J. L. Palacios^I; J. M. Balthazar^II; R. M. L. R. F. Brasil^III

^IUniversidade Estadual de Campinas Faculdade de Engenharia Mecânica Departamento de Projeto Mecânico – UNICAMP C .P. 6122 13083-970 Campinas, SP. Brazil jfelix@fem.unicamp.br
^IIUniversidade Estadual Paulista Instituto de Geociências e Ciências Exatas Departamento de Estatística, Matemática Aplicada e Computação – UNESP C. P. 178 13500-230 Rio Claro, SP. Brazil and visiting Professor at Universidade Estadual de Campinas Departamento de Projeto Mecânico C. P. 6122 13083-970 Campinas, SP. Brazil jmbaltha@rc.unesp.br
^IIIUniversidade de São Paulo, Escola Politécnica Departamento de Estrutura e Fundações – USP C. P. 61546 9524-970 São Paulo, SP. Brazil rmlrdfbr@usp.br

ABSTRACT

Keywords: Internal resonance, saturation phenomenon, averaging method, non-ideal system

Introduction

Over the last years, the vibrations of the linear systems have been studied exhaustively. Recently, significant contributions have been made to the theory of vibrations of non-linear systems. Nevertheless, special kinds of vibrations, arising from the interaction of the system with the energy source, can not yet be completely explained by means of current theory. It is convenient to introduce two new concepts: an ideal energy source and a non-ideal energy source, as follows.

Nomenclature

A = cross sectional area, m²
A_j = parameter, dimensionless ]]> a_j = Amplitudes, dimensionless
â = related to voltage applied across the armature, N m
B = constant, m^-1
= related to a type of motor, Nm/s
C = constant, m^-1
c = modal linear viscous damping, Ns/m
E = Young modulus,N/m²
g = gravity, m/s²
H = resisting torque of motor, Nms
h = length of the column, m ]]> I = second moment of area of the beams, kgm²
I_m = moment of inertia of the rotor, kgm²
k = stiffness, N/m
L = length of the beam, m
M = mass of the motor, kg
M_m = driving torque of the motor, Nm
m = mass, kg
m₀ = unbalance mass, kg
q = generalized coordinate, dimensionless
r = eccentricity, m ]]> u = horizontal displacement, m
v = vertical displacement, m

Greek Symbols

b_j= parameter, rad
D = parameter, rad/s
e = small parameter, dimensionless
j = angular displacement, rad.
r = density, kg/m³
s_j = detuning parameters,rad/s
x_j = phases, rad. ]]> W = natural frequency of non-ideal motor, rad/s
w = natural frequency of modes, rad/s

Subscripts

b = relative to beam
c = relative to column
m = relative to motor
0 = relative to unbalanced mass
1 = relative to the horizontal displacement
2 = relative to the vertical displacement
3 = relative to the angular displacement

An ideal energy source is one that acts on the vibrating system, but does not experience any reciprocal influence from the system. A non-ideal source is one that acts on a vibrating system and at the same time experiences a reciprocal action from the system. Changes in the parameters of the system may be accomplished by changing the working conditions of the energy source. These interactions may become especially active when the energy source has very limited power and they will be more visible in the resonance regions. That is, we assume that the difference between the natural frequency w of the system and the frequency of the exciting force (for example, a DC motor) W is small, i.e., a detuning parameter s = w - W is small.

In an ideal system, we assume that a motor mounted on a structure requires a certain input (Power) to produce a certain output (RPM) regardless of the motion of the structure. If we consider the same system as non-ideal, this may be not the case. Hence, we are interested in what happens to the motor, input, and output, as the response of the system changes.

Vibrating problems with a limited power supply have been investigated by a number of researchers. Kononenko (1969) devoted an entire text to this subject, Nayfeh and Mook (1979) present an overview of different theories up to 1979 and Balthazar et al. (1999) and Balthazar et al. (2001) present a complete review of these kinds of problems up to 2001. Further contributions to non-ideal problems were presented in books of Blekhman (1953) and Evan-Iwanowski (1976) and papers by Prof. Dimentberg (1994, 1997).

Barr and Macwanell (1971) studied a simple portal frame under support motion, but nonlinear elastic forces were not taken into account. A study of nonlinear oscillations of portal frames under a single ideal harmonic excitation can be found in Mazzilli and Brasil (1995). Recently a study of nonlinear oscillations of portal frames under several ideal loads can be found in Brasil (1999). The non-ideal case appears in Brasil and Balthazar (2000), Brasil, Palacios and Balthazar (2000) and Brasil, Palacios and Balthazar (2001).

Averaging methods have been in use since the time of Lagrange and Laplace. The methods include the Krylov-Bogoliubov method, the generalized method of averaging, the Krylov-Bogolioubov-Mitropolsky method, and averaging using canonical variables or Lie transforms. Relevant references on this subject include Bogoliubov and Mitropolsky (1961), Mitropolsky (1967), Nayfeh (1973,1981). Many examples of applications of the method of averaging are provided by Nayfeh (1973, Chapter 5).

Sethna (1965), and Haxton and Barr (1972) used the method of averaging to analyze primary resonance’s of systems governed by equations with quadratic nonlinearities when one natural frequency is twice another. They investigated primary resonances of both the first and second modes. When w₂ » 2w₁ and W » w₂, where W is the excitation frequency, and the w_j are the linear natural frequencies, they found a saturation phenomenon. A first preliminary announcement of this paper was done in Palacios, et al. (2001).

The main goal of this paper is to present a reasonably simple analytical method for the study of elastic portal frame foundation for a non-ideal energy source. In particular, we use the Bogoliubov averaging method (BAM), and study its ability to construct a satisfactory approximate solution, which will by compared with the results obtained by means of numerical integration. To find the saturation phenomenon we choose the physical and geometric properties of the portal frame to tune the natural frequencies of the two first modes into a 1:2 internal resonance (w₂ » 2w₁) and the non-ideal excitation frequency is near of the second natural frequency (W » w₂ ). The driving torque of the motor has been taken as the characteristic of the DC motor (energy source).

Dynamical Model of the System

The portal frame has two columns clamped in their bases with height h, cross-sectional area A_c, second moment of area I_c. The horizontal beam is pinned to the columns at both ends with length L, cross-sectional area A_b and second moment of area I_b. The members are of linear elastic material with Young’s modulus E and volume density r.

The foundation is modeled as a two-degree-of-freedom system. The coordinate q₁ is related to the horizontal displacement in the sway mode (with natural frequency w₁) and q₂ to the mid-span vertical displacement of the beam in the first symmetrical mode (with natural frequency w₂). The two dimensionless generalized coordinates of this model are

where u₂ is the lateral displacement of the mid-span section of the beam and n₂ is its vertical displacement. The linear stiffness of the columns and of the beam associated to these modes K_c and K_b can be evaluated by a Rayleigh-Ritz procedure from cubic trial functions. The deflections of the columns and beam are as follows:

for the beam

u and n describe the static deflections of a cantilever beam with a concentrated force applied to its free end and a simply supported beam with a concentrated forced applied at its mid point, respectively.

Because of the postulated inextensibility, the following relations can be written

where the constants C and B are obtained from the same cubic trial functions whose values are and .

An unbalanced non-ideal motor is placed at mid-span of the beam. The angular displacement of its rotor is given by

It has total mass M, its rotor has moment of inertia I_m and carries and unbalanced mass m₀ at a distance r form the axis. The characteristic driving torque of the motor M_m () and the resisting torque H (), for each given power level, are assumed to be known, either from the manufacturer or from previous experiments.

The horizontal and vertical displacement of the unbalanced mass m₀ are

The kinetic energy of the foundation T_f is

]]>

where

The kinetic energy of the non-ideal motor T_m is

where

The potential energy of the foundation V_f is

The potential energy of the non-ideal motor V_m is

The potential energy of the system v in generalized coordinates to cubic terms is

The equations of motion in the configuration space are obtained upon substituting the kinetic and potential energy expressions into the Lagrange’s equation,

where Q_j are generalized non-conservative forces that consist of

where c₁ and c₂ are modal linear viscous damping. After some algebraic manipulations, the equations of motion are:

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where

We now consider real limited power supply motors. For simplicity, their characteristic curves of the DC motor are assumed to be straight lines of form

Note that the parameter â is related to the voltage applied across the armature that will be the control parameter and is the constant for each model of motor considered.

An Analytical Solution

When e =0, the solutions of Eq. (17) can be expressed to be

subjected to the constraints

where A₁, A₂, b₁, b₂ and D are constants, which are sometimes referred to as parameters.

When e ¹ 0, we assume that the solution of Eq. (17) is still given by Eq. (19) but with time varying A₁, A₂, b₁, b₂ and D, that is, A₁=A₁(t), A₂=A₂(t), b₁ = b₁(t), b₂ = b₂(t) and D = D(t).

Differentiating q₁, q₂ of Eq. (19) leads to

Taking into account Eq. (20) and (21) we obtain

Next, differentiating of Eq. (20) leads to

We restrict our attention to a narrow band of frequencies around the natural frequency w₂ introducing the detuning parameter s₂ and detuning parameter s₁ in the presence of internal resonance:

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We substitute Eq. (19) and (23) into the equations of motion (17), use Eq. (22) and some trigonometric identities, keep up to O(e) terms, and obtain

To solve equations (25) we use the Bogoliubov Averaging Method as presented in Kononenko, (1969). According to this perturbation method, we can write, to first approximation

where:U_1j = U_1j (t, W, a₁, a₂, x₁, x₂), j = 1...5, are slowly changing periodic function of time. To find solutions for a₁, a₂, x₁, x₂, W in the first approximation we average the right side of (25):

The above autonomous differential equations (referred to as the averaged system), Eq. (28), determine the amplitudes a₁(t), a₂(t), phases x₁ (t), x₂ (t), and non-ideal excitation frequency W (t) of the first order approximations of the generalized responses of Eq. (17).

To the first approximation, the solution of Eq. (17) is given by

where the a_j, x_j and W are governed by Eq. (28).

Constant solutions or equilibrium solutions or fixed points of Eq. (28) are obtained by setting and =0. The result is

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Numerical Simulations Results

The model of Eq. (17) and (28) are solved using a fourth-five order Runge-Kutta-Fehlberg integration algorithm of Burden and Faires (1993) with variable time step. The basic data for the portal frame-non-ideal motor system are shown in Table 1.

Note that the values of Table 1 were also chosen to allow for an internal resonance condition (w₂ » 2w) for the foundation where w₂=156.77 rad/s and w₁= 78.37 rad/s.

In the first simulation, shown in Fig. 2, we consider the time responses of the amplitudes a₁ and a₂ obtained of Eq. (28) for various values of the control parameter â, namely, 0.35, 0.38, 0.41, 0.43 Nm. We show the non-trivial solutions of a₂ and trivial solutions ofv a₁for â=0.35, 0.43 Nm and for time t Î (0,5) È (10, 20) seconds, that is, the response of system is linear when the excitation frequency W is below/above the second natural frequency w₂. We also show the non-trivial solutions of a₂ and a₁for â=0.38, 0.41 Nm and for time t Î (5, 10) seconds, that is, the response of system is nonlinear when the excitation frequency W is near/captured of the second natural frequency w₂. In this case, the saturation appears in the energy transference from a higher frequency mode to a lower frequency mode.

In the second simulation, shown in Fig. 3, we compare the analytical approximate solution with the numerical integration to verify if the Bogoliubov averaging method is a valid tool to approximate solutions of this non-ideal system. We show in the phase space () a comparison between the analytical approximate solution (29) and the numerical solution of (17): circles represent the numerical solution and crosses represent the approximation solution for the control parameter values â, namely, 0.35, 0.38, 0.41 and 0.43 Nm.

In order to complete the results obtained here, we show in Fig. 4 the approximate solution of the horizontal displacement q₁ and vertical displacement q₂ of the non-ideal system using (29) for the control parameter value â= 0.41Nm.

Canstant Solutions of the Averaged System

In order to verify the trivial and nontrivial solutions that will be the linear and nonlinear response of the system, respectively, and the saturation phenomenon obtained in the dynamic solutions of the averaged system (28), we determine the constant solutions of (30)-(34).

Case I: we analyze constant solutions when a₁= 0 and a₂¹ 0 . In this case, (32) and (33) become

Solving these equations for the a_j , x_j and W, we have

of Eq. (34) become

These equations determine the excitation frequency of the motor.

Using these solutions and recalling that a₁ = 0, we rewrite (29) as

which is essentially the steady state of the linear solutions of (17) (Nayfeh, (2000)).

Case II: we analyze constant solutions when a₁ ¹ 0 and a₂ ¹ 0. Dividing (30) and (31) by a₁ ¹ 0 yields

where

In this case, the non-ideal system response is given by (29), where the a_j, x_j and W are constants given by (44) and (37). This periodic response is nonlinear.

As the system has internal resonance, s₁= 0, and if s₂= 0, Eq. (44) shows that a₂ is proportional to w₂m₁/a₅ and is independent of W (the so-called saturation phenomenon), and a₁ is proportional to and is dependent of W. Finally, we verify analytically the saturation phenomenon of results of Eq. (28) (see Fig. 2).

In the third simulation, shown in Fig. 5, we solve numerically Eq. (28) applying the Newton-Raphson method. In Fig. 5 (a) and (b) we show a typical response-control parameter curve and a typical frequency-response respectively â Î (0.30,0.50) Nm where the jump and saturation phenomenon is clearly observable. For increasing values of the control parameter and non-ideal excitation frequency we observe a discontinue transition from the trivial solution to a finite steady-state periodic response: circles represent the amplitudes a₁ and triangles represent the amplitudes a₂.

Conclusions

We have investigated the nonlinear vibration of a portal frame foundation for a non-ideal motor using the Bogoliubov averaging method in the resonance region W » w₂ and internal resonance conditions W₂ » 2 w₁.

We found the saturation phenomenon between the first two vibration modes considered to study system motion, in the passage through resonance region. It is shown the influence of the internal resonance, the presence of the quadratic nonlinearities terms in the equations of motion and interaction of the non-ideal excitation with the foundation response in primary resonance region. We verify the saturation phenomenon by analytical procedures using constant solutions of the averaged system. Various researchers suggest using this theory based in the saturation phenomenon to implement a nonlinear active control (saturation control) to suppress the structural responses. Future work by the authors will apply this saturation control to a non-ideal system.

The comparison of numerical results of the equations of motion Eq. (17) and averaged equations Eq. (31) were carried out, and we conclude that the Bogoliubov averaging method is an excellent tool to study the characteristic of motion of a non-ideal system.

Acknowledgements

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Article received April, 2001
Technical Editor: Atila P. Silva Freire