Multiple scales analysis of nonlinear oscillations of a portal frame foundation for several machines

Multiple Scales Analysis of Nonlinear Oscillations of a Portal Frame Foundation for Several Machines

Reyolando M. L. R. F. Brasil

Departamento de Engenharia de Estruturas e Fundações

Escola Politécnica C.P. 61548,

Universidade deSão Paulo

05424-930, São Paulo, Brazil

rmlrdfbr @ usp.br

Abstract

An analytical study of the nonlinear vibrations of a multiple machines portal frame foundation is presented. Two unbalanced rotating machines are considered, none of them resonant with the lower natural frequencies of the supporting structure. Their combined frequencies is set in such a way as to excite, due to nonlinear behavior of the frame, either the first anti-symmetrical mode (sway) or the first symmetrical mode. The physical and geometrical characteristics of the frame are chosen to tune the natural frequencies of these two modes into a 1:2 internal resonance. The problem is reduced to a two degrees of freedom model and its nonlinear equations of motions are derived via a Lagrangian approach. Asymptotic perturbation solutions of these equations are obtained via the Multiple Scales Method.

Keywords: Nonlinear Oscillations, Structures, Perturbation Methods.

Introduction

The nonlinear vibrations of frames have been investigated by a number of researchers. In one of the earliest, Barr and Mc Wannell (1970) studied a frame under support motion, but nonlinear elastic forces were not taken into account. Yet, these are extremely important and affect qualitatively and quantitatively the analysis. Brasil and Mazzilli (1990) studied the related problem of a framed machine foundation of similar geometry. They recast the problem, considering both inertial and elastic nonlinear effects, including that of the geometric stiffness of the columns and geometric imperfections, such as the elastic deformations of the frame, before the excitation would come into action. Some other studies of nonlinear oscillations of other portal frames will be found in Brasil and Mazzilli (1993) and Mazzilli and Brasil (1995).

Here, another related problem of considerable practical importance is presented. Two unbalanced rotating machines are considered, none of them resonant with the lower natural frequencies of the supporting structure. Their combined frequencies is set in such a way as to excite, due to nonlinear behavior of the frame, either the first anti-symmetrical mode (sway) or the first symmetrical mode. The physical and geometrical characteristics of the frame are chosen to tune the natural frequencies of these two modes into a 1:2 internal resonance.

Mathematical model

The portal frame of Fig. 1 is considered in the analysis. It has two columns clamped in their bases with h height and constant cross section of A_c area and I_c moment of inertia. The horizontal beam is pinned to the columns at both ends with L length and constant cross section of A_b area and I_b moment of inertia. A linear elastic material is considered whose Young modulus is E and density is r.Two machines of mass m are placed on the top of the columns, rotating at angular speeds W₁ and W₂. Small masses mo at distance e of their axes will render forces due to unbalance of amplitude S_o =m₀e. M is the mass of a dead weight at mid span of the beam.

Fig. 1 The portal frame model

The structure will be modeled as a two-degree-of-freedom system related to the sway mode (with natural frequency w₁) and the first symmetrical mode (with natural frequency w₂). The two nondimensional generalized coordinates of this model are chosen to be

(1)

(2)

where u(x) is the lateral displacement of the axis of the left column in the sway mode, and v(x) is the vertical displacement of the axis of the beam in the first symmetric mode, x being the coordinates of the axes of the struts.

The linear stiffness of the columns and of the beam associated to these modes can be evaluated by a Rayleigh-Ritz procedure from the trial functions

(3)

(4)

obtaining

(5)

(6)

Nomenclature

A_b = cross section area of the beam

A_c = cross section area of the columns

B = constant related to shortening of the beam

C = constant related to shortening of the columns

D₀ = partial derivative in T0

D²_O= second partial derivative in T0

E = Young's modulus

= vector of nonconservative applied forces

I_b = cross section moment of inertia of beam

I_c = cross section moment of inertia of columns

L = length of beam

M = dead weight mass

N_r = generalized non conservative force

Q_i = full generalized modal displacement

Q_i⁰= static part of modal generalized displacement

= vector of position of point masses

S_o = amplitude of excitation due to unbalance of motor

T = time

T₀ = time scale

T₁ = time scale

U = strain energy

V = potential energy

W_c = work of conservative forces

a_r = amplitude of modal motions

g = gravity

h = height of columns

k_i = modal stiffness

k_b = stiffness of beam

k_c = stiffness of column

m = mass of motor

m_o = unbalanced mass

e = eccentricity

q_r = dynamic part ofgeneralized modal displacements

= modal generalized velocity

= modal generalized acceleration

s₀ = scaled amplitude of excitation

u(x) = horizontal displacements

v(x) = vertical displacements

x = axial coordinate of a memberA Temperature Predicting Model for Manufacturing Processes Requiring Coiling

The geometric nonlinearity is introduced by considering the shortening due to bending of the columns and of the beam as

(7)

(8)

where the constants are obtained from the same functions (3) and (4) as

(10)

(11)

The initial static equilibrium configuration of the system can be derived of its total potential energy

(12)

where U is the strain energy and -W_c is the potential energy of the conservative forces. An approximation of these to cubic terms in th_e chosen coordinates can be written as

(13)

(14)

where the mass of the struts themselves has been neglected. Thus,

(15)

The static equilibrium coordinates and come from the conditions for stationary total potential energy:

(16)

(17)

This is a nonlinear system of algebraic equations for which approximated solutions can be obtained using a straightforward expansion

(18)

(19)

In this paper, the parameter e will be related to the small vertical static mid span displacement of the beam:

(20)

Collecting terms of same order one gets

(21)

(22)

(23)

(24)

When the dynamic loading is considered, q₁ and q₂ small generalized displacements about the static equilibrium configuration will take place_, such that

(25)

(26)

Thus, the total potential energy will take the form:

(27)

where

(28)

(29)

Taking the velocities of the lumped masses in terms of the generalized coordinates, the kinetic energy T can be approximated, to cubic terms, as:

(30)

(31)

where

(33)

(34)

To derive the equations of motion, the generalized Lagrange's Equations

(35)

The right hand sides stand for the generalized nonconservative forces applied to the system, which can be derived of the expression

(36)

where is the resultant nonconservative applied force at each k mass point (k = 1 to M, in this model M = 3) and is the correspondent time varying position vector of the point as shown in Fig. 2.

Fig. 2 Applied nonconservative forces

If modal linear viscous damping and is assumed too, we obtain the following generalized nonconservative forces:

(37)

(38)

From (35) the equations of motion of this model are:

(39)

(40)

where

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

and the squares of the frequencies of the modes associated to the two generalized coordinates are

(49)

(50)

Asymptotic perturbation solutions

In this section, the method of Multiple scales is used to determine uniform first order expansions for the solutions of Eqs. (39)-(40) of the form:

(51)

(52)

where the time scales T₀ = t and T₁ = et are considered.

Then, the time derivatives become

(53)

(54)

where , and

After substitution of Eqs. (51) and (53)-(54) into Eq. (39) and separation of coefficients of like powers of e on both sides one gets:

Order e

(55)

where , whose solution is of the form

(56)

where

(57)

and c.c. stands for the complex conjugate.

Order e²

(58)

Performing similar operations for Eq. (40) one gets:

Order e

(59)

whose solution is of the form

(60)

Order e²

(61)

Internal Resonance and Combination External Resonance with the Second Mode

In this subsection, the case of internal resonance w₂» 2w₁ between the two first modes and combination external resonance W₁ + W₂ » 2w₁ with the second mode is considered. The frequencies of operation of the motors, (1 and (2 are supposed to be away from the two lower natural frequencies of the supporting structure.

Introducing detuning parameters s₁ and s_o these conditions can be cast in the form

(62)

(63)

Upon introduction of Eqs. (56) and (62)-(63)into Eq. (58) one obtains

(64)

where [NST] stands for non secular terms.

In a similar fashion, the introduction of Eqs. (60) and (62)-(63) in (61) renders

(65)

The solvability conditions are

(66)

(67)

Introducing polar notation

(68)

(69)

and separating imaginary and real parts, one obtains the following set of first order equations:

(70)

(71)

(72)

(73)

where

(74)

(75)

For equilibrium solutions , and to render the system autonomous.

Thus,

(76)

(77)

(78)

(79)

Possible solutions are:

1. For a₁= 0 in Eqs. (78)-(79) squared and added

(80)

This is the linear solution.

2. For a₁ ¹ 0, canceling a₁ in Eqs. (76)-(77),squared and added

(81)

(82)

(83)

Using these results in Eqs_. (78)-(79) squared and added one gets a quadratic equation in a₁ rendering the solution

(84)

where

(85)

(86)

(87)

A stability analysis of this solution can be _done according to Nayfeh and Mook (1979). Two limit values are defined as

(88)

(89)

• If G₁ > 0 and f¹ < F < f², there are two roots, being stable that associated to the positive sign in (84), unstable the other;

• If G₁ > 0 and F > f², there is only the root associated to the positive sign in (84) and it is stable;

• If G₁£ 0 and F > f², there is only the root associated to the positive sign in (84) and it is stable.

Thus, to first approximation, a nonlinear solution, subjected to the above stability conditions, is

(90)

(91)

(92)

If there is no internal resonance, the solution is essentially that of the corresponding linear problem.

Internal resonance and combination external resonance with the first mode

In this subsection, the case of internal resonance w₂ » 2w₁ between the two first modes and combination external resonance W₁ + W₂» w₁ with the first mode is considered. The frequencies of operation of the motors, W₁ and W₂ are supposed to be away from the two lower natural frequencies of the supporting structure.

Introducing detuning parameters s₁ and s_o, these conditions can be cast in the form

(93)

(94)

Upon introduction of Eqs. (62) and (93)-(94)into Eq. (58) one obtains

(95)

In a similar fashion, the introduction of Eqs. (60) and (93)-(94) in (61)renders

(96)

The solvability conditions are

(97)

(98)

Introducing polar notation, as in Eqs. (68)-(69), and separating imaginary and real parts, one obtains the following set of first order equations:

(99)

(100)

(101)

(102)

where

(103)

(104)

For equilibrium solutions, ,and to render the system autonomous.

Thus,

(105)

(106)

(107)

(108)

Squaring and adding Eqs. (107)-(108) it is possible to obtain a relationship between a₁ and a₂ in the form of

(109)

where

(110)

and

(111)

(112)

Squaring and adding Eqs. (105)-(106), the following cubic equation for a₂ is obtained:

(113)

where

(114)

Thus, to first approximation

(115)

(116)

If there is no internal resonance, the solution is essentially that of the corresponding linear problem.

Conclusions

An analytical study of the nonlinear vibrations of a multiple machines portal frame foundation was presented. Two unbalanced rotating machines were considered, none of them resonant with the lower natural frequencies of the supporting structure. Their combined frequencies was set in such a way as to excite, due to nonlinear behavior of the frame, either the first anti-symmetrical mode (sway) or the first symmetrical mode. The physical and geometrical characteristics of the frame were chosen to tune the natural frequencies of these two modes into a 1:2 internal resonance. The problem was reduced to a two degrees of freedom model and its nonlinear equations of motions were derived via a Lagrangian approach.

Asymptotic perturbation solutions of these equations were obtained via the Multiple Scales Method. Some unexpected resonant motions, not predicted by linear theory and potentially dangerous were detected.

Acknowledgements

The author acknowledges support by FAPESP, Fundação de Apoio a Pesquisa do Estado de São Paulo, and CNPq. He also wishes to thank Prof. Dean T. Mook of Virginia Polytechnic Institute for his valuable help and great kindness.

Presented at DINAME 97 - 7^th International Conference on Dynamic Problems in Mechanics, 3 - 7 March 1997, Angra dos Reis, RJ, Brazil. Technical Editor: Agenor de Toledo Fleury.

• Barr, A.D.S. and McWannell, D.C., 1971,"Parametric instability in structures under support motion", Journal of Sound and Vibration, Vol. 14, no. 4, p. 491-509.
• Brasil, R.M.L.R.F. and Mazzilli, C.E.N., 1990, "Nonlinear vibrations in machine framed foundations", Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, (in Portuguese), Vol.6 (1), p. 147-158.
• Brasil, R.M.L.R.F. and Mazzilli, C.E.N., 1993, "A general FEM formulation of nonlinear dynamics applied to accessing the statical loading effect upon the dynamic response of planar frames", Applied Mechanics Reviews, Vol.46 (11), part 2, p. 110-117.
• Mazzilli, C.E.N. and Brasil, R.M.L.R.F., 1995, "Effect of static loading on the nonlinear vibrations of a three-time redundant portal frame: analytical and numerical studies", Nonlinear Dynamics, Vol.8, pp. 347-366.
• Nayfeh, A.H.; Mook, D.T., 1979, "Nonlinear oscillations" John Wiley, New York.

Publication Dates

• Publication in this collection
  11 Oct 2001
• Date of issue
  Dec 1999