Dynamics and bifurcations of a coupled column-pendulum oscillator

Journal of Sound and Vibration (1995) 182(3), 393–413

DYNAMICS AND BIFURCATIONS OF A COUPLED COLUMN–PENDULUM OSCILLATOR G. M  A. E Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas 79409, U.S.A. (Received 1 October 1993, and in final form 25 February 1994) This study deals with the dynamics of a large flexible column with a tip mass–pendulum arrangement. The system is a conceptualization of a vibration-absorbing device for flexible structures with tip appendages. The bifurcation diagrams of the averaged system indicate that the system loses stability via two distinct routes; one leading to a saddle-node bifurcation, and the other to the Hopf bifurcation, indicating the existence of an invariant torus. Under the change of forcing amplitude, these bifurcations coalesce. This phenomenon has important global ramifications, in the sense that the periodic modulations associated with the Hopf bifurcation tend to have an infinite period, a strong indicator of existence of homoclinic orbits. The system also possesses isolated solutions (the so-called ‘‘isolas’’) that form isolated loops bounded away from zero. As the forcing amplitude is varied, the isolas appear, disappear or coalesce with the regular solution branches. The response curves indicate that the column amplitude shows saturation and the pendulum acts as a vibration absorber. However, there is also a frequency range over which a reverse flow of energy occurs, where the pendulum shows reduced amplitude at the cost of large amplitudes of the column. The experimental dynamics shows that the periodic motion gives rise to a quasi-periodic response, confirming the existence of tori. Within the quasi-periodic region, there are windows containing intricate webs of mode-locked periodic responses. An increase in the force amplitude causes the tori to break up, a phenomenon similar to the onset of turbulence in hydrodynamics.

1. INTRODUCTION

A large variety of mechanical structures may be modelled as flexible beams, with appendages attached along the span. For instance, as a model for an ‘‘autoparametric’’ vibration absorber, Haxton and Barr [1] studied a flexible column with tip mass fixed to a heavy block undergoing parametric vibration. Yoshizawa et al. [2] investigated the response of a simply supported beam, traversed by a heavy body with a pendulum, as an example of vibration of bridges. In recent years, large flexible structures with appendages have found widespread application in aerospace, due to the availability of durable lightweight materials. Generally, these lightweight materials have low internal damping and, in the absence of other forms of external energy dissipation mechanisms, such as air resistance, these structures are susceptible to large amplitude motions. The vibrations may be initiated by orbit or altitude maneuvers, deployment of satellites or weapons, or impacting meteorites and other space debris. In this paper, a new model for vibration absorbing device is proposed. This model of the large flexible structure consists of a flexible column, with an appendage consisting of a mass–pendulum attached to its tip. This type of device may find widespread application in, for instance, large space stations, where heavy sensitive equipment may have to be moved by long flexible arms. Any small motion of the base results in large amplitude 393 0022–460X/95/180393 + 21 $8.00/0

7 1995 Academic Press Limited

.   . 

394

vibrations of the arm and the payload, causing damage to the equipment. To absorb severe vibrations of the mass, a simple pendulum is attached, that, when properly tuned, will be capable of absorbing a large portion of the energy. In outer space applications, the effect of reduced gravity, which acts as a restoring force for the pendulum, may be compensated for by incorporating a torsional spring. 2. MATHEMATICAL FORMULATION

3.1.      The structure under investigation consists of a slender column, rigidly clamped at the base. The tip of the column consists of an appendage that comprises a lumped mass, and to this mass a pendulum is attached, as shown in Figure 1. The device is subjected to an external excitation along the undeformed axis of the column. It is assumed that the column behaves like an Euler–Bernoulli beam. The beam has a length L and stiffness EI. Oxy is the reference frame in the undeformed state. j is the deformed elastic axis of the beam. u(j, t) and v(j, t) are axial and transverse deflections with respect to the deformed axis at j, and at time instant t. Let u(j, t) be the angle that the beam section dj makes with the Ox-axis, at j = s. From the inextensibility condition (1 − u')2 + v'2 = 1, (1) one can write the bending moment at j = s as v0 . (2) z(1 − v'2) The moment of the forces in the u-direction acting on the beam element dj, located at j is M(s) = EI

Mu =

g

L

{rA(u¨ − g)}

s

g

j

sin u(h, t) dh dj

s

zXXcXXv zXXXcXXXv Force

Moment arm

Figure 1. A column with a tip appendage.

(3)

– 

395

Similarly, the moment of forces in the v-direction is Mv = −

g

g

L

(rAv¨ + cv˙ )

s

j

cos u(h, t) dh dj

(4)

s

zXXcXXvzXXXcXXXv Force

Moment arm

where c is the damping for the beam. The moment of the forces action along the u-direction at j = L is given as

g

MuL = [(M + m)(u¨ − g) + ml(f 2 cos f + f sin f)]j = L

L

sin u(h, t) dh.

(5)

s

zXXcXXv

zXXXXXXXXXcXXXXXXXXXv Force

Moment arm

The moment of the forces acting along the v-direction at j = L is

g

MvL = [(M + m)v¨ + ml(f cos f − f 2 sin f]j = L

L

cos u(h, t) dh

(6)

s

zXXcXXv

zXXXXXXXXcXXXXXXXXv Force

Moment arm

In equations (5) and (6), m and l are the mass and length of the pendulum, respectively, and f is the angular displacement, measured counterclockwise positive. Differentiating equations (2)–(6) twice with respect to s, and equating the terms resulting from the bending moment to those from inertia and the externally applied load, yields the equation for column dynamics as EI

$

%

v2 v03 + 3v'v0v1 3v'2v03 + 2 1/2 + 2 3/2 (1 − v' ) (1 − v' ) (1 − v'2)5/2

+ (1 − v'2)1/2(rAv¨ + cv˙ ) +

+

1 1s

6$g g 0 L

rA

s

j

0

v'v0 (1 − v'2)1/2

g

L

(rAv¨ + cv˙ ) dj

s

1

%7

v˙ '2 + v'v¨ ' v'2 + v˙ '2 dh dj v' 2 1/2 + (1 − v' ) (1 − v'2)3/2

+

1 v'v0 {[rA(x¨g − g)(L − s)]v'} + [(M + m)v¨ + ml(f cos f − f 2 sin f]j = L 1s (1 − v'2)1/2

+

1 1s

6$

g0 j

(M + m)

0

1 % 7

v˙ '2 + v'v¨ ' v'2 + v˙ '2 dh 2 1/2 + (1 − v' ) (1 − v'2)3/2

v'

j=L

1 {[(M + m)(x¨g (t) − g) + ml(f 2 cos f + f sin f)]j = L v'} = 0. (7) 1s Equation (7) is in terms of the transverse deformation, v(s, t), only. The effects of the axial deformation u(s, t) are related to v(s, t), via the inextensibility condition (1). While deriving equation (7), the non-linear terms arising due to curvature and inertia have been retained. The equation of motion for the pendulum is +

f +

$

%

d v¨ (L, t) g u¨ (L, t) [f + u (L, t)] + cos f + + sin f = 0, ml 2 l l l

(8)

where d is the damping at the pivot of the pendulum. If one includes the effect of the tip

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396

mass alone, then by dropping the non-linear terms, the following equation is obtained: EIv2 + rAv¨ + cv˙ +

6$ g rA

1 1s

%7

L

x¨ (j, t) dj + rA(x¨g − g)(L − s)M(x¨ )j = L + M(x¨g − g) v' = 0,

s

(9)

which is identical to that obtained by Handoo and Sundararajan [3]. In the above equation, x¨ (j, t) is the axial acceleration. Next, the hybrid parameter integro-differential equation (7) is transformed into an ordinary differential equation. This task is usually achieved by utilizing the weighted residual methods, such as the Galerkin procedure. This procedure is equivalent to projecting the dynamics on to a much lower, but finite-dimensional manifold. The dimension of this manifold depends on the number of modes that are excited. 2.2.   In applying the Galerkin method, one starts with the assumption that the solution v(s, t) can be represented by the modal series, and that the spatial and temporal variables are separable; thus, n

n Q a,

v(s, t) = s ryj (s)zj (t),

(10)

j=1

where r is some scaling factor and yj (s) belong to a set of orthonormal functions, usually a solution of the corresponding linear eigenvalue problem. zj (t) are the unknown time modulations of the corresponding eigenfunctions or modes. In truncating the modal series, it is implicitly assumed that as one includes higher modes, the solution v(s, t) converges to the true solution. The Galerkin procedure consists of substituting equation (10) into the partial differential equation, multiplying the resulting equation by yj (s) and integrating over the domain. Then, using orthogonality of the linear eigenfunctions and certain other relationships obtained by integration by parts, we obtain a set of ordinary differential equations, defining the time evolution zj (t) of the corresponding linear modes. One has thus transformed a PDE defined on infinite-dimensional function space to an ODE defined on a finite-dimensional Euclidean space. In applying the Galerkin method to the present problem, the modal series will be truncated at the first mode, so that v(s, t) = ry(s)z(t).

(11)

It has been confirmed in laboratory observations that this is a physically viable restriction, particularly if the excitation frequency is close to some integral multiple of the first natural frequency. Substituting equation (11) into the partial differential equation (7) and orthogonalizing the error with respect to the eigenfunction, we obtain the following pair of ODE’s (the details of the procedure are in reference [4]): z¨ + zz + (v12 + G12 x¨g )z + az˙z 2 + bzz˙ 2 + (g1 + G2 x¨g )z 3 + m1 (f 2 cos f + f sin f)z + m2 (f cos f − f 2 sin f)z 2 + m3 (f cos f − f 2 sin f)z 3 = 0,

$

f + a1 z¨ cos f + v22 +

%

x¨g + b1 (z˙ 2 + zz¨ ) sin f + z1 f + z2 z˙ = 0. l

(12) (13)

–  397 Introducing the displacements and the velocities as the coordinates of the phase space, assuming the excitation to be of the form x¨g (t) = f0 cos vt, and re-scaling the time vt:t, z'1 = z2 z'2 = −

01 6 $ 01 $ 01 01 % $0 1 % 7 v1 v

z1 + −f1 cos tz1 − a1 z1 − a2 + a3 v1 v

−bz1 z22 − a1 m1

v2 v

+a7 z1 z2 f1 + f'1 = f2 , f'2 = −

01 v2 v

v1 v

2

2

− m2

v1 v

+ f3 cos t z12 f1 + a5 z12 f2 + a6 z1 f1 f2

2

+ f4 cos t z1 f12 − m1 z1 f22 ,

6

2

f1 + −f4 cos tf1 − b1 z12 z2 − b2 z1 z2 f1 − b3 z2 − b4 z12 f2 − b5 z12 f1 f2

$

− b12 − b11

$0 1 v1 v

+ f2 cos t z13 + a4 z12 z2

2

$ 01 % $01 01 % $0 1 %7

−b6 f2 − b7 z2 f12 + b8 z1 z22 − b1 z22 f1 + b9 + b10

+a1

%

2

01 v1 v

%

2

+ f6 cos t z12 f1 −

%

2

a1 v1 2 v

+ f8 cos t z1 + b14 z1 f22 + 16

v2 v

v1 v

2

2

+ b13

+ f5 cos t z13 v2 v

2

+ f7 cos t z1 f12

2

+ f4 cos t f13 .

(14)

Equation (14) is the Taylor expansion of the vector field about the equilibrium, and contains cubic order terms. The coefficients in equations (12), (13) and (14) result from the Galerkin modal truncation and Taylor series expansion, respectively, and are given in reference [4]. 2.3.     – To obtain the averaged dynamical system, equation (14) is rewritten as

01 01

z'1 = z2 ,

z'2 = −

f'1 = f2 ,

f'2 = −

v1 v

v2 v

2

z1 + eF(z1 , z2 , f1 , f2 , t), 2

f1 + eG(z1 , z2 , f1 , f2 , t),

(15)

where z $ R4, and smooth functions F and G are 2p-periodic in t. e is some small parameter, sometimes a system parameter such as damping, coupling between the modes or forcing, more often introduced artificially to stress the fact that the perturbations (mostly the non-linearities and excitation terms) are small. Introduce a small detuning parameter ed into the system to indicate that one is interested in obtaining the averaged dynamics, while passing through the (m, n) resonance region. The integers m and n indicate external and internal frequency ratios, respectively, i.e., mv1 1 v and nv1 1 v2 . v1 , v2 and v correspond to the column natural frequency, the pendulum natural frequency and the frequency of excitation, respectively. More precisely, let v12 = v 2/m 2 − ed/m 2;

v22 = n 2v 2/m 2 − en 2d/m 2.

(16)

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398

Substituting equation (16) in equation (15) gives a detuned (m, n) system as

F z'1 J G G G z'2 G G G= G f'1 G G G f f'2 j

F 0 G G− 12 G m G 0 G f 0

0J

1

0

0

0

0

0

0

n2 − 2 m

G 0G G 1G G 0j

Fz1J F J 0 G G G G Gz2G GF(z1 , z2 , f1 , f2 , t, d)G G G +e G G, 0 Gf1G G G G G G G ff2j fG(z1 , z2 , f1 , f2 , t, d)j

z' = Az + ef(z, t, d).

(17)

To transform the system (17) into the standard form (see Guckenheimer and Holmes [5]), we use the van der Pol invertible transformation t F G cos m z1 G z2 1 t − sin G f m m

t m

01

J G y1 1 tG y − cos G 2 m mj

nt F G cos m f1 G n nt f2 − sin G m f m

nt J m G y3 . n nt G y − cos G 4 m mj

01

−sin

−sin

01 01

(18)

Introduce equation (18) in equation (17): y' = eF(t)−1f(F(t)y, t, d),

(19)

where F(t) is the transformation in equation (18). Following the method of averaging and transforming the non-dimensional time t to the slow time et, i.e., t  et, and integrating the right side of equation (19) over (0, 2p), we obtain the detuned (m, n) averaged system y' = (y, f d),

(20)

where (y) f =

1 2p

g

2p

F(t)−1f(F(t)y, t, d) dt.

(21)

0

The averaging is performed over the slow time. The integration indicated in equation (21) was performed on MAPLE. The coefficients of the averaged system generated by MAPLE for (2,1/2) resonance are given by Mustafa [4]. The averaged system (20) represents an approximation to the Poincare´ map of the original time-dependent system (17). Hence, the steady state solutions of the averaged system correspond to 2p/v-periodic solutions of the original system. The periodic solutions of the averaged system indicate the existence of invariant tori in the phase space of the original system. Furthermore, one can also relate the bifurcations of the averaged system to that of the original system. Under certain conditions (see Guckenheimer and Holmes [5]), if the averaged system undergoes a saddle-node or a Hopf bifurcation at a certain parameter value, then the Poincare´ map of the time-dependent system undergoes a similar bifurcation close to the parameter value (for e small).

– 

399

2.4.    Any study of the dynamics of the averaged system (20) entails obtaining its steady states. One is therefore faced with the task of computing the roots of the averaged vector field f(y, d) = 0;

y $ R4, d $ IUR

and

f :R4 × R:R4;

(22)

in particular, how these steady states are created, destroyed or change stability as the control parameter (d) is varied. In this paper, the steady state solutions of the averaged system (20) are obtained by the pseudo-arclength continuation algorithm due to Keller [6] (also Keller [7]). The computational details are given in Parker and Chua [8] and Kubicˇek and Marek [9]. The bifurcation diagrams are obtained by considering the detuning parameter d as the bifurcation parameter. Changing the parameter d corresponds to varying the excitation frequency v about the resonance under consideration. The averaged system is four-dimensional, i.e., y = (y1 , y2 , y3 , y4 ); to draw the solution diagrams, the norm of the solution vector is split in two. The first half, r1 = zy12 + y22 , corresponds to the beam amplitude and the second half, r2 = zy32 + y42 , to the pendulum amplitude, respectively.

3. RESULTS AND DISCUSSION

3.1.   This section contains details of the averaged dynamics of the column–pendulum oscillator for the chosen external and internal resonance combination (2, 1/2), where the excitation frequency is twice the natural frequency of the first column mode, and the linearized pendulum frequency is one-half that of the first column mode. The results obtained apply to the system with following parameters: EI = beam flexural rigidity = 4·0096 × 107 N − mm2; rA = beam density = 3·133 × 10−4 kg/mm, L = length of the beam = 432·0 mm, M = tip mass = 0·204 kg, and m = pendulum mass = 0·0544 kg. With these parameters, the natural frequency v1 of the column with tip mass was calculated as 2·264 Hz. Setting the pendulum frequency v2 = 1/2v1 the length of the pendulum was calculated as l = 193·84 mm. The scaling factor r, which corresponds to the arbitrary constant in the linear eigenvalue problem, was set at 25·0; this corresponds to approximately a 1 inch of tip displacement of the column. With these values of the parameters, the numerical values of the coefficients of the ordinary differential equations were computed and are given in Mustafa [4]. The linear damping coefficients for the beam and the pendulum z and z1 , respectively, were determined experimentally using logarithmic decrement, and came out to be 0·165 and 0·170, respectively. To obtain the steady state solution diagrams, the damping coefficients (z, z1 ) and the forcing amplitude f0 were varied, grouped as follows: (1) case I, (z, z1 ) = (0·165, 0·17), 3·0 E f0 E 10·0; (2) case II, (z, z1 ) = (0·165, 0·25), 3·0 E f0 E 10·0; (3) case III, (z, z1 ) = (0·10, 0·170), 2·0 E f0 E 12·0. 3.2.  : (z, z1 ) = (0·165, 0·170), 3·0 E f0 E 10·0. For this case, a brief description of the overall dynamics is given. Cases II and III deal with specific phenomena. From the application point of view, a very significant behavior observed is the saturation of the beam mode. The bifurcation diagram for f0 = 3·0 is shown in Figure 2. The arrows indicate the direction of solution; physically, it corresponds to sweeping from a higher frequency to a lower one, about the resonance at d = 0. Referring to Figure 2, soon after the beam mode is excited, it tends to saturate. On the other hand, the pendulum mode continues to grow. This behavior indicates that after the beam mode

400

.   . 

Figure 2. The solution diagram for (z, z1 ) = (0·165, 0·170), f0 = 3·0. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2).

saturates, the additional input energy is spilled over to the pendulum, which continues to show increase in amplitude. Furthermore, the beam mode shows a decrease with the increase in the pendulum amplitude. Based on this observation, it is clear that for f0 = 3·0 and −0·23 Q d Q 0·2, the pendulum acts as a vibration absorbing device. However, there is also a region of reversed action, in which the pendulum amplitude shows a sharp decrease, accompanied by a sharp increase in the beam response. The ‘‘reversed’’ flow of energy causes the system to become unstable via a saddle-node, indicated by S1 in Figure 2; the unstable solution is indicated by a dashed line. At S1, the beam response shows a sudden increase, while that of the pendulum experiences a sudden decrease. Between S1 and S2, d $ (−0·2, −0·262), the system has multiple solutions, two stable separated by one unstable solution. To which stable solution the column–pendulum oscillator settles depends on whether the initial conditions lie in the basin of attraction of one stable solution or the other. Clearly, it is not advisable to set the operating range between S1 and S2, where the system shows dependence on initial conditions, hysteresis and jumps. When f0 is increased beyond 3·05, the root locus crosses the imaginary axis transversely, in accordance with the Hopf bifurcation theorem, giving rise to periodic solution, with period 2p/=b=, where b is the purely imaginary eigenvalue. The bifurcation diagram for f0 = 4·0 is shown in Figure 3; H represents the Hopf bifurcation point, and marks the beginning of a branch of periodic solutions. Since the continuation algorithm traces the steady state solutions, one is unable to follow this branch of periodic solutions emanating from H in Figure 3. One of the periodic solutions for f0 = 3·10 is shown in Figure 4. This closed orbit indicates the existence of an invariant torus in the phase space of the original system. The motion on this torus has two frequencies; one with time period 2p/v, and the other generated by the Hopf bifurcation with period 2p/=b=. The motion is periodic if v/=b= is a rational number p/q, and is quasi-periodic if the frequency ratio is irrational. With the increase in the forcing amplitude, the Hopf bifurcation point H and the saddle-node S1 located close to it move about on the solution curve. The details of this movement will be made clear in case III. At f0 = 4·0, the root locus becomes tangent to

– 

401

Figure 3. The solution diagram for (z, z1 ) = (0·165, 0·170), f0 = 4·0. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2); w, Hopf bifurcation point (H).

the imaginary axis, at the origin, indicating the coalescence of two primary bifurcations; a steady state bifurcation corresponding the eigenvalue 0 (i.e., a saddle-node) and a Hopf bifurcation corresponding to the eigenvalue (+ib, −ib). This coalescence makes the saddle-node, S1, a degenerate singularity of the averaged vector field. There is some work due to Langford [10], relating the interaction of steady state and Hopf bifurcation, indicating that these generic co-dimension one bifurcations can interact non-linearily to create very complex dynamics, such as quasi-periodicity, phase-locking, intermittency, homoclinic orbits, appearance of invariant tori and various type of aperiodic and chaotic behavior. The bifurcation diagrams obtained for case I, indicate that the Hopf bifurcation exits for 3·05 Q f0 Q 4·3, with detuning parameter range d $ (−0·2649, −0·3712). For this parameter range, between the saddle-nodes S1 and S2, the averaged system possesses multiple solutions; two stable solutions, and two types of unstable solution (one due to the saddle-node, and the other created by the Hopf bifurcation).

Figure 4. The periodic solution as a result of Hopf bifurcation for (z, z1 ) = (0·165, 0·170), f0 = 3·10. (a) (y1 , y2 )-plane, (b) (y3 , y4 )-plane.

402

.   . 

Another significant behavior of the averaged system is the creation of isolated solutions. An increase in the forcing amplitude beyond 4·18 causes the solution curves to break into two. One corresponding to the primary solution which starts and ends at the zero solution, and the second forming an isolated closed curve, bounded away from the zero solution. Such solutions have been termed ‘‘isolas’’ in the bifurcation literature. As a parameter is varied, such as the forcing amplitude, the isolas change shape and orientation or collapse at a point, called the isola center. Details are deferred to case II. Isolas are an elusive set of solutions, since they are difficult to detect. In most bifurcation problems, one normally begins by a known solution and follows its evolution as a control parameter is varied. Unless at least one point on an isola is known, a continuation algorithm cannot detect isolated solutions. Kubicˇek, Stuchl, and Marek [11] give an algorithm for detecting isolas. As f0 is increased, the isola re-unites with the primary solution. For case I, isolas exist for 4·18 Q f0 Q 6·58. Within this parameter range, the averaged system possesses isolated solutions, in addition to multiple solutions. Details of isolated solutions are given in case II. 3.3.  : (z, z1 ) = (0·165, 0·25), 3·0 E f0 E 10·0 For this case, the behavior of the isolated solutions is discussed. At some critical value of the parameter f0 = 4·875, the primary solution curve starts to break up at the saddle-node S1, shown in Figure 5. An increase in the forcing amplitude beyond this critical value causes the isola to re-orient and undergo bifurcations. For the beam, an increase in f0 causes the isolated solution to elongate, while remaining mostly inside of the primary solution. For the pendulum, the isolated solution shrinks and moves away from the primary solution. The isola undergoes its first bifurcation when the Hopf bifurcation coalesces with the saddle-node, resulting in a degenerate saddle, and accompanies a distortion of the isola, shown in Figure 6, for f0 = 8·0. Note the burst of saddle-nodes on the isolated solution. The primary solution does not show any qualitative change. Further increase in f0 makes S4 approach the primary curve and, finally referring to Figure 7, at

Figure 5. The solution diagram for (z, z1 ) = (0·165, 0·25), f0 = 4·875. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2); w, Hopf bifurcation point (H). Formation of isola.

– 

403

Figure 6. The solution diagram for (z, z1 ) = (0·165, 0·25), f0 = 8·0. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2, S3, S4).

Figure 7. The solution diagram for (z, z1 ) = (0·165, 0·170), f0 = 6·62. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2, S3, S4). The re-unification of the isola with the primary solution.

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.   . 

Figure 8. The solution diagram for (z, z1 ) = (0·165, 0·170), f0 = 10·0. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2, . . . , S6). ‘‘Softening’’ behavior at higher forcing amplitudes.

some critical value of f0 , two curves coalesce, marking the re-unification of the isolated solution with the primary curve. An interesting observation is that the formation and the re-unification of the isolated solution take place at different locations on the primary curve. For higher values of the forcing amplitude, beam and the pendulum show a ‘‘softening’’ behavior by creating two additional saddle-nodes and bending backwards, as is shown in Figure 8. 3.4.  : (z, z1 ) = (0·10, 0·170), 2·0 E f0 E 12·0 In this subsection, some details of how the Hopf bifurcation points are created and the mechanism under which these coalesce with the saddle-nodes are presented. The Hopf bifurcation point for this case first appears at f = 2·3, as a result of the root locus becoming tangent to the imaginary axis. In the neighborhood of the point tangency H, the root locus lies entirely on the left of the imaginary axis. Increasing the forcing amplitude causes a part of the curve to intersect transversely with the imaginary axis. In fact, there are four points of intersection, two above and two below the real axis. The original Hopf point H has split in two, H1 and H2 as shown in Figure 9, f0 = 2·50. The dashed line between H1 and H2 represents a branch of unstable steady state solution; however, a branch of periodic solutions exists. This branch of unstable steady state solution; however, a branch of periodic solutions exists. This branch of periodic solutions emanates from H1 and ends at H2. It is highly probable that this branch of periodic solutions undergoes bifurcation itself. If such is the case, then there are several possibilities, as discussed in the following paragraphs. The bifurcational behavior of the periodic solution depends on the eigenvalues (the Floquet multipliers) of the monodromy matrix. For averaged dynamics of the column–pendulum oscillator, there will be four multipliers (l1 ,l2 , l3 and l4 ). The multiplier associated with the periodic orbit is always 1·0; let l4 = 1·0. If the moduli of the rest of the three multipliers is less than 1, then the periodic orbit is stable. If any of these

– 

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Figure 9. The solution diagram for (z, z1 ) = (0·10, 0·170), f0 = 2·5. (a) Column response; (b) pendulum response. ——, Stable; – – – , unstable; q, saddle-nodes (S1, S2); w, Hopf bifurcation points (H1, H2). The arrows on S1, H1 and H2 indicate the direction of movement of these points as f0 is increased.

Figure 10. Variation of the time period Tperiod of the Hopf bifurcated solutions with (a) the forcing amplitude f0 and (b) the detuning parameter d.

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.   . 

multipliers crosses the unit circle, the periodic solution becomes unstable. The type of instability depends on how these multipliers cross the unit circle. For this particular problem, there are three different ways in which the stable periodic solution can lose stability. If one of the multipliers leaves the unit circle along the real axis through −1, then the originally stable T-periodic solution becomes unstable, and a subharmonic periodic solution of period 2T, called the period-doubling bifurcation, is created. Under the change of parameter d, the system can show a sequence of period-doubling bifurcations, the so-called Feigenbaum scenario. If a multiplier leaves the unit circle along the real axis through +1, the stable solution gives rise to a saddle-cycle or flip bifurcation. Finally, if a complex conjugate pair of multipliers leave the unit circle, the originally stable T-periodic solution gives rise to a quasi-periodic solution lying on an invariant two-torus, sometimes referred to as the torus bifurcation. As in the case of period-doubling bifurcations, the system can show a sequence of torus bifurcations. Each such bifurcation

Figure 11. Root loci for (a) f0 = 2·3, (b) f0 = 2·7, (c) f0 = 2·85 and (d) f0 = 4.257. The points H, H1, H2 and S1 represent bifurcation points on the solution diagrams.

– 

407

introduces a new frequency into the solution, thus increasing the dimension of the invariant torus by one. Whereas there is no theoretical upper limit on the number of period-doubling bifurcations, Ruelle and Takens [12] suggest that only two bifurcations to invariant tori are generically possible. A third bifurcation makes the torus structurally unstable, and gives rise to a strange attractor (a term coined by Ruelle and Takens). The Ruelle–Takens scenario is the most widely accepted model for the onset of turbulence in fluids. The Ruelle–Takens conjuncture has proved to be of great significance in the experimental dynamics of the column–pendulum oscillator, and will be discussed in the next section. The above discussed behavior is likely to occur between H1 and H2 (Figure 9), the points at which the periodic solutions start and end. Increasing f0 causes the Hopf bifurcation points H1 and H2 to move apart, while the Hopf point H2 and the saddle-node S1 approach each other, as indicated in the insets of Figure 9. Finally, at f0 1 2·85, these two points coalesce. Some possible outcomes of such a coalescence have already been discussed (refer to Langford [10] for details). The periodic solution branch that originates from H1 has possibly undergone several subharmonic as well as torus-adding bifurcations, finally culminating at S1, which may be thought of as a periodic solution of infinite period. As S1 and H2 approach each other, the periodic modulations associated with the Hopf bifurcation tend to have an infinite period, as is clear from Figure 10. For f0 e 2·85, there is only one Hopf point. Beyond f0 = 2·85, S1 and H move in the same direction; however, the speed of S1 on the solution curve is higher than that of H, and so finally, at f0 = 4·257, S1 catches up with H. Details of all this behavior are explained on the complex plane of eigenvalues of the Jacobian. Root loci for f0 = 2·3. 2·7, 2·85 and 4·257, respectively, as shown in Figure 11. In Figure 11a is shown the root locus for f0 = 2·3, when it becomes tangent to the imaginary axis. Small perturbations have a qualitatively significant effect on the dynamics, in the sense that either the curve moves away from the imaginary axis, in which case no bifurcation occurs, or it intersects the imaginary axis transversely, in accordance with the Hopf bifurcation theorem, and gives rise to a periodic solution. The latter situation is shown in Figure 11(b), for f0 = 2·7. H1 and H2 mark points at which the periodic solutions begin and end (referring to Figure 9). An increase in f0 causes the point H2 to move closer to the origin, where S1 is located. At f0 = 2·85, H2 and S1 coalesce as shown in Figure 11(c). For 2·3 Q f0 Q 2·85, the steady state solution loses stability due to a Hopf bifurcation at H1; moving along the arrows of Figure 9, H1 represents a supercritical Hopf bifurcation point. This branch of periodic solution that started at H1 ends at H2, which is a subcritical Hopf bifurcation point. This assignment of H1 and H2 is based on the assumption that locally the periodic branches lie to the left of H1 and to the right of H2. The notion is based on brute-force location of the periodic solutions in the neighborhood of H1 and H2, and seems to justify the assignment. Far away from H1 and H2, the branches of periodic solutions may wind back and forth. Referring to Figure 11(d), increasing f0 beyond 2·85 causes H1 to move closer to the origin and finally collapse at f0 = 4·257, when the root locus becomes tangent to the imaginary axis at the origin, where S1 is located. Here, the Jacobian has three zero eigenvalues. An important observation here is that for 2·85 Q f0 Q 4·257, a part of the root locus stays tangent to the origin. For this range of f0 , there is one Hopf point H1 and a saddle-node S1. The saddle-node S1 is not the ordinary ‘‘generic’’ bifurcation point, in the sense that it has already consumed one Hopf point H2, and is located at a point on the solution diagram where the root locus is tangent to the imaginary axis, making S1 a degenerate singularity of the averaged vector field. Since the nature of S1 does not change over this range of f0 , we expect very complex global behavior of the averaged system to persist. One can thus split the interval of existence of Hopf points for case III (2·3 Q f0 Q 4·257) in two

.   . 

408

sub-intervals. For the first sub-interval, 2·3 Q f0 Q 2·85, there are two Hopf points, H1 and H2. The branch of periodic solutions starts at H1 and ends at H2. The second sub-interval 2·85 Q f0 Q 4·257 contains a single Hopf point H1. For this sub-interval, the periodic branch that originates from H1 may end with a homoclinic orbit at S1.

4. REMARKS ON THE AVERAGED DYNAMICS

From the application point, it is observed that the column mode shows saturation and the pendulum response indicates relatively large amplitude motion. Over the range in which this behavior is observed, the pendulum acts as a viable absorber. However, there is also a response range over which a reversed flow of energy occurs and the pendulum response is drastically reduced; the column, on the other hand, shows a sudden increase in amplitude. These two types of distinct responses are separated by the first type of instability, caused by saddle-nodes. This type of instability gives rise to jumps and hysteresis in physical systems. Between the saddle-nodes, there are multiple solutions, some stable and others unstable. For low amplitudes of excitation, the solution curves are fairly well behaved. As the amplitude of excitation is increased, the second type of instability, caused by the Hopf bifurcation, sets in. This instability gives rise to quasi-periodic motions of the actual system. The results presented in this paper suggest that the two types of instabilities interact in a very complex manner. The numerical evidence suggests that as the saddle-nodes approach the Hopf bifurcation points, the periodic modulations associated with these points tend to have an infinite period. Such long-period solutions indicate that there is a homoclinic orbit in the neighborhood of the point at which these two types of instabilities coalesce. This coalescence sets the stage for a myriad of global phenomena, such as the existence of invariant tori, period-doubling cascades, intermittency and chaos, etc. These possibilities have to be explored. For the intermediate range of the forcing amplitude, in addition to the primary solution curves, a family of isolated solutions is formed. The isolated solutions form closed loops, bounded away from zero. As the forcing amplitude is increased, a part of the primary solution curve dissociates, forming the isolated loops called ‘‘isolas’’. The isolas wiggle and move with the increase in the forcing amplitude, and finally re-unite with the primary solutions. Since isolas are far removed they are not detected, unless the initial condition happens to lie in the basin of attraction of such solutions. A summary of the ranges of existence of Hopf points and isolated loops is given in Table 1. For higher amplitudes of forcing, the solution curves are very convoluted, and several stable and unstable branches of solutions criss-cross each other. All this complex behavior lies on the lower end of the frequency axis, for d Q 0. Furthermore, parts of the solution curves show a ‘‘softening’’ type behavior. This behavior is more dominant in the pendulum response.

T 1 Ranges of existence of Hopf points and isolas Case I Case II Case III

Hopf points

Isola

3·05 Q f0 Q 4·30 4·50 Q f0 Q 6·24 2·30 Q f0 Q 4·275

4·18 Q f0 Q 6·58 4·875 Q f0 Q 8·54 4·41 Q f0 Q 6·16

– 

409

Figure 12. The motion on T 2 for f0 = 725 mV. (a) Cover of T 2; (b) slice of T 2; (c) the first return map. The (5, 11)-periodic response.

Qualitatively speaking, the most significant behavior detected is the existence of quasi-periodic solutions due to the Hopf bifurcation, and the coalescence of these with the saddle-nodes. This type of motion forms the basis of the experimental dynamics of the column–pendulum oscillator. The experimental investigation may be looked upon as an extension of the averaged dynamics. Since the periodic solutions of the averaged system were not followed, one is unable to confirm the bifurcations associated with these solutions. However, there is no such limitation on the experimental system, and so one can study the quasi-periodic behavior of the experimental system in greater detail. 5. EXPERIMENTAL EVIDENCE OF QUASI-PERIODICITY AND ITS BREAK-UP

The details of the experimental dynamics of the column–pendulum oscillator are in Mustafa [4], and are the subject of a forthcoming paper. Here, a brief description of the quasi-periodic behavior is presented. The parameters chosen are the same as for the averaged system. In the averaging analysis, no internal detuning was assumed; however, in the experimental setting some detuning is inevitable, and so the measured frequency ratio is v2 /v1 1 0·48. The column displacement is measured by attaching a piezo film close to the base. The pendulum amplitude is measured by the opto-digital angular measurement system (see Ertas and Mustafa [13]). Assuming that only the first column mode is excited, the phase space of the experimental system is four-dimensional. The remaining two vectors are created by delaying the measured quantities, in accordance with the delay co-ordinate embedding theorem of Takens [14]. Experiments were performed for three amplitudes of excitation, f0 = 475, 650, and 750 mV, corresponding to the gain of the shaker table power amplifier. To illustrate how

410

.   . 

Figure 13. The motion on T 2 for f0 = 740 mV. (a) Cover of T 2; (b) slice of T 2; (c) the first return map. The quasi-periodic response.

Figure 14. The break-up of T 2 for f0 = 775 mV. (a) Cover of T 2; (b) slice of T 2; (c) the first return map.

– 

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Figure 15. Time histories and FFT spectra for (a) f0 = 725 mV, (b) f0 = 740 mV and (c) f0 = 775 mV.

the periodic motions give rise to the quasi-periodic ones, consider Figure 12. This figure shows the cover of the two-dimensional torus, for f0 = 725 mV, filled with a periodic orbit. The part of the orbit that runs off at the top re-appears at the bottom, and those that leave at the right re-enter at the left. The response is (m, n)-periodic if its orbit intersects the u1 -axis m times and the u2 -axis n times. Thus, the response in Figure 12(a) is (5, 11)-periodic. The parallel lines on the cover indicate a slice of the torus. For an (m, n)-periodic response, the slice of the torus shows m points; refer to Figure 12(b). The return map,

412

.   . 

Figure 12(c), is extracted from points lying on the slice, for an (m, n)-periodic orbit, f1m (u1 ) = u1 . For the quasi-periodic response, the cover of the torus becomes dense with orbits and the slices are smooth circles, representing cross-sections of the torus. The return maps are circle maps. Increasing the forcing amplitude to 740 mV, causes the (5, 11)-periodic window to expand and overlap with the neighboring periodic windows, giving rise to resonance overlap that consequently results in the destruction of the periodic response. The motion on the torus is quasi-periodic, the cover of which is now dense with orbits as shown in Figure 13(a). The slice of T 2 is a circle (Figure 13(b)) and the return map (Figure 13(c)) is a one-to-one circle map. A further increase in the forcing amplitude to 775 mV causes more frequencies to appear in the response, resulting in the break-up of the torus, and the corresponding non-invertibility of the return map (refer to Figure 14). This phenomenon is further illustrated by Figure 15, which shows the time histories and the FFT spectra. It is clear that, with the increase in the forcing amplitude, more frequencies creep into the response. In conclusion, the observed behavior of the experimental oscillator provides ample evidence that the underlying dynamics is that of the map of a torus to itself. The oscillator shows rich dynamics, consisting of quasi-periodic motions, interrupted by webs of mode-locked resonant periodic windows. With each window, one can associate integers (m, n) that characterize the periodic response. An increase in the forcing amplitude causes these periodic windows to interact, consequently destroying the torus. The dynamics observed is similar to the hydrodynamical instability in the Taylor–Couette flow in concentrically located cylinders (see Swinney and Gollub [15]). The breakup of the torus in Taylor–Couette flow is related to onset of turbulence, in accordance with the Ruelle–Takens–Newhouse criterion (refer to Newhouse, Ruelle and Takens [16]).

REFERENCES 1. R. S. H and A. D. S. B 1972 Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 94, 119–125. The autoparametric vibration absorber. 2. M. Y, I. K and Y. T 1984 Bulletin of the Japan Society of Mechanical Engineers 27, 779–785. Flexural vibration of a simply supported beam under the action of a moving body with a pendulum. 3. K. L. H and V. S 1971 Journal of Sound and Vibration 18, 45–53. Parametric instability of a cantilevered column with end mass. 4. G. M 1992 Ph.D. Dissertation, Texas Tech University, Lubbock, Texas. Dynamics and bifurcations of a column–pendulum oscillator: theory and experiment. 5. J. G and P. H 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York: Springer-Verlag. 6. H. B. K 1977 in Applications of Bifurcation Theory (P. H. Rabinowitz, editor). New York: Academic Press. Numerical solution of bifurcation and nonlinear eigenvalue problems. 7. H. B. K 1987 Lectures on Numerical Methods in Bifurcations Problems. New York: Springer-Verlag. 8. T. S. P and L. O. C 1989 Practical Numerical Algorithms for Chaotic Systems. New York: Springer-Verlag. 9. M. K˘ and M. M 1983 Computational Methods in Bifurcation Theory and Dissipative Structures. New York: Springer-Verlag. 10. W. F. L 1983 in Nonlinear Dynamics and Turbulence (G. Barenblatt, G. Iooss and D. D. Joseph, editors). London: Pitman. A review of interactions of Hopf and steady state bifurcations. 11. M. Kˇ, I. S. S and M. M 1982 Journal of Computational Physics 48, 106–116. ‘‘Isolas’’ in solution diagrams. 12. D. R and F. T 1971 Communications in Mathematical Physics 64, 167–192; 23, 343–344. On the nature of turbulence.

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13. A. E and G. M 1992 Experimental Techniques 16, 33–35. Real-time response of the simple pendulum: an experimental technique. 14. F. T 1981 in Dynamical Systems and Turbulence (D. S. Rand and L.-S. Young, editors). Berlin: Springer-Verlag. Detecting strange attractors in turbulence. 15. H. L. S and J. P. G 1978 Physics Today 31, 41–50. The transition to turbulence. 16. S. E. N, D. R and F. T 1978 Communications in Mathematical Physics 64, 35–40. Occurrence of strange axiom A attractors near quasiperiodic flows on T m, m e 3. APPENDIX A: AVERAGED DYNAMICAL SYSTEMS

The averaged dynamical system, equation (20), in a compact form, is given by a a a a a a a a y'a = c1d dy1 + c2d dy2 + c3d dy3 + c4d dy4 + c1000 y1 + c0100 y2 + c0010 y3 + c0001 y4 a a a a a +c1110 y1 y2 y3 + c1101 y1 y2 y4 + c1011 y1 y3 y4 + c0111 y2 y3 y4 + c2100 y12 y2 a a a a a a +c2010 y12 y3 + c2001 y12 y4 + c3000 y13 + c0120 y2 y32 + c0102 y2 y42 + x0210 y22 y3 a a a a a a +c0201 y22 y4 + c0300 y23 + c0012 y3 y42 + c0021 y32 y4 + c0030 y33 + c0003 y43 , a for a = 1, 2, 3, 4, where ckd are the coefficients of dyk , and d is the detuning parameter. The a non-zero coefficients cijkl generated by MAPLE for the (2, 1/2) resonance are given in Mustafa [4].