Relative tability of Mathieu Equation in Third Zone N. Mahmoudian, Ph.D. tudent, Nina.Mahmoudian@ndsu.nodak.edu Tel: 701.231.8303, Fax: 701.231-8913 G. Nakhaie Jazar, Assistant Professor, Reza.N.Jazar@ndsu.nodak.edu Tel: 701.231.8303, Fax: 701.231-8913 M. Rastgaar Aagaah, Ph.D. tudent, aagaah.rastgaar@ndsu.nodak.edu Tel: (701) 231-8839, Fax: (701) 231-8913 M. Mahinfalah, Associate Professor, M.Mahinfalah@ndsu.nodak.edu Tel: (701) 231-8839, Fax: (701) 231-8913 Department of Mechanical Engineering and Applied Mechanics North Dakota tate niversity, Fargo, 58105-5285, A Keywords: Mathieu stability diagram, Energy-Rate method, Parametric vibrations, tablewells. ABTRACT Mathieu stability diagram is made by transient curves for π and 2π-periodic responses of Mathieu equation, && x+ a x 2b x cos(2 t) = 0. sing a new method, called Energy-Rate, an algorithm is presented to find the stability diagram numerically for large values of parameters. A function is define to be used as a gauge function to compare the relative strength of stability of different points of a stable zone. The third stable region of Mathieu stability diagram, is analysed as an example for 2πperiodic chart. Two wells of stability are detected within each stable region to be the most stable points. Time response analysis of at different levels of the value of Energy-Rate function depicts the common and different characteristics of points within a stable region. The stability diagram of the angular Mathieu equation dx && x+ a x 2b x cos( 2t) = 0 x& = (1) dt in which a and b are constant parameters, is depicted in Figure 1. It is made by plotting the relationship between a and b for π and 2π-periodic solutions of Equation (1). The parameters a and b control the stability of the equation. Depending on the value of a and b, the response of the equation can be stable, unstable, or periodic. The curves shown in Figure 1 are called transition or periodic curves since they are on the boundary of stable and unstable regions. In this report we introduce and apply the Energy-Rate method to answer this question that how important are different points within a stabile zone of Mathieu equation. Investigating the Figure 1, indicates that the stable zones are bounded with two transition curves starting 2 from integer roots of a n =0, n=0, 1, 2, 3,. Let us name the stable zones as first, for the zone bounded between transient curves started from a=0,1; second, for the zone bounded between transient curves started from a=1,4, and so on. The third stable zone, which is bounded between curves starting from 4 and 9 will be analysed as an example. It will be shown that there are two canons of stability to be the most stable and safest point, separated by another periodic curve. Page 1 of 5
nstable ce 5 ce 4 se 6 ce 3 se 5 ce 0 se 1 ce 1 se 2 ce 2 se 3 se 4 table Figure 1. Mathieu stability diagram based on the numerical values, generated by McLachlan (1947). ENERGY-RATE METHOD Consider the following differential equation where ( ) (,, ) && x + f x = g x x& t (2) ( & ) = ( & + ) g x,x,t g x,x,t T. (3) g ( 00,,t) = 0. (4) The single variable function f(x), and the nonlinear time dependent function depending on a set of parameters which control the behaviour of the system. The Equation (2) is a model of a unit mass attached to a conservative spring, acted upon by a nonconservative force g x,x,t &. The free motion of the system is governed by && x+ f x =. ( ) ( g x,x,t & ) ( ) 0 In general, the applied force g( x,x,t & ) generates or absorbs energy, depending on the value of parameters, time, position and velocity of the particle. If f ( x) = f ( x) is a restoring force and g( x,x,t & ) is a periodic function of time, then we should expect a tendency to oscillate (Esmailzadeh, Mehri, and Jazar 1996). 1 2 Defining kinetic energy, T ( x & ) = 2 x&, potential V( x) = f( x) dx, and mechanical energy functions E = T( x& ) + V( x), we may write an integral of energy as follows. d d 1 2 E& = ( E) = x& + f ( x) dx = x g ( x, x, dt dt 2 & & t) (5) are Page 2 of 5
This expression represents the rate of mechanical energy generated or absorbed by the term g( x,x,t & ), during the motion. When E & < 0 for a set of parameters and a nonzero response x(t), then, E decreases along the path of x(t) and the effect of g x,x,t & resembles damping. In this case, energy is withdrawn ( ) from the system, to produce a general decrease in amplitude until the Energy-Rate function runs out and a new solution make E & = 0. On the other hand, as long as E & > 0 for a set of parameters and a nonzero response x(t), the amplitude increases and x(t) runs away. Applying a suitable numerical integration can show if a set of parameters belong to a stable or unstable region; since time derivative of mechanical energy over one period must be zero for conservative and autonomous systems in a steady state periodic cycle (zemplinska- tupnicka, 1990). TABILITY RFACE OF THE MATHIE EQATION In order to find the value of Energy-Rate for Mathieu equation, the Energy-Rate function is solved numerically for 0<t<T, T=2π, to include both π-periodic and 2π-periodic solutions. sing the Energy-Rate function, a value will be assigned to every single point of the parameter space to generates a surface of weighted stability. The parameter space of Mathieu equation is a-b plane. Figure 2 depicts the surface of the Energy-Rate function for Mathieu equation (1). E av b a Figure 2. The 2π-stability surface of Mathieu equation The stability surface is cut by Eav = 4. ince the value of Energy-Rate grows rapidly, the pattern of Mathieu stability chart can be seen in the top intersection plane. Mathieu stability chart shown in Figure 1 can be assumed as the intersection of the stability surface with E = 0. Appearance of two wells and a ridge within each stability valley of 2π-stability av Page 3 of 5
mountain is interesting. The ridge touches the zero plane indicating that there is another 2πstability curve embedded in each stable zone. THE THIRD TABLE ZONE In order to use the Energy-Rate function as a gauge to investigate the distribution of strength of stability, the third stable zone is analysed. A three dimensional view of 2π-stability surface in the third zone is shown in Figure 3(a) and contours of Energy-Rates are plotted in Figure 3(b). E av b Figure 3(a). The third stability zone of 2π-stability surface of Mathieu equation a Figure 3(b). Contours of Energy-Rates of the third stability zone of 2π-stability surface of Mathieu equation Page 4 of 5
REFERENCE Alhargan, F. A., (1996), A Complete Method for the Computations of Mathieu Characteristic Numbers of Integer Orders, IAM Review, 38(2), 239-255. Alhargan, F. A., (2000), Algorithms for the Computation of All Mathieu Functions of Integer Order, ACM Transaction on Mathematical oftware, 26(3) 390-401. Bickley, W. G., (1945), The tabulation of Mathieu equations, Mathematical Tables and Other Aids to Computation, 1(11), 409-419. Bolotin, V. V., (1964), The Dynamic tability of Elastic ystems, Holden-Day, an- Francisco, A. Butcher, E.A., and inha,. C., (1995), On the Analysis of Time -Periodic Nonlinear Hamiltonian Dynamical ystems, AME Design Engineering Technical Conferences DE 84-1, Vol. 3, part A, 375-386. Cesari, L., (1964), Asymptotic Behaviour and tability Problems in Ordinary Differential Equations, econd Edition, Academic Press, New York, A. Colonius, F., and Kliemann, W., (1995), tability of Time Varying ystems, AME Design Engineering Technical Conferences DE. 84-1, Vol. 3, part A, 365-373. Esmailzadeh E., Mehri B., and Nakhaie Jazar G., (1996), "Periodic olution of a econd Order, Autonomous, Nonlinear ystem", Nonlinear Dynamics, No. 10, pp. 307-316. Esmailzadeh E. and Nakhaie Jazar G., "Periodic olution of a Mathieu-Duffing Type Equation", International Journal of Nonlinear Mechanics, 32(5), p. 905-912, 1997. Goldstein,., (1927), Mathieu Functions, Transaction of the Cambridge Phil. ociety, vol. 23, p. 303-336. Guttalu R.., and Flashner H., (1995), An Analytical tudy of tability of Periodic ystems by Poincare Mappings, AME Design Engineering Technical Conferences DE 84-1, Vol. 3, part A, 387-398. Luo, A. C. J., (2001), Chaotic Motion in the Generic eparatrix Band of a Mathieu-Duffing Oscillator with a Twin-Well Potential, Journal of ound and Vibration, 248(3), 521-532. Mathieu, E., (1868), Memoire sur le mouvement vibratoire d'une membrane de forme elliptique, J. Math. Pure Appl. 13, 137. McLachlan, N. W., (1947), Theory and Application of Mathieu Functions, Clarendon, Oxford. P., Reprinted by Dover, New York, 1964. Meirovitch, L., (1967), Analytical Methods in Vibrations, Macmillan, New York. Nakhaie Jazar, G., (1997), Analysis of Nonlinear Parametric Vibrating ystems, Ph. D. Thesis, Mech. Eng. Dept., harif niversity of Technology. Nayfeh, A. H., and Mook, D. T., (1979), Nonlinear Oscillation, John Wiley, New York, A. Rand, R., and Hastings, R., (1995), A Quasiperiodic Mathieu Equation, AME Design Engineering Technical Conferences DE 84-1, Vol. 3, part A, 747-758. Richards, J. A., (1983), Analysis Periodically Time Varying ystems, pringer-verlag, New York. zemplinska-tupnicka, W., (1990), The Behaviour of Nonlinear Vibrating ystems, Kluwer publishers. Zounes, R.., and Rand, R. H., (2001), Global Behaviour of a Nonlinear Quasiperiodic Mathieu Equation, Proc. of DETC01, AME 2001 Design Engineering Technical Conference, Pittsburgh, PA, ept. 2001, p. 1-11. Page 5 of 5