Abstract
We apply the Bogoliubov Averaging Method to the study of the vibrations of an elastic foundation, forced by a Nonideal energy source. The considered model consists of a portal plane frame with quadratic nonlinearities, with internal resonance 1:2, supporting a direct current motor with limited power. The nonideal excitation is in primary resonance in the order of onehalf with the second mode frequency. The results of the averaging method, plotted in time evolution curve and phase diagrams are compared to those obtained by numerically integrating of the original differential equations. The presence of the saturation phenomenon is verified by analytical procedures.
Internal resonance; saturation phenomenon; averaging method; nonideal system
On nonideal and nonlinear portal frame dynamics analysis using bogoliubov averaging method
J. L. Palacios^{I}; J. M. Balthazar^{II}; R. M. L. R. F. Brasil^{III}
^{I}Universidade Estadual de Campinas Faculdade de Engenharia Mecânica Departamento de Projeto Mecânico UNICAMP C .P. 6122 13083970 Campinas, SP. Brazil jfelix@fem.unicamp.br
^{II}Universidade Estadual Paulista Instituto de Geociências e Ciências Exatas Departamento de Estatística, Matemática Aplicada e Computação UNESP C. P. 178 13500230 Rio Claro, SP. Brazil and visiting Professor at Universidade Estadual de Campinas Departamento de Projeto Mecânico C. P. 6122 13083970 Campinas, SP. Brazil jmbaltha@rc.unesp.br
^{III}Universidade de São Paulo, Escola Politécnica Departamento de Estrutura e Fundações USP C. P. 61546 9524970 São Paulo, SP. Brazil rmlrdfbr@usp.br
ABSTRACT
We apply the Bogoliubov Averaging Method to the study of the vibrations of an elastic foundation, forced by a Nonideal energy source. The considered model consists of a portal plane frame with quadratic nonlinearities, with internal resonance 1:2, supporting a direct current motor with limited power. The nonideal excitation is in primary resonance in the order of onehalf with the second mode frequency. The results of the averaging method, plotted in time evolution curve and phase diagrams are compared to those obtained by numerically integrating of the original differential equations. The presence of the saturation phenomenon is verified by analytical procedures.
Keywords: Internal resonance, saturation phenomenon, averaging method, nonideal 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 nonlinear 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 nonideal energy source, as follows.
Nomenclature
A = cross sectional area, m^{2}
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/m^{2}
g = gravity, m/s^{2}
H = resisting torque of motor, Nms
h = length of the column, m
I = second moment of area of the beams, kgm^{2}
I_{m} = moment of inertia of the rotor, kgm^{2}
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
m_{0} = 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/m^{3}
s_{j}= detuning parameters,rad/s
x_{j} = phases, rad.
W = natural frequency of nonideal 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 nonideal 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 nonideal, 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 nonideal problems were presented in books of Blekhman (1953) and EvanIwanowski (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 nonideal 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 KrylovBogoliubov method, the generalized method of averaging, the KrylovBogolioubovMitropolsky 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 resonances 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 w_{2}» 2w_{1} and W » w_{2}, 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 nonideal 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 (w_{2}» 2w_{1}) and the nonideal excitation frequency is near of the second natural frequency (W » w_{2} ). The driving torque of the motor has been taken as the characteristic of the DC motor (energy source).
Dynamical Model of the System
Let's consider the nonideal system model, which includes a direct current (DC) motor with limited power, operating on a portal plane frame foundation (Fig. 1). The excitation of the system is limited by the characteristic of the energy source. Vibration of the system depends on the motion of the motor, and the energy source motion depends on vibration of the system, as well. Then, coupling of the vibrating portal frame and the DC motor takes place.
The portal frame has two columns clamped in their bases with height h, crosssectional area A_{c}, second moment of area I_{c}. The horizontal beam is pinned to the columns at both ends with length L, crosssectional area A_{b} and second moment of area I_{b}. The members are of linear elastic material with Youngs modulus E and volume density r.
The foundation is modeled as a twodegreeoffreedom system. The coordinate q_{1} is related to the horizontal displacement in the sway mode (with natural frequency w_{1}) and q_{2} to the midspan vertical displacement of the beam in the first symmetrical mode (with natural frequency w_{2}). The two dimensionless generalized coordinates of this model are
where u_{2} is the lateral displacement of the midspan section of the beam and n_{2} 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 RayleighRitz procedure from cubic trial functions. The deflections of the columns and beam are as follows:
for the columns
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 nonideal motor is placed at midspan 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 m_{0} 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 m_{0} are
The kinetic energy of the foundation T_{f} is
where
The kinetic energy of the nonideal motor T_{m} is
where
The kinetic energy of the system T in generalized coordinates to cubic terms is
The potential energy of the foundation V_{f} is
The potential energy of the nonideal 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 Lagranges equation,
where Q_{j} are generalized nonconservative forces that consist of
where c_{1} and c_{2} are modal linear viscous damping. After some algebraic manipulations, the equations of motion are:
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
The motions described by Eq. (17) near resonance region will be studied by the BAM. To apply the BAM, we first use the method of variation of parameters.
When e =0, the solutions of Eq. (17) can be expressed to be
subjected to the constraints
where A_{1}, A_{2}, b_{1}, b_{2} 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 A_{1}, A_{2}, b_{1}, b_{2} and D, that is, A_{1}=A_{1}(t), A_{2}=A_{2}(t), b_{1} = b_{1}(t), b_{2} = b_{2}(t) and D = D(t).
Differentiating q_{1}, q_{2} 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 w_{2} introducing the detuning parameter s_{2} and detuning parameter s_{1} in the presence of internal resonance:
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, a_{1}, a_{2}, x_{1}, x_{2}), j = 1...5, are slowly changing periodic function of time. To find solutions for a_{1}, a_{2}, x_{1}, x_{2}, W in the first approximation we average the right side of (25):
and after integration we obtain
The above autonomous differential equations (referred to as the averaged system), Eq. (28), determine the amplitudes a_{1}(t), a_{2}(t), phases x_{1} (t), x_{2} (t), and nonideal 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
Numerical Simulations Results
The model of Eq. (17) and (28) are solved using a fourthfive order RungeKuttaFehlberg integration algorithm of Burden and Faires (1993) with variable time step. The basic data for the portal framenonideal motor system are shown in Table 1.
Note that the values of Table 1 were also chosen to allow for an internal resonance condition (w_{2}» 2w) for the foundation where w_{2}=156.77 rad/s and w_{1}= 78.37 rad/s.
Dynamic Solutions of the Averaged System
In the first simulation, shown in Fig. 2, we consider the time responses of the amplitudes a_{1} and a_{2} 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 nontrivial solutions of a_{2} and trivial solutions ofv a_{1}for 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 w_{2}. We also show the nontrivial solutions of a_{2} and a_{1}for 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 w_{2}. 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 nonideal 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 q_{1} and vertical displacement q_{2} of the nonideal 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 a_{1}= 0 and a_{2}¹ 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 a_{1} = 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 a_{1}¹ 0 and a_{2}¹ 0. Dividing (30) and (31) by a_{1}¹ 0 yields
Hence,
where
In this case, the nonideal 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, s_{1}= 0, and if s_{2}= 0, Eq. (44) shows that a_{2} is proportional to w_{2}m_{1}/a_{5} and is independent of W (the socalled saturation phenomenon), and a_{1} 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 NewtonRaphson method. In Fig. 5 (a) and (b) we show a typical responsecontrol parameter curve and a typical frequencyresponse 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 nonideal excitation frequency we observe a discontinue transition from the trivial solution to a finite steadystate periodic response: circles represent the amplitudes a_{1} and triangles represent the amplitudes a_{2}.
Conclusions
We have investigated the nonlinear vibration of a portal frame foundation for a nonideal motor using the Bogoliubov averaging method in the resonance region W » w_{2} and internal resonance conditions W_{2 } » 2 w_{1}.
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 nonideal 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 nonideal 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 nonideal system.
Acknowledgements
The authors acknowledge support by FAPESP, Fundação de Apoio à Pesquisa do Estado de São Paulo. The second and third authors also thank CNPq, Conselho Nacional de Pesquisas. Both are Brazilian Research Funding Agencies.
Kononenko, V., 1969, "Vibrating Systems with Limited Power Supply", Illife.
Article received April, 2001
Technical Editor: Atila P. Silva Freire
 Balthazar, J.M., Mook D.T., Weber H.I., Fenili A., Belato D., De Mattos M.C. and Wieczorek S., 1999, "On Vibrating Systems with a Limited Power Supply and their Applications", Brazilian Seminar of Analysis, State University of Campinas, Short Course, pp. 137227.
 Balthazar, J.M, Brasil R.M.L.R.F., Weber H.I., Fenili A., Belato D., Felix J.L.P., "Recent Results on Vibrating Problems with Limited Power Supply", in 6^{th} conference on Dynamical Systems Theory and Aplications, Lodz, December 1012, 2001, pp 2750.
 Barr, DC. and DC. Mcwannell, 1971, "Parametric Instability in Structures under Support Motion", Journal of Sound of Vibration V. 14, N 4, pp. 491509.
 Blekhman, I.I., 1953, "SelfSynchronization of Certain Vibratory Devices", Eng. Trans., 16.
 Bogoliubov, N.N., and Y.A. Mitropolsky, 1961, "Asymptotic Methods in the Theory of Nonlinear Oscillations", Gordon and Breach, New York, 20.
 Brasil, R.M.F.L, and Balthazar J.M., 2000, "Nonlinear Oscillations of a Portal Frame Structure Excited by a Nonideal Motor, Proceedings of 2000 2nd International Conference Control of Oscillations and Chaos, Edited by F.L. Chernousko, A. L. Fradlov, Volume 2 of 3, St. Petersburg, Russia, pp. 275278.
 Brasil, R.M.F.L, Palacios J.L and Balthazar J.M., 2000, "Some comments on Numerical analysis of Nonlinear Vibrations of a Civil Structure Induced by a NonIdeal Source", Proceedings of Computational Methods in Engineering 2000, CD ROM, 21^{st} Iberian Latin American Congress on Computational Methods in Engineering, December 6 8, Rio de Janeiro, Brazil.
 Brasil, R.M.F.L, Palacios J.L and Balthazar J.M., 2001, "On the Nonlinear Dynamic Behavior of a NonIdeal Machine Foundation: Numerical Simulations", Proceedings of Third International conference on Nonlinear Dynamics, Chaos, Control, and Their Applications in Engineering Sciences, Campos do Jordão, Brazil, July 31august 4, pp. 326354.
 Brasil, R.M.F.L, 1999, "Multiple Scales Analysis of Nonlinear Oscillations of A Portal Frame Foundations for Several Machines", Journal of the Brazilian Society of Mechanical Sciences, Vol. 21, Nº 4, pp. 641654.
 Burden, R.L. and Faires J.D., 1993, "Numerical Analysis", PWS Publishing Company, Boston.
 Dimentberg M.F., Chapdelaine J., Norton R.L., Harrison R., 1994, "Passage Through Critical Speed with Limited Power by Switching System Stiffness", In Nonlinear and Stochastic Dynamics, A.K. Bajai, N.S. Namachchivaya and R.I. Ibrahim Editors, AMD, (192), DE 78, pp. 5767.
 Dimentberg M.F., McGovern L., Norton R.L., Chapdelaine J., Harrison R., 1997, "Dynamic of an Unbalanced Shaft Interacting with a Limited Power Supply", Nonlinear Dynamics (13), pp. 171187.
 Haxton, R.S. and Barr A.D.S., 1972, "The Autoparametric Vibration Absorber", Journal of engineering Industry 94, pp. 119125.
 Mazzilli, C.E.N. and Brazil R.M.F.L., 1995, " Effect of static Loading on the Nonlinear Vibrations of a Threetime Redundant Portal Frame: Analytical and Numerical Studies", Nonlinear Dynamics, Vol. 8, pp. 347366.
 Mitropolsky, Y.A., 1967, "Averaging Method in Nonlinear Mechanics", Int. J. Nonlinear Mech. 2, pp. 6996
 Nayfeh, A. H., 1973, "Perturbation Methods", WileyInterscience, New York.
 Nayfeh, A. H. and Mook, D. T., 1979, "Nonlinear Oscillations" , WileyInterscience, New York.
 Nayfeh, A. H., 1981, "Introduction to Perturbation Techniques", WileyInterscience, New York.
 Nayfeh, A.H., 2000, "Nonlinear Interactions: Analytical, Computational, and Experimental Methods", John Wiley & Sons, INC.
 Palacios, J.L., Balthazar J.M. and Brasil R.M.F.L, 2001, "On NonIdeal Dynamics of Nonlinear Portal Frame Analysis Using Averaging Method", Proceedings of the IX DINAME, 59 March, FlorianópolisSC Brazil, Edited by J.J. Spindola, E. M.O. Lopes and F.V.S. Bazan, pp. 143147.
 Palacios, J.L., Balthazar J.M. and Brasil R.M.L.R.F., 2001, "On Nonideal Vibrations of a Portal Frame: Analysis of the Passage Through Resonance, Proceedings of the XVI COBEM, 2630 November, Uberlândia, MG, Brasil, CDROM (9 pgs.).
 Sethna, P.R., 1965, "Vibration of Dynamical Systems with quadratic nonlinearities", Journal Applied Mechanics 32, pp. 576582.
Publication Dates

Publication in this collection
08 Sept 2003 
Date of issue
Nov 2002
History

Accepted
Apr 2001 
Received
Apr 2001