ISSN 1068798X, Russian Engineering Research, 2011, Vol. 31, No. 12, pp. 1190–1193. © Allerton Press, Inc., 2011. Original Russian Text © A.P. Sergiev, V.A. Achkasov, A.S. Dolgikh, A.I. Es’kov, 2011, published in Vestnik Mashinostroeniya, 2011, No. 12, pp. 26–29.

Design Principles for Elastic Elements of Vibrational Supports A. P. Sergieva, V. A. Achkasova, A. S. Dolgikhb, and A. I. Es’kovc a

Staryi Oskol Technological Institute, Moscow Institute of Steel and Alloys, Staryi Oskol b OOO DALLAO Proekt cOOO ManakAvto, Staryi Oskol email: [email protected]

Abstract—A threedimensional system performing oscillations according to the conicalpendulum law is considered. A means of calculating elastic elements that will ensure stable oscillations of the system is pro posed. DOI: 10.3103/S1068798X11120240

Threedimensional pendulumtype vibrational systems (shown in the figure) are effective in the fin ishing of small parts. The most promising means of improving such systems with oscillation by the coni calpendulum law is to use flexible links in an elastic suspension system, so as to intensify the machining process and eliminate the undesirable dynamic influ ence of the gyroscopic torque. Model research on a threedimensional pendu lumtype vibrational system shows that oscillation by the conicalpendulum law requires the following ratio of the spring’s transverse and longitudinal rigidity [1]

O2

Ky

(1)

Kx

Y 1 O h1

In addition, stability and longevity of the spring must be ensured when the vibrational system operates beyond resonance. Existing methods of spring calcu lation do not permit the determination of structural parameters such that the required conditions are satis fied [2]. Therefore, we need to develop a new method of calculating the elastic elements of threedimen sional pendulumtype vibrational systems. The eigenfrequency ω0 of the system must be con siderably less than the working frequency ωw, and the system must operate in a stable zone beyond reso nance, where the amplitude of the oscillations is prac tically independent of their damping. The dynamic coefficient of the system is as follows, according to [3]

X

O1

2

h

K y = 2K x .

reduced mass of the system; Kc is the total rigidity of the elastic shock absorbers (springs); b is the resistance of the medium. Obviously, at constant mass mc, the system passes through resonance less vigorously with decrease in the total rigidity Kc of the springs and with greater dissipation (greater b).

mv ε 3 4

5

1 β = . ω 2 1 – ⎛ 0 ⎞ ⎝ ω w⎠

ω

At a working frequency ωw ≥ 5ω0, we find that β = 1.02. In other words, the amplification of the oscilla tions is no more than 2%. The dependence of the dynamic coefficient on the system parameters was established in [4] β = m c K c /b, where mc is the 1190

Threedimensional pendulumtype vibrational system: (1) working container; (2) elastic element; (3) unbalanced vibrator; (4) shaft; (5) flexible drive; O, coordinate origin; O1, system’s center of mass; O2, theoretical mass of conical pendulum’s suspension; mv and ε, mass and eccentricity of unbalanced vibrator; h and h1, distance from level of elastic suspension to center of mass and to point of application of the perturbing force, respectively.

DESIGN PRINCIPLES FOR ELASTIC ELEMENTS OF VIBRATIONAL SUPPORTS

The eigenfrequency of the oscillating system may be expressed as ω0 =

K c /m c .

(2)

The reduced mass of the system is m c = m el + k ad m w , where mel is the total mass of the oscillating elements in the system; kad is the addedload coefficient (kad = 1 is only assumed to find the static shrinkage of the springs and the eigenfrequency of the system in the preresonance zone); mw is the mass of the working load (components and filler). The static shrinkage of the spring under the action of the reduced load mass is δ st = m c /K c .

(3)

The mass mw of the working load has little influence on δst. In other words, it must be approximately equal to the mass of all the oscillating elements in the sys tem: mw ≤ mel. If we select springs of rigidity corresponding to Eq. (2), the condition ωw > 5ω0 will be satisfied both in operation and when idling, if mel ≈ mw. In other words, the system will operate stably beyond resonance, with β ≈ 1. In addition, correct choice of the static flexure δst.sp under the action of mw determines the position (h in the figure) of the vibrator shaft in the loaded and unloaded system. With considerable flexure δst.sp, mutual bonding of the turns of the spring is possible during system operation. In the given model, the total longitudinal rigidity Kx of the springs consists of the equal rigidities of all eight springs in the system. The longitudinal rigidity of each spring is K x = K c /8,

(4)

parasitic oscillations. The correctly chosen spring should have sufficient margin of stability. The spring will remain stable if its margin of stability satisfies the condition η = H0/D ≤ 2.5, according to [2]. Here H0 is the height of the undeformed spring. For cylindrical spring with wire of round cross sec tion, the stability condition takes the form [5] 4πGJ p i max = 1.13  . 2 H mg Here Jp is the polar moment of inertia of the wire cross section and H = H0 – δst = it0 – δst or H = it

D C spr =  ≤ 8. d

12EJ p K y =  , 3 Hχ

4

(6)

where G is the shear modulus; d is the wire diameter; i is the number of working turns; D is the mean diameter of the turns. To minimize the transmission of the oscillations to the system’s base, the spring rigidity must be a mini mum. With increase in the number of turns, the rigid ity of the spring will be reduced. However, the rigidity of a spring operating in compression cannot be infi nitely reduced by increasing the number of turns for a specified oscillating mass, since the spring becomes unstable with some number imax of turns. In addition, with a large number of turns, the spring will have a very large eigenfrequency spectrum, which may give rise to Vol. 31

(9)

where E is the elastic modulus; the coefficient χ depends on the inclination α of the deformed spring and Poisson’s ratio μ 2

The longitudinal rigidity of the cylindrical helical spring is [3]

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(8)

In operation, the springs undergo preliminary static compression. Therefore, in calculating the transverse rigidity, the height of the deformed spring must be taken into account. To determine the spring rigidity perpen dicular to its axis, we use an approximate formula writ ten for an integer number of turns i [5]

2 + μ cos α χ = . 2 sin α

(5)

Gd K x = 3 , 8iD

(7)

is the height of the deformed spring; t0 and t are the pitch values of the undeformed and deformed spring, respectively. Another condition of spring stability is a constraint on the compression index [6]

and each spring experiences the reduced mass m = m c /8.

1191

(10)

To ensure long life of the springs with an alternating dynamic load, it is expedient to use lowrigidity springs with i = 5–6, in which the rigidity is reduced by increas ing the spring diameter [4]. Low rigidity results in con siderable static shrinkage of the springs (50–70 mm), whereas the working amplitude of the forced container oscillations rarely exceeds 5 mm. In other words, the spring deformation under the action of the dynamic load is 10–15 times less than the static flexure. This means that, in calculating the strength of elastic ele ments, we may focus, with acceptable loss of accuracy, on static loads and ensure practically infinite spring life. Otherwise, spring fracture due to fatigue will set in after operation for 100–120 h. The permissible stress is

No. 12

τ = 1.2 8mgD   ≤ 300 MPa. 3 πd 2011

(11)

1192

SERGIEV et al.

We now substitute Eqs. (6) and (9) into Eq. (1), as well as the value of i found from Eq. (7). (To this end, we preliminarily determine t from the formula tanα = t/(πD), where α is the spring’s inclination.) Then, tak ing into account that the polar moment of inertia for a round cross section is Jp = πd4/64, we obtain 3E H = D  . 4Gχ tan α

(12)

On the other hand, we know that H = ηD – δ st .

(13)

Taking account of Eqs. (4)–(7) and the value of tan α, we find from Eq. (3) that 3

8mgHD δ st =  . 4 πGd tan α

(14)

Selecting Cspr and substituting d and H from Eqs. (8) and (12), respectively, into Eq. (14), we finally obtain 4

8mgG spr 0.75E δ st =   . πGD tan α Gχ tan α

(15)

Then, taking account of Eqs. (17) and (14), we write Eq. (15) in the form 4

8mgG spr 0.75E 0.75E  = ηD –  D   . πG tan α Gχ tan α Gχ tan α

(16)

Equation (16) depends on two unknowns: D and α. On that basis, we may determine all the parameters of the spring. The spring’s diameter D is determined from Eq. (11) by the substitution of d from Eq. (8). After the substitution of D, η, and Cspr, we only have one unknown α in Eq. (16). In practice, its solution relies on certain assumptions. Since the spring’s inclination in the system is small, we may assume that sinα = tanα and cosα = 1. Then Eq. (10) takes the form 2 + μ . χ =  2 tan α

(17)

Substituting Eq. (17) into Eq. (16), we obtain 4

8mgG spr 1.5E  = ηD –  1.5E . D    G(2 + μ) πG tan α G ( 2 + μ ) Hence 4 1.5E 8mgC spr  G ( 2 + μ) tan α =  . 2⎛ 1.5E ⎞ πGD η –  ⎝ G ( 2 + μ )⎠

Having determined tanα, we find t = πDtanα, and than we may determine χ, H, δst, i, and H0 from Eqs. (17), (13), and (15). Finally, from Eq. (7), we find t0 and we can then calculate α0 = arctant0/(πD).

This method of calculating the elastic elements in vibrational systems permits the selection of the dimen sions of cylindrical helical springs characterized by sufficient stability, optimal rigidity, and minimal trans mission of the oscillations to the system’s base. Having calculated the rigidity of the elastic ele ments in the working springs, we may determine the oscillation amplitude of the system’s base, which rests on shock absorbers [5]. Assuming that the spring deformation is equal to the working amplitude Aw of the container’s oscillations, we obtain the force acting through the elastic elements on the supporting plate F1 = Kc Aw .

(18)

On the other hand, the oscillations of the support ing plate on shock absorbers creates a force on the shock absorbers 2

F 2 = Wm su = A su ω w m su ,

(19)

where W is the vibrational acceleration; msu and Asu are, respectively, the mass and oscillation amplitude of the supporting plate. Equating Eqs. (18) and (19), we obtain the oscilla tion amplitude of the supporting plate Aw Kc A su =  . 2 m su ω w

(20)

It is evident from Eq. (20) that the oscillation amplitude of the supporting plate is directly propor tional to the working amplitude Aw and the total spring rigidity Kc and inversely proportional to the mass msu of the supporting plate and the square of the working frequency ωw. Hence, as the base becomes more mas sive and the spring becomes less rigid, the transmission of oscillations to the shock absorbers will be less. Tak ing account of the damping of the oscillations in rub ber shock absorbers (30–40%), the oscillation ampli tude of the base is A su K sh A ba = C dam  , 2 m ba ω w where Cdam = 0.6–0.7 is the damping coefficient of the oscillations in the shock absorbers; Kam is the rigidity of the rubber shock absorbers; mba is the mass of the base. The selected rigidity of the shock absorbers is such that the eigenfrequency of the base on the shock absorbers is several times less than the working fre quency. The damping coefficient Cdam of the oscilla tions in the shock absorbers depends on the structure. If the vibrational system is mounted in a production building on a onepiece base, then mba Ⰷ mc and Aba 0. The proposed method has been used in the design of the VPM50, VPM100, and VPM200 industrial

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DESIGN PRINCIPLES FOR ELASTIC ELEMENTS OF VIBRATIONAL SUPPORTS

vibrational machines. Tests show that these machines ensure optimal vibrational machining. 3.

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4.

5.

6.

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Elements in Machines and Instruments), Moscow: Mashinostroenie, 1980. Den Hartog, J.P., Mechanical Vibrations, New York: McGrawHill, 1934. Sergiev, A.P., Calculation of the Elastic Elements in Vibrational Systems from Specified Technological Param eters, Proizv.Tekhn. Byull., 1963, no. 5, pp. 33–37. Nikolenko, G.I., Theory of ShockAbsorber Calcula tion for Vibrational Machines, Cand. Sci. Dissertation, Moscow, 1954. Astaf’ev, V.D., Spravochnik po raschetu tsilindricheskikh vintovykh pruzhin szhatiya–rastyazheniya (Handbook on the Calculation of Cylindrical Helical Compres sion–Tension Screws), Moscow: Mashgiz, 1960.

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