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2 Analysis techniques, Three broad categories of techniques for analyzing nonlinear systems are. a heuristic techniques like Galerkin methods including harmonic balance. b asymptotic techniques including the methods of averaging and multiple scales and. c rigorous mathematical results about dynamical systems. This introduction will concentrate on the first two categories. 2 1 Convergent asymptotic and heuristic, To make the later discussion more meaningful let us distinguish between the terms convergent. asymptotic and heuristic, A convergent series dependent on a parameter say is one where if we fix and take more. and more terms the sum converges to the correct answer An asymptotic series dependent on a. parameter say small is one where if we take a fixed number of terms and take smaller and. smaller the sum gets more and more accurate Convergent series need not be asymptotic and vice. In harmonic balance there is a periodic solution we wish to approximate That periodic solution. has a convergent Fourier series representation However in the application of harmonic balance with. many terms we obtain equally many coupled usually nonlinear equations in terms of the coefficients. see below In practice harmonic balance is often used with only a few harmonics usually with. excellent results but never any formal advance guarantees of how accurate the solution will be with. a given number of terms included In this sense harmonic balance is a heuristic method. We now discuss these methods in more detail, 2 2 Galerkin methods and harmonic balance. The basic Galerkin method is now described using a simple boundary value problem. x x 3t 0 with x 0 x 2 0, The exact solution is x 3t sin t As an approximation we assume say x ak sin 2kt.
Substituting into the governing equation we obtain a nonzero quantity r t called the residual We. make r t orthogonal to the assumed basis functions i e set. r t sin 2kt dt 0 for k 1 2 N, The above process called a Galerkin projection yields N equations for the N unknown ak s which. upon solution give the approximate solution The approximation to 3 terms is. x sin 2t sin 4t sin 6t, which has an error 0 024 More terms yield more accuracy. See e g E J Hinch Perturbation Methods Cambridge University Press 1991. Note that for this linear ODE the equations for the unknown ak s are linear and algebraic. while for general nonlinear ODE s these will be nonlinear algebraic equations see below For. partial differential equations in time and space the approximation will typically be of the form. ak t k x where the k are functions of space chosen to suit the problem e g satisfy boundary. conditions, The technique of Harmonic Balance is a specialized application of the Galerkin method to find. periodic solutions in vibration problems There are several slightly different versions of the method. Here we consider unforced undamped conservative problems e g. We start with say x A sin t B sin 3 t Note that the unknown appears in the functions. sin t and sin 3 t and so there are actually three unknowns in the two term approximation Sub. stituting into the differential equation multiplying in turn by sin t and sin 3 t and integrating in. each case from 0 to 2 and then equating to zero the Galerkin projection we obtain. A 2 3A3 4 3A2 B 4 3AB 2 2 0, 9B 2 A3 4 3A2 B 2 3B 3 4 0. Treating the indeterminate A as a parameter we obtain 0 8869A and B 0 04482A. Variations of the above method are used as the problem changes. Harmonic balance with a few terms usually gives good approximations to periodic solutions For. example some numerical results for the above nonlinear oscillations of Eq 1 as compared with the. two term harmonic balance calculation given above are shown in Fig 1 Oscillations at four different. amplitudes are shown and the figure appears to have four different curves Each of these curves is. in fact two superimposed and nearly indistinguishable curves one solid one dash dot The small. difference between the solid numerical and dash dot harmonic balance is visible towards the right. side of the figure for larger t, The results show that the two term harmonic balance solution is very accurate The strong.
dependence of frequency on amplitude is also clearly seen. 2 3 A first look at asymptotic techniques, Asymptotic techniques depend on some parameter in the problem being very small or very large. which is the same thing on taking reciprocals In the limit as the small parameter becomes zero the. problem should be analytically tractable The basic ideas can be demonstrated using the following. root finding example, x6 x 1 0 2, where 0 1 If 0 x 1 is the only root For nonzero that root is perturbed to. x 1 6 2 51 3 O 4, The O 4 above represents a quantity that is no bigger than some finite constant times 4 as goes. For 6 0 Eq 2 has five other large roots obtainable via a singular perturbation scheme One. of them is, 1 3 14 2 5, x 1 5 1 5 O 3 5, The two asymptotic approximations above are useful for sufficiently small. 0 2 4 6 8 10 12 14 16 18 20, Figure 1 Solutions for Eq 1 Solid line numerical Dashdot harmonic balance can be viewed as.
slightly distinct from solid line for larger times. 2 4 Averaging and multiple scales, The method of averaging is a specialized asymptotic technique for systems of the form. x f x t 1 3, Here we assume f x t f x t T for all x t An approximation to the solution is found by. solving the simpler equation, x f0 x where f0 x f x t dt. Nonlinear oscillators e g, x x x 1 x2 4, are not directly amenable to averaging but they can be put in that form via a change of variables to. x A t sin t t along with the added constraint equation x A t cos t t In this form. the asymptotic method of averaging has been widely used to study a variety of weakly nonlinear. oscillators that are slightly perturbed versions of the harmonic oscillator x x 0. For illustration Eq 4 yields the two equations, A A 2 A3 8 A cos 2t 2 2 A3 cos 4t 4 8.
sin 2t 2 2 A2 sin 2t 2 4 A2 sin 4t 4 8, Finally by first order averaging higher order averaging is possible but not done here we get. A A 2 A3 8 and 0, The above two equations show that A 0 is an unstable equilibrium all other solutions slowly but. eventually approach A 2 assuming A 0 and the phase of the oscillation remains steady at. least at first order, The method of multiple scales also applicable to Eq 3 involves an additional issue namely the. identification and removal of secular terms as illustrated below for Eq 4 using two time scales. Let t be the actual time and t be a slow time Assume x x t Now. x and x 2 2 O 2, Using subscripts t and to denote partial derivatives with respect to these quantities we have. xtt x 2x t xt 1 x2 O 2, Assuming a solution of the form x x0 x1 we obtain.
x0 tt x0 x1 tt x1 2x0 t x0 t 1 x20 O 2, Collecting terms at leading order we obtain. x0 tt x0 0, which has the general solution x0 A sin t Substituting this at the next order we obtain. dropping the explicit dependence of A and on and using primes to denote a derivative. x1 tt x1 A3 cos 3t 3 4 2A A A3 4 cos t 2A sin t, In the above equation the solution for x1 can contain t sin t and t cos t effectively the. same as t sin t and t cos t These secular terms make the approximation break down by the time. t O 1 The validity of the expansion can be extended by removing the secular terms which. can be done here by requiring that the coefficients of the sine and cosine in the forcing be zero i e. 2A A A3 4 0 and 2A 0 Noting that A A etc we find the evolution of A and. are governed at this order of approximation by the same equations as obtained by averaging. A A 2 A3 8 and 0 5, 3 The phase plane, Our study of entrainment in section 9 will involve the use of a popular and powerful idea from. nonlinear dynamics the idea of the phase space The essential idea is described below. Consider a system of two equations, x f x y and y g x y.
Sometimes instead of plotting x and y individually versus t we just plot x versus y If say x rises. monotonically from 0 to 1 as t increases while y rises from 1 to 3 during the same time then on. Figure 2 a A linear damped forced system b A nonlinear system The spring has a free length. L0 h c A nonlinear two degree of freedom system Mass m1 is constrained to move frictionlessly. in the vertical direction while mass m2 moves in the horizontal direction Gravity is neglected for. simplicity, the x versus y plane we have a single curve that goes from the point 0 1 to 1 3 The x y. plane is called the phase plane In a more general case with n dependent variables we would have. an n dimensional phase space, Looking at solutions in the phase space has the disadvantage of losing detailed information about. the exact way in which x and y vary with time However it has the obvious advantage of reducing. the dimensionality of the system by one the solution goes from a curve in the three dimensional. x y t space to the two dimensional x y plane In addition there are other advantages involving. geometrical ideas about various types of solutions and how they behave For example if x and y. approach constant values then the graphs of x and y versus t are horizontal lines but in the phase. plane the graph of x versus y approaches a point Similarly if x and y are periodic functions with. some period T and with some phase difference between them then in the phase plane we see a. closed curve Interested readers will find many excellent books available on nonlinear dynamics and. topics touched upon in these notes are discussed properly in such books A representative sample of. references is provided at the end, 4 Multiple solutions. A damped linear system such as sketched in Fig 2 a governed by the linear differential equation. mx cx kx f t, has a uniquely defined long term behaviour after transients die out For example consider. x 0 3 x x sin 3 2 t 6, 0 5 10 15 20 25 30, Figure 3 Solutions for Eq 6 converge to the same long time behaviour regardless of initial conditions.
Two different solutions for two different initial conditions are shown to converge to the same long. time solution in Fig 3, In contrast consider the system shown in Fig 2 b with the spring s free length L0 greater than. h Now it is clear that this nonlinear system will have three equilibrium positions one at x 0. which will be unstable while one stable position at some nonzero positive x and another reflected. one for negative x This simple example shows that it is possible for general deterministic nonlinear. systems to have more than one steady state solution in response to the same inputs but of course. with different initial conditions, This system is not analyzed here in detail other examples of multiple solutions will soon be. In practical engineering examples of multiple solutions are encountered in a variety of situations. A few examples are provided below, Buckling Beyond a certain load the structure has more than one equilibrium the nominal. equilibrium loses stability and new stable equilibrium positions appear This is related to the. system in Fig 2 b, Whirling of shafts at near and possibly beyond critical speed A non whirling solution still. exists but is now unstable, Resonances in nonlinear systems When the forcing frequency is near the linear natural fre.
quency there can be more than one possible stable steady state solution This example will be. covered again under jumps, Machine tool chatter Under certain operating conditions the cutting tool might chatter a lot. poorer surface finish or very little there is more than one stable steady state solution. Systems with dry friction Some systems with dry friction for small forcing near resonance. can have two solutions one with large amplitude and one without vibrations. 5 Forced vibrations via harmonic balance, Consider the damped nonlinear forced system given by. x cx x ax3 F sin t 0 7, We will study this system using single term harmonic balance Let us assume x A sin t B cos t. The assumption is that the solution is dominated by a response at the same frequency though not. at the same phase as the forcing The assumption is exactly true for the linear system with a 0. and approximately true for reasonable values of a and most values of This single harmonic. approximation is sufficient for the purposes of this section. Substituting into the equation of motion and using some trigonometric identities such as sin3 x. 3 sin x sin 3x 4 we obtain, 2 A sin t 2 B cos t c A cos t c B sin t A sin t B cos t 41 aA3 sin 3 t. 34 aA3 sin t 43 aA2 B cos t 34 aA2 B cos 3 t 43 aAB 2 sin 3 t 34 aA sin tB 2. 41 aB 3 cos 3 t 34 aB 3 cos t F sin t negligible terms. Multiplying by sin t or cos t integrating w r t t from 0 to 2 and then setting them equal. to zero is equivalent to simply picking out the coefficients of sin t or cos t respectively and setting. them equal to zero This gives, A 2 cB A aA3 aAB 2 F 0.
B 2 cA B aA2 B aB 3 0, The solutions to the two simultaneous equations above provide a fairly accurate picture of the. dynamics of the system in Eq 7, 5 1 Unforced undamped case. If we put c 0 and F 0 then we obtain an approximate solution to the unforced undamped. system for which, B 0 and 4 3aA2, The above approximate result tells us that for undamped unforced periodic oscillations the fre. quency of oscillations depends on the amplitude The graph of A versus i e with amplitude. along the vertical axis is usually called a backbone curve because of its shape In this system. A Brief Introduction to Nonlinear Vibrations Anindya Chatterjee Mechanical Engineering Indian Institute of Science Bangalore anindya100 gmail com February 2009 I have used these in the past in a lecture given at RCI Hyderabad as well as during a summer program at IISc organized by the now defunct Nonlinear Studies Group 1 General comments Vibration phenomena that might be modelled

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