- Email: [email protected]
Non-linear behavior in a discretely forced oscillator
International Journal of Non-Linear Mechanics 42 (2007) 744 – 753 www.elsevier.com/locate/nlm
Non-linear behavior in a discretely forced oscillator R.B. Davis ∗ , L.N. Virgin Pratt School of Engineering, Duke University, Durham, NC 27708, USA Received 20 November 2006; received in revised form 31 January 2007; accepted 20 February 2007
Abstract The simulated and experimental responses of a rigid-arm pendulum driven by an external impactor are considered. Here, impact occurs if the trajectory of a rotating impactor intersects that of the pendulum. Using the rotation rate of the impactor as the control parameter, experimental trials have demonstrated much of the dynamic behavior predicted by numerical simulations. The system exhibits chatter (i.e., multiple impacts within a single forcing period), sticking (i.e., contact between the pendulum and the impactor for non-negligible amounts of time), high-order periodicity, and behavior suggestive of chaos. A new convention for classifying periodic motions as well as insights regarding the nature of the coefficient of restitution (COR) in an experimental impacting system are also presented. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete forcing; Non-smooth characteristic; Non-linear oscillations; Impact oscillator; Coefficient of restitution; Chatter
1. Introduction Systems subject to non-smooth (i.e., discontinuous) internal or external forces are known to exhibit a wide range of dynamic behavior. Numerical and experimental studies of mechanical systems with discontinuities in their internal (i.e., stiffness or damping) characteristics are numerous (see e.g., [1–4]). A second class of problems arises when a system’s internal forces are smooth, but the external forcing is effectively non-smooth. Such a system can be realized by a relatively lightweight particle or structure undergoing repeated collisions with a moving obstacle. This class can be further divided into two sub-classes. In the first sub-class, the obstacle’s motion can be considered smooth, while in the second sub-class, it may not. The theory governing systems belonging to the first sub-class has been well investigated, but there has historically been less effort to experimentally validate the results. In this study, we discuss the simulated and experimental responses of a novel system that can be characterized by the first sub-class. The system consists of a simple pendulum that is driven by collisions with a smoothly rotating obstacle. Systems belonging to the second sub-class have received recent theoretical attention [5]. ∗ Corresponding author.
E-mail address: [email protected] (R.B. Davis). 0020-7462/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2007.02.012
Repeated collisions with a moving obstacle can initiate a complex interaction between the state of the system and that of the obstacle. A classic example of this is a small ball bouncing on a massive vibrating table. Numerous researchers have considered this problem in both theoretical [6–9] and experimental contexts [10]. Tufillaro et al. [11] provide an extended discussion of the bouncing ball problem as an introduction to many of the central concepts of non-linear dynamics. The authors illustrate how changes to a control parameter (either the amplitude or frequency of the table’s motion) can lead to a variety of dynamic behaviors including chattering, sticking, periodic motion, and chaos. In the presence of a dynamic obstacle, impact will tend to occur across a continuous spatial envelope, which broadens the range of possible dynamic behavior. Contrast this with more conventional impact systems where a harmonically driven oscillator will undergo collisions with an obstacle that is fixed in space. Thus, impact occurs if and when the oscillator achieves a single spatial position. This impact condition can be interpreted as a discontinuity in the oscillator’s stiffness [2]. An object undergoing repeated collisions with a moving obstacle is a situation arising in a variety of natural and engineered settings. The research related to a ball bouncing on a moving table was first motivated by the problem of noise generation in machinery. Other physical examples of interest include wind
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
gusts striking flexible structures and marine vessels colliding with floating platforms. The particular system considered here is especially analogous to milling processes. The impactor can be representative of the rotating milling tool while the pendulum corresponds to the workpiece. The issue of chatter and instability in milling processes is an important concern that has received considerable attention (see e.g., [12–16]). The system of interest is also somewhat reminiscent of the “kicked rotor” [17], a simple (albeit physically impractical) configuration often used to demonstrate complex dynamics in a Hamiltonian system. The kicked rotor is a simple undamped pendulum allowed to rotate through all angles. As it rotates, a gravitational field is periodically turned on and off. This results in intriguing dynamic behavior that has been shown to have applications to quantum mechanics [18]. The remainder of this paper will describe the experimental set-up as well as the modeling and simulation techniques that are employed to capture its dynamic features. Section 4 presents the simulated and experimental results for a variety of qualitative behaviors. Considerations related to the nature of the coefficient of restitution (COR) in an experimental system are also discussed. These results are followed by general conclusions and the suggestion of a system that may merit future consideration.
2. Experimental system Fig. 1(a) depicts a schematic of pendulum/impactor system and Fig. 1(b) is a photo of the experimental realization. The
745
experimental set-up consists of an aluminum pendulum of length L = 31.80 cm that is allowed to pivot with a small amount of friction through the use of two journal bearings. A steel impacting tab is embedded in an aluminum sleave fixed to the shaft of a General Electric Zero-Max variable speed electric motor. The motor delivers a rated torque of 2.82 N m. The distance from the center of the motor shaft to the tip of the impacting tab is R = 2.87 cm. The vertical distance between the pendulum pivot and the motor shaft is d = 34.02 cm. The envelope in which impact may occur assumes the shape of an asymmetrical lens, the boundaries of which can be described by the points (denoted A and B) where the arcs traced by the pendulum and by the impactor intersect. The experimental system is designed such that the horizontal offset between the pendulum pivot and motor shaft, , is zero. The pendulum pivot and the motor shaft are each coupled to a Bourns Electronics precision potentiometer (model number 6639S-1-103) to simultaneously measure the angular displacement of each component. Voltage is supplied to the potentiometers via an HP 6228B dual DC power supply. The potentiometer output is collected with Brüel & Kjær’s PULSE data acquisition software. The natural frequency of the pendulum is measured from a free decay signal to be fn = 1.16 Hz. For motions of typical amplitude, the pendulum viscous damping ratio is estimated to be = 0.04 by application of the logarithmic decrement method. (The effective value of is found to be higher for small amplitude motion.) The motor rotational frequency can be controlled reliably over a range of 0–5 Hz with a resolution of approximately 0.02 Hz.
Fig. 1. (a) The system schematic (side view) and (b) the experimental realization (front view).
746
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
3. Modeling and Simulation The equation modeling the angular position, , of a simple viscously damped pendulum under free vibration is well known ¨ + 2n ˙ + 2n sin = 0,
(1)
where is the viscous damping ratio and n is the angular natural frequency. Eq. (1) is non-linear due to the sin() term, but can be easily linearized provided the pendulum undergoes only small amplitude oscillations. While the experimental system generally experiences oscillations that may be considered small, and while Eq. (1) can be solved analytically through the use of elliptic integrals, the discrete-forcing that is of interest here has led us to simulate the system by solving Eq. (1) numerically. The experimentally determined values of n = 7.29 rad/s and = 0.04 are used as inputs to Eq. (1) in all of the simulations. 3.1. Detecting and classifying impact The fact that impact may occur across a continuous spatial envelope complicates the issue of impact detection. Both bodies residing within the impact envelope at a given time is a necessary but insufficient condition for impact. We can express the impact condition by recognizing that impact will occur when the horizontal and vertical distances between the two bodies vanish L˜ sin pi − R˜ sin di = 0, (2) L˜ cos pi − d + R˜ cos di = 0, where pi and di are the angular positions of the pendulum and the impactor at impact. L˜ and R˜ are a modified pendulum length and impactor radius, respectively. These modified parameters are necessary because due to the overlap of trajectories, the two bodies do not generally impact at their tips. In fact, tip-to-tip impact will only occur at the envelope boundary points A and B. Eqs. (2) become tractable if we account for the observation that when di is negative, the tip of the pendulum will strike the impactor at some point below the impactor’s tip. This point is said to be located a distance R˜ from the motor shaft. Thus, L˜ = L and Eqs. (2) become L sin pi − R˜ sin di = 0, L cos pi − d + R˜ cos di = 0,
(3)
provided di 0. The second of Eqs. (3) is used to solve for R˜ and the result, R˜ = (d − L cos pi )/ cos di , is substituted into the first of Eqs. (3) to give pi =
(d − L) tan(di ) L
for di 0,
(4)
where small angle approximations (sin(pi )≈pi , cos(pi )≈1) are used since the pendulum angles associated with the extreme boundaries of the impact envelope are only about ±2◦ . A similar procedure is applied when pi is positive. In this case, the tip of the impactor will strike the pendulum at some distance L˜ from the pendulum pivot. Eqs. (2) can thus be rewritten
for positive di values by allowing R˜ = R. Using one rewritten equation to solve for L˜ and substituting the result into the other equation gives pi =
R sin(di ) (d − R cos(di ))
for di > 0,
(5)
where again the small angle approximations have been used. The domains of applicability associated with Eqs. (4) and (5) are reversed in the case of catch-up impacts (see below). The relationship between pi and di is also determined experimentally by bringing the pendulum and impactor into contact with each other at different points across the impact envelope. A straight line is subsequently fit to the resulting data set. A typical experimental impact relationship is shown in Fig. 2(a) along with the theoretical relationship given by Eqs. (4) and (5). The differences between the two curves are due to the finite thicknesses associated with both the pendulum and the impactor. These thicknesses are also responsible for the slight asymmetry of the experimental impact envelope (i.e., the magnitudes of di associated with points A and B are somewhat different). All simulations are generated using an experimentally determined impact relationship. The fitted line defining this relationship is used as an “event” condition within Matlab’s ODE45 (fourth–fifth order Runge–Kutta) numerical integration scheme. When the trajectories of the pendulum and the impactor satisfy the equation of the line, an impact is signaled and the integration is stopped. It then becomes necessary to classify the type of impact. Four qualitatively dissimilar impact classifications have been observed in the experimental system and are summarized graphically in Fig. 2(b). By far the most common impact is termed a head-on impact. Here, the pendulum and the impactor collide while traveling in opposite directions. The pendulum subsequently rebounds in the direction opposite of its approach. A second possibility is for the pendulum to be impacted while traveling in the same direction as the impactor. This push impact (akin to pushing a child on a swing) does not change the sign of the pendulum’s velocity, just its magnitude. A third impact type occurs when the pendulum and the impactor collide near the outer boundaries of the impact envelope. This impact, termed a clip, does not change the direction of the pendulum; it merely slows it down. In theory, clip impacts should only occur at the discrete points A and B. In the experimental system, however, these impacts are observed to occur over small continuous envelopes located near points A and B. This is believed to be the result of two factors: (1) the finite thickness of the impactor. Because the impactor does have an associated thickness, it is possible for the pendulum to scrape across the top of the impactor if the two bodies are located within a certain small region. (2) There is some slight bending and recoil of the impactor at impact. This bending and recoil effectively increases the size of the two envelopes over which clip impacts can occur. The boundaries of the two clip envelopes are estimated experimentally and considered in the simulation. The two clip envelopes combine for about 5% of the total size of the overall impact envelope.
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
747
3 (-) (+)
(-)
B
2
(+)
θpi (Degrees)
1 ωd 0
ωd
“Head-on”
“Push”
-1 (-)
(-) -2
(+)
(+)
A -3 -50 -40 -30 -20 -10 0
ωd 10 20 30 40 50
ωd
“Clip”
θdi (Degrees)
“Catch-up”
Fig. 2. (a) Theoretical (dashed) and experimental (solid line fitted to discrete data points) impact detection relationship. (b) Graphical depiction of the four experimentally observed impact scenarios.
The fourth experimentally observed impact scenario is termed a catch-up. It occurs when the pendulum strikes the back of the impactor thus changing the pendulum’s velocity from positive to negative. Since this type of impact is rare and is only observed in transient responses, it is not included in any of the simulations. It is also noted that a fifth type of impact is theoretically possible, but has never been observed experimentally. This fifth impact type is a combination of the catch-up and clip impacts and occurs when the pendulum catches up to the back of the impactor, but just clips the top of it. Thus, the pendulum’s direction of travel is not changed. This type of impact is not considered in the simulations.
most impacts, the modified angular velocity, ˙ d , is quite close to the actual angular velocity, d . The COR, K, is given by the ratio of post- and pre-impact velocities K =−
− − + + mp L˙ p + md R ˙ d = mp L˙ p + md R ˙ d ,
(6)
where the + and − superscripts denote the pre- and post-impact velocities and md is the mass of the impactor. Since the impact is generally oblique and only a component of the impact force acts in the direction of the pendulum’s degree of freedom, ˙ d represents a modified form of the actual angular velocity of the impactor. It can be shown from geometry that ˙ d = d cos(pi + di ).
(7)
For an impact envelope of the size considered here cos(pi +di ) attains values between approximately 0.75 and unity. Thus for
− − ˙ d − ˙ p
.
(8)
Using Eqs. (6) and (8) the post-impact velocities of the two bodies can be written as + ˙ p =
3.2. Modeling impact Once an impact has been detected and classified, it becomes necessary to reinitialize the equation of motion by assigning an appropriate post-impact velocity. We first consider impacts of the head-on and push variety and write the law of conservation of angular momentum in terms of the angular velocity of the pendulum, ˙ p , and the modified angular velocity of the impactor, ˙ d ,
+ + ˙ d − ˙ p
+ ˙ d =
− − (mp L − Kmd R)˙ p + md R(1 + K)˙ d
mp L + m d R − − mp L(1 + K)˙ p + (md R − Kmp L)˙ d
mp L + m d R
,
(9)
.
(10)
Assuming that the motion of the impactor is not affected by the + − impact (i.e., md → ∞ and ˙ d = ˙ d = ˙ d ) Eq. (9) becomes + − ˙ p = ˙ d + K(˙ d − ˙ p ).
(11)
Eq. (11) is valid for both head-on and push type collisions. Because they are observed to occur rarely and only during transient motion, catch-up impacts are not included in the simulation. However, if they were to be included, the post-impact velocity would be given by the opposite of Eq. (11) due to the reversed direction of the impact force associated with catch-up impacts. Eq. (11) (or its opposite) is valid for head-on, push, and catch-up impacts because in each of these cases the impactor effectively acts like a barrier to the pendulum’s motion. This is not true for clip impacts where the impactor does not block the motion of the pendulum; it merely decreases the pendulum’s velocity. Based on this observation, the pendulum’s post-impact velocity in the case of a clip impact is
748
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
assumed to be + − ˙ p = Kc ˙ p ,
(12)
where Kc is the COR associated with a clip impact (a value of Kc = 0.2 is estimated experimentally and used in the simulation). Eqs. (11) and (12) are used to reinitialize the velocity of the pendulum after impact is detected and classified. The use of Eq. (11) requires that K be specified. Since K is a parameter that not only depends upon the two materials undergoing impact, but also the relative velocity of their approach, an attempt is made to estimate K experimentally. To this end, the pendulum is released from various angles such that it may collide with the vertically positioned (static) impactor. By taking the ratio of the measured post- and pre-impact velocities, K is computed for each angle. These data are plotted against the relative velocity of approach, vrel , and a best-fit cubic expression is determined. This velocity-dependent best-fit expression is used to prescribe K in the simulation. However, it is well known that laboratory measurements of the COR are generally fraught with uncertainty. Impact is a complex phenomenon where energy is converted to heat, sound, and structural vibrations within the colliding bodies [19]. In order to account for this uncertainty, a random component is added to the velocity-dependent COR, K(vrel ) ˜ rel ) = K(vrel ) + ( − ), K(v
(13)
where is a random component characterized by a mean value . In all simulations, = 0.085 and varies from 0 to 0.17. The empirical expression for K(vrel ) is given by 3 2 − 0.6478vrel + 0.5492vrel + 0.1328. K(vrel ) = 0.2015vrel
(14) ˜ rel ) range from approximately 0.16 to Typical values of K(v 0.34. The effect that the random component has on the simulation results is discussed in Section 4.2. 4. Results and discussion As the driving frequency of the impactor is varied, the pendulum demonstrates a wide range of dynamic behavior. It is convenient to define a parameter equal to the angular velocity of the rotating impactor normalized by the natural frequency of the pendulum (i.e., ≡ d /n ). The various dynamic responses that are encountered as is increased are now discussed. 4.1. Low frequency responses For values of that are much less than unity, a chattering effect can be observed in which the impactor and the pendulum impact many times as the impactor makes a single pass through the impact envelope. During chatter, the relative velocity between the impactor and the pendulum decreases with each successive impact until it eventually vanishes. At that point,
sticking occurs as the impactor pushes the pendulum to the end of the impact envelope where the two bodies separate. The pendulum then executes free decay oscillations and depending upon the value of and its damping ratio, the pendulum may in fact come to rest before the impactor reenters the impact envelope. Several researchers have characterized chattering by a P(p, q) convention where p is the number of impacts occurring over q forcing periods [20–22]. Two experimental low frequency responses are shown in Fig. 3. In part (a), only sticking behavior can be observed. Any chattering that may occur when the two bodies first come into contact cannot be observed visually or by inspection of the time series. Part (b) is an example of a P(5,1) chattering response. The impact envelope is denoted in Fig. 3 by the dashed lines. Note that in both cases the pendulum returns to rest before the impactor reenters the impact envelope. From various observations of the experimental system it is clear that a variety of chattering responses are possible. Indeed, the evolution of different (periodic or chaotic) chattering behavior in response to a changing control parameter is an interesting problem in its own right and has been investigated theoretically by Zhao [20].
4.2. Periodic responses As increases, the number of impacts per single pass through the impact envelope reduce and the maximum amplitude of the response increases. Eventually, each impact will tend to drive the pendulum outside of the impact envelope thus making it possible for the rotating impactor to not collide with the pendulum on a given pass through the impact envelope. Out of this condition arise many possible responses including a variety of periodic behavior. Fig. 4 displays the time series and time-lag embedded phase portraits associated with a sampling of experimental low-order periodic responses. To more clearly illustrate the form of the phase portrait, a low-pass tenth-order Butterworth filter with cut-off frequency of 20 Hz is applied to the experimental data. This filter removes electrical and other high frequency noise associated with the structural vibration of the pendulum arm and is used to produce all experimental phase portraits contained herein. All experimental time series data presented remain unfiltered. All phase portraits (as well as the Poincaré sections to be discussed later) use a time-lag equal to one fourth the free period of the pendulum. Since the system undergoes free oscillations in between impacts, this proves to be an effective choice. The free period of the pendulum is 1/fn = 0.86 s, thus the time-lag in all phase portraits and Poincaré sections is T = 0.22 s. Again the impact envelope is indicated in the times series by dashed lines. In all cases, transients are allowed to decay prior to data acquisition. According to the convention introduced in Section 4.1 the responses depicted in Fig. 4(a)–(c) would all be denoted P(1,1). However, these responses are clearly qualitatively dissimilar. A second more detailed convention might therefore be useful in the classification of individual responses. We propose a new convention that compactly captures the salient feature of
3 2 1 0 -1 -2 -3
θ(t) (Degrees)
θ(t) (Degrees)
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
0
1
2
3
4
5
6
3 2 1 0 -1 -2 -3 0
Time (Seconds)
749
1
2
3
4
5
6
Time (Seconds)
Fig. 3. Experimental time series of (a) sticking response at = 0.042 and (b) P(5,1) chattering response at = 0.17.
5
5 0
0
-5
Ω = 0.45, P(4,1,1)
-5 0
5
10
15
5
0
5
-5
0
5
-5
0
5
-5
0
5
-5
0
5
-5
0
5
5 0
0 Ω = 0.94, P(2,1,1)
-5 0
-5 5
10
15
5
Ω = 1.73, P(1,1,1)
-5 0
5
10
15
5 0 Ω = 0.82, P(6,2,2)
-5 0
θ(t-Δ t) (Degrees)
5
0
θ(t) (Degrees)
-5
0 -5
5 0 -5
5
10
15
5 5 0
0 Ω = 0.86, P(5,1,2)
-5 0
-5 5
10
15
5
5
0
0
-5
-5
Ω =1.31, P(3,1,2)
0
5
10 Time (Seconds)
15
θ(t) (Degrees)
Fig. 4. Experimental time series and the corresponding phase portraits for a variety of low-order periodic responses. Impact locations are indicated on the phase portraits with arrows. Phase portraits cycle in a counter-clockwise direction due to the nature of the time-lag embedding. The unique P(p, q, n) identifier associated with each response is indicated in the bottom portion of each time series plot.
a given response: P(n, p, q) where n represents the number of (non-impact) extrema (i.e., changes in direction) occurring over q forcing periods. As before, p is the number of impacts that occur in q forcing periods. Using this convention each of the responses shown in Fig. 4 is given a unique identifier.
For > 2, high-order (i.e., q > 3) responses are observed in both simulation and experiment. Experimental time series, phase portraits, and power spectra corresponding to P(6,2,7) and P(8,2,9) are shown in Fig. 5. It is noted that the range of frequencies associated with the P(8,2,9) motion is small, and
750
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753 20 θ(t-Δ t) (Degrees)
θ(t) (Degrees)
20 10 0 -10
5
10
0 -10 -20 -20
-20 0
10
15
-10
0
10
20
θ(t) (Degrees)
Time (Seconds) Power (Radians2)
10-1 10-5 10-9 10-14 0
1
2
3
4
5
6
ω/ωd 20 θ(t-Δ t) (Degrees)
θ(t) (Degrees)
20 10 0 -10
5
10
15
Power (Radians2)
-10
-10
0
10
20
θ(t) (Degrees)
Time (Seconds) 10
0
-20 -20
-20 0
10
-1
10-5 10-9 10-14 0
1
2
3
4
5
6
ω/ωd
Fig. 5. Experimental time series, phase portraits, and power spectra corresponding to (a) P(6,2,7) response at = 2.32 and (b) P(8,2,9) response at = 2.27. The phase portraits again cycle in a counter-clockwise direction.
locating this motion experimentally is difficult (at least for the physical realization being considered). The strong periodicity of the response is also evident in the power spectra where the response frequency, is normalized by the drive frequency, d . The spectra are produced by means of a conventional Fast Fourier transform (FFT) algorithm. An intriguing feature related to the COR in an experimental system can be illustrated by the P(6,2,7) motion. Fig. 6 depicts three (500 point) Poincaré sections associated with this motion. Part (a) is a simulated Poincaré section generated using ˜ rel ) = K(vrel )). a COR without a random component (i.e., K(v
Part (b) is created from a simulation that includes a random component which (in the extreme) can produce values of ˜ rel ) that are about 36% higher or lower than K(vrel ). (Note K(v ˜ rel ) in Eq. (13), the maximum extent that given the form of K(v to which the random component contributes to the value of ˜ rel ) varies with vrel . Thus, the 36% value given above is K(v only valid for the P(6,2,7) motion under consideration. In the general limiting case where vrel → 0, the random component ˜ rel ) that are up to 64% used here can produce values of K(v higher or lower than K(vrel ).) Part (c) is generated from experimental data. Observe the agreement between the simulated
0
10
20
20
θ(tk-Δ T) (Degrees)
20 15 10 5 0 -5 -10 -15 -20 -20 -10
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
10 0 -10 -20 -20 -10
θ(tk) (Degrees)
0
10
20
751
20 10 0 -10 -20 -20 -10
0
10
20
θ(tk) (Degrees)
θ(tk) (Degrees)
Fig. 6. Poincaré sections of P(6,2,7) motion at = 2.32. Five hundred Poincaré points are plotted in all cases. (a) Simulated Poincaré section with = = 0. (b) Simulated Poincaré section with = 0.085, min = 0.0, and max = 0.17. (c) Experimental Poincaré section. 10
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
10 5 0 -5
5 0 -5 -10 -10
-10 -10
-5
0
5
10
Power (Radians2)
Power (Radians2)
10-2 10-6 10-10 10-14 1
2
3
0
5
10
θ(tk) (Degrees)
θ(tk) (Degrees)
0
-5
4
5
10-2 10-6 10-10 10-14
6
0
1
2
3
ω/ωd
4
5
6
ω/ωd
Fig. 7. (a) Simulated and (b) experimental Poincaré sections as well as (c) simulated and (d) experimental power spectra for the response at = 1.51. 20
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
20 10 0 -10 -20 -20
-10
0
10
10 0 -10 -20 -20
20
10-2 10-6 10-10 10-14 0
1
2
3 ω/ωd
-10
0
10
20
θ(tk) (Degrees)
Power (Radians2)
Power (Radians2)
θ(tk) (Degrees)
4
5
6
10-2 10-6 10-10 10-14 0
1
2
3
4
5
6
ω/ωd
Fig. 8. (a) Simulated and (b) experimental Poincaré sections as well as (c) simulated and (d) experimental power spectra for the response at = 2.43.
752
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753 20 15
10 θ(tk) (Degrees)
θ(tk) (Degrees)
10
5 0 -5
5 0 -5 -10 -15
-10
-20
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
0
1.6
0.5
1
1.5
2
2.5
3
2
2.5
3
Ω
Ω 20
15
15
10 θ(tk) (Degrees)
θ(tk) (Degrees)
20
10 5 0
5 0 -5
-5
-10
-10
-15
-15
-20 0
-20 2.2
2.25
2.3
2.35
2.4
2.45
0.5
1
2.5
1.5 Ω
Ω Fig. 10. (a) Simulated and (b) experimental swept bifurcation diagrams. Fig. 9. Simulated bifurcation diagrams over selected ranges of .
Poincaré section in part (b) and the experimental one in part (c). The shape and directional nature of the seven point clusters in each plot correspond strongly. This agreement seems to indicate an intrinsic variability associated with the COR in an experimental impact system. Nevertheless, studies involving impact oscillators which model the COR as a constant (i.e., neither velocity dependent nor partially random) have demonstrated excellent agreement between simulation and experiment (see e.g., [2]). The fact that the system of interest here involves the impact of two dynamic bodies both traveling at relatively high speeds (vis-à-vis other simple impacting systems) may be responsible for bringing COR variability further into focus. It is noted that the simulated Poincaré sections in Fig. 6 are produced without consideration of clip impacts. All other simulations contained herein employ the detection and modeling of this type of impact. Furthermore, the same , min , and max values used to generate Fig. 6(b) are used in all simulated responses yet to be discussed. 4.3. Chaotic responses The system exhibits responses that are suggestive of chaos in both simulation and experiment. Here we characterize chaos as a principally broadband response containing some underlying order that may be revealed by means of a Poincaré section. Figs. 7 and 8 depict simulated and experimental Poincaré
sections and power spectra associated with two such responses. Good qualitative agreement exists between the simulated and experimental Poincaré sections with each section displaying a fingered appearance characteristic of responses that undergo low velocity grazing impacts [1,23]. The power spectra in both cases are largely broadband, though the response does possess noticeably more frequency content at certain discrete frequencies. (This is particularly true for the power spectra associated with the response in Fig. 8.) In both cases, however, strong agreement exists between the simulated and experimental power spectra. 4.4. Bifurcation diagrams Many of the responses highlighted in the previous sections can be observed more generally through the use of bifurcation diagrams. Fig. 9(a) depicts a simulated bifurcation diagram spanning values from 1.2 to 1.6 while Fig. 9(b) spans values from 2.2 to 2.5. In both cases, is incremented by 0.001 and the initial position and velocity of the pendulum are set to zero with each new increment. (This is consistent with how the constant experimental responses displayed in Sections 4.1–4.3 are obtained.) Fig. 9 confirms some of the qualitative agreement that exists between the simulation and experiment. For example, the response suggestive of chaos at = 1.51 can be observed in Fig. 9(a) while the P(6,2,7) response at =2.32 is evident in Fig. 9(b).
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
Simulated and experimental bifurcation diagrams representing a slow sweep through the control parameter () are also generated and displayed in Fig. 10. The experimental sweep is achieved by coupling a second motor to the speed control dial of the primary motor. Through the use of gears and by allow˙ ≈ ing the second motor to run very slowly, a sweep rate of 0.0009 Hz is achieved. It therefore takes about 56 min to sweep from = 0 to 3. The fact that the experimental sweep does not necessarily exhibit responses consistent with the experimental responses displayed in Sections 4.2 and 4.3 suggests that coexisting attractors may exist in the experimental system. While the two diagrams exhibit similar features, qualitative disagreements between the simulated and experimental swept bifurcation diagrams do exist. For example, the simulation suggests the onset of a chaotic window near = 1.2 while the experiment does not show evidence of such a transition until = 1.5. With respect to the second chaotic window near =2.25, much better agreement exists. 5. Conclusions Here we have begun to explore the dynamic behavior of a novel and (deceptively) simple impacting system. The motor drive frequency is the control parameter of interest, but the spatial overlap between the pendulum and the impactor, , could also be investigated as such. While the system seems to be quite sensitive to both COR and damping ratio, (two parameters that are notoriously difficult to measure in a laboratory setting) strong qualitative agreement between the simulated and experimental responses is achieved. It is noted that the system seems to exhibit a self-regulating aspect in the sense that the total energy added to the system is maintained within certain bounds. Imparting strong forces on the pendulum will promote large excursions from the impact envelope potentially producing long inter-impact intervals. Conversely, weak impact forces increase the likelihood that the pendulum will reside within the impact envelope upon the return of the rotating impactor. It is also interesting to note that pendulum does not demonstrate a typical resonant response at driving frequencies near its natural frequency. Instead, such driving frequencies lead to a relatively low amplitude response generated by weak push impacts. Resonant behavior does tend to occur at driving frequencies that are approximately two times the pendulum natural frequency, however. In that case, strong head-on impacts effectively trap the pendulum on one half of its plane of motion. Discrete multi-degree-of-freedom systems or continuous systems that are driven by interactions with a moving obstacle can be imagined. A flexible cantilevered beam struck by pulses emitted periodically from an air puffer device is one such example. However, the single degree-of-freedom system is
753
advantageous here as it eliminates the possibility of higherorder modes obscuring otherwise salient features of the dynamic response.
References [1] J.M.T. Thompson, R. Ghaffari, Chaotic dynamics of an impact oscillator, Phys. Rev. A 27 (1983) 1741–1743. [2] P.V. Bayly, L.N. Virgin, An experimental study of an impacting pendulum, J. Sound Vib. 164 (1993) 364–374. [3] C.J. Budd, F. Dux, Chattering, related behaviour in impact oscillators, Philos. Trans. R. Soc. London, A 347 (1994) 365–389. [4] V.G. Oancea, T.A. Laursen, Investigations of low frequency stick-slip motion: experiments numerical modelling, J. Sound Vib. 213 (1998) 577–600. [5] C.J. Budd, P.T. Piiroinen, Corner bifurcations in non-smoothly forced impact oscillators, Physica D 220 (2006) 127–145. [6] P.J. Holmes, The dynamics of repeated impacts with a sinusoidally vibrating table, J. Sound Vib. 84 (1982) 173–189. [7] R.M. Everson, Chaotic dynamics of a bouncing ball, Physica D 19 (1986) 355–383. [8] A. Mehta, J. Luck, Novel temporal behavior of a nonlinear dynamical system: the completely inelastic bouncing ball, Phys. Rev. Lett. 65 (1990) 393–396. [9] C.N. Bapat, S. Sankar, N. Popplewell, Repeated impacts on a sinusoidally vibrating table reappraised, J. Sound Vib. 108 (1986) 99–115. [10] P. Piera´nski, Jumping particle model. Period doubling cascade in an experimental system, J. Physics (Paris) 44 (1983) 573–578. [11] N. Tufillaro, T. Abbott, J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, New York, 1992. [12] J. Tlusty, F. Ismail, Special aspects of chatter in milling, ASME J. Vib. Acoust. Stress Reliab. Des. 105 (1983) 24–32. [13] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling, part I: general formulation, ASME J. Dyn. Sys. Meas. Control 120 (1998) 22–30. [14] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling, part II: applications to common milling systems, ASME J. Dyn. Sys. Meas. Control 120 (1998) 31–36. [15] M.X. Zhao, B. Balachandran, Dynamics stability of milling process, Int. J. Solids Struct. 38 (2001) 2233–2248. [16] G. Stepán, R. Szalai, B.P. Mann, P.V. Bayly, T. Insperger, J. Gradiek, E. Govekar, Nonlinear dynamics of high-speed milling-analyses, numerics experiments, ASME J. Vib. Acoust. 127 (2005) 197–203. [17] B.V. Chirikov, A universal instability of many dimensional oscillator systems, Phys. Rep. 52 (1979) 263. [18] G. Casti, B.V. Chirkov, J. Ford, F.M. Izraelev, Stochastic behavior of a quantum pendulum under a periodic perturbation, in: G. Casti, J. Ford (Eds.), Stochastic Behavior in Classical Quantum Hamiltonian Systems, Lecture Notes in Physics, vol. 93, Springer, Berlin, 1979. [19] J.L. Meriam, Dynamics, second ed., SI Version, Wiley, New York, 1975. [20] X. Zhao, C.K. Reddy, A.H. Nayfeh, Nonlinear dynamics of an electrically driven impact, Nonlinear Dyn. 40 (2005) 227–239. [21] A.P. Ivanov, Bifurcations in impact systems, Chaos, Solitons, Fract. 7 (1996) 1615–1634. [22] F. Peterka, Bifurcations transition phenomena in an impact oscillator, Chaos, Solitons, Fract. 7 (1996) 1635–1647. [23] P.T. Piiroinen, L.N. Virgin, A.R. Champneys, Chaos period-adding: experimental numerical verification of the grazing bifurcation, J. Nonlinear Sci. 14 (2004) 383–404.
Non-linear behavior in a discretely forced oscillator R.B. Davis ∗ , L.N. Virgin Pratt School of Engineering, Duke University, Durham, NC 27708, USA Received 20 November 2006; received in revised form 31 January 2007; accepted 20 February 2007
Abstract The simulated and experimental responses of a rigid-arm pendulum driven by an external impactor are considered. Here, impact occurs if the trajectory of a rotating impactor intersects that of the pendulum. Using the rotation rate of the impactor as the control parameter, experimental trials have demonstrated much of the dynamic behavior predicted by numerical simulations. The system exhibits chatter (i.e., multiple impacts within a single forcing period), sticking (i.e., contact between the pendulum and the impactor for non-negligible amounts of time), high-order periodicity, and behavior suggestive of chaos. A new convention for classifying periodic motions as well as insights regarding the nature of the coefficient of restitution (COR) in an experimental impacting system are also presented. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete forcing; Non-smooth characteristic; Non-linear oscillations; Impact oscillator; Coefficient of restitution; Chatter
1. Introduction Systems subject to non-smooth (i.e., discontinuous) internal or external forces are known to exhibit a wide range of dynamic behavior. Numerical and experimental studies of mechanical systems with discontinuities in their internal (i.e., stiffness or damping) characteristics are numerous (see e.g., [1–4]). A second class of problems arises when a system’s internal forces are smooth, but the external forcing is effectively non-smooth. Such a system can be realized by a relatively lightweight particle or structure undergoing repeated collisions with a moving obstacle. This class can be further divided into two sub-classes. In the first sub-class, the obstacle’s motion can be considered smooth, while in the second sub-class, it may not. The theory governing systems belonging to the first sub-class has been well investigated, but there has historically been less effort to experimentally validate the results. In this study, we discuss the simulated and experimental responses of a novel system that can be characterized by the first sub-class. The system consists of a simple pendulum that is driven by collisions with a smoothly rotating obstacle. Systems belonging to the second sub-class have received recent theoretical attention [5]. ∗ Corresponding author.
E-mail address: [email protected] (R.B. Davis). 0020-7462/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2007.02.012
Repeated collisions with a moving obstacle can initiate a complex interaction between the state of the system and that of the obstacle. A classic example of this is a small ball bouncing on a massive vibrating table. Numerous researchers have considered this problem in both theoretical [6–9] and experimental contexts [10]. Tufillaro et al. [11] provide an extended discussion of the bouncing ball problem as an introduction to many of the central concepts of non-linear dynamics. The authors illustrate how changes to a control parameter (either the amplitude or frequency of the table’s motion) can lead to a variety of dynamic behaviors including chattering, sticking, periodic motion, and chaos. In the presence of a dynamic obstacle, impact will tend to occur across a continuous spatial envelope, which broadens the range of possible dynamic behavior. Contrast this with more conventional impact systems where a harmonically driven oscillator will undergo collisions with an obstacle that is fixed in space. Thus, impact occurs if and when the oscillator achieves a single spatial position. This impact condition can be interpreted as a discontinuity in the oscillator’s stiffness [2]. An object undergoing repeated collisions with a moving obstacle is a situation arising in a variety of natural and engineered settings. The research related to a ball bouncing on a moving table was first motivated by the problem of noise generation in machinery. Other physical examples of interest include wind
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
gusts striking flexible structures and marine vessels colliding with floating platforms. The particular system considered here is especially analogous to milling processes. The impactor can be representative of the rotating milling tool while the pendulum corresponds to the workpiece. The issue of chatter and instability in milling processes is an important concern that has received considerable attention (see e.g., [12–16]). The system of interest is also somewhat reminiscent of the “kicked rotor” [17], a simple (albeit physically impractical) configuration often used to demonstrate complex dynamics in a Hamiltonian system. The kicked rotor is a simple undamped pendulum allowed to rotate through all angles. As it rotates, a gravitational field is periodically turned on and off. This results in intriguing dynamic behavior that has been shown to have applications to quantum mechanics [18]. The remainder of this paper will describe the experimental set-up as well as the modeling and simulation techniques that are employed to capture its dynamic features. Section 4 presents the simulated and experimental results for a variety of qualitative behaviors. Considerations related to the nature of the coefficient of restitution (COR) in an experimental system are also discussed. These results are followed by general conclusions and the suggestion of a system that may merit future consideration.
2. Experimental system Fig. 1(a) depicts a schematic of pendulum/impactor system and Fig. 1(b) is a photo of the experimental realization. The
745
experimental set-up consists of an aluminum pendulum of length L = 31.80 cm that is allowed to pivot with a small amount of friction through the use of two journal bearings. A steel impacting tab is embedded in an aluminum sleave fixed to the shaft of a General Electric Zero-Max variable speed electric motor. The motor delivers a rated torque of 2.82 N m. The distance from the center of the motor shaft to the tip of the impacting tab is R = 2.87 cm. The vertical distance between the pendulum pivot and the motor shaft is d = 34.02 cm. The envelope in which impact may occur assumes the shape of an asymmetrical lens, the boundaries of which can be described by the points (denoted A and B) where the arcs traced by the pendulum and by the impactor intersect. The experimental system is designed such that the horizontal offset between the pendulum pivot and motor shaft, , is zero. The pendulum pivot and the motor shaft are each coupled to a Bourns Electronics precision potentiometer (model number 6639S-1-103) to simultaneously measure the angular displacement of each component. Voltage is supplied to the potentiometers via an HP 6228B dual DC power supply. The potentiometer output is collected with Brüel & Kjær’s PULSE data acquisition software. The natural frequency of the pendulum is measured from a free decay signal to be fn = 1.16 Hz. For motions of typical amplitude, the pendulum viscous damping ratio is estimated to be = 0.04 by application of the logarithmic decrement method. (The effective value of is found to be higher for small amplitude motion.) The motor rotational frequency can be controlled reliably over a range of 0–5 Hz with a resolution of approximately 0.02 Hz.
Fig. 1. (a) The system schematic (side view) and (b) the experimental realization (front view).
746
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
3. Modeling and Simulation The equation modeling the angular position, , of a simple viscously damped pendulum under free vibration is well known ¨ + 2n ˙ + 2n sin = 0,
(1)
where is the viscous damping ratio and n is the angular natural frequency. Eq. (1) is non-linear due to the sin() term, but can be easily linearized provided the pendulum undergoes only small amplitude oscillations. While the experimental system generally experiences oscillations that may be considered small, and while Eq. (1) can be solved analytically through the use of elliptic integrals, the discrete-forcing that is of interest here has led us to simulate the system by solving Eq. (1) numerically. The experimentally determined values of n = 7.29 rad/s and = 0.04 are used as inputs to Eq. (1) in all of the simulations. 3.1. Detecting and classifying impact The fact that impact may occur across a continuous spatial envelope complicates the issue of impact detection. Both bodies residing within the impact envelope at a given time is a necessary but insufficient condition for impact. We can express the impact condition by recognizing that impact will occur when the horizontal and vertical distances between the two bodies vanish L˜ sin pi − R˜ sin di = 0, (2) L˜ cos pi − d + R˜ cos di = 0, where pi and di are the angular positions of the pendulum and the impactor at impact. L˜ and R˜ are a modified pendulum length and impactor radius, respectively. These modified parameters are necessary because due to the overlap of trajectories, the two bodies do not generally impact at their tips. In fact, tip-to-tip impact will only occur at the envelope boundary points A and B. Eqs. (2) become tractable if we account for the observation that when di is negative, the tip of the pendulum will strike the impactor at some point below the impactor’s tip. This point is said to be located a distance R˜ from the motor shaft. Thus, L˜ = L and Eqs. (2) become L sin pi − R˜ sin di = 0, L cos pi − d + R˜ cos di = 0,
(3)
provided di 0. The second of Eqs. (3) is used to solve for R˜ and the result, R˜ = (d − L cos pi )/ cos di , is substituted into the first of Eqs. (3) to give pi =
(d − L) tan(di ) L
for di 0,
(4)
where small angle approximations (sin(pi )≈pi , cos(pi )≈1) are used since the pendulum angles associated with the extreme boundaries of the impact envelope are only about ±2◦ . A similar procedure is applied when pi is positive. In this case, the tip of the impactor will strike the pendulum at some distance L˜ from the pendulum pivot. Eqs. (2) can thus be rewritten
for positive di values by allowing R˜ = R. Using one rewritten equation to solve for L˜ and substituting the result into the other equation gives pi =
R sin(di ) (d − R cos(di ))
for di > 0,
(5)
where again the small angle approximations have been used. The domains of applicability associated with Eqs. (4) and (5) are reversed in the case of catch-up impacts (see below). The relationship between pi and di is also determined experimentally by bringing the pendulum and impactor into contact with each other at different points across the impact envelope. A straight line is subsequently fit to the resulting data set. A typical experimental impact relationship is shown in Fig. 2(a) along with the theoretical relationship given by Eqs. (4) and (5). The differences between the two curves are due to the finite thicknesses associated with both the pendulum and the impactor. These thicknesses are also responsible for the slight asymmetry of the experimental impact envelope (i.e., the magnitudes of di associated with points A and B are somewhat different). All simulations are generated using an experimentally determined impact relationship. The fitted line defining this relationship is used as an “event” condition within Matlab’s ODE45 (fourth–fifth order Runge–Kutta) numerical integration scheme. When the trajectories of the pendulum and the impactor satisfy the equation of the line, an impact is signaled and the integration is stopped. It then becomes necessary to classify the type of impact. Four qualitatively dissimilar impact classifications have been observed in the experimental system and are summarized graphically in Fig. 2(b). By far the most common impact is termed a head-on impact. Here, the pendulum and the impactor collide while traveling in opposite directions. The pendulum subsequently rebounds in the direction opposite of its approach. A second possibility is for the pendulum to be impacted while traveling in the same direction as the impactor. This push impact (akin to pushing a child on a swing) does not change the sign of the pendulum’s velocity, just its magnitude. A third impact type occurs when the pendulum and the impactor collide near the outer boundaries of the impact envelope. This impact, termed a clip, does not change the direction of the pendulum; it merely slows it down. In theory, clip impacts should only occur at the discrete points A and B. In the experimental system, however, these impacts are observed to occur over small continuous envelopes located near points A and B. This is believed to be the result of two factors: (1) the finite thickness of the impactor. Because the impactor does have an associated thickness, it is possible for the pendulum to scrape across the top of the impactor if the two bodies are located within a certain small region. (2) There is some slight bending and recoil of the impactor at impact. This bending and recoil effectively increases the size of the two envelopes over which clip impacts can occur. The boundaries of the two clip envelopes are estimated experimentally and considered in the simulation. The two clip envelopes combine for about 5% of the total size of the overall impact envelope.
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
747
3 (-) (+)
(-)
B
2
(+)
θpi (Degrees)
1 ωd 0
ωd
“Head-on”
“Push”
-1 (-)
(-) -2
(+)
(+)
A -3 -50 -40 -30 -20 -10 0
ωd 10 20 30 40 50
ωd
“Clip”
θdi (Degrees)
“Catch-up”
Fig. 2. (a) Theoretical (dashed) and experimental (solid line fitted to discrete data points) impact detection relationship. (b) Graphical depiction of the four experimentally observed impact scenarios.
The fourth experimentally observed impact scenario is termed a catch-up. It occurs when the pendulum strikes the back of the impactor thus changing the pendulum’s velocity from positive to negative. Since this type of impact is rare and is only observed in transient responses, it is not included in any of the simulations. It is also noted that a fifth type of impact is theoretically possible, but has never been observed experimentally. This fifth impact type is a combination of the catch-up and clip impacts and occurs when the pendulum catches up to the back of the impactor, but just clips the top of it. Thus, the pendulum’s direction of travel is not changed. This type of impact is not considered in the simulations.
most impacts, the modified angular velocity, ˙ d , is quite close to the actual angular velocity, d . The COR, K, is given by the ratio of post- and pre-impact velocities K =−
− − + + mp L˙ p + md R ˙ d = mp L˙ p + md R ˙ d ,
(6)
where the + and − superscripts denote the pre- and post-impact velocities and md is the mass of the impactor. Since the impact is generally oblique and only a component of the impact force acts in the direction of the pendulum’s degree of freedom, ˙ d represents a modified form of the actual angular velocity of the impactor. It can be shown from geometry that ˙ d = d cos(pi + di ).
(7)
For an impact envelope of the size considered here cos(pi +di ) attains values between approximately 0.75 and unity. Thus for
− − ˙ d − ˙ p
.
(8)
Using Eqs. (6) and (8) the post-impact velocities of the two bodies can be written as + ˙ p =
3.2. Modeling impact Once an impact has been detected and classified, it becomes necessary to reinitialize the equation of motion by assigning an appropriate post-impact velocity. We first consider impacts of the head-on and push variety and write the law of conservation of angular momentum in terms of the angular velocity of the pendulum, ˙ p , and the modified angular velocity of the impactor, ˙ d ,
+ + ˙ d − ˙ p
+ ˙ d =
− − (mp L − Kmd R)˙ p + md R(1 + K)˙ d
mp L + m d R − − mp L(1 + K)˙ p + (md R − Kmp L)˙ d
mp L + m d R
,
(9)
.
(10)
Assuming that the motion of the impactor is not affected by the + − impact (i.e., md → ∞ and ˙ d = ˙ d = ˙ d ) Eq. (9) becomes + − ˙ p = ˙ d + K(˙ d − ˙ p ).
(11)
Eq. (11) is valid for both head-on and push type collisions. Because they are observed to occur rarely and only during transient motion, catch-up impacts are not included in the simulation. However, if they were to be included, the post-impact velocity would be given by the opposite of Eq. (11) due to the reversed direction of the impact force associated with catch-up impacts. Eq. (11) (or its opposite) is valid for head-on, push, and catch-up impacts because in each of these cases the impactor effectively acts like a barrier to the pendulum’s motion. This is not true for clip impacts where the impactor does not block the motion of the pendulum; it merely decreases the pendulum’s velocity. Based on this observation, the pendulum’s post-impact velocity in the case of a clip impact is
748
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
assumed to be + − ˙ p = Kc ˙ p ,
(12)
where Kc is the COR associated with a clip impact (a value of Kc = 0.2 is estimated experimentally and used in the simulation). Eqs. (11) and (12) are used to reinitialize the velocity of the pendulum after impact is detected and classified. The use of Eq. (11) requires that K be specified. Since K is a parameter that not only depends upon the two materials undergoing impact, but also the relative velocity of their approach, an attempt is made to estimate K experimentally. To this end, the pendulum is released from various angles such that it may collide with the vertically positioned (static) impactor. By taking the ratio of the measured post- and pre-impact velocities, K is computed for each angle. These data are plotted against the relative velocity of approach, vrel , and a best-fit cubic expression is determined. This velocity-dependent best-fit expression is used to prescribe K in the simulation. However, it is well known that laboratory measurements of the COR are generally fraught with uncertainty. Impact is a complex phenomenon where energy is converted to heat, sound, and structural vibrations within the colliding bodies [19]. In order to account for this uncertainty, a random component is added to the velocity-dependent COR, K(vrel ) ˜ rel ) = K(vrel ) + ( − ), K(v
(13)
where is a random component characterized by a mean value . In all simulations, = 0.085 and varies from 0 to 0.17. The empirical expression for K(vrel ) is given by 3 2 − 0.6478vrel + 0.5492vrel + 0.1328. K(vrel ) = 0.2015vrel
(14) ˜ rel ) range from approximately 0.16 to Typical values of K(v 0.34. The effect that the random component has on the simulation results is discussed in Section 4.2. 4. Results and discussion As the driving frequency of the impactor is varied, the pendulum demonstrates a wide range of dynamic behavior. It is convenient to define a parameter equal to the angular velocity of the rotating impactor normalized by the natural frequency of the pendulum (i.e., ≡ d /n ). The various dynamic responses that are encountered as is increased are now discussed. 4.1. Low frequency responses For values of that are much less than unity, a chattering effect can be observed in which the impactor and the pendulum impact many times as the impactor makes a single pass through the impact envelope. During chatter, the relative velocity between the impactor and the pendulum decreases with each successive impact until it eventually vanishes. At that point,
sticking occurs as the impactor pushes the pendulum to the end of the impact envelope where the two bodies separate. The pendulum then executes free decay oscillations and depending upon the value of and its damping ratio, the pendulum may in fact come to rest before the impactor reenters the impact envelope. Several researchers have characterized chattering by a P(p, q) convention where p is the number of impacts occurring over q forcing periods [20–22]. Two experimental low frequency responses are shown in Fig. 3. In part (a), only sticking behavior can be observed. Any chattering that may occur when the two bodies first come into contact cannot be observed visually or by inspection of the time series. Part (b) is an example of a P(5,1) chattering response. The impact envelope is denoted in Fig. 3 by the dashed lines. Note that in both cases the pendulum returns to rest before the impactor reenters the impact envelope. From various observations of the experimental system it is clear that a variety of chattering responses are possible. Indeed, the evolution of different (periodic or chaotic) chattering behavior in response to a changing control parameter is an interesting problem in its own right and has been investigated theoretically by Zhao [20].
4.2. Periodic responses As increases, the number of impacts per single pass through the impact envelope reduce and the maximum amplitude of the response increases. Eventually, each impact will tend to drive the pendulum outside of the impact envelope thus making it possible for the rotating impactor to not collide with the pendulum on a given pass through the impact envelope. Out of this condition arise many possible responses including a variety of periodic behavior. Fig. 4 displays the time series and time-lag embedded phase portraits associated with a sampling of experimental low-order periodic responses. To more clearly illustrate the form of the phase portrait, a low-pass tenth-order Butterworth filter with cut-off frequency of 20 Hz is applied to the experimental data. This filter removes electrical and other high frequency noise associated with the structural vibration of the pendulum arm and is used to produce all experimental phase portraits contained herein. All experimental time series data presented remain unfiltered. All phase portraits (as well as the Poincaré sections to be discussed later) use a time-lag equal to one fourth the free period of the pendulum. Since the system undergoes free oscillations in between impacts, this proves to be an effective choice. The free period of the pendulum is 1/fn = 0.86 s, thus the time-lag in all phase portraits and Poincaré sections is T = 0.22 s. Again the impact envelope is indicated in the times series by dashed lines. In all cases, transients are allowed to decay prior to data acquisition. According to the convention introduced in Section 4.1 the responses depicted in Fig. 4(a)–(c) would all be denoted P(1,1). However, these responses are clearly qualitatively dissimilar. A second more detailed convention might therefore be useful in the classification of individual responses. We propose a new convention that compactly captures the salient feature of
3 2 1 0 -1 -2 -3
θ(t) (Degrees)
θ(t) (Degrees)
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
0
1
2
3
4
5
6
3 2 1 0 -1 -2 -3 0
Time (Seconds)
749
1
2
3
4
5
6
Time (Seconds)
Fig. 3. Experimental time series of (a) sticking response at = 0.042 and (b) P(5,1) chattering response at = 0.17.
5
5 0
0
-5
Ω = 0.45, P(4,1,1)
-5 0
5
10
15
5
0
5
-5
0
5
-5
0
5
-5
0
5
-5
0
5
-5
0
5
5 0
0 Ω = 0.94, P(2,1,1)
-5 0
-5 5
10
15
5
Ω = 1.73, P(1,1,1)
-5 0
5
10
15
5 0 Ω = 0.82, P(6,2,2)
-5 0
θ(t-Δ t) (Degrees)
5
0
θ(t) (Degrees)
-5
0 -5
5 0 -5
5
10
15
5 5 0
0 Ω = 0.86, P(5,1,2)
-5 0
-5 5
10
15
5
5
0
0
-5
-5
Ω =1.31, P(3,1,2)
0
5
10 Time (Seconds)
15
θ(t) (Degrees)
Fig. 4. Experimental time series and the corresponding phase portraits for a variety of low-order periodic responses. Impact locations are indicated on the phase portraits with arrows. Phase portraits cycle in a counter-clockwise direction due to the nature of the time-lag embedding. The unique P(p, q, n) identifier associated with each response is indicated in the bottom portion of each time series plot.
a given response: P(n, p, q) where n represents the number of (non-impact) extrema (i.e., changes in direction) occurring over q forcing periods. As before, p is the number of impacts that occur in q forcing periods. Using this convention each of the responses shown in Fig. 4 is given a unique identifier.
For > 2, high-order (i.e., q > 3) responses are observed in both simulation and experiment. Experimental time series, phase portraits, and power spectra corresponding to P(6,2,7) and P(8,2,9) are shown in Fig. 5. It is noted that the range of frequencies associated with the P(8,2,9) motion is small, and
750
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753 20 θ(t-Δ t) (Degrees)
θ(t) (Degrees)
20 10 0 -10
5
10
0 -10 -20 -20
-20 0
10
15
-10
0
10
20
θ(t) (Degrees)
Time (Seconds) Power (Radians2)
10-1 10-5 10-9 10-14 0
1
2
3
4
5
6
ω/ωd 20 θ(t-Δ t) (Degrees)
θ(t) (Degrees)
20 10 0 -10
5
10
15
Power (Radians2)
-10
-10
0
10
20
θ(t) (Degrees)
Time (Seconds) 10
0
-20 -20
-20 0
10
-1
10-5 10-9 10-14 0
1
2
3
4
5
6
ω/ωd
Fig. 5. Experimental time series, phase portraits, and power spectra corresponding to (a) P(6,2,7) response at = 2.32 and (b) P(8,2,9) response at = 2.27. The phase portraits again cycle in a counter-clockwise direction.
locating this motion experimentally is difficult (at least for the physical realization being considered). The strong periodicity of the response is also evident in the power spectra where the response frequency, is normalized by the drive frequency, d . The spectra are produced by means of a conventional Fast Fourier transform (FFT) algorithm. An intriguing feature related to the COR in an experimental system can be illustrated by the P(6,2,7) motion. Fig. 6 depicts three (500 point) Poincaré sections associated with this motion. Part (a) is a simulated Poincaré section generated using ˜ rel ) = K(vrel )). a COR without a random component (i.e., K(v
Part (b) is created from a simulation that includes a random component which (in the extreme) can produce values of ˜ rel ) that are about 36% higher or lower than K(vrel ). (Note K(v ˜ rel ) in Eq. (13), the maximum extent that given the form of K(v to which the random component contributes to the value of ˜ rel ) varies with vrel . Thus, the 36% value given above is K(v only valid for the P(6,2,7) motion under consideration. In the general limiting case where vrel → 0, the random component ˜ rel ) that are up to 64% used here can produce values of K(v higher or lower than K(vrel ).) Part (c) is generated from experimental data. Observe the agreement between the simulated
0
10
20
20
θ(tk-Δ T) (Degrees)
20 15 10 5 0 -5 -10 -15 -20 -20 -10
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
10 0 -10 -20 -20 -10
θ(tk) (Degrees)
0
10
20
751
20 10 0 -10 -20 -20 -10
0
10
20
θ(tk) (Degrees)
θ(tk) (Degrees)
Fig. 6. Poincaré sections of P(6,2,7) motion at = 2.32. Five hundred Poincaré points are plotted in all cases. (a) Simulated Poincaré section with = = 0. (b) Simulated Poincaré section with = 0.085, min = 0.0, and max = 0.17. (c) Experimental Poincaré section. 10
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
10 5 0 -5
5 0 -5 -10 -10
-10 -10
-5
0
5
10
Power (Radians2)
Power (Radians2)
10-2 10-6 10-10 10-14 1
2
3
0
5
10
θ(tk) (Degrees)
θ(tk) (Degrees)
0
-5
4
5
10-2 10-6 10-10 10-14
6
0
1
2
3
ω/ωd
4
5
6
ω/ωd
Fig. 7. (a) Simulated and (b) experimental Poincaré sections as well as (c) simulated and (d) experimental power spectra for the response at = 1.51. 20
θ(tk-Δ T) (Degrees)
θ(tk-Δ T) (Degrees)
20 10 0 -10 -20 -20
-10
0
10
10 0 -10 -20 -20
20
10-2 10-6 10-10 10-14 0
1
2
3 ω/ωd
-10
0
10
20
θ(tk) (Degrees)
Power (Radians2)
Power (Radians2)
θ(tk) (Degrees)
4
5
6
10-2 10-6 10-10 10-14 0
1
2
3
4
5
6
ω/ωd
Fig. 8. (a) Simulated and (b) experimental Poincaré sections as well as (c) simulated and (d) experimental power spectra for the response at = 2.43.
752
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753 20 15
10 θ(tk) (Degrees)
θ(tk) (Degrees)
10
5 0 -5
5 0 -5 -10 -15
-10
-20
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
0
1.6
0.5
1
1.5
2
2.5
3
2
2.5
3
Ω
Ω 20
15
15
10 θ(tk) (Degrees)
θ(tk) (Degrees)
20
10 5 0
5 0 -5
-5
-10
-10
-15
-15
-20 0
-20 2.2
2.25
2.3
2.35
2.4
2.45
0.5
1
2.5
1.5 Ω
Ω Fig. 10. (a) Simulated and (b) experimental swept bifurcation diagrams. Fig. 9. Simulated bifurcation diagrams over selected ranges of .
Poincaré section in part (b) and the experimental one in part (c). The shape and directional nature of the seven point clusters in each plot correspond strongly. This agreement seems to indicate an intrinsic variability associated with the COR in an experimental impact system. Nevertheless, studies involving impact oscillators which model the COR as a constant (i.e., neither velocity dependent nor partially random) have demonstrated excellent agreement between simulation and experiment (see e.g., [2]). The fact that the system of interest here involves the impact of two dynamic bodies both traveling at relatively high speeds (vis-à-vis other simple impacting systems) may be responsible for bringing COR variability further into focus. It is noted that the simulated Poincaré sections in Fig. 6 are produced without consideration of clip impacts. All other simulations contained herein employ the detection and modeling of this type of impact. Furthermore, the same , min , and max values used to generate Fig. 6(b) are used in all simulated responses yet to be discussed. 4.3. Chaotic responses The system exhibits responses that are suggestive of chaos in both simulation and experiment. Here we characterize chaos as a principally broadband response containing some underlying order that may be revealed by means of a Poincaré section. Figs. 7 and 8 depict simulated and experimental Poincaré
sections and power spectra associated with two such responses. Good qualitative agreement exists between the simulated and experimental Poincaré sections with each section displaying a fingered appearance characteristic of responses that undergo low velocity grazing impacts [1,23]. The power spectra in both cases are largely broadband, though the response does possess noticeably more frequency content at certain discrete frequencies. (This is particularly true for the power spectra associated with the response in Fig. 8.) In both cases, however, strong agreement exists between the simulated and experimental power spectra. 4.4. Bifurcation diagrams Many of the responses highlighted in the previous sections can be observed more generally through the use of bifurcation diagrams. Fig. 9(a) depicts a simulated bifurcation diagram spanning values from 1.2 to 1.6 while Fig. 9(b) spans values from 2.2 to 2.5. In both cases, is incremented by 0.001 and the initial position and velocity of the pendulum are set to zero with each new increment. (This is consistent with how the constant experimental responses displayed in Sections 4.1–4.3 are obtained.) Fig. 9 confirms some of the qualitative agreement that exists between the simulation and experiment. For example, the response suggestive of chaos at = 1.51 can be observed in Fig. 9(a) while the P(6,2,7) response at =2.32 is evident in Fig. 9(b).
R.B. Davis, L.N. Virgin / International Journal of Non-Linear Mechanics 42 (2007) 744 – 753
Simulated and experimental bifurcation diagrams representing a slow sweep through the control parameter () are also generated and displayed in Fig. 10. The experimental sweep is achieved by coupling a second motor to the speed control dial of the primary motor. Through the use of gears and by allow˙ ≈ ing the second motor to run very slowly, a sweep rate of 0.0009 Hz is achieved. It therefore takes about 56 min to sweep from = 0 to 3. The fact that the experimental sweep does not necessarily exhibit responses consistent with the experimental responses displayed in Sections 4.2 and 4.3 suggests that coexisting attractors may exist in the experimental system. While the two diagrams exhibit similar features, qualitative disagreements between the simulated and experimental swept bifurcation diagrams do exist. For example, the simulation suggests the onset of a chaotic window near = 1.2 while the experiment does not show evidence of such a transition until = 1.5. With respect to the second chaotic window near =2.25, much better agreement exists. 5. Conclusions Here we have begun to explore the dynamic behavior of a novel and (deceptively) simple impacting system. The motor drive frequency is the control parameter of interest, but the spatial overlap between the pendulum and the impactor, , could also be investigated as such. While the system seems to be quite sensitive to both COR and damping ratio, (two parameters that are notoriously difficult to measure in a laboratory setting) strong qualitative agreement between the simulated and experimental responses is achieved. It is noted that the system seems to exhibit a self-regulating aspect in the sense that the total energy added to the system is maintained within certain bounds. Imparting strong forces on the pendulum will promote large excursions from the impact envelope potentially producing long inter-impact intervals. Conversely, weak impact forces increase the likelihood that the pendulum will reside within the impact envelope upon the return of the rotating impactor. It is also interesting to note that pendulum does not demonstrate a typical resonant response at driving frequencies near its natural frequency. Instead, such driving frequencies lead to a relatively low amplitude response generated by weak push impacts. Resonant behavior does tend to occur at driving frequencies that are approximately two times the pendulum natural frequency, however. In that case, strong head-on impacts effectively trap the pendulum on one half of its plane of motion. Discrete multi-degree-of-freedom systems or continuous systems that are driven by interactions with a moving obstacle can be imagined. A flexible cantilevered beam struck by pulses emitted periodically from an air puffer device is one such example. However, the single degree-of-freedom system is
753
advantageous here as it eliminates the possibility of higherorder modes obscuring otherwise salient features of the dynamic response.
References [1] J.M.T. Thompson, R. Ghaffari, Chaotic dynamics of an impact oscillator, Phys. Rev. A 27 (1983) 1741–1743. [2] P.V. Bayly, L.N. Virgin, An experimental study of an impacting pendulum, J. Sound Vib. 164 (1993) 364–374. [3] C.J. Budd, F. Dux, Chattering, related behaviour in impact oscillators, Philos. Trans. R. Soc. London, A 347 (1994) 365–389. [4] V.G. Oancea, T.A. Laursen, Investigations of low frequency stick-slip motion: experiments numerical modelling, J. Sound Vib. 213 (1998) 577–600. [5] C.J. Budd, P.T. Piiroinen, Corner bifurcations in non-smoothly forced impact oscillators, Physica D 220 (2006) 127–145. [6] P.J. Holmes, The dynamics of repeated impacts with a sinusoidally vibrating table, J. Sound Vib. 84 (1982) 173–189. [7] R.M. Everson, Chaotic dynamics of a bouncing ball, Physica D 19 (1986) 355–383. [8] A. Mehta, J. Luck, Novel temporal behavior of a nonlinear dynamical system: the completely inelastic bouncing ball, Phys. Rev. Lett. 65 (1990) 393–396. [9] C.N. Bapat, S. Sankar, N. Popplewell, Repeated impacts on a sinusoidally vibrating table reappraised, J. Sound Vib. 108 (1986) 99–115. [10] P. Piera´nski, Jumping particle model. Period doubling cascade in an experimental system, J. Physics (Paris) 44 (1983) 573–578. [11] N. Tufillaro, T. Abbott, J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, New York, 1992. [12] J. Tlusty, F. Ismail, Special aspects of chatter in milling, ASME J. Vib. Acoust. Stress Reliab. Des. 105 (1983) 24–32. [13] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling, part I: general formulation, ASME J. Dyn. Sys. Meas. Control 120 (1998) 22–30. [14] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling, part II: applications to common milling systems, ASME J. Dyn. Sys. Meas. Control 120 (1998) 31–36. [15] M.X. Zhao, B. Balachandran, Dynamics stability of milling process, Int. J. Solids Struct. 38 (2001) 2233–2248. [16] G. Stepán, R. Szalai, B.P. Mann, P.V. Bayly, T. Insperger, J. Gradiek, E. Govekar, Nonlinear dynamics of high-speed milling-analyses, numerics experiments, ASME J. Vib. Acoust. 127 (2005) 197–203. [17] B.V. Chirikov, A universal instability of many dimensional oscillator systems, Phys. Rep. 52 (1979) 263. [18] G. Casti, B.V. Chirkov, J. Ford, F.M. Izraelev, Stochastic behavior of a quantum pendulum under a periodic perturbation, in: G. Casti, J. Ford (Eds.), Stochastic Behavior in Classical Quantum Hamiltonian Systems, Lecture Notes in Physics, vol. 93, Springer, Berlin, 1979. [19] J.L. Meriam, Dynamics, second ed., SI Version, Wiley, New York, 1975. [20] X. Zhao, C.K. Reddy, A.H. Nayfeh, Nonlinear dynamics of an electrically driven impact, Nonlinear Dyn. 40 (2005) 227–239. [21] A.P. Ivanov, Bifurcations in impact systems, Chaos, Solitons, Fract. 7 (1996) 1615–1634. [22] F. Peterka, Bifurcations transition phenomena in an impact oscillator, Chaos, Solitons, Fract. 7 (1996) 1635–1647. [23] P.T. Piiroinen, L.N. Virgin, A.R. Champneys, Chaos period-adding: experimental numerical verification of the grazing bifurcation, J. Nonlinear Sci. 14 (2004) 383–404.