Lecture Notes on Nonlinear Vibrations

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Lecture Notes on Nonlinear Vibrations Richard H. Rand Dept. Theoretical & Applied Mechanics Cornell University Ithaca NY 14853 rhr2cornell.edu http://www.tam.cornell.edu/randdocs/ version 52 Copyright 2005 by Richard H. Rand 1R.Rand Nonlinear Vibrations 4 1PhasePlane The differential equation describing many nonlinear oscillators can be written in the form:   2 d x dx + f x, =0 (1) 2 dt dt A convenient way to treat eq.(1) is to rewrite it as a system of two first order o.d.e.’s: dx dy = y, =−f(x, y)(2) dt dt Eqs.(2) may be generalized in the form: dx dy = F(x, y), = G(x, y)(3) dt dt A point which satisfies F(x, y)=0 and G(x, y) = 0 is called an equilibrium point. The solution to (3) may be pictured as a curve in the x-y phase plane passing through the point of initial conditions (x ,y ). Each time a motion passes through a given point (x, y), its direction is always 0 0 the same. This means a given motion may not intersect itself. A periodic motion corresponds to a closed curve in the x-y plane. In the special case that the first equation of (3) is dx/dt = y,as in the case of eqs.(2), the motion in the upper half-plane y 0 must proceed to the right, that is, x must increase in time fory 0, and vice versa for y 0. 1.1 Classification of Linear Systems An important special case of the general system (3) is the general linear system: dx dy =ax +by, =cx +dy (4) dt dt We may seek a solution to eqs.(4) by setting x(t)=A exp(λt)and y(t)=B exp(λt). For a nontrivial solution, the following determinant must vanish:     a−λb     2  =0 ⇒ λ − tr λ+det = 0 (5)     cd− λ where tr = a+d is the trace, and det = ad−bc is the determinant of the associated matrix. The eigenvalue λ is given by    2 tr tr λ = ± − det (6) 2 2 If det 0, then (6) shows that there are two real eigenvalues, one positive and one negative. This type of linear system is called a saddle. An example of a saddle is provided by the equation: 2 d x − x=0 (7) 2 dtR.Rand Nonlinear Vibrations 5 2 If det 0and tr 4 det, then there are still two real eigenvalues, but both have the same sign as the trace tr. If tr 0, then both eigenvalues are positive and the solution becomes unbounded as t goes to infinity. This linear system is called an unstable node. The general solution is a linear combination of the two eigensolutions, and for large time the eigensolution corresponding to the larger eigenvalue dominates. Similarly, if the trace tr 0, we have a stable node. An example of a stable node is provided by the overdamped oscillator: 2 d x dx +3 + x=0 (8) 2 dt dt 2 If det 0and tr 4 det, then there are two complex eigenvalues with real part equal to tr/2. Euler’s formula shows us that the resulting motion will involve an oscillation as well as exponential growth or decay. If the trace tr 0 we have unbounded growth and the linear system is called an unstable spiral or focus. Similarly, if the trace tr 0, we have a stable spiral or focus. An example of a stable spiral is provided by the underdamped oscillator: 2 d x dx + + x=0 (9) 2 dt dt If det 0 and tr = 0, then there are two pure imaginary eigenvalues. The corresponding linear system is called a center. An example of a center is provided by the simple harmonic oscillator: 2 d x + x = 0 (10) 2 dt All the foregoing results can be summarized in a diagram in which the determinant det is plotted on the horizontal axis, while the trace tr is plotted on the vertical axis. 1.2 Lyapunov Stability Suppose that we have an equilibrium point P :(x ,y ) in eqs.(3). And suppose further that we 0 0 want to characterize the nature of the behavior of the system in the neighborhood of point P. A tempting way to proceed would be to Taylor-expand F and G about (x ,y ) and truncate the 0 0 series at the linear terms. The motivation for such a move is that near the equilibrium point, the quadratic and higher order terms are much smaller than the linear terms, and so they can be neglected. A convenient way to do this is to define two new coordinates ξ and η such that ξ = x− x,η = y− y (11) 0 0 Then we obtain dξ ∂F ∂F dη ∂G ∂G = ξ + η +···, = ξ + η +··· (12) dt ∂x ∂y dt ∂x ∂y where the partial derivatives are evaluated at point P and where we have used the fact that F and G vanish at P since it is an equilibrium point. The eqs.(12) are known as the linear variational equations. Now if we were satisfied with the linear approximation given by (12), we could apply the classifi- cation system described in the previous section, and we could identify a given equilibrium pointR.Rand Nonlinear Vibrations 6 as a saddle or a center or a stable node, etc. This sounds like a good idea, but there is a problem with it: How can we be assured that the nonlinear terms which we have truncated do not play a significant role in determining the local behavior? As an example of the sort of thing that can go wrong, consider the system:   3 2 d x dx −  + x=0, 0 (13) 2 dt dt This system has an equilibrium point at the origin x=dx/dt=0. If linearized in the neighborhood of the origin, (13) is a center, and as such exhibits bounded solutions. The addition of the nonlin- ear negative damping term will, however, cause the system to exhibit unbounded motions. Thus the addition of a nonlinear term has completely changed the qualitative nature of the predictions based on the linear variational equations. In order to use the linear variational equations to characterize an equilibrium point, we need to know when they can be trusted, that is, we need sufficient conditions which will guarantee that the sort of thing that happened in eq.(13) won’t happen. In order to state the correct conditions we need a couple of definitions: Definition:Amotion M is said to be Lyapunov stable if given any  0, there exists a δ 0 such that if N is any motion which starts out at t=0 inside a δ-ball centered at M,thenitstays in an -ball centered at M for all time t. In particular this means that an equilibrium point P will be Lyapunov stable if you can choose the initial conditions sufficiently close to P (inside a δ-ball) so as to be able to keep all the ensuing motions inside an arbitrarily small neighborhood of P (inside an -ball). A motion is said to be Lyapunov unstable if it is not Lyapunov stable. Definition: If in addition to being Lyapunov stable, all motions N which start out at t=0inside a δ-ball centered at M (for some δ), approach M asymptotically as t→∞,then M is said to be asymptotically Lyapunov stable. Lyapunov’s theorems: 1. An equilibrium point in a nonlinear system is asymptotically Lyapunov stable if all the eigen- values of the linear variational equations have negative real parts. 2. An equilibrium point in a nonlinear system is Lyapunov unstable if there exists at least one eigenvalue of the linear variational equations which has a positive real part. Definition: An equilibrium point is said to be hyperbolic if all the eigenvalues of its linear varia- tional equations have non-zero real parts. Note that a center is not hyperbolic. Also, from eq.(6), any linear system which has det = 0 is not hyperbolic.R.Rand Nonlinear Vibrations 7 Thus Lyapunov’s theorems state that if the equilibrium is hyperbolic then the linear variational equations correctly predict the Lyapunov stability in the nonlinear system. (Note that in the second of Lyapunov’s theorems, it is not necessary for the equilibrium to be hyperbolic since the presence of an eigenvalue with positive real part implies instability even if it is accompanied by other eigenvalues with zero real part.) 1.3 Structural Stability If an equilibrium point is hyperbolic, then we saw that the linear variational equations correctly represent the nonlinear system locally, as far as Lyapunov stability goes. But more can be said. For a hyperbolic equilibrium point, the topology of the linearized system is the same as the topol- ogy of the nonlinear system in some neighborhood of the equilibrium point. Specifically, for a hyperbolic equilibrium point P, there is a continuous 1:1 invertible transformation (a homeo- morphism) defined on some neighborhood of P which maps the motions of the nonlinear system to the motions of the linearized system. This is called Hartman’s theorem. A related idea is that of structural stability. This idea concerns the relationship between the dynamics of a given dynamical system, say for example eqs.(3), and the dynamics of a neighboring system, for example: dx dy = F(x, y)+ F (x, y), = G(x, y)+ G (x, y) (14) 1 1 dt dt where  is a small quantity and where F and G are continuous. A system S is said to be 1 1 structurally stable if all nearby systems are topologically equivalent to S. Specifically, eqs.(3) are structurally stable if there exists a homeomorphism taking motions of (3) to motions of (14) for some . Note the similarity between Lyapunov stability and structural stability: Both involve a given dy- namical object, and both are concerned with the effects of a perturbation off of that object. For example in the case of Lyapunov stability, the object could be the equilibrium point x=dx/dt=0 in eq.(13), and the perturbation could be a nearby initial condition. In the case of structural stability, the object could be the simple harmonic oscillator (10), and the perturbation could be 3 the addition of a small term such as−(dx/dt) , giving eq.(13). From this example, we can see that if a system S has an equilibrium point which is not hyper- bolic, then S is not structurally stable. Another common feature which can prevent a system from being structurally stable is the presence of a saddle-saddle connection. In fact it is possible to characterize all structurally stable flows on the phase plane. To do so, we need another Definition:Apointissaidtobe wandering if it has some neighborhood which leaves and never (as t→∞) returns to intersect its original position. Now it is possible to state Peixoto’s theorem for flows on the plane which are closed and bounded (that is, which are compact). Such a system is structurally stable if and only if: 1. the number of equilibrium points and periodic motions is finite, and each one is hyperbolic;R.Rand Nonlinear Vibrations 8 2. there are no saddle-saddle connections; and 3. the set of nonwandering points consists only of equilibrium points and periodic motions. 1.4 Examples Example 1.1 The plane pendulum. 2 d x g + sinx = 0 (15) 2 dt L Equilibria: y =0,x=0,π By identifying x = π with x = −π we see that the topology of the phase space is a cylinder S× R. 2 y g First integral: − cosx =constant 2 L Example 1.2 Pendulum in a plane which is rotating about a vertical axis with angular speed ω. 2 d x g 2 + sinx− ω sinx cos x = 0 (16) 2 dt L g Equilibria: y =0,x=0,π and also cos x = 2 ω L g For real roots, 1, illustrating a pitchfork bifurcation. 2 ω L 2 2 y g ω First integral: − cosx + cos 2x =constant 2 L 4 Example 1.3 Pendulum with constant torque T. 2 d x g T + sinx = (17) 2 2 dt L mL T Equilibria: sin x = mgL T For real roots, 1, illustrating a fold or saddle-node bifurcation. mgL 2 y g T First integral: − cosx− x =constant 2 2 L mL 1.5 Problems Problem 1.1 Volterra’s predator-prey equations. We imagine a lake environment in which a certain species of fish (prey) eats only plankton, which is assumed to be present in unlimited quantities. Also present is a second species of fish (predators) which eats only the first species. Let x=number of prey and let y=number of predators. The model assumes that in the absence of interactions, the prey grow without bound and the predators starve:R.Rand Nonlinear Vibrations 9 dx dy = ax, =−by (18) dt dt As a result of interactions, the prey decrease in number, while the predators increase. The number of interactions is modeled as xy. The final model becomes: dx dy = ax− cxy, =−by + dxy (19) dt dt where parameters a,b,c,d are positive. a. Find any equilibria that this system possesses, and for each one, determine it’s type. dy dy dx b. Using = ÷ , obtain an exact expression for the integral curves. dx dt dt c. Sketch the trajectories in the phase plane. d. Is this system structurally stable? Explain your answer. Problem 1.2 Zhukovskii’s model of a glider. Imagine a glider operating in a vertical plane. Let v=speed of glider and u=angle flight path makes with the horizontal. In the absence of drag (friction), the dimensionless equations of motion are: dv du 2 =− sinu, v =−cosu + v (20) dt dt a. Using numerical integration, sketch the trajectories on a slice of the u-v phase plane between −πuπ, v 0. dv dv du b. Using = ÷ , obtain an exact expression for the integral curves. du dt dt c. Using your result in part b, obtain an exact expression for the separatrix in this system. d. What does the flight path of the glider look like for motions inside the separatrix versus motions outside the separatrix? Sketch the glider’s flight path in both cases. Problem 1.3 Malkin’s error. In his book “Theory of Stability of Motion” (1952), I.G. Malkin presents an example of a physical problem in which the linear variational equations do not predict stability correctly. His analysis is restated below for your convenience. In fact, there is a mistake in his argument. Your job is to find it. The periodic motions of a pendulum are certainly Lyapunov unstable (since the period of per- turbed motions differs from the period of the unperturbed motion, etc.) However, the linearized equations predict stability: 2 d θ The governing equation is +sinθ = 0. A periodic solution θ = f(t) will correspond to the 2 dt df dθ initial condition θ(0) = f(0) = α, (0) = (0) = 0. (The pendulum is released from rest at an dt dtR.Rand Nonlinear Vibrations 10 2 d f angle α.) Note that (0) =− sin f(0) =− sinα. 2 dt Consider the linearized stability of θ = f(t). Set θ = f(t)+x(t) and linearize the eq. on x to get 2 d x + x cos f(t) = 0. Consider the perturbed motion defined by the initial condition x(0) = 0, 2 dt dx (0) = β. Malkin shows that “for a sufficiently small value of β it i.e. x(t) will remain smaller dt than any preassigned quantity.” 2 2 d f df df d Since f(t)satisfies +sinf =0, f(t) satisfies (differentiating) ( )+ cosf =0. But this 2 2 dt dt dt dt df df equation on (t) has the same form as the linearized equation on x(t). Since the function (t) dt dt 2 df d df d f satisfies the initial conditions (0) = 0, ( )(0) = (0) =− sinα, it follows from uniqueness 2 dt dt dt dt β df df that x(t)=− (t). Now since (t) is bounded, x(t) can be made as small as desired for all sinα dt dt t by choosing β small enough. Ha 1.6 Appendix: Lyapunov’s Direct Method Lyapunov’s Direct Method offers a procedure for investigating the stability of an equilibrium point without first linearizing the differential equations in the neighborhood of the equilibrium. Using this approach, Lyapunov was able to prove the validity of the linear variational equations. As an introduction to the method, consider the simple example: dx dx 1 2 =−x , =−x (21) 1 2 dt dt It is obvious that the origin in this system is an asymptotically stable equilibrium point since we know the general solution, x = c exp(−t),x = c exp(−t) (22) 1 1 2 2 Ignoring this knowledge, consider the function: 2 2 V (x ,x )= x + x (23) 1 2 1 2 being the square of the distance from the origin in the x -x phase plane. Now consider the 1 2 derivative of V with respect to time t: dV dx dx 1 2 =2x +2x (24) 1 2 dt dt dt Substituting eqs.(21), we see that along the trajectories of the system, dV 2 2 =−2x − 2x ≤ 0 (25) 1 2 dt Thus V must decrease as a function of time t, that is, the distance of a point on a trajectory from the origin must decrease in time. Since there is no place at which such a point can get stuck (since dV/dt = 0 only at the origin), we have shown that all solutions must approach theR.Rand Nonlinear Vibrations 11 origin as t→∞, which is to say that the origin is asymptotically stable. The approach in this example can be generalized by inventing an appropriate Lyapunov function V (x ,x ) for a given problem. Without loss of generality, we may assume that the equilibrium 1 2 point in question lies at the origin (since a simple translation will move it to the origin if it isn’t already there.) In all cases we will require that: 1) V and its first partial derivatives must be continuous in some neighborhood of the origin, and 2) V (0, 0) = 0. For a general system dx dx 1 2 = f (x ,x ), = f (x ,x ) (26) 1 1 2 2 1 2 dt dt we shall be concerned with dV/dt along trajectories. As in the example, we will compute this as: dV ∂V dx ∂V dx ∂V ∂V 1 2 = + = f (x ,x )+ f (x ,x ) (27) 1 1 2 2 1 2 dt ∂x dt ∂x dt ∂x ∂x 1 2 1 2 We present the following three theorems without proof: Theorem 1: If in some neighborhood of the origin, V is positive definite while dV/dt≤ 0, then the origin is Lyapunov stable. Theorem 2: If in some neighborhood of the origin, V and−dV/dt are both positive definite, then the origin is asymptotically Lyapunov stable. Theorem 3: If in some neighborhood of the origin, dV/dt is positive definite, and if V can take on positive values arbitrarily near the origin (but not necessarily everywhere in some neighborhood of the origin), then the origin is Lyapunov unstable. Using these theorems, the validity of the linearized variational equations can be established under appropriate conditions on the eigenvalues. Suppose the system is written in the form: dx dx 1 2 = ax + bx + F (x ,x ), = cx + dx + F (x ,x ) (28) 1 2 1 1 2 1 2 2 1 2 dt dt where F (x ,x )and F (x ,x ) are strictly nonlinear, i.e. they contain quadratic and higher 1 1 2 1 1 2 T T order terms. Writing this in vector form, where x=x x and F =F F , 1 2 1 2 dx = Ax + F(x) (29) dt Transforming to eigencoordinates y,weset x = Ty,where T is a matrix which has the eigenvec- tors of A as its columns, and obtain: dy −1 −1 = T ATy + T F(Ty)= Dy + G(y) (30) dt where D is a diagonal matrix (the theorem also holds if D is in Jordan form), and where −1 G = T F is strictly nonlinear in y.R.Rand Nonlinear Vibrations 12 Theorem 4: x = 0 is asymptotically Lyapunov stable if all the eigenvalues of A have negative real parts. Take V = y y ¯ + y y ¯,where y¯ is the complex conjugate of y.Then V so defined is certainly 1 1 2 2 i positive definite. For asymptotic stability we need to show that−dV/dt is positive definite. dV dy ¯ dy dy ¯ dy 1 1 2 2 = y + y ¯ + y + y ¯ (31) 1 1 2 2 dt dt dt dt dt Now we have that dy dy ¯ i i ¯ ¯ = λ y + G and = λ y ¯ + G (32) i i i i i i dt dt so that (31) becomes dV ¯ ¯ =(λ + λ ) y y ¯ +(λ + λ ) y y ¯ + cubic and higher order nonlinear terms (33) 1 1 1 1 2 2 2 2 dt which gives dV − =−2Re(λ ) y y ¯ −2Re(λ ) y y ¯ + cubic and higher order nonlinear terms (34) 1 1 1 2 2 2 dt Thus in some neighborhood of the origin, the cubic and higher order nonlinear terms in (34) are dominated by the quadratic terms, which themselves are positive definite if Re(λ ) 0for i i=1, 2. Thus−dV/dt is positive definite and the origin is asymptotically stable by Theorem 2. In a similar way we can use Theorem 3 to prove Theorem 5: x = 0 is Lyapunov unstable if at least one eigenvalue of A has positive real part. The idea of the proof is the same as for Theorem 4, except now take V = y y ¯ − y y ¯ if, for 1 1 2 2 example, Re(λ ) 0and Re(λ ) 0. (The case where Re(λ ) = 0 is more complicated and we 1 2 2 omit discussion of it.) An excellent reference on Lyapunov’s Direct Method is “Stability by Liapunov’s Direct Method with Applications” by J.P.LaSalle and S.Lefschetz, Academic Press, 1961.R.Rand Nonlinear Vibrations 13 2 The Duffing Oscillator The differential equation 2 d x 3 + x + αx =0, 0 (35) 2 dt is called the Duffing oscillator. It is a model of a structural system which includes nonlinear restoring forces (for example springs). It is sometimes used as an approximation for the pendu- lum: 2 d θ g + sinθ = 0 (36) 2 dt L 3 θ √ 5 Expanding sinθ = θ− + O(θ ), and then setting θ = x, 6   2 3 d x g x 2 + x−  =0( ) (37) 2 dt L 6  g Now we stretch time with z = t, L 2 3 d x x 2 + x−  =0( ) (38) 2 dz 6 which is (35) with α =−1/6. In order to understand the dynamics of Duffing’s equation (35), we begin by writing it as a first order system: dx dy 3 = y, =−x− αx (39) dt dt For a given initial condition (x(0),y(0)), eq.(39) specifies a trajectory in the x-y phase plane, i.e. the motion of a point in time. The integral curve along which the point moves satisfies the d.e. dy 3 dy −x− αx dt = = (40) dx dx y dt Eq.(40) may be easily integrated to give 2 2 4 y x x + + α = constant (41) 2 2 4 Eq.(41) corresponds to the physical principle of conservation of energy. In the case that α is positive, (41) represents a continuum of closed curves surrounding the origin, each of which rep- resents a motion of eq.(35) which is periodic in time. In the case that α is negative, all motions which start sufficiently close to the origin are periodic. However, in this case eq.(39) has two √ additional equilibrium points besides the origin, namely x = ±1/ −α, y = 0. The integral curves which go through these points separate motions which are periodic from motions which grow unbounded, and are called separatrices (singular: separatrix).

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