Singularities in soft-impacting systems

Physica D 241 (2012) 553–565

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Physica D journal homepage: www.elsevier.com/locate/physd

Singularities in soft-impacting systems Soumya Kundu a , Soumitro Banerjee b , James Ing c , Ekaterina Pavlovskaia c,∗ , Marian Wiercigroch c a

Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbour, USA

b

Indian Institute of Science Education & Research, Kolkata, Mohanpur Campus, Nadia-741252, WB, India

c

Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, UK

article

info

Article history: Received 22 April 2011 Received in revised form 29 September 2011 Accepted 21 November 2011 Available online 28 November 2011 Communicated by G. Stepan Keywords: Impact oscillator Grazing Nonsmooth system

abstract In this paper, the character of the normal form map in the neighbourhood of a grazing orbit is investigated for four possible configurations of soft impacting systems. It is shown that, if the spring in the impacting surface is relaxed, the impacting side of the map has a power of 3/2, but if the spring is pre-stressed the map has a square root singularity. The singularity appears only in the trace of the Jacobian matrix and not in the determinant. Under all conditions, the determinant of the Jacobian matrix varies continuously across the grazing condition. However, if the impacting surface has a damper, the determinant decreases exponentially with increasing penetration. It is found that the system behaviour is greatly dependent upon a parameter m, given by 2ω0 /ωforcing , and that the singularity disappears for integer values of m. Thus, if the parameters are chosen to obtain an integer value of m, one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non-impacting mode to an impacting mode with increasing excitation amplitude. The above result has been tested on an experimental rig, which showed a persistence of a period-1 orbit across the grazing condition for integer values of m, but an abrupt transition to a chaotic orbit or a high-period orbit for non-integer values of m. Finally, through simulation, it is shown that the condition for vanishing singularity is not a discrete point in the parameter space. This property is valid over a neighbourhood in the parameter space, which shrinks for larger values of the stiffness ratio k2 /k1 . © 2011 Elsevier B.V. All rights reserved.

1. Introduction Many dynamical systems are encountered in nature and in engineering, where there is a possibility of impact occurrence between elements of the system. Rich in nonlinear dynamical behavior, these impacting systems have been studied extensively over the years. In the mid-seventies of the last century, Feigin (e.g. [1]) unveiled their bifurcation structure, and a comprehensive treatment of such systems can be found in [2]. Whiston [3] and Chillingworth [4] have provided solid theoretical background to investigate impact systems. In the papers by Peterka and Vacik [5], Ivanov [6] and Lenci and Rega [7], a transition to chaos and its stabilization was investigated. Luo and his co-workers [8–10] extensively studied the nonlinear behaviour of various impacting systems showing period doubling and Hopf bifurcations. The other notable work includes investigations on multi-sliding bifurcation in a two-degree of freedom impact oscillator [11], maps and border

∗

Corresponding author. E-mail addresses: [email protected] (S. Kundu), [email protected] (S. Banerjee), [email protected] (J. Ing), [email protected] (E. Pavlovskaia), [email protected] (M. Wiercigroch). 0167-2789/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2011.11.014

collision bifurcations [12–15] and numerical studies comparing dynamics of soft and hard impacts [16]. Through these studies, it is now known that impacting systems can display incredibly rich dynamics, and diverse types of bifurcations have been observed in them. Of particular interest to our study are the bifurcations induced by grazing. When studying these bifurcations, the most fruitful approach has been to derive the form of the Poincaré map in the neighbourhood of the grazing orbit. In 1991, Nordmark [17] showed that the Poincaré map close to the grazing condition involves the square root of the penetration depth (the distance the mass would have gone beyond the position of the wall, if the wall were not there). Close to the grazing condition, the square root term causes the Jacobian to assume infinite values as a result of which there is an infinite local stretching in the state space. This feature is called the square root singularity. In most practical impacting systems, the impact is not really instantaneous, as the wall is expected to have some compliance. There are also systems where the wall is deliberately cushioned by a spring-damper support to soften the impact. The nature of the Poincaré map for such systems has not been studied directly, though an important clue was provided by the studies on stick-slip systems. Dankowicz and Nordmark [18] showed that if the vector field is smooth close to the grazing condition, then the leading

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S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

(b)

(c)

(d)

Fig. 1. (a) Soft impacting system with secondary constraint provided by an unprestressed spring attached to the wall (System-A); (b) with secondary constraint provided by an unprestressed spring and damper attached to the wall (System-B); (c) with secondary constraint provided by a prestressed spring attached to the wall (System-C ); (d) with secondary constraint provided by a prestressed spring and damper attached to the wall (System-D).

order term in the Taylor expansion of the map has a power of 3/2, which results in the map being smooth. There can be various kinds of soft-impact systems differing in the nature of the surface to which the mass makes impact. If the wall is cushioned by only a compliant element (Fig. 1(a)), the vector field is actually smooth. If there is a damper in addition to the spring (Fig. 1(b)), the vector field is smooth only in the immediate vicinity of the grazing point, but not elsewhere. If the spring is prestressed (Fig. 1(c)), the vector field is non-smooth on the impacting surface irrespective of the impacting velocity. There can also be a system with a pre-stressed spring as well as a damper (Fig. 1(d)). The near-grazing behaviour of these four types of soft-impact systems has been studied numerically and experimentally [19]. The main purpose of this paper is to show that the square root singularity appears only in the trace of the Jacobian matrix at one side of the border, and that it disappears under some conditions of the system parameters. This observation may be of technical value in designing practical systems where the abrupt onset of largeamplitude chaotic attractor (which is caused by the square root singularity) is potentially harmful. The paper is organized as follows. In Section 2, we describe the four types of soft impacting systems, and their mathematical models. Then we present the form of the Poincaré map in the neighbourhood of a grazing orbit. For the sake of completeness we include the derivation in Section 3, because the systematic derivation of the zero-time discontinuous map (ZDM) for soft impacting systems has not been presented anywhere so far. In Section 4 we investigate the Jacobian to infer how the determinant and the trace vary across the grazing condition, and show that the singularity disappears if the damped frequency of the oscillator is an integral multiple of half of the forcing frequency. In Section 5 we present the results of the experiments on a laboratory-scale softimpact oscillator, and show that the abrupt onset of chaos can be avoided if the parameters are tuned to satisfy the condition given in Section 4. 2. The soft impact oscillator In this paper we analytically investigate the character of the Poincaré map close to grazing in soft impact systems with four different configurations, as shown in Fig. 1, adopting the approached developed in [20] for hard impact systems. In each of the variants of soft impact systems, the mass M is attached to a spring of stiffness k1 and a damper of damping coefficient R1 . It is being acted upon by a periodic force G(t ) having a time period of T = (2π /ωforcing ), ωforcing being the excitation frequency. u is the displacement of the mass measured from its equilibrium position towards an elastic restraint placed at a

distance σ = L2 − L1 from the equilibrium position of the mass (L1 and L2 are the equilibrium distances of the mass and the constraint from some reference point, as shown in Fig. 1). The evolution of its state vector x = (u, u˙ , t mod T )T (inclusion of a time variable in the state vector makes the system an autonomous one) is governed by a set of ordinary differential equations (ODEs). The ODE defining the motion of the mass before the impact with the wall is the same in all the four variants, while the ODE changes when the mass makes an impact. The mass makes contact with the wall when u > σ , and breaks contact when the contact force becomes zero. The expression for the contact force is different for the four system configurations. In general we can write

 x˙ =

F1 (x), F2 (x),

if x ∈ Σα if x ∈ Σβ

(1)

where Σα = {x : H (x) > 0 ∪ Hβ (x) < 0} and Σβ = {x : H (x) ≤ 0 ∩ Hβ (x) ≥ 0}. H (x) = σ − u is a smooth function giving the distance between the mass and the secondary constraint and Hβ (x) is another smooth function giving the force applied on the mass from the wall when they are in contact:

 −k   2 −k2 Hβ (x) = −   k2 −k2

H (x) , H (x) + R2 u˙ , H (x) + k2 d, H (x) + k2 d + R2 u˙ ,

sys-A sys-B sys-C sys-D.

(2)

The transition from non-contact (Σα ) to contact (Σβ ) region is demarcated by the boundary H (x) = 0 while the transition from contact to non-contact region, near grazing, could be demarcated either by the boundary Hβ (x) = 0 as in system-B or by the boundary H (x) = 0 as in system-A, C and D (we assume that the force due to prestressing of the spring is larger than that due to friction). If we consider the state-space of the system, an impacting orbit viewed near the impacting boundary would look as in Fig. 2. Also the functions F1 (x) and F2 (x) are assumed to be smooth up to a sufficiently high order. The normal velocity v(x) is defined as the rate at which the trajectory approaches the impact boundary. It is given by

v(x) :=

dH (x) dt

 ∂ H (x) dx Hx F1 (x), = = Hx F2 (x), ∂ x dt

if x ∈ Σα if x ∈ Σβ .

Here the subscript x implies partial derivative with respect to x. Similarly the normal acceleration a(x) of the flow with respect to the boundary is a(x) :=



(Hx F1 )x F1 (x), (Hx F2 )x F2 (x),

if x ∈ Σα if x ∈ Σβ .

S. Kundu et al. / Physica D 241 (2012) 553–565

Fig. 2. The switching manifold in the state space, an impacting trajectory (black) and a grazing trajectory (blue). The red line indicates the boundary between Σα and Σβ . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

555

Fig. 3. The zero-time-discontinuity mapping (ZDM) near grazing for soft-impact system.

3. Grazing and discontinuity mapping Let us now look at the smoothness property of the system near the boundary. The governing equations of the system are ODEs and are individually smooth. But it is important to know what the smoothness character of the system flow is near the boundary. To do so, we obtain the algebraic relation between the two flow equations on either side of the boundary. For system-A, the expressions of F1 (x) and F2 (x) are F1 (x) = F1 (u, u˙ , t mod T )   u˙ = (−R1 u˙ − k1 u + G(t )) /M F2 (x) = F2 (u, u˙ , t mod T )

=

u˙ (−R1 u˙ − k1 u − k2 (u − σ ) + G(t )) /M 1

 .

(3)

Thus for system-A we can write 0





k2 (σ − u) /M 0

F2 (x) − F1 (x) =

.

Since H (x) = (σ − u), this gives

 F2 (x) = F1 (x) + Q H (x);

Q =

0 k2 /M 0

 .

(4)

For System-B, the corresponding equation is 0

 F2 (x) − F1 (x) =



(k2 (σ − u) − R2 u˙ ) /M 0

which again yields, near grazing (˙u ≈ 0),

 F2 (x) ≈ F1 (x) + Q H (x);

Q =

H (x∗ ) = 0

v(x∗ ) = 0 a(x∗ ) = a∗ > 0.

1



We take a closer look at the grazing event which occurs when the orbit just touches the impact boundary. This event can be expressed mathematically as follows (where x = x∗ is the grazing point):

0 k2 /M 0

 .

(5)

But for system-C and system-D, the relation between F1 (x) and F2 (x) is significantly different. While the vector field in systems A and B changes continuously across the switching manifold in the neighbourhood of the grazing point, for systems C and D the change will be discontinuous. For system-C and system-D we can write

(x), F2 (x) = F1 (x) + Q

(6)

where,

  0     (k2 (σ − u − d)) /M ,   0 (x) =  Q   0    (k2 (σ − u − d) − R2 u˙ ) /M ,   0

(x) both are smooth 3 × 1 vectors. Q and Q

sys-C

sys-D

(7)

H (x) is assumed to be well defined at x∗ , i.e., Hx (x∗ ) ̸= 0. Let us now briefly review the current approach, originated from Nordmark [17,21], in obtaining the discrete-time Poincaré map in the neighbourhood of the grazing orbit. In this approach one observes the state discretely at instants spaced by time period T of the forcing function. The mapping of a point in the state space to another point in the state space after a time T gives the Poincaré map. For the part of the flow that does not have any impact with the discontinuity boundary the mapping can be derived from the ODE x˙ = F1 (x) of Eq. (1). Whenever there is an impact with the boundary, the other ODE x˙ = F2 (x) comes into action and there is a discontinuity in the flow. The discontinuity near grazing is of particular interest. Special kinds of mapping have been proposed to account for this discontinuity [21]. In the current situation the zero-time discontinuity mapping (ZDM) is dealt with. Let us consider the situation as shown in Fig. 3. There is an orbit which grazes the discontinuity boundary Σ at a point x∗ at some point of time t = 0. Now consider a perturbed initial condition xi , so that the resulting trajectory (xi x1 x2 x4 ) crosses the switching manifold, but lies close to the grazing orbit. Let us back-trace the trajectory, governed by the ODE as in Eq. (1), from the point x2 to the point x3 such that the time taken by the trajectory to reach x2 from xi is the same as would have been taken by the flow to reach x2 from x3 . Thus, we can consider the system’s dynamics as if the switching boundary was not there. In that case we have to assume an instantaneous jump of the state from xi to x3 . The ZDM is defined as the mapping xi → x3 . For the purpose of deriving the ZDM, we break an impacting trajectory in the close neighbourhood of the grazing orbit into three parts, namely one just before the impact (xi → x1 , Fig. 4(a)), during the impact (x1 → x2 , Fig. 4(b)) and after the impact (x2 → x4 , Fig. 4(c)). Let the solution of the ODE governing the motion of the mass starting from an initial state xi be given by ϕk (t , xi ) (where k = 1 in the non-impacting region and k = 2 in the impacting region), i.e., ϕk (0, xi ) = xi . From Fig. 4, assuming xi is very close to x∗ , i.e., ∥xi − x∗ ∥ = ϵ ≈ 0, and using Taylor series expansion with respect to time and the assumption of smoothness of the system functions F1 (x) and F2 (x) up to sufficiently high order, we obtain: dϕ 1 ( t , x i ) 

  

δ0 dt t =0   2 3  d2 ϕ1 (t , xi )   δ0 + d ϕ1 (t , xi )  +   2 3 dt 2 dt

ϕ1 (δ0 , xi ) = ϕ1 (0, xi ) +

t =0

⇒ x1 = xi + F1 (xi )δ0 + F˙1 (xi )

δ02 2

+ F¨1 (xi )

δ03 6

δ0 3 t =0

+ ···.

6

+ ···

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S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

(b)

(c)

Using similar treatment as in Eq. (9), we can write H (x2 ) = H (ϕ2 (δ ′ , x5 )) dH (ϕ2 (t , x5 )) 

= H (ϕ2 (0, x5 )) +

Fig. 4. The breakup of the impacting trajectory: (a) the time taken to reach x1 from xi is δ0 ; (b) the time taken to reach x2 from x1 is δ1 ; (c) the total time taken to reach x2 from x3 would have been δ0 + δ1 .

∗

Since ∥xi − x ∥ ≈ 0, we get xi ≈ x1 − F1∗ δ0 − F˙1

∗ δ0

2

2

∗δ − F¨1 0 − · · · . 3

6

(8)

Here the superscript ‘∗’ indicates that the corresponding function value is evaluated at the grazing point x∗ . Let us now concentrate on the situation when the mass has already made the impact (Fig. 4(b)). In Fig. 4(b), the point x5 is the farthest point (given by v(x5 ) = 0) reached by the trajectory in the impacting region. Since we are looking very close to the grazing condition, the time spent by the mass inside the impacting region is very short compared to the time period of the forcing function. If δ is the time taken to reach x5 from x1 , one can evolve the same equation for a duration of −δ starting from x5 to reach x1 . Thus we can write x1 = ϕ2 (−δ, x5 ), and thus,

= H (x5 ) − v(x5 )δ + a(x5 )

2

− ···.

(9)

Now, H (x) is a negative number in the impacting side, and hence H (x5 ) is a negative number. Let us denote it as the minimum value of H , Hmin = H (x5 ) = −y2 . Since H (x1 ) = 0, v(x5 ) = 0 and a(x5 ) ≈ a(x∗ ) = a∗ , we get 0 ≈ −y2 + a

2 ∗δ

2

δ≈

2 a∗

2

t =0

+ ···

(δ ′ )2

+ ···. (14) 2 The rest of this section will be discussed in three sub-sections— first one addressing the derivation of ZDM for system-A, second one focusing on system-B, and the third sub-section on systems-C and D. We will discuss the case of system-A in detail and present the other cases in short. 3.1. Zero-time discontinuity map: system-A Focusing on system-A, and recalling that H (x5 ) = −y2 and v(x5 ) = 0, we obtain from Eqs. (12) and (14):

 δ ≈ ′

2 a∗

y

⇒ δ′ ≈ δ

(using Eq. (10))

(15)

and using Eq. (11), we can also write



∗ δ1

∗ δ1

3

2

+ ···. (17) 2 6 Let us now look at the scenario after the impact is complete (Fig. 4(c)). The time taken by the trajectory to reach x2 from the virtual point x3 would have been the same as that taken to reach x2 from xi . Thus we can write (using the smoothness assumption): + F¨2

x2 = x3 + F1 (x3 )(δ0 + δ1 ) + F˙1 (x3 )

(δ0 + δ1 )2 2

(δ0 + δ1 ) + F¨1 (x3 ) + ··· 3

6

∗ (δ0 + δ1 ) ⇒ x3 ≈ x2 − F1∗ (δ0 + δ1 ) − F˙1

2

− ···.

y.

− F¨1

(10)

In Fig. 3 the point x2 is the state at the boundary of the transition from contact to non-contact region. We denote the time taken from x5 to x2 as

δ = δ1 − δ. ′

(11)

From our discussion in Section 2, we remember that the boundary of the transition from contact to non-contact is given by H (x) = 0 in systems-A, C and D, and by Hβ (x) = 0 in system-B. Hence in systems-A, C and D we have H (x2 ) = 0

(12)

∗ (δ0

+ δ1 ) 3

2

− ···.

(18)

6 Using expression Eq. (17) in Eq. (18): x3 ≈ x1 + F2 ∗ δ1 + F˙2

− F˙1  ≈

∗ (δ0

∗ δ1

+ δ1 ) 2

2

2

∗δ + F¨2 1 + · · · − F1∗ (δ0 + δ1 ) 3

6

− F¨1

∗ (δ0

+ δ1 )3

− ···  3 2 ∗δ ∗δ x1 − F1∗ δ0 − F˙1 0 − F¨1 0 − · · · 2

2

6

6

∗1

− F˙1∗ δ0 δ1 − F¨1 (δ0 δ12 + δ02 δ1 ) − · · · 2

∗

∗

+ (F2 − F1 )δ1 + (F˙2 − F˙1 ) ∗

∗

δ12 2

∗

∗

+ (F¨2 − F¨1 )

δ13 6

+ ···

1

≈ xi − F˙1∗ δ0 δ1 − F¨1∗ (δ0 δ12 + δ02 δ1 ) − · · ·

and in system-B we have

2

Hβ (x2 ) = 0

⇒ H (x2 ) =

dt 2

= H (x5 ) + v(x5 )δ ′ + a(x5 )

Ignoring the higher order terms, we obtain



(δ ′ )2

x2 = x1 + F2 ∗ δ1 + F˙2

2

δ2

d2 H (ϕ2 (t , x5 )) 

  

δ′

t =0

δ1 ≈ 2

 dH (ϕ2 (t , x5 ))   (−δ) = H (ϕ2 (0, x5 )) +  dt t =0  2 d2 H (ϕ2 (t , x5 ))  (−δ)  + + ···  2 t =0

dt

2 y. (16) a∗ Using Taylor series expansion, similarly as in Eq. (8), we obtain

H (x1 ) = H (ϕ2 (−δ, x5 ))

dt

+

  

∗ ∗ δ ∗ ∗ δ + (F2∗ − F1∗ )δ1 + (F˙2 − F˙1 ) 1 + (F¨2 − F¨1 ) 1 + · · · . 2

R2 u˙

(using Eq. (2)) k2 R2 v(x2 )

= −

k2

.

(13)

3

2 6 It should be noted here that the point xi could be selected very close to the impacting surface so that the condition δ0 ≪ δ1 is satisfied. Then we can ignore the higher order terms in δ0 and obtain

S. Kundu et al. / Physica D 241 (2012) 553–565

∗

∗

x3 ≈ xi + (F2∗ − F1∗ )δ1 + (F˙2 − F˙1 )

δ

2 1

2

557

∗ δ ∗ + (F¨2 − F¨1 ) + · · · . 3 1

6

(19) ∗

Using the relation in Eq. (4) at the grazing point x , we get: F2 = F1∗ + Q H (x∗ ) = F1∗ ∗

(20)

˙ (x) F˙2 (x) = F˙1 (x) + Q H F˙2 (x∗ ) = F˙1 (x∗ ) + Q v(x∗ ) ∗ ∗ ⇒ F˙2 = F˙1

(21)

Fig. 5. A grazing periodic orbit and a combined stroboscopic map.

¨ (x) F¨2 (x) = F¨1 (x) + Q H ˙

∗

∗

⇒ F¨2 = F¨1 + Q a∗ .

(22)

Hence using Eqs. (20)–(22) in Eq. (19) we obtain x3 ≈ xi + Q a∗

(δ1 )3

+ ···.

(23)

6 Using Eq. (16), it yields x3 ≈ xi +

8 3

 Q

a∗



8

2

≈ xi + Q

y

2 a∗

3

(−Hmin )3/2 .

(24)

2

As δ ′ is negligibly small, we can ignore the second order term and obtain k2 y2 a∗ R 2

Then, δ1 ≈

≈

∗ a 2

a∗

x3 ≈ xi +

3 1

 Q

y+ ∗ a R2

(∵ y is negligibly small).

(25)

≈ xi + Q 3

2 a



y3 ∗

2 a∗

2 a∗

∗ Q

 −Hmin .

(27)

3/2

(−Hmin )

.

(26)

3.3. Zero-time discontinuity map: system-C and D As in sub-Section 3.1, using Eq. (10), (11), (12) and (14), we obtain Eq. (16). The Eqs. (17)–(19) presented in sub-Section 3.1 ∗ = hold true for both systems-C and D, as well. Introducing Q ∗  Q (x ) and substituting Eq. (6) in Eq. (19) we get:

(28)

where P1 is the map that takes a point on the Poincaré plane and maps it to the discontinuity boundary Σ by evolution through the ODE x˙ = F1 (x) in (1), and P2 is the map that takes a point on the discontinuity boundary Σ and maps it back to the Poincaré plane via the same ODE. Thus, if xi is located on a periodic orbit, we must have Ps (xi ) = xi . Now we investigate the elements of the Jacobian. In the earlier works on border collision bifurcations [12,13], a coordinate transformation was applied to a 2-dimensional Jacobian matrix, J , to express it in the normal form, J , as follows: J =

The Eqs. (17)–(19) presented in sub-Section 3.1 hold true for system-B as well (and for systems-C and D, too). And since near grazing Eq. (5) is the same as Eq. (4), following similar arguments as in Eqs. (20)–(22), we again arrive at Eq. (23). Using Eq. (25) in Eq. (23) we obtain the ZDM for system-B: 1

+ ···

4. Investigating the Jacobian for singularity



k2 y2

y

6

a∗

Ps (x) = P2 ◦ ZDM ◦ P1 (x),

R2 ∗ ′ δ′ a δ ≈ −y2 + a∗ . k2 2

2

δ13 x=x∗

Let us consider an orbit which has an intersection with the discontinuity boundary very close to the grazing orbit, as shown in Fig. 5. The stroboscopic Poincaré map in this case is

Using Taylor series expansion about the point x5 , we can write v(x2 ) = v(x5 ) + a∗ δ ′ + O(δ ′ 2 ). Thus from Eq. (13), H (x2 ) ≈ − R2 a∗ δ ′ /k2 . Hence from Eq. (14), we write



2

 ¨ (x)  +Q

Eq. (27) gives the ZDM for systems-C and D.

3.2. Zero-time discontinuity map: system-B

δ′ ≈

δ12

∗ δ1 + H .O.T ≈ xi + Q  2 ∗  y; (using Eq. (15)) ≈ xi + 2 Q ≈ xi + 2

Eq. (24) gives the form of the ZDM for system-A.

−

x=x∗



3

 

(x∗ )δ1 + Q (x) x3 ≈ xi + Q

tr(J ) −det(J )



1 0

(29)

where tr(J ) and det(J ) are the trace and the determinant of the two-dimensional Jacobian matrix J , respectively. Since the trace and the determinant are invariant under coordinate transformation, the character of the dynamics around a fixed point (a periodic orbit, seen on a Poincaré plane) is given by just these two numbers. Suppose the Poincaré plane has the coordinates (z , w) in discrete time. The local map in the neighbourhood of the border becomes:

     zn 1      J L wn + µ 0 , zn+1 =     wn+1   J R zn + µ 1 , wn 0

for zn ≤ 0 (30) for zn > 0

where (zn , wn ) and (zn+1 , wn+1 ) are the consecutive iterates on the Poincaré plane. The subscripts L and R stand for the left half and the right half of the phase space, respectively, separated by the borderline zn = 0 and µ is some parameter. In the present case one of the halves would represent points on the Poincaré section corresponding to non-impacting orbits, while the other half would represent the points for impacting orbits. The

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S. Kundu et al. / Physica D 241 (2012) 553–565

borderline would represent points from where the orbits graze the switching surface. The normal form structure Eq. (30) suggests that while the objective is to look at the elements of the Jacobian, the focus should be on its trace and determinant. Let N1 and N2 be the linearized forms of the maps P1 and P2 (used in Eq. (28)) about the nominal points (corresponding to the grazing periodic orbit) x0 and x∗ , respectively, i.e.,

  N1 = , dx x=x0

  and N2 = . dx x=x∗

dP1 

dP2 

JZDM = I + 4

−2Hmin a∗

2 kM

QHx∗ .

−2Hmin

JZDM = I +

d

(32)

And using Eq. (27) we get that for system-C and system-D 2 Q˜ ∗ Hx∗ . √ a∗ −Hmin

JZDM = I −

(33)



dt

  0 δ u( t ) δ u˙ (t ) = −ωn 2 0

Nτ = e−ζ ωn τ

From Eq. (4) we have

∂ H (x) = Hx = ∂x



∂H ∂u

= 0 and ∂H ∂ u˙

∂H ∂t



= −1. Hence  = −1

0

JZDM = I + 4

 = 

1 0 0

0 1 0

1

0 1 0

0 0 , 1

α

=

0



α=−

4k2



−2Hmin a∗

M

JZDM = I +



1

α

=

0

−2Hmin a∗

0 1 0

0 0 , 1

JZDM =

1

α 0

0 1 0

0



0

α=

B(τ ) D(τ ) 0

0 0 1

 (40)

B(τ ) = sin(ω0 τ )/ω0 1 C (τ ) = − ω0 sin(ω0 τ ) 1 − ζ2

(35)

 ω0 = ωn 1 − ζ 2 .

4.1. Investigating the determinant for singularity From Eq. (34), the determinant of the normal form map near grazing, in all the variants (i.e., systems A, B, C and D), is

|J | = |N2 ∥JZDM ∥N1 | = |N2 | |N1 |

α=−



A(τ ) C (τ ) 0

sin(ω0 τ ) A(τ ) = cos(ω0 τ ) +  1 − ζ2

and

QHx∗



0 0 , 1



ζ

k2



M

−2Hmin a∗

.

(36)

For systems C and D, Q˜ (x) is dependent on x, but close to grazing it becomes Q˜ ∗ = (0 − k2 d/M 0)T . Thus, using Eq. (33) we get the expressions for system-C and system-D as



(39)

sin(ω0 τ ) D(τ ) = cos(ω0 τ ) −  1 − ζ2

and from Eq. (32) we have for system-B,





ζ



−2Hmin

QHx∗ a∗     0 0  −2Hmin 0 +4 k2 /M −1 ∗ a 1 0



with

0 .

For systems A and B, we write Q = (0 k2 /M 0)T . Hence from Eq. (31) we have for system-A,



0 δ u( t ) 0 δ u˙ (t ) , 0 0

0

J = N2 · JZDM · N1 . ∂H ∂u

.

M

1 −2ζ ωn 0

0

0

Let us now consider the Jacobian of the combined stroboscopic map near grazing which is: ∂H ∂ u˙

k

   δ u(τ ) δ u0 δ u˙ (τ ) = Nτ δ u˙ 0

where

(34)

and ωn =

with δ u(0) = δ u0 , δ u˙ (0) = δ u˙ 0 . Following simple algebraic steps and assuming complex conjugate eigenvalues of the matrix in Eq. (39), the solution to the above problem can be expressed as

 



The variational equation needs to be solved to obtain the perturbed flow δ p(τ ) = (δ u(τ ), δ u˙ (τ ), 0)T . Solving the variational equation amounts to solving the first-order differential equations

(31)

QHx∗ .

a∗

(38)

where ζ is the damping factor and ωn is the natural frequency of oscillation. For a system with spring constant k, mass M, and damping coefficient R,

ζ = √

Using Eq. (26) we obtain the Jacobian of the zero-time discontinuity map for system-B as



δ u¨ + 2ζ ωn δ u˙ + ωn 2 δ u = 0,

R

Using Eq. (24) we obtain the Jacobian of the zero-time discontinuity map for system-A as



Let (p(t ) + δ p(t )) be a perturbed orbit also lying in Σα . The variations δ u(t ), δ u˙ (t ) satisfy a differential equation of the form

k2 d M



2

−a ∗ H

(37) min

where α is the term that contains the singularity. Let us now concentrate on the general form the matrices N1 and N2 will take. For that, let us consider a trajectory in Σα over a time period t as p(t ) = (u(t ), u˙ (t ), t mod T )T .

since |JZDM | = 1, from (35)–(37). Since the singularity is only in the ZDM, and not in the matrices N1 and N2 , we conclude that the determinant of the normal form map does not contain the square root singularity, and remains invariant in the immediate neighbourhood of the grazing orbit. But there is another aspect with the variation of determinant. Although the determinant does not show any singularity near the grazing, in the case of system-B and system-D (where the impacting wall is attached to both a spring and a damper) the determinant decreases as the orbit increasingly penetrates into the impacting region [19]. In the pursuit of explaining this, we take a closer look at the expression of the Jacobian matrix J allowing space for the event of a non-grazing impact as we tend to move away from the grazing condition. Let us assume that the orbit now makes a finite velocity impact with the boundary and stays in the impacting side for a very short (but positive) duration ∆ = δ1 (with reference to Fig. 4(b)). Also let the orbit make the impact after a time τ0 , so that after coming

S. Kundu et al. / Physica D 241 (2012) 553–565

out of the impacting region, it takes time (T − τ0 − δ1 ) to return to the Poincaré plane (see Fig. 5). Then modifying Eq. (34) we can write the new combined Jacobian as J = N2 · S21 · Nδ1 · S12 · N1

(41)

S21 = I +

Hx F1 Hx F2

.

S12 =

1 s 0

0 1 0

0 0 1





−ζ ωn τ0

n11 n13 0

S21 =

 N2 = N(T −τ0 ) = e

0 0 1



n21 n23 0

(46) n22 n24 0

0 0 1

 (47)

1 −s 0

0 1 0

0 0 1

J = e

−ζ ωn T

(42)

=e

−ζ ωn T

n21 n23 0

n22 n24 0

0 0 1

n21 + α n22 n23 + α n24 0



1

0 1 0

α 0 n22 n24 0

0 0 1

n11 n13 0

n12 n14 0

n11 n13 0

n12 n14 0

0 0 . 1



0 0 1



0 0 1





Simple algebraic manipulation yields the trace as

0, −k2 d/ (M u˙ ) , −(k2 d + R2 u˙ )/ (M u˙ ) ,



for system-A and B for system-C for system-D.

Tr(J ) = e−ζ ωn T {n21 n11 + n22 n13 + n23 n12 + n24 n14 (43)

It is important to notice that for each of the system variants the determinant of the saltation matrix is unity, i.e., |S12 | = |S21 | = 1. From Eq. (40) we observe that |Nτ | = e−ζ ωn τ . As per our notations, N1 and N2 are nothing but Nτ0 and N(T −τ0 −δ1 ) defined in Eq. (40). By our definition, δ1 = 0 for a non-impacting orbit. Hence the determinant, Det1 , of the combined Jacobian for all the variants, till the grazing impact will be nothing but Det1 = e−ζ ωn (T −τ0 ) · e−ζ ωn τ0 −R1 T /(2M )

=e

+ α (n22 n11 + n24 n12 ) + 1} .

(using Eq. (38))

(44)

n11 = cos(ω0 τ0 ) +  n12 =

ζ 1 − ζ2

ω0

sin{ω0 (T − τ0 )}

ω0

n24 = cos{ω0 (T − τ0 )} − 

−

(R1 +R2 )δ1

2M =e ·e −R1 T /(2M ) −R2 δ1 /(2M ) =e ·e

−

·e

R1 τ0 2M

n22 n11 + n24 n12 =

(45)

 √ 2

Going back to Eqs. (16) and (25), δ1 = 2

a∗

m=

−Hmin for systems-A,

 √ 2 −Hmin for system-B. This gives a∗

 Det1 ,       R2 2    exp − 2M a∗ −Hmin Det1 , Det1 ,        R2 2   −Hmin Det1 , exp − ∗ M

a

1 − ζ2

sin{ω0 (T − τ0 )}.

sin(ω0 T )

ω0

ω0



= sin 2π

ωforcing



1

ω0

.

(49)

Let the number m be defined as

= e−R2 δ1 /(2M ) · Det1 .

C & D and δ1 =

ζ

Substituting, we get

Det2 = |N(T −τ0 −δ1 ) | · |S21 | · |Nδ1 | · |S12 | · |Nτ0 | R1 (T −τ0 −δ1 ) 2M

sin(ω0 τ0 )

sin(ω0 τ0 )

and the determinant, Det2 , for the finite-velocity impact will be

−

(48)

The expression for the trace of the Jacobian Tr(J ), in Eq. (48), shows that the singularity term α has a coefficient e−ζ ωn T (n22 n11 + n24 n12 ). Let us take a closer look at this coefficient. Using Eqs. (40), (46) and (47) we get

n22 =

= e−ζ ωn T

Det2 =

n12 n14 0

−ζ ωn (T −τ0 )





where s=

N 1 = N τ0 = e



Following simple algebraic manipulations, we obtain the expressions for the saltation matrices



Now we can proceed to obtain the expression of the trace of the Jacobian in Eq. (34). We let

where the expressions for nij , i ∈ {1, 2}& j ∈ {1, 2, 3, 4}, can be obtained from Eq. (40). Substituting Eq. (35) (or (36) or (37)), (46) and (47) in Eq. (34), we get

(F2 − F1 )Hx (F1 − F2 )Hx

4.2. Investigating the trace for singularity



where, S12 and S21 are the saltation matrices for the transitions from non-contact to contact and from contact to non-contact phases, respectively. The matrix Nδ1 is the linearized form of the map that takes a point on the orbit just after the impact and maps it to a point on the orbit just before it leaves the impacting region. The saltation matrices can be obtained using the relations [21] S12 = I +

559

(sys-A) (sys-B) (sys-C ) (sys-D).

Thus, for system-B and system-D, where R2 ̸= 0, the determinant after grazing decreases exponentially with increasing penetration. But for system-A and system-B, where R2 = 0, it remains unaltered, i.e., Det2 = Det1 . This explains the numerical results obtained in [19].

2ω0 ωforcing

(50)

where ωforcing is the frequency of the periodic forcing function G(t ). It follows that n22 n11 + n24 n12 ̸= 0,

∀ non-integer m.

(51)

Eq. (51) implies that when m is non-integer, the coefficient of α in the expression of the trace equation (48) of the Jacobian of the stroboscopic map must be a non-zero entity. Thus the singularity in α survives, and hence a square root singularity must occur in the trace of the Jacobian. This result also has a very interesting corollary: that the singularity must vanish for integer values of m. This implies that if the frequency is chosen to satisfy this condition, the singularity will disappear, and there will be no stretching of the phase space in the neighbourhood of a grazing orbit. We will address this issue specifically in next section where this result will be verified experimentally.

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S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

(b)

Fig. 6. Schematic of the experimental rig with prestressed secondary support. The oscillating mass is marked by number 1, leaf springs providing the primary stiffness by number 2, prestressed secondary constraint by 3, accelerometers by 4, Eddy current probe by 5, and arch applying the prestress by 6. (b) The overall restoring force of the beam: the experimental measurements (black line) and the fitted model curve (green line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

Fig. 7. (a) Bifurcation diagram for the system with preload for a frequency of 11.57 Hz, corresponding to m = 3. The red line indicates the amplitude where grazing occurs. (b) Phase portraits show the pre-grazing orbit at A = 0.82, and two impacting orbits at A = 0.83 and 0.92. The red lines mark the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

5. Experimental validation Experimental validation was sought using a rig similar to that detailed in [22], with the addition of an arch to provide a controllable prestress to the secondary spring. This allowed testing of System-A and System-C shown in Fig. 1(a) and (c) respectively. A schematic of the rig is presented in Fig. 6(a). In this configuration the arch (marked in Fig. 6(a) by number 6) clamps the beam so that as soon as the mass touches the beam, the full force of the beam is felt. The level of this prestress is controlled by placement of spacers beneath the arch. The overall restoring force of the beam was measured quasistatically using an INSTRON machine, and a multiparameter nonlinear minimization was used to simultaneously obtain parameters for the stiffness ratio, gap and prestress from the following relationship: restoring force is equal to k1 u when there is no contact and k1 u + k2 (u − σ + d) during the contact. The experimental results and model fit are shown in Fig. 6(b).

For both setups, the frequency of excitation was set to integer values of m from Eq. (50) to probe the disappearance of the singularity, and part way between these values to demonstrate the effect of the singularity on the system response at grazing. Bifurcation diagrams were then constructed using amplitude as the branching parameter so that, at grazing, the value of m could be precisely controlled. The methodology, sampling rates, smoothing etc. are the same as those detailed in [22]. First System-C was investigated. For this system, the stiffness ratio was reduced to the minimum possible for the rig, so that the nonlinearity was mainly due to the force discontinuity resulting from the prestress. It was ensured that the prestress was sufficiently small that the time of contact was finite and measurable, so that the impact was still yielding. The measured parameters are as follows: ξ = R1 /(M ωn ) = 0.015, M = 1 kg, √ fn = 21π ωn = 21π k1 /M = 17.35 Hz, k2 /k1 = 3.64. The gap, given by σ = L2 − L1 , was 0.64 mm, and the prestress was 0.78 mm.

S. Kundu et al. / Physica D 241 (2012) 553–565

561

(a)

(b)

Fig. 8. (a) Bifurcation diagram demonstrating the persistence of the periodic solution for the system with preload at a frequency of 8.68 Hz, corresponding to m = 4. The red line indicates the amplitude where grazing occurs. (b) Phase portraits show the pre-grazing period-1 orbit at A = 1.89, and two impacting orbits at A = 1.9 and A = 2.04. The red lines mark the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

Fig. 9. (a) Bifurcation diagram constructed for the system with preload at a frequency of 9.0 Hz, corresponding to m ∼ 3.85. The red line indicates the amplitude where grazing occurs. (b) A phase portrait is shown for a non-impacting response at A = 1.7 mm, and Poincaré sections for impacting chaotic responses at A = 1.71 mm, A = 1.85 mm and A = 1.89 mm. The red lines mark the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

For the prestressed case some particular difficulties prevented all values of m from being tested. m = 1 and 2 resulted in a very large amplitude response which could potentially damage the rig, and these results are omitted from the paper. Two cases are therefore presented, Fig. 7 shows results for m = 3 and Fig. 8 shows results for m = 4. In both cases the period-1 response persists across grazing and into the impacting regime. In contrast, Fig. 9 presents results for m ∼ 3.85, and at grazing there is a clear band of chaos which persists for the rest of the amplitudes under investigation. Fig. 10 shows results for m ∼ 2.72. In this case there is a narrow band of chaos followed by a period-5 response, after

which a further band of chaos is followed by a period-3 and then a period-1 response. This confirms that for System-C experimental results match theory and the chaotic response close to grazing is absent for certain values of the frequency. Next System-A was considered. Although this system does not have a square root singularity, if the stiffness ratio of the two springs is high, its behaviour shows signatures of stretching close to the grazing condition. This stretching behaviour is known to result in bifurcation near grazing, and often a narrow band of chaos [23], which was also found to disappear for integer m. Therefore experimental verification for this was also sought.

562

S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

(b)

Fig. 10. (a) Bifurcation diagram for the system with preload at a frequency of 12.714 Hz, corresponding to m ∼ 2.72. The red line indicates the amplitude where grazing occurs. (b) A phase portrait is shown for a nonimpacting response at A = 0.5 mm, a Poincaré section for an impacting chaotic response at A = 0.51 mm, a phase portrait for a period-5 response at A = 0.55 mm, and another for a period-3 response at A = 0.57 mm. The red lines mark the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

(c)

(d)

Fig. 11. Bifurcation diagrams constructed for the system without prestress (System-A) at frequencies corresponding to integer values of m, demonstrating the persistence of the local period-1 solution. Diagrams are for (a) m = 1, (b) m = 3, (c) m = 4 and (d) m = 5 (σ = 0.55 mm). Red lines on the graphs indicate the grazing amplitudes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

For this system the parameters were found to be as follows; ξ = 0.015, M = 1 kg, fn = 11.38 Hz, k2 /k1 = 33 and the gap (σ = L2 − L1 ) was 0.55 mm. For integer values of m, Fig. 11 clearly shows that the period-1 response is persistent throughout the parameter range tested, and narrow band chaos and boundary crisis bifurcations are absent. We have not presented the results for m = 2, because this corresponds to resonance and since the system is lightly damped the amplitude of the linear system was too large to be able to test this condition. The final step in experimentally verifying the special nature of trajectories near integer values of m is to demonstrate that the period-1 response away from this condition is subject to stretching

and boundary crisis. Figs. 12 and 13 show bifurcation diagrams for m = 2.4 and 3.4 respectively. Clearly in both considered cases the period-1 solution loses stability at grazing, and the resulting bifurcation is one of the those detailed in [23–25], which appear abruptly after grazing due to the stretching behaviour close to this condition. Specifically, for m = 2.4 (see Fig. 12) at grazing there is a boundary crisis and the response jumps discontinuously to a co-existing period-3 attractor observed both experimentally and in simulation. This is followed by a narrow band of chaos and a period-2 response, before the period-1 motion regains stability. The co-existing attractors are shown in black and red in Fig. 12(a) and (b), and the trajectories and Poincaré sections are presented in

S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

563

(b)

(c)

(d)

Fig. 12. (a) Experimental and (b) numerical bifurcation diagrams constructed for the un-prestressed case (System-A) for fn = 21π ωn = 9.49 Hz or m = 2.4 (σ = 0.55 mm). The stable coexisting attractors are shown by black and red points. (c) Experimental trajectories (black lines) and Poincaré maps (red points) recorded for the amplitudes A = 0.230, 0.2565, 0.282 and 0.450 mm (from left to right respectively). (d) Numerically obtained trajectories (black lines) and Poincaré maps (red points) for A = 0.230, 0.2565, 0.272 and 0.450 mm (from left to right respectively). Red lines on plots (c) and (d) indicate the location of the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12(c) and (d) showing period-1 response at A = 0.230 mm, period-3 response at A = 0.2565 mm, narrow band of chaos at A = 0.282 mm (experimental plot) and A = 0.272 mm (numerical plot), and period-2 response at A = 0.450 mm. It is worth mentioning here, that special care was taken to record both co-existing attractors experimentally. After the nonimpacting period-1 attractor jumps to the period-3 attractor at grazing, the obtained period-3 orbit was followed for increasing amplitudes and the results are shown in Fig. 12(a) in black. When the whole diagram was recorded, the excitation amplitude was set up close to grazing again so that the system settled on the same period-3 orbit. In some cases it was necessary to try a few different initial conditions to ensure that the required response was achieved. Then, this period-3 orbit was followed for decreasing amplitude with the results shown in red in Fig. 12(a). The same procedure was implemented when recording the coexisting attractors shown in Fig. 13(a). Fig. 13 shows a different scenario, namely a saddle–node bifurcation at grazing resulting in a loss of stability of the local period-1 non-impacting orbit, and a jump in the response, in this case to a narrow band of chaos and subsequently to a distant impacting period-1 orbit which co-exists with the non-impacting orbit due to a fold. The co-existing attractors are shown in black and red in Fig. 13(a) and (b), and the connecting unstable orbit is presented in green in Fig. 13(b). A period doubling route to chaos

bounds a chaotic window. Representative trajectories and Poincaré sections of the responses obtained experimentally and numerically are shown in Fig. 13(c) and (d) respectively, i.e. period-1 response near grazing amplitude at A = 1.02 mm, impacting period-1 orbit at A = 1.10 mm, period-2 orbit at A = 1.29 mm (experimental plot) and A = 1.36 mm (numerical plot) and chaotic attractor for A = 1.65 mm. 6. Neighbourhood of the singularity The derivation in the last section seems to indicate that the singularity vanishes for discrete values of the parameters but says nothing about the neighbourhood of these discrete points. However, in order to be able to meet the condition for the disappearance of the singularity in the experimental system, which is under the influence of parameter uncertainties and noise, there must be some neighbourhood of the point for which the coefficients of the singularity term are small, and the period-1 orbit still remains stable. The size of this neighbourhood is expected to depend on the parameters of the impact, especially the ratio of stiffnesses. We have run some numerical simulations for system-A shown in Fig. 1(a) where pure stiffness support is considered. Parameter plots are presented in Fig. 14 showing simulations of the system response for various stiffness ratios, k2 /k1 , for the initial conditions

564

S. Kundu et al. / Physica D 241 (2012) 553–565

(a)

(b)

(c)

(d)

Fig. 13. (a) Experimental and (b) numerical bifurcation diagrams constructed for the un-prestressed case (System-A) for fn = 21π ωn = 6.74 Hz or m = 3.4 (σ = 0.55 mm). The stable coexisting attractors are shown by black and red points. The unstable attractor calculated numerically is shown in green in the plot (b). (c) Experimental trajectories (black lines) and Poincaré maps (red points) recorded for the amplitudes A = 1.02, 1.10, 1.29 and 1.65 mm (from left to right respectively). (d) Numerically obtained trajectories (black lines) and Poincaré maps (red points) for A = 1.02, 1.10, 1.36 and 1.65 mm (from left to right respectively). Red lines on plots (c) and (d) indicate the location of the discontinuous boundary. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

which correspond to a grazing trajectory for median values of frequency and amplitude considered. Each pixel is coloured according to the period of the attractor (if one exists), and the vertical dotted lines denote the frequencies for which m is integer. The firm curved line indicates the parameters for which grazing conditions are met for a period-1 orbit, and at the intersection of this curve and the vertical lines, the condition for the disappearance of the stretching is met. It is clear from the figures that there exists a neighbourhood for which the period-1 orbit retains stability. The parameter regions for the appearance of high periodic orbits and chaos show finite-width gaps close to the grazing line. These gaps are smaller in width for larger stiffness ratios, and are placed around the vertical dotted lines which correspond to the integer values of m. It was shown in [23] that the disappearance of the local period-1 orbit is related to stretching in the neighbourhood of the grazing orbit and the appearance of co-existing unstable orbits which are pulled into the vicinity of the local orbit. This phenomenon appears to be absent for grazing trajectories close to integer m, but are present for other parameter values. 7. Conclusion In this paper we have investigated the character of the normal form map in the neighbourhood of a grazing orbit for four possible configurations of soft impacting systems. We have shown that, if the spring in the impacting surface is relaxed, the impacting

side of the map has a power of 3/2, but if the spring is prestressed the map has a square root singularity. We also noted, that in the case when the spring is relaxed but the stiffness ratio is very large, the system shows the signatures of stretching for non-zero impact velocity, and the behaviour becomes similar to a system with square root singularity. The singularity appears only in the trace of the Jacobian matrix and not in the determinant. Under all conditions the determinant of the Jacobian matrix varies continuously across the grazing condition. However, if the impacting surface has a damper, the determinant decreases exponentially with increasing penetration. We have found that the system behaviour is greatly dependent upon a parameter m, given by 2ω0 /ωforcing , and that the singularity disappears for integer values of m. Thus, if the parameters are chosen to obtain an integer value of m, one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non-impacting mode to an impacting mode with increasing excitation amplitude. The above result has been tested on an experimental rig for both a continuous and a discontinuous restoring force characteristic, which showed a persistence of period-1 orbit across the grazing condition for integer values of m, but an abrupt transition to a chaotic orbit or a high-period orbit for non-integer values of m. Finally through simulation we have shown that the condition for vanishing singularity is not really a discrete point in the parameter space. This property is valid over a neighbourhood in the parameter space, which shrinks in width for larger values of the stiffness ratio k2 /k1 .

S. Kundu et al. / Physica D 241 (2012) 553–565

(b)

Amplitude

(a)

565

3.00

3.00

2.25

2.25

1.50

1.50

0.75

0.75

0.00

0.4

0.6

0.8

1.0

1.2

0.4

0.6

0.4

0.6

0.8

1.0

1.2

0.8 1.0 Frequency

1.2

(d)

Amplitude

(c)

0.00

3.00

3.00

2.25

2.25

1.50

1.50

0.75

0.75

0.00

0.4

0.6

0.8 1.0 Frequency

1.2

0.00

Period-1

Period-5

Period-9

Period-14

Period-2

Period-6

Period-10

Chaos

Period-3

Period-7

Period-11

Period-4

Period-8

Period-12

Fig. 14. Numerical parameter plots for stiffness ratios (a) 2, (b) 10, (c) 33 and (d) 100, with the frequency condition corresponding to integer values of m marked by dotted lines.

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