=0.004 m, a =250m/s2,
g = 5 rad/s |
|||
|
a-view of the oscillations in the plane of the ring at t= |
b - general view
of natural and forced vibrations in the plane of the ring at t = 0.3 s,D1R=0.006cos(3j)× ×(2cos(103380×t)-1)-0.27726×106×cos(3j)(5cos(103380×t)-103380sin(5t) |
||
|
c- the spatial view
for t = 0.3 s Figure
6 - Longitudinal radial strain state ring when n=3,q1=0.002 m, a=200m/s2,g=5rad/s |
|||
|
a-view
of the oscillations in the plane of the ring
at t = |
b-
general view of natural and forced vibrations in the plane of the ring at t = 0.0005s D1R=0.0024cos(12j)×(2cos(413510t)-1)-0.351×10-6× × cos(12j)(5cos(413510t)-413510 sin(5t)) |
||
|
c-
the spatial view for t = 0.0005 s Figure
7-Longitudinal radial strain state ring when n=12,q1=0.0002m,a =4050 m/s2,
g = 5 rad/s |
|||
The presented
numerical and analytical calculations can be argued that the
longitudinal-radial deformation depends strongly on the number of their own form.
Free vibration form are crucial in comparison with forced at high frequencies,
where n
. The elastic line deflections
of the ring at a lower load modes (internal axial pressures) become apparent at
low frequencies, with increasing load speed to the speed of propagation of a
traveling wave in the material.
Images of the natural forms in the ring plane in Figures 3a, 5a, 6a for n = 1, 2, 3 fully correspond
to the forms of motion, presented in [2].
References
1. Dokukova,
N.A. One method for solving the dynamics of bending and twisting of the ring./N.
A. Dokukova, E.N. Kaftaikina, P.N. Konon// The modern science formation -2011:
The material for the 7 international scientific and practical conference,
Praha, 27 September-6 October 2011,- Praha: Publishing House «Education and
Science» s.r.o., 2011.-. T. 11. - P. 3-9.
2. V.L Biderman.
The theory of mechanical vibrations.-M.: Vyssh.shkola.1980. 408 p.
3. A.
N. Tikhonov, A. A. Samarskii Tikhonov, Samarsky. Equations of mathematical
physics. Macmillan,
1963 - 765 page