INVESTIGATION OF NATURAL VIBRATIONS OF NEEDLES EMBEDDED IN AN ELASTIC BASE Текст научной статьи по специальности «Физика»

UDC 534-16

Varlamova A.V. Investigation of natural vibrations of needles

embedded in an elastic base

Varlamova Alla

Ph. D, Associate Professor of the Department of Mechanical Engineering Technology of the Dimitrovgrad Engineering and Technological Institute of the National Research Nuclear University MEPhI

Abstract. In this paper, small oscillations of needles on an elastic base are considered, taking into account dissipation in the case of a suddenly applied force, an equation of motion and its solution are proposed, the dynamism coefficient and its maxima are determined.

Keywords: oscillations, equation of motion, needle, base, dynamism coefficient.

DOI 10.54092/25421085_2021_11 _145

Рецензент: Сагитов Рамиль Фаргатович, кандидат технических наук, доцент, заместитель директора по научной работе в ООО «Научно-исследовательский и проектный

институт экологических проблем», г. Оренбург

1. Introduction.

The aim of this work is to study the vibrations of needles embedded in an elastic base. To determine the dynamic coefficients in the first approximation, we represent the needle in the form of an elastic non-inertial beam, rigidly fixed at one end, carrying a concentrated mass at the point of application of the shock load.

2. Materials and methods

The equation of small oscillations of such a system, taking into account the dissipation in the case of a suddenно applied force, will be as follows [1]:

m ■ y + k ■ y + c ■ y = PH (1)

or

7 P

y + 2■ n ■y + p ■y = — (2)

m

where m is the mass of the needle brought to the point of impact;

k is the damping coefficient associated with the stiffness coefficient by the relation:

K — ¥

2np (3)

n = — ; p = - - natural vibration frequency

2m V m

Absorption coefficient y we assume approximately 0.62. The solution of equation (1) has the form:

p

y = em (Cj cos pt + C2 sin pt) +—

mp (4)

The first two equations describe damped natural oscillations. Differentiate (4):

y = ce"' (-p sin(pt) - n cos(pt)) + c2e~nt (h cos(pt) - n sin(pt)) (5)

To define arbitrary constants, we have the initial conditions for t=0 y=0, y = 0, whence

P n P

C\ — ^ ,c — "

Then (5) is written as:

P

y — 2

mp

2 ' 2 2 mp p mp

n

1 - e~nt co s (pt) +—s in(pt) p

(6)

P

It is easy to see that-^ -represents a static movement of mass, hence

mp

n

Kmax- = 1 + e""(cos(pt) ^sin(pt)) (7)

Its successive maxima are determined at the following time points t =l—, i = 1,2,3..:

k = 1 + e p

kdr max = 1 + e

Taking into account the introduced notation and dependence (3), the maximum values

of the dynamic coefficient are calculated using the formula:

y

kdr max = 1 + e4 (8)

n

—in

Since the time spent by the needles in the impact zone t =(12 ... 60) s is large compared to the period of natural vibrations, the termination of this force does not differ from the removal of static load during static deflection of the needle. Therefore, dynamic phenomena are not of interest at this point.

Let us now proceed to the determination of the dynamic coefficient from the impact on the need^ .

The force pulse from their impact on the needle:

To

5 = Py J f (t)dt, (9)

0

where f(t) is the pulse shape, t0 -time of impact.

We assume f(t)=1 -a rectangular pulse, so

* = S (10)

static deflection of the needle under the action of force p :

p p

y=p=A (11)

C mp

Substituting (10) and (11) in the expression for dynamic deflection, we obtain:

J, = Kdy = A, (12)

mp

where S0 = KdS (13)

PT0

The homogeneous differential equation of vibrations is represented in the form:

y + 2ny + p2y = 0 (14)

Its solution, which looks like:

y = e~nt (cj cos (pt) + c2sin(pt)), (15)

it is a free oscillation with attenuation. Expression (15) under the action on the needle of a certain mass mbmoving with a linear velocity V, that is, the force pulse S=mbv, must satisfy the initial conditions for t=0 y=0, mby = s . Given them, we find arbitrary constants:

C = 0; c2 =

mp

Now (15) takes the form:

s k p T

y = _<L e- sin(pt) = e-" sin(pt) = kdyce"' sin(pt), (16)

mp pT0mp

where does the dynamic coefficient come from?

kdy = kder"' sin(p0

(17)

As before, to find its maximum, we equate the first derivative (17) to zero

kdy = kde~nt ( p cos(pt) - n sin(pt)) = 0 (18)

This equality is possible when

p cos(pt)-n sin(pt) = 0 (19)

3. Results and Discussion

Let's set aside a vector of value p from the origin pby the angle pt from the

abscissa axis in a counterclockwise direction, then a vector of value n it will be positioned at a right angle in the clockwise direction from the first one. The resulting vector is a projection

on the x-axis of the vector modulo = Tp2^" and composing the angle with the a-b-sciss axis

( r w

n

- pt + arctg —

I p jj

. As a result, the latter equation can be represented in the equivalent form

p2 + n2 cos^- pt + arctg—j = 0 (20)

-(2j -1)n + arctgn Its solution t —-2-p , j=1,2,....

n n 3n n ---+ arctg— ---+ arctg—

There are two roots in the first period ^ =—2-p , t2 =—2-p

Substitute them in the expression

K

dy

■Kde [-sinpt(p2 - n2)-2npcospt\

(21)

Since p>n, it should be noted that k^ < o when tl.

4 j - 3 n —--n + arctg—

Therefore, when t = —--p the dynamic coefficient has maxima .

p

Transforming (21), we arrive at the formula for the sequence of maxima of the dynamic coefficient when hitting the needle:

K

dy max?'

1 + e ■

V

4 j-3 n

—l---n+arcfc- !

— \ V 2_pi (

sin

- 4 j + 3 n

-n + arctg—

2 p

^

or

K

dy max?'

. 4 i-3 —

. wl--n=arctg—

— \ V 2 4n

1 = e 4

, 4 j - 3 —

sinl--n + arctg—

1 2 4n

In the case of consecutive (K+1) impacts that occur in the case under consideration ,

the dynamic coefficient can be

can be obtained by superposing functions (5.39) with

T P

different reference points j. If it is equal to an integer, a pulse resonance occurs:

K

dy max

1 + e 4 ^ „ 4n\ 2 gg 4n 0

I'

sinl -

V /1

4j - 3

n + arctg —-r0TP )

4n

The highest of the highs will be in the lastpeepod. Its value is greaterthe closer the ratio

TP

of the shock period to the period of natural vibrations is to 1.

e

2

References

1. Pisarenko, G. S. Matchmaker on the resistance of materials [Text]: textbook.posobie / G. S.Pisarenko [et al.]. - Kiev.:Naukova dumka,1988-736 p..

2. https://www.dissercat.com/content/razrabotka-i-issledovanie-igolchatoi-garnitury-chesalnykh-mashin

3. Varlamova A.V. On the determination of the bedding coefficients of the base of needle sets. [Text]:/ Innovations in mechanical engineering. Collection of materials.- Penza, 2001-p.72-75.