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Transition from soliton to chaotic motion during impact of a nonlinear elastic structure
Chaos,
Soi,,orrs
Pergamon
& Fracrals Printed
Vol. 1. No. 2. pp. 275-283, 1994 Elsevier Science Lfd Britain. All nghts rescued OY60-0779/94$6.00 + .W
m Great
Transition from Soliton to Chaotic Motion During Impact of a Nonlinear Elastic Structure M. A. DAVIES and F. C. MOON Department
of Mechanical
and Aerospace
Engineering,
Cornell
University,
Ithaca,
NY 14853, USA
(Received 25 March 1993)
Abstract-The existence of a transition from soliton-like motion to chaotic motion in a periodic structure with buckling nonlinearity is demonstrated. An experiment consisting of nine elastic oscillators coupled with buckling sensitive elastica was constructed. The experiment showed a strong demonstrated by the comparison of its response to several near sensitivity to initial conditions, identical impact loads. A numerical model based on a modified Toda lattice was developed and verified the observed sensitivity to initial conditions. Contour plots of energy flow in the numerical model suggest that this sensitivity can be related to a transition from relatively ordered wave motion. immediately following the impact, to disordered chaotic motions at a later time. These findings are relevant to space structures, ship and aircraft structures which are subjected to large dynamic loads.
INTRODUCTION are found in space structures, ship and aircraft structures where the Periodic structures form a lattice-like periodic structure. frame and the skin (or the bays in a space structure) Nonlinear effects in such structures enter naturally through the coupling of in-plane and out-of-plane deformations. In the past decade, nonlinear effects in structures have been shown to lead to dynamics which are very sensitive to initial conditions and hence difficult to predict with conventional numerical structural codes. In this study we examine how wave-like deformations, due to both impact and modal excitation, propagate through a nonlinear periodic structure (Fig. 1). Experiments and numerical simulation are used to show that solitary wave dynamics in a coupled-cell lattice system with a finite number of cells can evolve into a complex spatial pattern with chaos-like dynamics. The experiment consists of nine elastic oscillators coupled with buckling sensitive elastica. We use color contour plots as a new technique for visualizing energy flow in these structures. The results clearly show that the buckling nonlinearity produces numerous complex phenomena. An understanding of these phenomena is necessary if the dynamics and ultimately the failure modes of the structure are to be predicted. This problem is also a benchmark to assess the prediction capabilities of new and existing finite element and other structural dynamics codes. EXPERIMENT To simulate a periodically reinforced shell or plate-like structure, nine elastic oscillators, each one composed of a mass and a cantilevered beam, are coupled together with spring steel elastic buckling elements as shown in Fig. 2. The location of each mass can be measured with a strain gage pair at the base of each cantilevered beam. The state of each buckling element is monitored by strain gages placed directly on the member. The
M. A. D.~UES and F. C. MOON
276
Fig. 1. Sketch
impact pendulum
of periodically
reinforced
plate and shell structures.
shaker -\
Fig. 2. Experimental structure consisting of nine aluminum masses members and to each other through spring steel eccentric
coupled nonlinear
to ground through buckling elements.
cantilevered
experiment is excited either sinusoidally, with an electromagnetic shaker table, or transiently, with an impact pendulum. Preliminary experimental data indicate that the system exhibits a strong sensitivity to initial conditions. Figure 3 shows the comparison of eight separate but nearly identical impacts using strain data from the buckling element nearest
Soliton
6 8 Impacts
to chaotic
motion
transition
277
(Experiment)
4
-6 -8
0.00
0.01
8 Impacts
0.02
Time
0.03
(s)
(Experiment)
6 4
-6 0.08
0.09
Time Fig. 3. Evidence
for sensitivity
0.10
This data to initial conditions in experiment. buckling element nearest the free end.
0.11
comes
from
0.12
strain
gages
on the
M. A. DAVIES and F. C. MOON
278
o o4 8 Impacts(Init Vel - Z.o,Var 3.5%) I' " [I -1 ’ 11 11 ’ *, 11 lI,II II,II
If
E -0.04 si -0.06
0.04
0.02 fz 12
0.00
g h -0.02 a ii -0.04
L -
J
si -0.06
-
I,‘
22
I
,I,
I,
23
1.
24
Fig. 5. Numerical simulation of lattice dynamics for large impact case showing high sensitivity to initial conditions (m = 0.157 kg, y = 362 N/m, b = 725 m-l, k = 117 N/m, c = 0.01 Ns/m, v9 (0) = -2.0 m/s).
Soliton
Fig. 6. Energy
contour
to chaotic
for the case shown
Fig. 8. Energy
contour
motion
transition
in Fig. 5 showing
the breakdown
for the case considered
in Fig. 7.
of solitary
wave order.
Soliton
to chaotic
motion
transition
281
the impact. Immediately following the impact [Fig. 3(a)], the eight strain signals are nearly identical. Later [Fig. 3(b)], they are uncorrelated. This indicates that the dynamics of the system are highly sensitive to differences in the initial impact, that is, the system is sensitive to initial conditions.
NUMERICAL
Modified
MODELING
Toda lattice
The system was modeled as a one-dimensional lattice of nine masses with nonlinear nearest neighbor interactions (Fig. 4). The deformation of the shallow arch structures between the masses can be determined from the equations for the elastica with initial curvature. The nonlinear interaction forces produced by the elastica were modeled as springs that stiffen rapidly in tension while softening in compression. A relation for the force as a function of the relative displacement x, which is close to that of the elastica solution, is an exponential force law of the form, (1) F(x)
= X (ebx - 1)
Here x > 0 represents tension, and x < 0 represents compression. When y and b are greater than zero, this law produces the expected stiffening behavior. The dynamics of atomic lattices with nonlinear force interactions of this form have been studied extensively by Toda [l]. In this work nonlinear periodic and solitary wave solutions are analytically derived for this system. Using the exponential force law, (l), Newton’s equation of motion for the ith mass is,
3 d2t
=_
’
mb
(eW,+l-1,)
_ eb(v~,-l)
)
_ kx + 2 I m m
where y is the small motion interaction stiffness, k is the stiffness of the beams, c is the linear damping between masses and to ground, and b is the nonlinear stiffening parameter. The boundary conditions are x0 = 0 at the fixed-end and x,+i = x,, at the free-end. When k = 0, c = 0 equation (2) is identical to the nonlinear lattice of Toda [l] which has been used to study solitons in molecular solid state physics. Using data from the experiment, values for the equation parameters were estimated; y = 362 N/m, m = 0.157 kg, b = 725 m-‘, k = 117 N/m, c = 0.01 Ns/m. Using these values, a lattice of nine masses was examined through numerical integration. Impacts were simulated by giving the free-end mass (mass 9) an initial velocity with larger impacts corresponding to larger initial velocities. For low force impacts, the nonlinearity was less
Fig. 4. Lattice
model
of nonlinear
structure.
282
M. A. DAVIES and F. C. MOON
important and there was virtually no sensitivity to initial conditions. However, for larger impacts, strong sensitivity was observed. Figures 5(a) and (b) show a comparison of eight slightly different impacts with an initial end-mass velocity of 2 m/s + 3.5%. For this relatively high impact velocity, Fig. 5(b) shows substantial spreading of the mass 9 position trajectories over time and hence pronounced sensitivity to initial conditions. Nonlinear
energy
waves
For the case mentioned above, an energy contour plot of the dynamics was generated. The plot was created by dividing the lattice into cells, with each cell containing one mass, one nonlinear spring (buckling element), and one linear spring (cantilevered beam). The energy in each cell was calculated over time and the information was combined in the color contour plot (Fig. 6). Here red represents high cellular energy and blue represents low cellular energy. The existence of solitary waves is evident in the narrow fast moving energy pulses seen in Fig. 6. The energy pulse dynamics appears relatively ordered immediately following the impact but quickly degenerates into disordered chaos-like motion for longer times. Numerical simulations have also shown the existence of modal trading phenomena [2, 31. Figure 7 shows a plot of the time evolution of the approximate modal energies for a case where all the energy was initially in mode 5. Unlike the linear case, the modal energies do not remain constant. Rather, mode 5 appears to trade energy intermittently with modes 4 and 6. An energy contour plot of this case (Fig. 8) shows that the modal trading manifests itself as the appearance of a localized energy wave.
DISCUSSION
In predicting the effect of large dynamic loads on periodic structures such as space processes such as buckling are trusses, airplane fuselages, and ship hulls, nonlinear important. The results presented above reveal some of the phenomena that can occur. It has been shown that, in some cases, nonlinearity can lead to spatially and temporally complex energy distributions in the structure. Sensitivity to initial conditions may make the dynamic behavior difficult or impossible to predict and statistical measures might need to be developed. Understanding these phenomena is crucial for predicting dynamic failure, Furthermore, this understanding may damage patterns, and damage extent in structures. lead to new structural designs that use nonlinear thinking and intuition to improve performance above the levels that could be attained using exclusively linear techniques. Extension of this work to elasto-plastic structures is the next step in our research. Earlier work on unpredictable, chaotic dynamics in an elasto-plastic oscillator have been reported by Symonds and Yu at Brown [4], and Poddar et al. at Cornell [5, 61. The addition of elasto-plasticity in a periodic structural lattice introduces hysteretic nonlinearities, similar to dry friction, which are difficult to model in conventional codes. We are planning in future experiments to replace the elastic elements with highly plastic materials and to assess the damage predictability under impact loading.
CONCLUSIONS
In this short discussion paper, new findings on the dynamics reported using experimental and numerical techniques.
of nonlinear
structures
are
Soliton
to chaotic
motion
transition
283
TIME (a 7.’ rig.
7I
Approximate
modal energy content vs time for lattice simulation. Initial energy is exclusively (m = 0.157 kg, y = 362 N/m, 6 = 725 m-‘, k = 117 N/m, c = 0.0).
in mode
5
1. Impact dynamics in periodically
reinforced structures appear to be extremely sensitive to small initial variations in the impact parameters. 2. Nonlinear wave-like deformation patterns (solitons) with localized energy density travel through the structure. 3. Soliton-like waves sometimes disappear into complex spatial patterns only to reemerge at a later time as energy intensive wave-like spatial patterns. 4. Modal excitation of small spatial wavelengths show a nonlinear coupling to neighboring modes which act to produce localized wave-like energy intensive deformations that propagate through the structure. REFERENCES 1. M Toda, Theory of Nonlinear Lattices, 2nd Ed. Springer, Berlin (1989). 2. E Fermi, J. Pasta and S. Ulam, Collected Papers of Enrico Fermi, Vol. II, p. 978. University of Chicago Press, Chicago (1965). of energy in nonlinear systems, J. Math. Phys. 2, 387 (1961). 3. J. Ford, Equipartition behavior in a problem of elastic-plastic beam dynamics, ASME 4. P. S. Symonds and T. X. Yu, Counterintuitive J. Appl. Mech. 52, 517 (1985). F. C. Moon and S. Mukherjee, Chaotic motion of an elasto-plastic beam, ASME J. Appl. Mech. 5. B. Poddar, 55, 185 (1988). 6. F. C. Moon, Chaotic and Fractal Dynamics, Wiley, New York (1992).
Soi,,orrs
Pergamon
& Fracrals Printed
Vol. 1. No. 2. pp. 275-283, 1994 Elsevier Science Lfd Britain. All nghts rescued OY60-0779/94$6.00 + .W
m Great
Transition from Soliton to Chaotic Motion During Impact of a Nonlinear Elastic Structure M. A. DAVIES and F. C. MOON Department
of Mechanical
and Aerospace
Engineering,
Cornell
University,
Ithaca,
NY 14853, USA
(Received 25 March 1993)
Abstract-The existence of a transition from soliton-like motion to chaotic motion in a periodic structure with buckling nonlinearity is demonstrated. An experiment consisting of nine elastic oscillators coupled with buckling sensitive elastica was constructed. The experiment showed a strong demonstrated by the comparison of its response to several near sensitivity to initial conditions, identical impact loads. A numerical model based on a modified Toda lattice was developed and verified the observed sensitivity to initial conditions. Contour plots of energy flow in the numerical model suggest that this sensitivity can be related to a transition from relatively ordered wave motion. immediately following the impact, to disordered chaotic motions at a later time. These findings are relevant to space structures, ship and aircraft structures which are subjected to large dynamic loads.
INTRODUCTION are found in space structures, ship and aircraft structures where the Periodic structures form a lattice-like periodic structure. frame and the skin (or the bays in a space structure) Nonlinear effects in such structures enter naturally through the coupling of in-plane and out-of-plane deformations. In the past decade, nonlinear effects in structures have been shown to lead to dynamics which are very sensitive to initial conditions and hence difficult to predict with conventional numerical structural codes. In this study we examine how wave-like deformations, due to both impact and modal excitation, propagate through a nonlinear periodic structure (Fig. 1). Experiments and numerical simulation are used to show that solitary wave dynamics in a coupled-cell lattice system with a finite number of cells can evolve into a complex spatial pattern with chaos-like dynamics. The experiment consists of nine elastic oscillators coupled with buckling sensitive elastica. We use color contour plots as a new technique for visualizing energy flow in these structures. The results clearly show that the buckling nonlinearity produces numerous complex phenomena. An understanding of these phenomena is necessary if the dynamics and ultimately the failure modes of the structure are to be predicted. This problem is also a benchmark to assess the prediction capabilities of new and existing finite element and other structural dynamics codes. EXPERIMENT To simulate a periodically reinforced shell or plate-like structure, nine elastic oscillators, each one composed of a mass and a cantilevered beam, are coupled together with spring steel elastic buckling elements as shown in Fig. 2. The location of each mass can be measured with a strain gage pair at the base of each cantilevered beam. The state of each buckling element is monitored by strain gages placed directly on the member. The
M. A. D.~UES and F. C. MOON
276
Fig. 1. Sketch
impact pendulum
of periodically
reinforced
plate and shell structures.
shaker -\
Fig. 2. Experimental structure consisting of nine aluminum masses members and to each other through spring steel eccentric
coupled nonlinear
to ground through buckling elements.
cantilevered
experiment is excited either sinusoidally, with an electromagnetic shaker table, or transiently, with an impact pendulum. Preliminary experimental data indicate that the system exhibits a strong sensitivity to initial conditions. Figure 3 shows the comparison of eight separate but nearly identical impacts using strain data from the buckling element nearest
Soliton
6 8 Impacts
to chaotic
motion
transition
277
(Experiment)
4
-6 -8
0.00
0.01
8 Impacts
0.02
Time
0.03
(s)
(Experiment)
6 4
-6 0.08
0.09
Time Fig. 3. Evidence
for sensitivity
0.10
This data to initial conditions in experiment. buckling element nearest the free end.
0.11
comes
from
0.12
strain
gages
on the
M. A. DAVIES and F. C. MOON
278
o o4 8 Impacts(Init Vel - Z.o,Var 3.5%) I' " [I -1 ’ 11 11 ’ *, 11 lI,II II,II
If
E -0.04 si -0.06
0.04
0.02 fz 12
0.00
g h -0.02 a ii -0.04
L -
J
si -0.06
-
I,‘
22
I
,I,
I,
23
1.
24
Fig. 5. Numerical simulation of lattice dynamics for large impact case showing high sensitivity to initial conditions (m = 0.157 kg, y = 362 N/m, b = 725 m-l, k = 117 N/m, c = 0.01 Ns/m, v9 (0) = -2.0 m/s).
Soliton
Fig. 6. Energy
contour
to chaotic
for the case shown
Fig. 8. Energy
contour
motion
transition
in Fig. 5 showing
the breakdown
for the case considered
in Fig. 7.
of solitary
wave order.
Soliton
to chaotic
motion
transition
281
the impact. Immediately following the impact [Fig. 3(a)], the eight strain signals are nearly identical. Later [Fig. 3(b)], they are uncorrelated. This indicates that the dynamics of the system are highly sensitive to differences in the initial impact, that is, the system is sensitive to initial conditions.
NUMERICAL
Modified
MODELING
Toda lattice
The system was modeled as a one-dimensional lattice of nine masses with nonlinear nearest neighbor interactions (Fig. 4). The deformation of the shallow arch structures between the masses can be determined from the equations for the elastica with initial curvature. The nonlinear interaction forces produced by the elastica were modeled as springs that stiffen rapidly in tension while softening in compression. A relation for the force as a function of the relative displacement x, which is close to that of the elastica solution, is an exponential force law of the form, (1) F(x)
= X (ebx - 1)
Here x > 0 represents tension, and x < 0 represents compression. When y and b are greater than zero, this law produces the expected stiffening behavior. The dynamics of atomic lattices with nonlinear force interactions of this form have been studied extensively by Toda [l]. In this work nonlinear periodic and solitary wave solutions are analytically derived for this system. Using the exponential force law, (l), Newton’s equation of motion for the ith mass is,
3 d2t
=_
’
mb
(eW,+l-1,)
_ eb(v~,-l)
)
_ kx + 2 I m m
where y is the small motion interaction stiffness, k is the stiffness of the beams, c is the linear damping between masses and to ground, and b is the nonlinear stiffening parameter. The boundary conditions are x0 = 0 at the fixed-end and x,+i = x,, at the free-end. When k = 0, c = 0 equation (2) is identical to the nonlinear lattice of Toda [l] which has been used to study solitons in molecular solid state physics. Using data from the experiment, values for the equation parameters were estimated; y = 362 N/m, m = 0.157 kg, b = 725 m-‘, k = 117 N/m, c = 0.01 Ns/m. Using these values, a lattice of nine masses was examined through numerical integration. Impacts were simulated by giving the free-end mass (mass 9) an initial velocity with larger impacts corresponding to larger initial velocities. For low force impacts, the nonlinearity was less
Fig. 4. Lattice
model
of nonlinear
structure.
282
M. A. DAVIES and F. C. MOON
important and there was virtually no sensitivity to initial conditions. However, for larger impacts, strong sensitivity was observed. Figures 5(a) and (b) show a comparison of eight slightly different impacts with an initial end-mass velocity of 2 m/s + 3.5%. For this relatively high impact velocity, Fig. 5(b) shows substantial spreading of the mass 9 position trajectories over time and hence pronounced sensitivity to initial conditions. Nonlinear
energy
waves
For the case mentioned above, an energy contour plot of the dynamics was generated. The plot was created by dividing the lattice into cells, with each cell containing one mass, one nonlinear spring (buckling element), and one linear spring (cantilevered beam). The energy in each cell was calculated over time and the information was combined in the color contour plot (Fig. 6). Here red represents high cellular energy and blue represents low cellular energy. The existence of solitary waves is evident in the narrow fast moving energy pulses seen in Fig. 6. The energy pulse dynamics appears relatively ordered immediately following the impact but quickly degenerates into disordered chaos-like motion for longer times. Numerical simulations have also shown the existence of modal trading phenomena [2, 31. Figure 7 shows a plot of the time evolution of the approximate modal energies for a case where all the energy was initially in mode 5. Unlike the linear case, the modal energies do not remain constant. Rather, mode 5 appears to trade energy intermittently with modes 4 and 6. An energy contour plot of this case (Fig. 8) shows that the modal trading manifests itself as the appearance of a localized energy wave.
DISCUSSION
In predicting the effect of large dynamic loads on periodic structures such as space processes such as buckling are trusses, airplane fuselages, and ship hulls, nonlinear important. The results presented above reveal some of the phenomena that can occur. It has been shown that, in some cases, nonlinearity can lead to spatially and temporally complex energy distributions in the structure. Sensitivity to initial conditions may make the dynamic behavior difficult or impossible to predict and statistical measures might need to be developed. Understanding these phenomena is crucial for predicting dynamic failure, Furthermore, this understanding may damage patterns, and damage extent in structures. lead to new structural designs that use nonlinear thinking and intuition to improve performance above the levels that could be attained using exclusively linear techniques. Extension of this work to elasto-plastic structures is the next step in our research. Earlier work on unpredictable, chaotic dynamics in an elasto-plastic oscillator have been reported by Symonds and Yu at Brown [4], and Poddar et al. at Cornell [5, 61. The addition of elasto-plasticity in a periodic structural lattice introduces hysteretic nonlinearities, similar to dry friction, which are difficult to model in conventional codes. We are planning in future experiments to replace the elastic elements with highly plastic materials and to assess the damage predictability under impact loading.
CONCLUSIONS
In this short discussion paper, new findings on the dynamics reported using experimental and numerical techniques.
of nonlinear
structures
are
Soliton
to chaotic
motion
transition
283
TIME (a 7.’ rig.
7I
Approximate
modal energy content vs time for lattice simulation. Initial energy is exclusively (m = 0.157 kg, y = 362 N/m, 6 = 725 m-‘, k = 117 N/m, c = 0.0).
in mode
5
1. Impact dynamics in periodically
reinforced structures appear to be extremely sensitive to small initial variations in the impact parameters. 2. Nonlinear wave-like deformation patterns (solitons) with localized energy density travel through the structure. 3. Soliton-like waves sometimes disappear into complex spatial patterns only to reemerge at a later time as energy intensive wave-like spatial patterns. 4. Modal excitation of small spatial wavelengths show a nonlinear coupling to neighboring modes which act to produce localized wave-like energy intensive deformations that propagate through the structure. REFERENCES 1. M Toda, Theory of Nonlinear Lattices, 2nd Ed. Springer, Berlin (1989). 2. E Fermi, J. Pasta and S. Ulam, Collected Papers of Enrico Fermi, Vol. II, p. 978. University of Chicago Press, Chicago (1965). of energy in nonlinear systems, J. Math. Phys. 2, 387 (1961). 3. J. Ford, Equipartition behavior in a problem of elastic-plastic beam dynamics, ASME 4. P. S. Symonds and T. X. Yu, Counterintuitive J. Appl. Mech. 52, 517 (1985). F. C. Moon and S. Mukherjee, Chaotic motion of an elasto-plastic beam, ASME J. Appl. Mech. 5. B. Poddar, 55, 185 (1988). 6. F. C. Moon, Chaotic and Fractal Dynamics, Wiley, New York (1992).