Reza N. Jazar Reza.Jazar@manhattan.edu Department of Mechanical Engineering Manhattan College New York, NY, A M. Mahinfalah Mahinfalah@msoe.edu Department of Mechanical Engineering Milwaukee chool of Engineering Milwaukee, WI, A N. Mahmoudian and Nina.Mahmoudian@ndsu.nodak.edu Department of Aerospace Engineering, Virginia Institute of Technology Blacksurg, VI, A Energy-Rate Method and taility Chart of Parametric Virating ystems The Energy-Rate method is an applied method to determine the transient curves and staility chart for the parametric equations. This method is ased on the first integral of the energy of the systems. Energy-Rate method finds the values of parameters of the system equations in such a way that a periodic response can e produced. In this study, the Energy-Rate method is applied to the following forced Mathieu equation: && y + hy& + ( 1 + cos( rt) ) y = sin ( rt) This equation governs the lateral viration of a microcanilever resonator in linear domain. Its staility chart in the -r plane shows a complicated map, which cannot e detected y perturation methods. Keywords: energy-rate method, staility chart, parametric virations M. A. Rastgaar Aagaah.Rastgaar@ndsu.nodak.edu Department of Mechanical Engineering Virginia Institute of Technology Blacksurg, VI, A Introduction 1 In a recent paper, an energy-ased method, called Energy-Rate method has een developed to investigate the staility chart of the parametric differential equations. The method has een introduced and applied to the Mathieu equation. && x+ a x x cos( t) = Application of Energy-Rate method has shown that there are new periodic curves in stale domain of a- plane that have not een detectale y conventional methods (Jazar, 4) (Butcher and inha, 1998). In this study, the Energy-Rate method will e utilized to detect the staility diagram of a forced parametric equation, and a function to e used as a gauge function for determining relative staility in parametric space. In general, a solution is unstale if it grows unoundedly as time approaches to infinity, while a solution is stale if it remains ounded as time approaches to infinity. A periodic solution is neutral and is considered as a special case of a stale solution (Hayashi, 1964). The term parametric equations or parametrically-excited oscillation is generally used for phenomena in which a parameter of the system is time-varying, usually periodically. In early experiments, such systems showed resonance when the variale parameter has a frequency equal to the doule of the natural frequency of the linearized system. Existence of periodic viration with frequencies equal to multiple of the principal resonance frequency is an essential characteristic of parametric equations. Parametric equations can e seen in many ranches of applied science. They have een the suject of a vast numer of investigations since the eginning of the last century (Whittaker and Watson, 197) (Van der Pol and trutt, 198). Investigation of parametric phenomena has een started since 1831 when Faraday produced wave motion in water y virating an immersed plate (Iwanowski, 1965). hortly afterwards, the importance of parametric phenomena was revealed y many scientists in the late 19th and the eginning of th century (Richards, 1983). The Mathieu equation expressed in the form: dx && x+ a x x cos( t) = x& = (1) dt is a special case of Hill equation, && x+ f(t) x =, is the simplest and the first well defined parametric differential equation, in which a and are constant parameters. Depending on the value of a and, Mathieu equation has ounded as well as unounded solutions. Because the Mathieu equation is linear, it s staility is not dependent on initial conditions. The values of the pairs (a, ) that produces periodic responses at the oundaries of stale and unstale regions, make a set of continuous curves in the parametric plane, as shown in Figure 1 (McLachlan1947). taility charts of timevarying systems, including Mathieu equation, have een investigated using a variety of methods and concepts, such as Lyapunov exponents, Poincaré maps, Liapunov-Floquet transformations, and perturative Hamiltonian normal forms (inha and Butcher, 1997) (Guttalu and Flashner, 1996) (Butcher and inha, 1995). Paper accepted April, 8. Technical Editor: Domingos Alves Rade. 18 / Vol. XXX, No. 3, July-eptemer 8 ABCM
Energy-Rate Method and taility Chart of T Γ= 1 1 Edt & x g ( x,x,t) dt T = T & &. (6) T ce se 1 ce1 se ce se 3 ce 3 se 4 ce 4 se 5 nstale tale Figure 1. Mathieu staility chart ased on the numerical values, generated y (McLachlan, 1947). Energy-Rate Method The Energy-Rate method can e applied to the following general parametric differential equation: ( ) ( ) && x+ f x + g x, x&, t = () where, f(x) is a single variale function, and g ( x,x,t & ) is a smooth periodic time varying function with the following conditions. g (,,t) =. (3) ( + ) = ( ) g x,x,t & T g x,x,t & (4) Furthermore, the functions f and g can depend on finite sets of parameters. Equation () may e assumed as a model of a unit mass attached to a spring, acted upon y a non-conservative force g ( x,x,t & ). Introducing the kinetic, potential, and mechanical energies for the system, T( x& ) = x& /, V( x) = f( x) dx, and E = T( x& ) + V( x), we may transform Equation () to an integral of energy as follows: & d d 1 = = & + ( ) = ( ) dt dt & & &. (5) E ( E) x f( x) x dt x g x, x, t & & &, which is equal to the time derivative of the mechanical energy of the system, represents the instantaneous rate of generated or asored energy y the applied g x,x,t &. Instantaneous generation or asorption of energy The function E = x g( x, x, t) force ( ) depends on the value of parameters, time, x, and x&. If the overall value of the rate of energy in an excitation period is negative for a set of parameters and a nonzero response x(t), the response of the system shrinks along the path of x(τ), t <τ< t + T, and the amplitude of oscillation decreases. On the other hand, if the overall value of the rate of energy in an excitation period is positive for a set of parameters and a nonzero response x(t), then the response of the system expands along the path of x(τ), t <τ< t + T, and the amplitude of oscillation increases. We define an Energy-Rate function as an average of the integral of the energy rate: ce 5 se 6 The Energy-Rate function is zero for free viration of a conservative system, as well as the steady state (T/n)-periodic response of the system (). Equation () includes two parameters, a and, so the parameter space is two-dimensional. The numer of parameters determines the dimension of the parameter space. Determination of staility diagram is carried-out y finding the pairs of (a, ) located on the oundary of stale regions. For a steady state T-periodic response, we pick a pair of (a, ) and integrate Equation () numerically to evaluate Γ. If Γ> then, (a, ) elongs to an unstale region where energy is eing inserted to the system. However, if Γ<, then (a, ) elongs to a stale region in which energy is eing extracted from the system. On the common oundary of these two regions, Γ=, and (a, ) elongs to a transition curve. Let us fix and search for a on a transition curve such that its left hand side is stale and its right hand side is unstale. o, if a point (a, ) shows that Γ is less than zero, the point is in the stale region. In such a case, increasing a will increase Γ. On the other hand, if Γ is greater than zero, the point is in the unstale region so, decreasing a will decrease Γ. The Energy-Rate function Γ can e used as a scale for converging a to an appropriate value, such that Γ ecomes zero. Applying this procedure, we find a point on a transient curve with a stale region on the left and an unstale region on the right. Varying y an increment and repeating the calculation, we will find another periodic point. The transient curve can e found when is varied enough to cover the entire domain of interest. It is often useful to have a prior understanding of ehavior of the system. Reversing the strategy and searching for transient curves whose right hand side is stale and left hand side is unstale, completes the staility diagram of the system. This procedure can e arranged in an algorithm to e set up in a computer program, as follows: tep 1 - set a tep - set, equal to some aritrary small value tep 3 - solve the differential equation numerically tep 4 - evaluate Γ tep 5 - decrease / increase a, if Γ > / Γ <, y a small increment tep 6 - the increment of a must e decreased if Γ (a i ) Γ (a i-1)< tep 7 - save a and when Γ <<1 tep 8 - while < final, increase and go to tep 3 tep 9 - reverse the decision in tep 5 and go to tep 1 An alternative method is to set a domain of interest in the parameter plane and then, evaluate Γ for the domain using a fine resolution. A two-dimensional matrix will e found for a twoparameter equation. Each element of the matrix would e the value of Γ for the associated pair of parameters. sing this matrix, a threedimensional surface can e plotted to illustrate the Energy-Rate function. This plot shows the relative staility of different points in the parameter plane. Intersection of the surface with plane Γ= designates the transition curves, while the domain elow it indicate stale regions, and the domain aove this plane indicate instale regions. The accuracy of this analysis depends on the resolution in the parametric plane. There are some advantages in Energy-Rate method over classical perturation methods. First, Energy-Rate method can e applied to nonlinear parametric equations as well as linear ones. J. of the Braz. oc. of Mech. ci. & Eng. Copyright 8 y ABCM July-eptemer 8, Vol. XXX, No. 3 / 183
econd, it can e applied to oth, small and large values of parameters. Third, the values of the parameters for a periodic response can e found faster and more accurately than in other methods. Fourth, application of the Energy-Rate method for detection of periodic solutions can e adjusted to detect any periodic response, not necessarily on the oundary of stale and instale regions. Fifth, the Energy-Rate function can e used as a gauge function to compare the relative staility of different points of parameter space. The greatest disadvantage of perturation methods is that they can usually e applied to very small values of the parameters. Also, the accuracy of the results of most asymptotic methods cannot e increased y increasing the degree of polynomials or the numer of terms in the pertured solution. In the next section we apply the Energy-Rate method to the Mathieu equation and to forced Mathieu type equation to show that the Energy-Rate method provides much more accurate staility diagram as compared to perturation methods. Γ Figure. Plot of π for Mathieu equation as a function of a for =1. taility Diagram of Mathieu Equation To show the applicaility of the Energy-Rate method, Mathieu equation is examined as the first example. Mathieu equation is a well investigated equation, and ecause of its linearity, there exist some effective methods for finding its staility diagram. Transition curves of Mathieu equation are π and π-periodic. The analytic solution of the equation on transient curves are availale and are called cosine and sine elliptic functions (Richards, 1983). The first three even and odd cosine and sine elliptic functions, ce n, se n, ce n+1, and se n+1 are shown in Figure 1. The unstale tongues are ounded with two transition curves starting from integer roots of a n =, n=, 1,, 3,. It can e shown that although the odd order cosine and sine elliptic functions are not symmetric with respect to, the overall staility diagram looks symmetric. ince the parametric resonance of Mathieu equation happens in periods T and T, the ound of integral may e defined from zero to T to include T and T periodic solutions. The Energy-Rate function for Mathieu Equation (1) would e a two-parameter equation. Γ Figure 3(a). taility surface of Mathieu Equation (1). a π 1 Γ =Γ ( a,) = ( x x cos( t )) dt π & (7) Consider a case in which =1, then Γ is only a function of a. This function is plotted in Figure using 1 points in the interval <a<45. It can e seen that the curve Γ intersects the a axis exactly at the same points where the line =1 intersects the transition curves of the Mathieu staility diagram (see Figure 1). Evaluating Γ over a domain of a and generates a three dimensional staility surface. Figure 3(a) displays the staility surface of Mathieu equation over the area < a< 1 and < < 1. Based on Energy-Rate method, wherever Γ is negative, Mathieu Equation (1) is stale, and wherever Γ is positive, it is unstale. The staility surface has een found y integrating over T=π. Therefore, the intersection of the staility surface with the plane Γ=, indicates π or π-periodic curves, ecause a π-periodic function is π-periodic as well. Figure 3() depicts a top view of the staility surface. It also illustrates the intersection curves of the staility surface with plane Γ=. 1 1 8 6 4-4 6 8 1 1 a Figure 3(). π-periodicity curves of Mathieu Equation (1). If a periodic curve separates stale and unstale regions, it is called a transition curve, otherwise it is a splitting curve. A splitting curve is a periodic curve emedded in a stale region. Figure 4 depicts the π and π-periodic transition curves of Mathieu equation, along with a splitting curve in each stale region (Jazar, 4). 184 / Vol. XXX, No. 3, July-eptemer 8 ABCM
Energy-Rate Method and taility Chart of 1 1 8 6 4 h=1. h=.5 h=. h= - 4 6 8 1 1 a Figure 4. π-periodicity curves of Mathieu Equation (1), intersection of the staility surface and plane Γ =. r Figure 5. Transient curves and instaility region for Equation (1). taility Diagram of A Forced Mathieu Type Equation As a second example, forced viration of a Mathieu type system will e studied in this section. Consider the following forced and parametrically excited equation: ( ( )) ( ) && y+ hy& + 1 + cos rt y = sin rt. (1) which is the nondimensionalized governing equation of a linearized model of a micro-electro-mechanical system (Luo, ) (Jazar, 6). The system is a cantilever resonator activated y an alternating electric field. Equation (1) is dependent on three independent parameters. Therefore the parameter space of the system is a three-dimensional r--h space. Assuming h=, the condition for parametric resonance, which states that n times of the forcing frequency is equal to twice of the natural frequency of the unforced system, would e 1 = nr,n N. Therefore, transition curves, if there is any, must start from r = 1/n,n N on the line =. The periodic curves corresponding to n=1 makes the principal instaility domain, if they are transition curves. Assuming a periodic solution, y = Y n n sin( nrt +ϕ), and = applying multiple scale perturation method provides the following relationship for the periodic condition with Y =. ( ( ) ) 1 r = h 4± h 4 4h 1 + 4, n N. () n Traditionally, the periodic condition is used to determine the periodic curves in parameter plane r-. The periodic curves need to e checked to determine if they are transition or splitting curves. The periodic curves () corresponding to n=1 are plotted in Figure 5 for different values of h. It is expected that two periodic curves come out of r=1 and make the principal instaility tongue. Time response of the undamped system for selected points (r, ) = (.9,.1), and (1.,.1), are found and shown in Figure 5. It is concluded that Equation () are transition curves, and the region surrounded y curves coming out of r=1 is unstale. Figure (5) also illustrates that the instaility domain shrinks y increasing the dimensionless damping, h. Applying Poincare-Lindstad method provides the following equations for no damping transient curves will e found 4 1 = 1 r + O(r ) 1 r 3 3 4 = + O( r ) (3) which are in agreement with Equation () and Figure 5. It must e mentioned that not only the curves derived from perturation analysis are not necessarily transition curves, the staility ehavior of the system is also questionale when parameters are close to a curve. Furthermore, we know that although Equations () and (3) might e used to determine the periodic curves and staility regions in r- space around =, they cannot e used to determine the gloal staility regions of the system. Defining the Energy-Rate function as follow: π /r r Γ= y( sin ( rt) hy y cos( rt) ) dt π &. (4) and applying Energy-Rate method provides an exact staility diagram. To present a etter view of staility diagram, we study the system in - plane instead of r- plane. However, it is also possile to apply a transformation rt =τ, and convert Equation (1) into the form: where, ( ( )) ( ) y + Hy + A+ Bcos τ y = Bsin τ y = dy/rdτ, H A = ( 1 ) /r, = h/ r, B = /r. Therefore, the Energy-Rate function may e defined as ( ( ) ( )) * πγ = π y Bsin rt Hy Bycos τ dτ. A three dimensional illustration of staility surface πγ /r is depicted in Figure 6(a) to provide a relative staility view. The staility surface is made y 5 1 points. The intersection of the staility surface with Γ= specifies the transition curves of the system that are shown in Figure 6(). Figure 6(c) shows a top view of the staility surface along with the staility diagram. The staility characteristic of stale and unstale regions of Figure 6() is clear J. of the Braz. oc. of Mech. ci. & Eng. Copyright 8 y ABCM July-eptemer 8, Vol. XXX, No. 3 / 185
y investigating Figures 6(a) or 6(c). It can also e determined y numerical solution of the equation for a sample point within each region. A deep and an elevated regions are associated to strong stale and an unstale region: a stale region for large values of around r=1, and an unstale region corresponding to large values of and. There is also an unstale region connected to =1 which is the principal instaility region that could e predicted y perturation methods approximately. Figure 6(c). Top view of the staility surface for the parametric Equation (1) along with transition curves. πγ/r Figure 6(a). taility surface for the parametric Equation (1). πγ/r Figure 7(a). Magnification of staility surface at the principal instaility region of the parametric Equation (1). Figure 6(). taility diagram for the parametric Equation (1). Because of the importance of the principal instaility region, a magnified view of the staility surface for <<.4 and 1<<.5 is shown in Figure 7(a). A etter view of the transition curves is depicted in Figure 7(). A top view of the staility surface along with the transition curves are also shown in Figure 7(c) to highlight the staility characteristic of Figure 7(). In order to etter understand the ehaviour of the system when parameters are around the principal instaility region, time responses of the system for a few selected points in different regions are shown in Figure 8. The examined points are marked in Figure 7(). Figure 7(). Transition curves around the principal instaility region of the parametric Equation (1). 186 / Vol. XXX, No. 3, July-eptemer 8 ABCM
Energy-Rate Method and taility Chart of Perturation method Energy-Rate method Figure 7(c). Top view of the staility surface of Equation (8) along with transition curves around the principal instaility region. Figure 9. Comparison of the result of perturation and Energy-Rate method close to the principal instaility region of Equation (8). Comparison of the results of perturation analysis in Equation (9) or (1) with the results of Energy-Rate method is shown in Figure 9. From a design point of view, the strong stale region related to r 1 and large values of are important. This is proaly the est area to set the parameters of the system governed y Equation (8). To clarify the oundary of the stale and unstale regions in this area, a three dimensional illustration of staility surface, and a two dimensional staility diagram are shown in Figures 1-1. πγ/r Y =.4 =1.5 h=. =.13 =1.8 Figure 1. Magnification of staility surface around the strong stale region of the parametric Equation (8). t =.37 =.4 =.15 =.45 =.5 =1. Figure 8. Time response of the parametric Equation (8) for some points of the parameter plane. Figure 11. Magnification of transition curves around the strong stale region of the parametric Equation (8). J. of the Braz. oc. of Mech. ci. & Eng. Copyright 8 y ABCM July-eptemer 8, Vol. XXX, No. 3 / 187
are also found y applying the Energy-Rate method. Threedimensional illustration of staility surface shows a sense of relative staility of parameter plane. The intersection of a zero plane with staility surface determines the periodic and transition curves of the system. The procedure is sometimes time consuming, ut it generates a more efficient and more accurate staility diagram. Figure 1. Top view of the staility surface of the parametric Equation (8) along with transition curves around the strong stale region. Conclusion The Energy-Rate function, Γ= x g( x,x,t) dt /T & & is used to T generate a three-dimensional staility surface for two parametric equations example. The value of the function can e determined numerically and compared to zero for a parametric equation such as && x+ f ( x) + g( x, x&, t, a, ) =. The Energy-Rate method may e used to investigate the staility of the parametric equations. Procedure of searching for periodic curves in the parameter space is presented as an algorithm. To validate the algorithm, it was applied to the Mathieu equation, and its staility chart was developed. The results are in agreement with a previously reported results. In the second example, the method was applied to a forced and parametrically excited equation, which is ased on the model of a physical system. Multiple time scale, and Poincaré-Lindstad perturation methods were applied and periodic curves for small values of parameters were found. The staility surface and staility diagram of the system References Butcher E.A., and inha.c., 1998, ymolic Computation of Local taility and Bifurcation urfaces for Nonlinear Time-Periodic ystems, Nonlinear Dynamics, 17, 1-1. 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. Guttalu, R.., and Flashner, H., 1996, taility Analysis of Periodic ystems y Truncated Point Mapping, Journal of ound and Viration, 189, 33-54. Hayashi, C., 1964, Nonlinear Oscillations in Physical ystems, Princeton. Iwanowski, E.R.M., 1965, On the Parametric Response of tructures, Applied Mechanics Review, 18(9), 699-7. Jazar, Reza N., 4, taility Chart of Parametric Virating ystems, sing Energy Method, International Journal of Nonlinear Mechanics, 39(8), pp. 1319-1331. Jazar, Reza N., 6, Mathematical Modeling and imulation of Thermoelastic Effects in Flexural Microcantilever Resonators Dynamics, accepted for pulication in the Journal of Viration and Control. Luo, A. C. J.,, Chaotic Motion in the Generic eparatrix Band of a Mathieu-Duffing Oscillator with a Twin-Well Potential, Journal of ound and Viration, 48(3), 51-53. McLachlan, N. W., 1947, Theory and Application of Mathieu Functions, Clarendon, Oxford. P., Reprinted y Dover, New York, 1964. Richards, J. A., 1983, Analysis Periodically Time Varying ystems, pringer-verlag, New York. inha.c., and Butcher E.A, 1997, ymolic Computation of Fundamental olution Matrices for Linear Time-Periodic Dynamical ystems, Journal of ound and Viration, 6(1), 61-85. Van der Pol, B.,, and trutt, M.J.O., 198, On the taility of the solution of Mathieu s Equation, The Philosophical Magazine, vol. 5, p. 18. Whittaker, E.T., and Watson, G.N., 197, A Course of Modern Analysis, Camridge niv. Press, 4th ed. 188 / Vol. XXX, No. 3, July-eptemer 8 ABCM