Natural oscillations of single span beam placed on cylindrical supports/Ant cilindriniu atramu padetos dviatrames sijos savieji virpesiai.

Kargaudas, V. ; Adamukaitis, N. ; Zmuida, M. 等

1. Introduction

Oscillations of a beam supported by a motionless hinges is a classical problem and have universally accepted solution. But in some cases beam can be simply placed on a support surface and line of contact will change its location. We assume the circular cylindrical shape of supports (Fig. 1). When radius r is small such support can be approximately replaced by hinge, but if r is much more than length of the beam, variation of the contact line can significantly influence oscillations of the beam. When alternation of the contact line position is significant and when it can be ignored is investigated in this paper.

It is proved that nonlinear mechanical systems with an analytical first integral allow periodic solutions which tend towards linear normal vibration modes as amplitudes tend to zero. Mechanical systems with soft nonlinearity are comprehensively covered by Kauderer [1]. Nonlinear statics and dynamics of a beam sliding on two knife edge supports is investigated by Somnay, Ibrahim [2] Somnay, Ibrahim, Banasic [3]. To simplify the dynamic modelling the mass is concentrated at the center of the beam and exact solution is given in terms of elliptic functions. Dynamic formulation for sliding beams that are deployed or retrieved through prismatic joint are presented by Vu-Quoc, Li [4]. The channel orifice is moving toward the beam, or beam can be sliding when joint is motionless. Geometrically similar problem is investigated by Turnbull, Perkin, Schulch [5], where a beam or string is contacting a circular surface of radius r. Behavior of frictional contact support of a vibrating beam is studied by Ahmadian, Jalali, Pourahmadian [6]. The frictional shear force at the support is identified using its nonlinear normal modes. The nonlinear normal vibration modes are discussed by Mikhlin [7]. Zajaczkowski, Lipinski, Yamada [8], [9] investigated stability of Euler-Bernoulli beams subjected to periodic sliding motions. The complex nature of instability is revealed.

In this paper damping is neglected. Dependence of velocity on deflections in a phase plane and dependence of deflection on time are investigated.

2. Equation of motion

The first mode of oscillating beam usually is of fundamental importance. If the line of a beam coincides with the sine curve deflections of the beam can be determined

u(x,t) + [u.sub.A] = [[u.sub.C](t) + [u.sub.A]]cos[[pi]x/l], (1)

where [u.sub.C](t) and [u.sub.A] are deflections of the midpoint C and tangent point A (Fig. 1). Positive displacement of A is assumed to be down: [u.sub.A] = r(1 - cos[[phi].sub.A]). If positive angle of the sine curve is clockwise then tan [phi] = -[du/dx] = -[[pi]/x] ([u.sub.C] + [u.sub.A]) sin [[pi]x/l] and tan [[phi].sub.A] = [pi][[u.sub.C] + [u.sub.A]/l]. As distance l = [l.sub.o] + 2r sin [[phi].sub.A] the exact equation is tan [[pi].sub.A] = [pi][[u.sub.c] + r (1 - cos [[phi].sub.A])/[l.sub.o] + 2r sin [[phi].sub.A]]. If deflections are small [absolute value of [[phi].sub.A]] [much less than] 1, then:

1 - cos [[phi].sub.A] [approximately equal to] [[sin.sup.2] [[phi].sub.A]/2] (2)

and sin [[phi].sub.A] [approximately equal to] [pi]q - [[pi].sup.2] [4 - [pi]/2][r/[l.sub.o]][q.sup.2], where q = [u.sub.c]/[l.sub.o]. [l.sub.o] is distance between tangent points in equilibrium position. Dynamic distance between the tangent points:

l = [l.sub.o] (1 + [alpha]q - [4 - [pi]/4] [[alpha].sup.2][q.sup.2]); [alpha] = 2[pi][r/[l.sub.o]] (3)

and dependence of any beam point is expressed from Eq. (1):

u(x, t)/[l.sub.o] = (q + [[pi]/4][[alpha].sup.2][q.sup.2]) cos [[pi]x/l] - [[pi]/4][alpha][q.sup.2]. (4)

Only of the second degree of the relative displacement [q.sup.2] will be taken into account in this investigation.

[FIGURE 1 OMITTED]

Kinetic energy of the beam [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where m is mass of the beam, [l.sub.v]--length of the beam and [l.sub.v] [greater than or equal to] l at any moment. Therefore [l.sub.v] - l = 2[DELTA][l.sub.v] [greater than or equal to] 0, where [DELTA][l.sub.v] is dependent on time cantilever length also assumed to be small. When velocity [??] = du/dt from Eq. (4) is calculated dependence of l on time Eq. (3) also should be considered. Kinetic energy of the cantilevers can be calculated assuming that it is a portion of the sine curve or as an absolutely rigid straight line: outcome within the set accuracy is the same. Kinetic energy of the beam:

T = [[m.sub.o][l.sup.3.sub.o]/2][[q.sup.2]/2] (1 + [c.sub.1][alpha]q - [c.sub.2][[alpha].sup.2][q.sup.2]), (5)

where [m.sub.o] = m/[l.sub.v] is mass per unit length,

[PI] = [pi] - 2 = 1.142, [c.sub.2] = 11 - [[pi]/2] - [53[[pi].sup.2]/60] = 0.7111.

Potential energy of the beam:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where EI is the stiffness and distance l is presented in Eq. (3). The first natural circular frequency of the simply hinged beam [lambda] = [[[pi].sup.2]/[l.sup.2.sub.o]][square root of EI/[m.sub.o]], therefore the potential energy Eq. (6) [PI] = [[m.sub.o][l.sup.3.sub.o]/2] [[lambda].sup.2] [[q.sup.2]/2](1 - 2[[c.sub.3]/3] [alpha]q + [[c.sub.4]/2] [[alpha].sup.2][q.sup.2]), where c = 3[6 - [pi]/4] = 2. 144, [c.sub.4] = [144 - 36[pi] + [[pi].sup.2]/8] = 2.548. Lagrange's equation for the beam is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Mechanical system is conservative in spite of the term, dependent on velocity.

3. Velocities and displacements

If radius of the cylinder r [right arrow] 0 and supports can be assumed as a hinges Eq. (7) is [??]+ [[lambda].sup.2]q = 0 and after substitution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the first integral is [[??].sup.2] + [[lambda].sup.2][q.sup.2] = [q.sup.2.sub.o] or

[[??].sup.2]/[[??].sup.2.sub.o] + [[q.sup.2]/[q.sup.2.sub.s]] = 1, [q.sub.s] = [[[??].sub.o]/[lambda]] = const. (8)

If relative displacement of the beam center point x = [q/[q.sub.s]] = [[lambda]/[[??].sub.o]]q is applied the velocity [dx/dt] = [+ or -][square root of 1 - [x.sup.2]] and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The positive sign presents solution x = sin x.

Nonlinear differential Eq. (7) when q is considered as a function of time can be transformed to the linear equation of [[??].sup.2] as function of variable q. Let independent variable is z = [alpha]q, the function w = [[??].sup.2] and Eq. (7) is multiplied by (1 + 2 [[c.sub.2]/[c.sub.1]]z). Then the equation of beam motion is:

[dw/dz] (1 + [b.sub.1]z + [b.sub.3][z.sup.2]) + [c.sub.1]w + 2[[lambda].sup.2]z (1 - [b.sub.2]z - [b.sub.4][z.sup.2]) = 0, (9)

where [b.sub.1] = [c.sub.1] + 2[c.sub.2]/[c.sub.1], [b.sub.2] = [c.sub.3] - 2[c.sub.2]/[c.sub.1], [b.sub.3] = [c.sub.2], [b.sub.4] = (2[c.sub.2][c.sub.3]/[c.sub.1])- [c.sub.4]. A particular solution of the inhomogeneous equation [w.sub.a] = [B.sub.o] + [B.sub.1]z + [B.sub.2][z.sup.2] + [B.sub.3][z.sup.3].

[[??].sup.2]/[[lambda].sup.2] = 1 - [x.sup.2] - [c.sub.1][a.sub.s]x + [[eta].sub.o][a.sub.s][x.sup.3] + [[c.sub.1][[theta].sub.1]/2] [a.sup.2][x.sup.2] - [[eta].sub.1][a.sup.2.sub.s][x.sup.4], (10)

satisfies Eq. (9) if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but the term 3[b.sub.3][B.sub.3] [z.sup.4] is neglected and [[theta].sub.1] = [b.sub.1] + [c.sub.1], [[theta].sub.2] = 2[b.sub.1] + [c.sub.1], [[theta].sub.3] = 3[b.sub.1] + [c.sub.1]. Solution of this linear equations is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Particular solution (10) is expressed [w.sub.a] = [B.sub.o] (1 - [c.sub.1]z) + + 2[[lambda].sup.2][z.sup.2] ([[kappa].sub.2] + [[kappa].sub.3]z). Homogeneous differential Eq. (9) can be expressed as:

dw/dz (1 + [s.sub.1]z)(1 + [s.sub.2]z) + [c.sub.1]w = 0, (11)

where 1 + [b.sub.1] z + [b.sub.3][z.sup.2] = (1 + [s.sub.1]z + [s.sub.2]z), therefore [s.sub.1] and [s.sub.2] are roots of the equation [s.sup.2] - [b.sub.1]s + [b.sub.3] = 0. The exact solution of (11) is [w.sub.b] = [C.sub.o]F(z), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Derivative and the function are related by [c.sub.1]F(z) + (1 + [b.sub.1]z + [b.sub.3][z.sup.2]) F'(z) = 0. By differentiation of this identity derivatives of the higher order can be deduced and Taylor's formula gives F(z) = 1 - [c.sub.1]z + [c.sub.1][[theta].sub.1][z.sup.2]/2 - [c.sub.1][[eta].sub.3][z.sup.3]/6 + [c.sub.1][[eta].sub.4][z.sup.4]/24-- ..., where [[eta].sub.3] = [[theta].sub.1][[theta].sub.2] - 2[b.sub.3]. The general solution w = [C.sub.o]F (z) + [w.sub.a] (z) of the inhomogeneous Eq. (9) satisfies the initial value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. After substitution the general solution and the first integral is:

[[??].sup.2]/[[lambda].sup.2] = 1 - [x.sup.2] - [c.sub.1][a.sub.s]x + [[eta].sub.o][a.sub.s]x + [[c.sub.1][[theta].sub.1]/2][a.sup.2.sub.s] [x.sup.2] - [[eta].sub.1][a.sup.2.sub.s][x.sup.4], (12)

where [a.sub.s] = [alpha][q.sub.s] can be considered as the principal parameter to assess the magnitude of a term. Factors [c.sub.1], [[eta].sub.o] = ([[theta].sub.2] - [2[b.sub.3]/[[theta].sub.1])[1 + 2[[kappa].sub.2]/3] + 2[[kappa].sub.3], [c.sub.1][[theta].sub.1]/2, [[eta].sub.1] = [[[eta].sub.4]/[[theta].sub.1]][1 + 2[[kappa].sub.2]/12] and the variable x = q/[q.sub.s] takes the values approximately from the interval [-1, +1].

If [a.sub.s] = 0 equation coincides with Eq. (8), because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Second in importance are the two terms with factor [a.sub.s], next are two last terms with factor [a.sup.2.sub.s]. Roots of the polynomial:

P(x) = l - [c.sub.1][a.sub.s]x - (1 - [[c.sub.1][[theta].sub.1]/2] [a.sup.2.sub.s]) [x.sup.2] + [[eta].sub.o][a.sub.s][x.sup.3] - [[eta].sub.1][a.sup.2.sub.s][x.sup.4],(13)

depend on parameter [a.sub.s]. When [a.sub.s] = 0 we have quadratic equation and roots [x.sub.1] = -1, [x.sub.2] = +l. If [a.sub.s] = 0.001 the roots of Eq. (13) are [x.sub.1] = -0.993, [x.sub.2] = 1.007, [x.sub.3] = 26.15 + 37.18i, [x.sub.4] = 26.15-37.18i. If [a.sub.s] = 0.2 then [x.sub.1] = -0.889, [x.sub.2] = 1.118, [x.sub.3] = 1.193 + 1.942i, [x.sub.4] = 1.193 - 1.942i. It can be shown that P(x) [greater than or equal to] 0 if and only if [x.sub.1] [less than or equal to] x [less than or equal to] [x.sub.2], so. [x.sub.1] = [x.sub.min], [x.sub.2] = [x.sub.max] and do not exist real velocity solutions in other intervals. In Fig. 2 are depicted harmonic vibration dependence Eq. (8) and two approximations when [a.sub.s] and [a.sup.2.sub.s] are taken into account. The cubic equation P(x) = 0 is to be solved when the first approximation is examined and all three roots are real if [a.sub.s] [less than or equal to] [a.sub.s3] [approximately equal to] 0.1831. Displacement [x.sub.v] is when velocity of the beam [??] = [[??].sub.max] and depends on the parameter [a.sub.s]. All values [x.sub.v] < 0 (Table 1). If upper amplitude [A.sub.+] = [x.sub.v] - [x.sub.min] and lower amplitude [A.sub.-] = [x.sub.v] - [x.sub.min], the ratio [r.sub.A] = [A.sub.+]/[A.sub.-] approaches 1.5 when [a.sub.s] [approximately equal to] 0.15 even though the whole displacement [A.sub.+] + [A.sub.-] = 2[A.sub.o] [approximately equal to] 2 for every as. The displacement average [x.sub.0] = ([x.sub.max] + [x.sub.min])/2 is positive for all [a.sub.s]. When [a.sub.s] [less than or equal to] 0.l, difference between the first and the second approximations is insignificant. If [a.sub.s] [greater than or equal to] l.5, difference between the oscillations being studied and harmonic dependence of velocity on displacement is substantial.

[FIGURE 2 OMITTED]

4. Displacement as function of time

If terms with [a.sup.2.sub.s] are neglected and [a.sub.s] < [a.sub.s3] the polynomial Eq. (13) P(x) = [[eta].sub.o][a.sub.s], ([x.sub.3] - x) ([x.sub.2] - x) (x - [x.sub.1]) where all roots [x.sub.1], [x.sub.2], [x.sub.3] are real. Eq. (12) then can be presented dx/[square root of ([x.sub.3] - x)([x.sub.2] - x)(x - [x.sub.1])] = [square root of [[eta].sub.o][a.sub.s]][lambda]dt. Integration yields [10,11] [square root of [[eta].sub.o][a.sub.s]][lambda]t = 2/[square root of [x.sub.3] - [x.sub.1]] F([[phi].sub.i], [k.sub.i]), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is elliptic integral of the first kind, [sin.sup.2] [[phi].sub.i] = [[delta].sub.2]/[[delta].sub.1] x [[[x.sub.2] - x]/[[x.sub.3] - x]], [k.sup.2.sub.i] = [[delta].sub.1]/[[delta].sub.2], [[delta].sub.1] = [x.sub.2] - [x.sub.1], [[delta].sub.2] = [x.sub.3] - [x.sub.1]. If F([[phi].sub.i], [k.sub.i]) = [square root of [[eta].sub.o][a.sub.s][[delta].sub.2]] [lambda]t/2 [equivalent to] u then the invers function is the Jacobian elliptic sine: sn(u,[k.sub.i]) = sin[[phi].sub.i] = [square root of [[delta].sub.2]/[[delta].sub.1] x [[x.sub.2] - x]/[[x.sub.3] - x]]. The displacement as function of the time is deduced from this equation: x = [[delta].sub.1][x.sub.3][sn.sup.2]u - [[delta].sub.2][x.sub.2]/[[delta].sub.1]s[n.sup.2]u - [[delta].sup.2]. Initial values of the first two roots can be -1, +1, the first iteration step yields [x.sub.1] = -1 + [[[eta].sub.o] - [c.sub.1]/2] [a.sub.s], [x.sub.2] = 1 + [[[eta].sub.o] - [c.sub.1]/2] [a.sub.s]. The third root is approximated [x.sub.3] = l/([[eta].sub.o][a.sub.s]). Complete elliptic integral K = (1 + [[k.sup.2.sub.i]/4]) [[pi]/4] = (1 + [[[eta].sub.o][a.sub.s]/2]) [pi]/2, u = (1 + [[[eta].sub.o][a.sub.s]/2]) [[lambda]t/2], y = [pi]u/2K = [lambda]t/2, therefore the elliptic sine snu = sin y x (1 + 4[q.sub.i] [cos.sup.2] y), where elliptic Jakobi's parameter [q.sub.i] = exp (-[pi] K'/K) [approximately equal to] [1/2] 1 - [square root of [k'.sub.i]]/1 + [square root of [k'.sub.i]] [approximately equal to] [[eta].sub.o][a.sub.s]/2, [k'.sub.i] and K' are complementary module and complete elliptic integral. Substitution to (14) yields the first approximation:

x = [[[eta].sub.o][a.sub.s]/2] [a.sub.s] + cos [lambda]t, (14)

the shifted harmonic oscillations. Applying given above values we have [[eta].sub.o] = 2.532 and [[beta].sub.0] = ([[eta].sub.o] - [c.sub.1])/2 = 0.695, so [x.sub.0] = [[beta].sub.0] perfectly coincides with data given in Table 1 when [a.sub.s] [less than or equal to] 0.02 and coincides satisfactory when [a.sub.s] [less than or equal to] 0.l0. But asymmetry of the real oscillations, displayed by [x.sub.v], [r.sub.A] in Table 1, is not presented in Eq. (14).

The real values of [x.sub.0] < [[beta].sub.0] and solution of the second approximation (n = 4) better corresponds to the theoretical values, presented in Eq. (14).

The second approximation of the polynomial (13) has four roots, two of which are complex numbers: P(x) = [[eta].sub.1][a.sub.2.sub.s]G(x), G(x) = [[([x.sub.r] - x).sup.2] + [x.sup.2.sub.i]]([x.sub.2] - x) (x - [x.sub.1]), where [x.sub.3] = [x.sub.r] + i[x.sub.i], [x.sub.4] = [x.sub.r] - i[x.sub.i]. Eq. (12) can be expressed [12] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [x.sub.3] = [x.sub.r] + i[x.sub.i], [x.sub.4] = [x.sub.r] - i[x.sub.i]. Eq. (12) can be expressed [12] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Solution is:

x = [[[x.sub.2] + [x.sub.1]]/2] - [[[x.sub.2] - [x.sub.1]]/2] [[[v.sub.i] - cnu]/[1 - [v.sub.i]cnu]], (15)

where [v.sub.i] = tan [[[??].sub.2] - [[??].sub.1]/2] tan [[[??].sub.2] + [[??].sub.1]/2], u = [a.sub.s][square root of [u.sub.1]]/[[mu].sub.i]] [[lambda].sub.t] cnu [equivalent to] cn (u, [k.sub.i]) is Jacobian elliptic cosine. Dependence of relative displacement x = q/[q.sub.s] on ratio time t and period [tau], calculated from Eq. (15), is depicted in Fig. 3. The dashed line discloses difference between harmonic oscillation and solution (15).

[FIGURE 3 OMITTED]

Relative velocities [??]/[[??].sub.o] calculated from Eq. (12) indicates the maximal velocity time [t.sub.v]. Ratio of the deceleration duration to the acceleration duration [r.sub.[tau]] = ([t.sub.min] - [t.sub.v])/([t.sub.max] - [t.sub.v]) is presented in Table 2 and strongly depends on [a.sub.s], while ratio of the whole period [tau] with the period of harmonic oscillations [[tau].sub.0] is approximately constant. This corresponds with the first approximation (14). Dependence of the ratio [r.sub.[tau]] on [a.sub.s] is even greater than [r.sub.A] in Table 1.

5. Conclusions

When a single spanbeam is placed on cylindrical surface the contact line and length of the beam are periodically alternating. Nonlinear equation of the beam motion is reduced to linear parametric equation and solved in elliptic functions. The first approximation is harmonic oscillations about some center, elevated above the equilibrium position. Period of the oscillations is almost the same as when supports are hinges, but the half-periods and half-amplitudes in the upper and lower portions of the motion are significantly distorted. The level of distortion depends on the product [alpha]q = 2[pi] [ru.sub.C]/[l.sup.2.sub.o], where r is radius of the support cross section line, [u.sub.C]--amplitude of the beam center oscillations, [l.sub.o]--span in equilibrium position (Fig. 1). Therefore, if radius r is big and surface of the support cross section line is very close to straight line a small amplitudes can cause significant distortion of oscillations.

In a similar manner distortions of the oscillation regularity can appear when beam with fixed ends have some support surfaces tangent to the beam.

References

[1.] Kauderer, H. 1958. Nichtlineare Mechanik, Springer-Verlag, Berlin, 776 p. http://dx.doi.org/10.1007/978-3-642-92733-1.

[2.] Somnay, R.J.; Ibrahim, R.A. 2006. Nonlinear dynamics of a sliding beam on two supports under sinusoidal excitation, Sadhana 31(4): 383-397. http://dx.doi.org/10.1007/BF02716783.

[3.] Somnay, R.J.; Ibrahim, R.A. 2006. Nonlinear dynamics of a sliding beam on two supports under sinusoidal excitation, Journal of Vibration and Control 12(7): 685-712. http://dx.doi.org/10.1177/1077546306065855.

[4.] Vu-Quoc, L.; Li, S. 1995. Dynamics of sliding geometrically-exact beams: large angle maneuver and parametric resonance, Computer methods in applied mechanics and engineering 120: 65-118. http://dx.doi.org/10.1016/0045-7825(94)00051-N.

[5.] Turnbull, P.F.; Perkins, N.C.; Schultz, W.W. 1995. Contact-induced nonlinearity in oscillating belts and webs, Journal of Vibration and Control 1: 459-479. http://dx.doi.org/10.1177/107754639500100404.

[6.] Ahmadian, H.; Jalali, H.; Pourahmadian, F. 2010. Nonlinear model identification of a frictional contact support, Mechanical Systems and Signal Processing 24: 2844-2854. http://dx.doi.org/10.1016/j.ymssp.2010.06.007.

[7.] Mikhlin, Y.V. 2010. Nonlinear normal vibration modes and their applications, Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications, 151-170. Available from Internet: http://www.sbmac.org.br/dincon/trabalhos/PDF/invited /68092.pdf.

[8.] Zajaczkowski, J.; Lipinski, J. 1979. Instability of motion of a beam of periodically varying length, Journal of Sound and Vibration 63: 9-18. http://dx.doi.org/10.1016/0022-460X(79)90373-0.

[9.] Zajaczkowski, J.; Yamada, G. 1980. Further results on instability of the motion of a beam of periodically varying lengthy, Journal of Sound and Vibration 68: 173-180. http://dx.doi.org/10.1016/0022-460X(80)90462-9.

[10.] Gradshteyn, I.S.; Ryzhik, I.M. 2000. Table of Integrals, Series, and Products, 6th ed. San Diego, CA, Academic Press, 1164 p.

[11.] Abramowitz, M.; Stegun, I.A. 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover: 1044 p.

[12.] Erdelyi A.; et al. 1955. Higher Transcendental Functions, vol. 3, McGraw-Hill Book Company, New YorkToronto-London, 299 p.

Received January 15, 2014

Accepted April 18, 2014

V. Kargaudas *, N. Adamukaitis **, M. Zmuida ***

* Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: vytautas.kargaudas@ktu.lt

** Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: nerijus.adamukaitis@ktu.lt

*** Kaunas University of Technology, Studentu 48, 51367 Kaunas, Lithuania, E-mail: mykolas.zmuida@ktu.lt

cross ref http://dx.doi.org/10.5755/j01.mech.20.3.6365

Table 1

Dependence of average displacement [x.sub.v], [x.sub.0] and
ratio [r.sub.A] on as

[a.sub.s]                 n = 3

            [x.sub.0]   [x.sub.v]   [r.sub.A]

0.0100       0.0070      -0.0057     1.0257
0.0200       0.0140      -0.0114     1.0520
0.0500       0.0357      -0.0284     1.1361
0.1000       0.0778      -0.0559     1.3005
0.1500       0.1430      -0.0818     1.5326
0.2000

[a.sub.s]                 n = 4                 [[beta].sub.0]
                                                  [a.sub.s]

            [x.sub.0]   [x.sub.v]   [r.sub.A]

0.0100       0.0070      -0.0057     1.0256         0.0070
0.0200       0.0139      -0.0114     1.0519         0.0139
0.0500       0.0345      -0.0285     1.1343         0.0348
0.1000       0.0672      -0.0570     1.2820         0.0700
0.1500       0.0951      -0.0852     1.4364         0.104
0.2000       0.1148      -0.1130     1.5873

Table 2

Dependence of ratio of periods
[tau]/[[tau].sub.0] on [a.sub.s]

[a.sub.s]   [tau]/[[tau].sub.0]   [r.sub.[tau]]

0.01               1.000              1.031
0.02               1.001              1.073
0.05               1.003              1.187
0.10               1.009              1.398
0.20               1.044              1.764
0.30               0.950              2.000