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NON-LINEAR DYNAMIC RESPONSE OF SHALLOW ARCHES TO HARMONIC FORCING
Journal of Sound and Vibration (1996) 194(3), 353–367
NON-LINEAR DYNAMIC RESPONSE OF SHALLOW ARCHES TO HARMONIC FORCING K. B. B Moldyn, Inc. 955 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
C. M. K School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.
T. N. F School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, U.S.A. (Received 8 September 1994, and in final form 27 November 1995) The investigation of the dynamic response of a shallow arch to harmonic forcing is undertaken. The method of harmonic balance, coupled with a continuation scheme, is used to determine the solutions for an entire range of externally applied loading. Floquet analysis provides the requisite stability information, as well as information about the bifurcation points encountered in the solution. Results are presented for a range of loading at three excitation frequencies. In addition, the effect of the excitation frequency on the response for three fixed values of the magnitude of the loading is investigated. The dynamic response of the arch to harmonic forcing is shown to exhibit both symmetric and asymmetric solutions. Additionally, stable solutions are found that have a fundamental period of an integer multiple of the excitation period. Finally, regions of chaotic motion are observed. 7 1996 Academic Press Limited
1. INTRODUCTION
Buckled beams, shallow arches and spherical caps all exhibit geometrically-induced non-linear force–deflection curves, exhibiting both stable and unstable equilibrium states. Co-existing stable equilibrium states for a given external load correspond to different geometric configurations of the structural component [1]. With the application of sufficiently large loads, snap-buckling can occur, in which the structure suddenly jumps from one stable equilibrium configuration to another [2]. The loss of stability in the original equilibrium configuration is local in nature, and can be predicted through eigenvalue anlaysis of equations of motion which have been linearized about the equilibrium position. The geometric configuration of the structural component may also lose stability as a result of the application of a temporally varying load, leading to dynamic buckling [3]. This loss of stability may be global in nature, implying that perturbation analysis would not be capable of representing this type of dynamic behavior. For the case of shallow arches, Cheung and Babcock [4] found that the amplitude of the dynamic buckling load may be less in magnitude than the static buckling load. Additionally, Babcock [5] found that application of a load greater than that known to cause snap-buckling did not necessarily lead to snap-buckling. One of the earliest studies on the response of arches to dynamic forcing was undertaken by Eisley [6]. In this study, the first mode of the motion about a static equilibrium position 353 0022–460X/96/280353 + 15 $18.00/0
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was expressed as a single-degree-of-freedom (s.d.o.f.), Duffing-type equation with both quadratic and cubic non-linear terms. Since that time, several investigations into the dynamic response of arches have utilized a s.d.o.f. equation to describe the motion. Tseng and Dugundji [7] utilized the harmonic balance technique to analyze the response of an arch to small-amplitude forcing. A simple harmonic motion solution was found to provide a good estimate of the solution for small forcing, and for frequencies away from any possible harmonic resonances. Plaut and Hsieh [8] investigated the response of a shallow arch to large amplitude harmonic forcing. Using numerical integration, it was found that the response of the arch remained in the neighborhood of the stable equilibrium positions, or encompassed both the stable and unstable equilibrium positions. Researchers have also observed period-doubling behavior and chaotic motion when studying the response of arches to dynamic loading [9]. Leung and Fung [10] utilized a variation of the harmonic balance approach to determine one possible branch of the dynamic response of a Duffing oscillator to harmonic forcing over a range of operating parameters. Even the most recent studies have utilized numerical integration to determine the response of non-linear components to harmonic forcing, thus limiting the analysis to consider only stable responses [11, 12]. Many of the studies of non-linear dynamic response of structures have focused on the stable solutions to the equations of motion. As will be demonstrated in this work, the unstable solution trajectories provide additional insight into the understanding of the dynamic behavior of non-linear systems. Observing both the stable and unstable solution trajectories can explain the connection between co-existing stable responses, and can assist in finding stable and chaotic solutions with small basins of attraction. Additionally, it will be shown that the observation of the harmonic components of the response can provide a level of understanding of the response not attainable with solution techniques that focus on global characteristics of the system response. In this work, a method will be described and utilized that is capable of capturing the complex responses of a s.d.o.f. non-linear dynamic system subjected to harmonic loading. This method combines the techniques of harmonic balance, continuation and Floquet analysis. Subsequent applications of the analysis technique to a s.d.o.f. model of a shallow arch will demonstrate the additional insight into the behavior of non-linear dynamic systems attainable by observing both stable and unstable solution trajectories.
2. METHOD OF ANALYSIS
The selection of the particular model to study is driven by the desire to utilize the simplest model possible. However, the model must be capable of retaining the essential features of the geometrically controlled non-linear force–deflection relationship of a shallow arch. Thus the investigation of the dynamic response is undertaken using the two
Figure 1. Two rigid-link, s.d.o.f. arch model.
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T 1 Values of the normalized parameters used in the analysis I/(mL 2) cr /(mL 2) kr /(mL 2) 4ks /m L
0·5 0·6 6·0 400·0 0·98
rigid-link, s.d.o.f. model shown in Figure 1. The equation of motion for this model is written as
0
1
c k 4k 3 I u + (sin 2u)u 2 + r 2 u + r 2 u + s (L − cos u) sin u − cos 2u + 2 mL mL m 2 mL 2 =
F cos u cos vt, mL
(1)
where m and I represent the mass and inertia of the rigid links, L is the length of each link, cr and kr represent the angular velocity-proportional damper and the rotational spring, assumed unstretched for u = 0, ks represents the stiffness of the linear end spring, L = b/2L, b, a constant, is the span of the arch with the end spring undeflected, F is the magnitude of the external harmonic force of frequency v, and u is the physical angle of the link. This model is dynamically equivalent to a one-term Galerkin expansion of a continuous model of an arch [13]. As is evident in the equation of motion, the effect of the geometric non-linearities arises in the inertia, stiffness and forcing terms. Table 1 shows the value of the parameters considered for this study. Neglecting the time dependent terms in the equation of motion, the static load–deflection relation can be found, and is shown in Figure 2. It is evident that the system parameters are chosen such that there exist multiple equilibrium positions for a given load value. The load–deflection diagram shows the static snap-buckling points, as well as the three static equilibrium positions. Linear analysis shows that the outside equilibrium positions are stable, while the center equilibrium position is unstable. This is illustrated in Figure 2 where the solid line represents stable static equilibrium configurations and the dashed line represents unstable static equilibrium configurations.
Figure 2. Force–deflection diagram for the s.d.o.f. arch model: ——, stable; –––, unstable.
. . .
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In order to perform the dynamic analysis of the arch, the method of harmonic balance is employed, coupled with a continuation scheme. Here, an mT periodic solution is sought of the form
0
N
umT (t) = C0 + s Cj/m cos j j=1
1
2p t + cj/m , mT
(2)
where the fundamental period is T = 2p/v, and v is the frequency of the applied harmonic forcing. Substitution of equation (2) into the equation of motion (1) and balancing the coefficients of each of the harmonic terms produces a set of non-linear algebraic equations such as
0 1
2
F0 = 0,
Fj/m cos fj/m = − j
0 1
2p Cj/m cos cj/m , mT
2
Fj/m sin fj/m = − j
2p Cj/m sin cj/m , mT
(3)
with the Fourier coefficients and phase angles, Cj/m and cj/m , respectively, of the periodic response as the unknowns. A time history of the forcing function can be generated, and a fast Fourier transform (FFT) algorithm is utilized in the calculation of the Fourier coefficients, Fj/m , and the phase angles, fj/m , of the non-linear terms. A non-linear algebraic equation solver is used to determine Cj/m and cj/m . Since more than one solution is desired, the value of the magnitude of the applied harmonic load is systematically varied for subsequent solutions. If the results of this parametric variation yield turning points, a continuation scheme is employed in which the parameter being varied is the arc length of the path in the configuration space. Thus the parameter of interest, in this case the applied load, is allowed to pass unencumbered through turning points [17]. Alternatively, the frequency of the applied harmonic load can be chosen as the parameter of interest. The above techniques yield both stable and unstable solution branches, therefore it is desirable to determine the stability of the periodic solutions found. The stability of the solutions is determined through the application of Floquet theory to the linearized equations of motion corresponding to the periodic solution of interest. The eigenvalues of the monodromy matrix then yield the requisite stability information [18]. The Floquet multipliers also aided in the identification of the various bifurcation points found in the solution space. Additional details of the analysis technique can be found in reference [19]. Utilizing the above combination of harmonic balance and continuation techniques, the solution of the equation of motion can be represented by the coefficients of the Fourier series of u as a function of the applied load, F and forcing frequency, v. In analyzing the fundamental response trajectories, 10 coefficients (C0 , C1 , . . . , C10 ) and the associated phase angles (c0 , c1 , . . . , c10 ) were included in the solution process. Similarly, for the construction of subharmonic solution trajectories, the same 10 coefficients and phase angles representing the fundamental response were included, along with the intermediate coefficients (C1/2 , C3/2 , . . . , C19/20 ) and phase angles (c1/2 , c3/2 , . . . , c19/20 ) representing the subharmonics. Typically only the first three or four coefficients are required to characterize the solution. Additional insight into the behavior of the s.d.o.f. arch was obtained by the construction of phase–plane plots for various points along a solution trajectory. For the cases of periodic response, the phase–plane plots were constructed using all of the Fourier coefficients used in the harmonic balance solution. For the cases of chaotic response, the phase–plane plots were
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Figure 3. Selected Fourier coefficients for periodic motion of the arch for v/vn = 0·83. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
constructed using direct numerical integration of the equations of motion. Note that either technique will yield the same results for stable solutions. However, it can be difficult to find a set of initial conditions resulting in a given stable response if that particular response has a small basin of attraction. Only the harmonic balance technique can be used to construct phase–plane plots for unstable trajectories. As will be seen, understanding the characteristics of unstable solution branches can greatly add to the understanding of the dynamic response of a non-linear system. 3. RESULTS AND DISCUSSION
3.1. The results obtained in this study have been compared with those obtained from the numerical integration of the equations of motion. As will be seen, the qualitative aspects of the response are similar for a range of parametric variations. Other researchers [7–15] have presented detailed analyses on the behavior of arches. Consequently, the results presented herein will focus on the types of information that can be obtained by using harmonic balance and continuation techniques that may not available using other techniques such as numerical integration. Figure 3 shows the zero-, half- and first-order coefficients of the Fourier series as a function of the magnitude of the applied load. The applied forcing frequency is v/vn = 0·83, where vn is the undamped, linearized natural frequency of the system. Assuming a no-load equilibrium position of the arch u = a, vn can be expressed as vn2 =
kr + 4ks L 2(L cos a − cos 2a) . (3/2 − cos 2a)mL 2 + 2I
(4)
The dashed lines represent unstable solution trajectories, and the solid lines represent the
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. . .
stable solution trajectories. Selected details of the response plots are shown in Figure 4. The Fourier coefficients shown are those that contribute to understanding the qualitative aspects of the dynamic response. Note that there are many places where the solution trajectory reverses direction. The use of the continuation scheme in conjunction with the harmonic balance technique allows the solution process to trace the entire trajectory without having to stop and restart the solution process in a different direction. For small values of the nomalized force, f = F/mL, and C0 non-zero, the motion is in the neighborhood of one of the two stable equilibrium positions. It should be noted that two solutions exist, one about each of the equilibrium positions. This motion is asymmetric in the u–u phase plane. For the case of C0 being equal to zero, the motion is symmetric about the unstable equilibrium position of u = 0. The coefficient C1/2 is non-zero when motion exists for which the fundamental period is a 2n multiple of the forcing period. These concepts are further demonstrated by the u–u phase-plane plots shown in Figure 5. Viewing the plot of C1 in Figures 3 and 4, it is evident that the symmetric solution exists throughout the entire range of applied loading. As would be expected from the static results, for small levels of forcing the response about the saddle point u = u = 0 is unstable. For this range of forcing amplitude, the arch will vibrate about one of the stable equilibrium positions as evidenced by the non-zero C0 . Increasing the load from zero, the asymmetric branch has a turning point at A, and the first incidence of a discontinuity between the applied loading and the response is observed. This behavior is similar to that of the softening Duffing oscillator where there are two possible stable responses. The actual response observed is a function of the initial conditions imparted to the system. For this region, local analysis about the stable equilibrium position such as that used by Nayfeh and Mook [20], would show qualitatively similar results. As the loading is increased further, a period-doubling bifurcation is encountered on the asymmetric branch at point B (Figure 4(a)). At this point, the asymmetric period-one (1T) branch loses stability, and an asymmetric period-two (2T) branch arises. As stated above, the C1/2 coefficient is no longer zero in this region. A further increase in the load results in a period-doubling cascade in which the stable solutions only exist for 2n multiples of the fundamental period. Period doubling can lead to chaotic motion where no stable periodic solutions exist [21]. The solution trajectory for chaotic motion, f = 0·08, is shown in Figure 5. If the load is increased further, the period-doubling process reverses itself, and at point C (Figure 4(b)) the 2T solution branch rejoins the 1T branch, and a stable asymmetric period-one solution is again observed. Although only the solution trajectory for the first period-doubling bifurcation was shown here, it should be noted that the harmonic balance and continuation scheme utilized in this work allows for a systematic tracing of each of the period doubling solution
Figure 4. Selected details for the C1 coefficient of Figure 3. (a) detail 1; (b) detail 2; (c) detail 3. ——, stable; – – –, unstable.
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Figure 5. u − u phase–plane plots for selected values of the normalized load f for v/vn = 0·83. (a) f = 0·02, 1T; (b) f = 0·044, 2T; (c) f = 0·08, chaotic; (d) f = 0·6, 1T; (e) f = 0·7, 1T; (f) f = 0·85, 2T; (g) f = 3·5, chaotic; (h) f = 5·0, 1T.
360
. . .
trajectories. By tracing the entire trajectory, and observing changes in stability along the solution path, very detailed bifurcation diagrams similar to those shown by Abhyankar et al. [12] can be systematically constructed. At each period-doubling bifurcation, the additional non-linear algebraic equations containing the coefficients and phase angles required to represent the period-doubled solution can be added to the solutin set (equation (3)). Note that both the stable and unstable solution trajectories will be recorded allowing for an understanding of the entire solution trajectory which is not available using numerical integration techniques. Just beyond point C in Figure 4(b) the C0 term in the asymmetric response disappears, and the pair of asymmetric 1T branches merge to form the single, stable, symmetric branch at point D (Figure 4(b)). A further increase in load results in encountering another turning point, and the next stable section of the symmetric branch corresponds to motion encompassing all three equilibrium positions. As the loading is continually increased, it is observed that all of the asymmetric solutions arise from the symmetric solution branch at symmetry-breaking bifurcations. The symmetric solution loses stability at these points. Similarly, all of the 2T solution branches arise from the asymmetric solution branches at period doubling bifurcations, and are themselves asymmetric. Not all 2T solution branches undergo the period-doubling cascade and the eventual transition to chaos described above. As can be observed from the C1/2 plot in Figure 3, the 2T solutions can remain stable over their entire range of existence. All stable solutions for this range include motion about all three equilibrium positions. As shown in Figure 4(c), another range of loading results in a period-doubling cascade, and eventual chaotic motion. The phase–plane plot of the chaotic motion is shown in Figure 5 for f = 3·5. Figure 6 shows similar plots of the zero, half- and first-order coefficients for the case of the forcing frequency being twice that of the linearized natural frequency. As in the previous case, the only stable response for low load levels corresponds to motion about one of the two stable equilibrium positions. The C1/2 plot shows that there is again a 2T solution branch arising from a period-doubling bifurcation along the asymmetric response curve. The C1 coefficient for the 1T and 2T response curves is indistinguishable for the scale used in Figure 6. A further increase in the magnitude of the applied load results in the 1T and 2T solution branches rejoining, and the coalescence of the asymmetric and symmetric 1T solution branches at a symmetry-breaking bifurcation. A further increase in the magnitude of the applied load results in the symmetric response going through turning points that would result in snap-buckling between equilibrium solutions. Similarly, Figure 7 shows the plots of the coefficients of the Fourier series of u for the case of the excitation frequency being 0·6 times the linearized natural frequency. In comparison to the first case, the entire plot is shifted to the left, so that similar bifurcation points now occur at lower magnitudes of the applied forcing. As was seen in the first case, there is a clear distinction between the 1T and 2T solution branches at the lower levels of excitation. For large excitation levels, several more incidents of the symmetry-breaking bifurcation points along the symmetric solution paths are seen, along with the period-doubling bifurcations on the asymmetric solution paths leading to regions of chaotic motion. Plotting the phase behavior of the motion at various points along the solution paths for Figures 6 and 7 would result in phase portraits with features qualitatively similar to those seen in Figure 5. The static snap-buckling loads correspond to points at which the slope of the load–deflection diagram goes to zero (Figure 2). From Figure 3, it can be seen that for load values less than the static buckling load, all stable 1T solutions occur about one of the stable equilibrium positions. However, as is seen in Figure 7, there exist stable solutions
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for loading values less than the static buckling load that leave the neighborhood of a single equilibrium position. It appears that the static snap-buckling load alone does not provide an adequate criterion for the prediction of the stable periodic response of the arch. 3.2. In order to provide further insight into the effect of the excitation levels on the harmonic response of the s.d.o.f. arch, Figures 8–10 were constructed by varying the excitation frequency for a given value of the magnitude of the excitation. Figure 8 shows the C0 and C1 coefficients for the case of the forcing amplitude f = 0·01. As expected from the previous results, stable motion is observed about the equilibrium position, with the magnitude of the response being largest in the neighborhood of the linearized natural frequency. It is evident that for this small level of forcing amplitude, the effect of the non-linearities are virtually non-existent. Increasing the magnitude of the applied load by a factor of five results in the frequency response plot shown in Figure 9. The softening spring behavior observed by others (e.g. reference [6]) is seen here. Additionally, the effects of higher order resonances are observed.
Figure 6. Selected Fourier coefficients for periodic motion of the arch for v/vn = 2·0. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
362
. . .
Figure 7. Selected Fourier coefficients for periodic motion of the arch for v/vn = 0·5. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
As in all previous plots, the stable solutions are shown in bold. As evident from the non-zero C0 coefficient, the motion of the arch is restricted to the neighborhood of one equilibrium position. Tracing the response curves from large to small excitation frequencies, a period-doubling bifurcation is encountered just above the linearized natural frequency. This is similar to the behavior seen in the Duffing equation by Krousgrill and Zadoks [16] and Leung [22], who found period-doubling cascades to chaotic motion in
Figure 8. Selected Fourier coefficients for periodic motion of the arch for f = 0·01: ——, stable; – – –, unstable.
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Figure 9. Selected Fourier coefficients for periodic motion of the arch for f = 0·05: ——, stable; – – –, unstable.
similar regions of the frequency response curve. Indeed, numerical integration did not find any stable periodic solutions for that range of the excitation frequency. Doubling the forcing magnitude to a value of f = 0·1 results in a complete change in the characteristics observed in the frequency response curves. This is due to the forcing amplitude being larger than the static-buckling load. The frequency response plots for this loading condition are shown in Figure 10. Phase plane plots corresponding to specific values of the forcing frequency are shown in Figure 11. For high excitation frequencies, a stable response exists characterized by oscillations about one of the equilibrium positions. This response has a non-zero mean value as shown by branch A in Figure 10. Thus, high-frequency excitation tends to stabilize the motion of the arch about one static equilibrium position as shown by the corresponding phase–plane plot in Figure 11 (v = 1·5). Decreasing the forcing frequency results in a loss of stability due to a period doubling bifurcation at point A. This bifurcation initiates a period-doubling cascade to chaotic motion as the frequency is decreased. As shown in Figure 11, when the forcing frequency is decreased, an aperiodic response occurs about one equilibrium position (v = 1·4). This motion is unstable, and the trajectory is beginning to encompass the saddle point u − u = 0. Decreasing the frequency further results in chaotic motion encompassing all equilibrium positions. Note that for a significant range of forcing frequencies, no stable solutions are found along this branch. At point B, the 2T branch (not shown) will rejoin the main branch, which will in turn regain stability for a small range of the excitation frequency. Near the resonant frequency, both hardening and softening response curves can be seen. The hardening curve corresponds to branch C, for which the mean value of the response
Figure 10. Selected Fourier coefficients for periodic motion of the arch for f = 0·1. (a) C0 coefficient; (b) C1 coefficient. ——, stable; – – –, unstable.
364
. . .
Figure 11. u − u phase–plane plots for selected values of the forcing frequency v for f = 0·1. (a) v = 0·29; (b) v = 0·40; (c) v = 0·80; (d) v = 1·3; (e) v = 1·4; (f) v = 1·5.
is zero. Thus, the motion encompasses all three equilibrium states (v = 0·8 in Figure 11). Similar hardening responses for large amplitude motion have been reported by Plaut and Hsieh [8]. The behavior of the softening curve, branch A, was described in the previous paragraph. It is evident that branch A corresponds to the same solution branch as the curves shown in Figures 8 and 9. However, the curve no longer folds back under itself. Instead, the hardening and softening response curves intersect at point B. At this intersection, the mean value of the softening response curve (A) has gone to zero to correspond to the hardening response curve. The hardening response curve is only stable over a short range as well. A symmetry breaking bifurcation occurs at point C as indicated by the existence of a non-zero mean value branch in the C0 plot for this branch. The phase plane trajectory for this case is shown for v = 0·4 in Figure 11. Note that as in all previous cases, the asymmetric solution, branch
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D, rejoins the main symmetric, period-one solution curve. At point D on the asymmetric response branch, a period-doubling bifurcation occurs, and another cascade to chaotic motion is found. For small values of the applied frequency, f Q 0·4, only isolated regions of stable period-one motion were found. A typical example of this case is shown for v = 0·29 in Figure 11. For the unstable regions of the low frequency excitation, the motion is characterized by the arch snapping from one equilibrium state to the other with each change in direction of the applied load as shown in Figure 12. Superimposed on the large-amplitude motion are small oscillations about the equilibrium states. In order for the harmonic balance technique to accurately represent these solutions, a large number of coefficients is needed to capture the high frequency content of the response.
4. SUMMARY AND CONCLUSIONS
A technique combining harmonic balance, continuation and Floquet analysis has been applied to the study of a model of a shallow arch. This analysis technique allows for a detailed study of the solution trajectories of a non-linear dynamic system. Unlike numerical analysis techniques, both stable and unstable solution trajectories can be constructed allowing for a better understanding of the effects of the system parameters such as the magnitude and frequency of the applied load on the dynamic behavior of the system. Use of the technique in the construction phase–plane plots, time histories and bifurcation diagrams traditionally used in the analysis of non-linear systems was described. Perhaps most importantly, it was shown that simply observing the Fourier coefficients of the stable and unstable response trajectories provides the qualitative and quantiative aspects of the non-linear dynamic behavior that typcially requires the construction of phase–plane plots, time histories and bifurcation diagrams. In addition, this technique provides an understanding of the evolution of the solution as a function of the system parameters that may not be available using either numerical integration or a harmonic balance approach without a continuation scheme.
Figure 12. Displacement time history for v = 0·2 and f = 0·1.
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Using the solution techniques presented herein, it has been shown that a slight change of the values of the forcing frequency or magnitude can result in a large change in the response of the arch, both qualitatively and quantitatively. This large change in response can be of three types, each one resulting from the loss of stability along a given solution path. The first type of response change results from a jump in the magnitude of the response, while keeping qualitatively similar characteristics as seen at point A in Figure 3. The second type of transition commonly seen occurs at period-doubling bifurcations. Here, the magnitude of the response will undergo a slight change, while the character of the response will show the resulting period-doubling bifurcation. The other type of response change is the snap-buckling phenomena, where there is a large qualitative and quantitative difference in the response. Here the arch jumps from motion about one equilibrium state to motion about more than one equilibrium state, or vice versa. Thus, a large change in the magnitude and characteristic of the response is seen. Additionally, except for very low values of the applied excitation, the response plots show that there are ranges of forcing amplitude and frequency for which there exist no stable periodic solutions. In this range, chaotic motion encompassing all three equilibrium positions can be observed. For the values of the parameters studied, it is also shown that the static buckling load does not serve as an indicator of the form of the dynamic response. Comparing the results obtained by varying the forcing amplitude to those obtained by varying the forcing frequency, it is evident that much more information is available in the cases where the forcing amplitude is varied. Figures 8–10 show that even for wide ranges of the forcing frequency, the characteristics of the response do not change in a qualitative sense. The response tends to remain about the same set of equilibrium positions, making it impossible to infer any information about the snap-buckling behavior of the arch. Certainly many other qualitative and quantitative aspects of non-linear dynamic responses are typically associated with shallow arches. As with the responses shown herein, the observation of various types of non-linear dynamic response is highly dependent on the system parameters. The parameters chosen in this study resulted in several occurrences of period-doubling cascades. However, other responses such as significant second and third order superharmonic and 1/3 order subharmonic solutions were simply not observed. Investigation into the occurrence of these response characteristics will be the subject of future studies.
ACKNOWLEDGMENT
This work was supported in part by National Science Foundation Grant number MMS-9057082.
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NON-LINEAR DYNAMIC RESPONSE OF SHALLOW ARCHES TO HARMONIC FORCING K. B. B Moldyn, Inc. 955 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
C. M. K School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A.
T. N. F School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, U.S.A. (Received 8 September 1994, and in final form 27 November 1995) The investigation of the dynamic response of a shallow arch to harmonic forcing is undertaken. The method of harmonic balance, coupled with a continuation scheme, is used to determine the solutions for an entire range of externally applied loading. Floquet analysis provides the requisite stability information, as well as information about the bifurcation points encountered in the solution. Results are presented for a range of loading at three excitation frequencies. In addition, the effect of the excitation frequency on the response for three fixed values of the magnitude of the loading is investigated. The dynamic response of the arch to harmonic forcing is shown to exhibit both symmetric and asymmetric solutions. Additionally, stable solutions are found that have a fundamental period of an integer multiple of the excitation period. Finally, regions of chaotic motion are observed. 7 1996 Academic Press Limited
1. INTRODUCTION
Buckled beams, shallow arches and spherical caps all exhibit geometrically-induced non-linear force–deflection curves, exhibiting both stable and unstable equilibrium states. Co-existing stable equilibrium states for a given external load correspond to different geometric configurations of the structural component [1]. With the application of sufficiently large loads, snap-buckling can occur, in which the structure suddenly jumps from one stable equilibrium configuration to another [2]. The loss of stability in the original equilibrium configuration is local in nature, and can be predicted through eigenvalue anlaysis of equations of motion which have been linearized about the equilibrium position. The geometric configuration of the structural component may also lose stability as a result of the application of a temporally varying load, leading to dynamic buckling [3]. This loss of stability may be global in nature, implying that perturbation analysis would not be capable of representing this type of dynamic behavior. For the case of shallow arches, Cheung and Babcock [4] found that the amplitude of the dynamic buckling load may be less in magnitude than the static buckling load. Additionally, Babcock [5] found that application of a load greater than that known to cause snap-buckling did not necessarily lead to snap-buckling. One of the earliest studies on the response of arches to dynamic forcing was undertaken by Eisley [6]. In this study, the first mode of the motion about a static equilibrium position 353 0022–460X/96/280353 + 15 $18.00/0
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was expressed as a single-degree-of-freedom (s.d.o.f.), Duffing-type equation with both quadratic and cubic non-linear terms. Since that time, several investigations into the dynamic response of arches have utilized a s.d.o.f. equation to describe the motion. Tseng and Dugundji [7] utilized the harmonic balance technique to analyze the response of an arch to small-amplitude forcing. A simple harmonic motion solution was found to provide a good estimate of the solution for small forcing, and for frequencies away from any possible harmonic resonances. Plaut and Hsieh [8] investigated the response of a shallow arch to large amplitude harmonic forcing. Using numerical integration, it was found that the response of the arch remained in the neighborhood of the stable equilibrium positions, or encompassed both the stable and unstable equilibrium positions. Researchers have also observed period-doubling behavior and chaotic motion when studying the response of arches to dynamic loading [9]. Leung and Fung [10] utilized a variation of the harmonic balance approach to determine one possible branch of the dynamic response of a Duffing oscillator to harmonic forcing over a range of operating parameters. Even the most recent studies have utilized numerical integration to determine the response of non-linear components to harmonic forcing, thus limiting the analysis to consider only stable responses [11, 12]. Many of the studies of non-linear dynamic response of structures have focused on the stable solutions to the equations of motion. As will be demonstrated in this work, the unstable solution trajectories provide additional insight into the understanding of the dynamic behavior of non-linear systems. Observing both the stable and unstable solution trajectories can explain the connection between co-existing stable responses, and can assist in finding stable and chaotic solutions with small basins of attraction. Additionally, it will be shown that the observation of the harmonic components of the response can provide a level of understanding of the response not attainable with solution techniques that focus on global characteristics of the system response. In this work, a method will be described and utilized that is capable of capturing the complex responses of a s.d.o.f. non-linear dynamic system subjected to harmonic loading. This method combines the techniques of harmonic balance, continuation and Floquet analysis. Subsequent applications of the analysis technique to a s.d.o.f. model of a shallow arch will demonstrate the additional insight into the behavior of non-linear dynamic systems attainable by observing both stable and unstable solution trajectories.
2. METHOD OF ANALYSIS
The selection of the particular model to study is driven by the desire to utilize the simplest model possible. However, the model must be capable of retaining the essential features of the geometrically controlled non-linear force–deflection relationship of a shallow arch. Thus the investigation of the dynamic response is undertaken using the two
Figure 1. Two rigid-link, s.d.o.f. arch model.
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T 1 Values of the normalized parameters used in the analysis I/(mL 2) cr /(mL 2) kr /(mL 2) 4ks /m L
0·5 0·6 6·0 400·0 0·98
rigid-link, s.d.o.f. model shown in Figure 1. The equation of motion for this model is written as
0
1
c k 4k 3 I u + (sin 2u)u 2 + r 2 u + r 2 u + s (L − cos u) sin u − cos 2u + 2 mL mL m 2 mL 2 =
F cos u cos vt, mL
(1)
where m and I represent the mass and inertia of the rigid links, L is the length of each link, cr and kr represent the angular velocity-proportional damper and the rotational spring, assumed unstretched for u = 0, ks represents the stiffness of the linear end spring, L = b/2L, b, a constant, is the span of the arch with the end spring undeflected, F is the magnitude of the external harmonic force of frequency v, and u is the physical angle of the link. This model is dynamically equivalent to a one-term Galerkin expansion of a continuous model of an arch [13]. As is evident in the equation of motion, the effect of the geometric non-linearities arises in the inertia, stiffness and forcing terms. Table 1 shows the value of the parameters considered for this study. Neglecting the time dependent terms in the equation of motion, the static load–deflection relation can be found, and is shown in Figure 2. It is evident that the system parameters are chosen such that there exist multiple equilibrium positions for a given load value. The load–deflection diagram shows the static snap-buckling points, as well as the three static equilibrium positions. Linear analysis shows that the outside equilibrium positions are stable, while the center equilibrium position is unstable. This is illustrated in Figure 2 where the solid line represents stable static equilibrium configurations and the dashed line represents unstable static equilibrium configurations.
Figure 2. Force–deflection diagram for the s.d.o.f. arch model: ——, stable; –––, unstable.
. . .
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In order to perform the dynamic analysis of the arch, the method of harmonic balance is employed, coupled with a continuation scheme. Here, an mT periodic solution is sought of the form
0
N
umT (t) = C0 + s Cj/m cos j j=1
1
2p t + cj/m , mT
(2)
where the fundamental period is T = 2p/v, and v is the frequency of the applied harmonic forcing. Substitution of equation (2) into the equation of motion (1) and balancing the coefficients of each of the harmonic terms produces a set of non-linear algebraic equations such as
0 1
2
F0 = 0,
Fj/m cos fj/m = − j
0 1
2p Cj/m cos cj/m , mT
2
Fj/m sin fj/m = − j
2p Cj/m sin cj/m , mT
(3)
with the Fourier coefficients and phase angles, Cj/m and cj/m , respectively, of the periodic response as the unknowns. A time history of the forcing function can be generated, and a fast Fourier transform (FFT) algorithm is utilized in the calculation of the Fourier coefficients, Fj/m , and the phase angles, fj/m , of the non-linear terms. A non-linear algebraic equation solver is used to determine Cj/m and cj/m . Since more than one solution is desired, the value of the magnitude of the applied harmonic load is systematically varied for subsequent solutions. If the results of this parametric variation yield turning points, a continuation scheme is employed in which the parameter being varied is the arc length of the path in the configuration space. Thus the parameter of interest, in this case the applied load, is allowed to pass unencumbered through turning points [17]. Alternatively, the frequency of the applied harmonic load can be chosen as the parameter of interest. The above techniques yield both stable and unstable solution branches, therefore it is desirable to determine the stability of the periodic solutions found. The stability of the solutions is determined through the application of Floquet theory to the linearized equations of motion corresponding to the periodic solution of interest. The eigenvalues of the monodromy matrix then yield the requisite stability information [18]. The Floquet multipliers also aided in the identification of the various bifurcation points found in the solution space. Additional details of the analysis technique can be found in reference [19]. Utilizing the above combination of harmonic balance and continuation techniques, the solution of the equation of motion can be represented by the coefficients of the Fourier series of u as a function of the applied load, F and forcing frequency, v. In analyzing the fundamental response trajectories, 10 coefficients (C0 , C1 , . . . , C10 ) and the associated phase angles (c0 , c1 , . . . , c10 ) were included in the solution process. Similarly, for the construction of subharmonic solution trajectories, the same 10 coefficients and phase angles representing the fundamental response were included, along with the intermediate coefficients (C1/2 , C3/2 , . . . , C19/20 ) and phase angles (c1/2 , c3/2 , . . . , c19/20 ) representing the subharmonics. Typically only the first three or four coefficients are required to characterize the solution. Additional insight into the behavior of the s.d.o.f. arch was obtained by the construction of phase–plane plots for various points along a solution trajectory. For the cases of periodic response, the phase–plane plots were constructed using all of the Fourier coefficients used in the harmonic balance solution. For the cases of chaotic response, the phase–plane plots were
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Figure 3. Selected Fourier coefficients for periodic motion of the arch for v/vn = 0·83. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
constructed using direct numerical integration of the equations of motion. Note that either technique will yield the same results for stable solutions. However, it can be difficult to find a set of initial conditions resulting in a given stable response if that particular response has a small basin of attraction. Only the harmonic balance technique can be used to construct phase–plane plots for unstable trajectories. As will be seen, understanding the characteristics of unstable solution branches can greatly add to the understanding of the dynamic response of a non-linear system. 3. RESULTS AND DISCUSSION
3.1. The results obtained in this study have been compared with those obtained from the numerical integration of the equations of motion. As will be seen, the qualitative aspects of the response are similar for a range of parametric variations. Other researchers [7–15] have presented detailed analyses on the behavior of arches. Consequently, the results presented herein will focus on the types of information that can be obtained by using harmonic balance and continuation techniques that may not available using other techniques such as numerical integration. Figure 3 shows the zero-, half- and first-order coefficients of the Fourier series as a function of the magnitude of the applied load. The applied forcing frequency is v/vn = 0·83, where vn is the undamped, linearized natural frequency of the system. Assuming a no-load equilibrium position of the arch u = a, vn can be expressed as vn2 =
kr + 4ks L 2(L cos a − cos 2a) . (3/2 − cos 2a)mL 2 + 2I
(4)
The dashed lines represent unstable solution trajectories, and the solid lines represent the
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. . .
stable solution trajectories. Selected details of the response plots are shown in Figure 4. The Fourier coefficients shown are those that contribute to understanding the qualitative aspects of the dynamic response. Note that there are many places where the solution trajectory reverses direction. The use of the continuation scheme in conjunction with the harmonic balance technique allows the solution process to trace the entire trajectory without having to stop and restart the solution process in a different direction. For small values of the nomalized force, f = F/mL, and C0 non-zero, the motion is in the neighborhood of one of the two stable equilibrium positions. It should be noted that two solutions exist, one about each of the equilibrium positions. This motion is asymmetric in the u–u phase plane. For the case of C0 being equal to zero, the motion is symmetric about the unstable equilibrium position of u = 0. The coefficient C1/2 is non-zero when motion exists for which the fundamental period is a 2n multiple of the forcing period. These concepts are further demonstrated by the u–u phase-plane plots shown in Figure 5. Viewing the plot of C1 in Figures 3 and 4, it is evident that the symmetric solution exists throughout the entire range of applied loading. As would be expected from the static results, for small levels of forcing the response about the saddle point u = u = 0 is unstable. For this range of forcing amplitude, the arch will vibrate about one of the stable equilibrium positions as evidenced by the non-zero C0 . Increasing the load from zero, the asymmetric branch has a turning point at A, and the first incidence of a discontinuity between the applied loading and the response is observed. This behavior is similar to that of the softening Duffing oscillator where there are two possible stable responses. The actual response observed is a function of the initial conditions imparted to the system. For this region, local analysis about the stable equilibrium position such as that used by Nayfeh and Mook [20], would show qualitatively similar results. As the loading is increased further, a period-doubling bifurcation is encountered on the asymmetric branch at point B (Figure 4(a)). At this point, the asymmetric period-one (1T) branch loses stability, and an asymmetric period-two (2T) branch arises. As stated above, the C1/2 coefficient is no longer zero in this region. A further increase in the load results in a period-doubling cascade in which the stable solutions only exist for 2n multiples of the fundamental period. Period doubling can lead to chaotic motion where no stable periodic solutions exist [21]. The solution trajectory for chaotic motion, f = 0·08, is shown in Figure 5. If the load is increased further, the period-doubling process reverses itself, and at point C (Figure 4(b)) the 2T solution branch rejoins the 1T branch, and a stable asymmetric period-one solution is again observed. Although only the solution trajectory for the first period-doubling bifurcation was shown here, it should be noted that the harmonic balance and continuation scheme utilized in this work allows for a systematic tracing of each of the period doubling solution
Figure 4. Selected details for the C1 coefficient of Figure 3. (a) detail 1; (b) detail 2; (c) detail 3. ——, stable; – – –, unstable.
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Figure 5. u − u phase–plane plots for selected values of the normalized load f for v/vn = 0·83. (a) f = 0·02, 1T; (b) f = 0·044, 2T; (c) f = 0·08, chaotic; (d) f = 0·6, 1T; (e) f = 0·7, 1T; (f) f = 0·85, 2T; (g) f = 3·5, chaotic; (h) f = 5·0, 1T.
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. . .
trajectories. By tracing the entire trajectory, and observing changes in stability along the solution path, very detailed bifurcation diagrams similar to those shown by Abhyankar et al. [12] can be systematically constructed. At each period-doubling bifurcation, the additional non-linear algebraic equations containing the coefficients and phase angles required to represent the period-doubled solution can be added to the solutin set (equation (3)). Note that both the stable and unstable solution trajectories will be recorded allowing for an understanding of the entire solution trajectory which is not available using numerical integration techniques. Just beyond point C in Figure 4(b) the C0 term in the asymmetric response disappears, and the pair of asymmetric 1T branches merge to form the single, stable, symmetric branch at point D (Figure 4(b)). A further increase in load results in encountering another turning point, and the next stable section of the symmetric branch corresponds to motion encompassing all three equilibrium positions. As the loading is continually increased, it is observed that all of the asymmetric solutions arise from the symmetric solution branch at symmetry-breaking bifurcations. The symmetric solution loses stability at these points. Similarly, all of the 2T solution branches arise from the asymmetric solution branches at period doubling bifurcations, and are themselves asymmetric. Not all 2T solution branches undergo the period-doubling cascade and the eventual transition to chaos described above. As can be observed from the C1/2 plot in Figure 3, the 2T solutions can remain stable over their entire range of existence. All stable solutions for this range include motion about all three equilibrium positions. As shown in Figure 4(c), another range of loading results in a period-doubling cascade, and eventual chaotic motion. The phase–plane plot of the chaotic motion is shown in Figure 5 for f = 3·5. Figure 6 shows similar plots of the zero, half- and first-order coefficients for the case of the forcing frequency being twice that of the linearized natural frequency. As in the previous case, the only stable response for low load levels corresponds to motion about one of the two stable equilibrium positions. The C1/2 plot shows that there is again a 2T solution branch arising from a period-doubling bifurcation along the asymmetric response curve. The C1 coefficient for the 1T and 2T response curves is indistinguishable for the scale used in Figure 6. A further increase in the magnitude of the applied load results in the 1T and 2T solution branches rejoining, and the coalescence of the asymmetric and symmetric 1T solution branches at a symmetry-breaking bifurcation. A further increase in the magnitude of the applied load results in the symmetric response going through turning points that would result in snap-buckling between equilibrium solutions. Similarly, Figure 7 shows the plots of the coefficients of the Fourier series of u for the case of the excitation frequency being 0·6 times the linearized natural frequency. In comparison to the first case, the entire plot is shifted to the left, so that similar bifurcation points now occur at lower magnitudes of the applied forcing. As was seen in the first case, there is a clear distinction between the 1T and 2T solution branches at the lower levels of excitation. For large excitation levels, several more incidents of the symmetry-breaking bifurcation points along the symmetric solution paths are seen, along with the period-doubling bifurcations on the asymmetric solution paths leading to regions of chaotic motion. Plotting the phase behavior of the motion at various points along the solution paths for Figures 6 and 7 would result in phase portraits with features qualitatively similar to those seen in Figure 5. The static snap-buckling loads correspond to points at which the slope of the load–deflection diagram goes to zero (Figure 2). From Figure 3, it can be seen that for load values less than the static buckling load, all stable 1T solutions occur about one of the stable equilibrium positions. However, as is seen in Figure 7, there exist stable solutions
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for loading values less than the static buckling load that leave the neighborhood of a single equilibrium position. It appears that the static snap-buckling load alone does not provide an adequate criterion for the prediction of the stable periodic response of the arch. 3.2. In order to provide further insight into the effect of the excitation levels on the harmonic response of the s.d.o.f. arch, Figures 8–10 were constructed by varying the excitation frequency for a given value of the magnitude of the excitation. Figure 8 shows the C0 and C1 coefficients for the case of the forcing amplitude f = 0·01. As expected from the previous results, stable motion is observed about the equilibrium position, with the magnitude of the response being largest in the neighborhood of the linearized natural frequency. It is evident that for this small level of forcing amplitude, the effect of the non-linearities are virtually non-existent. Increasing the magnitude of the applied load by a factor of five results in the frequency response plot shown in Figure 9. The softening spring behavior observed by others (e.g. reference [6]) is seen here. Additionally, the effects of higher order resonances are observed.
Figure 6. Selected Fourier coefficients for periodic motion of the arch for v/vn = 2·0. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
362
. . .
Figure 7. Selected Fourier coefficients for periodic motion of the arch for v/vn = 0·5. (a) C0 coefficient; (b) C1 coefficient; (c) C1/2 coefficient. ——, stable; – – –, unstable.
As in all previous plots, the stable solutions are shown in bold. As evident from the non-zero C0 coefficient, the motion of the arch is restricted to the neighborhood of one equilibrium position. Tracing the response curves from large to small excitation frequencies, a period-doubling bifurcation is encountered just above the linearized natural frequency. This is similar to the behavior seen in the Duffing equation by Krousgrill and Zadoks [16] and Leung [22], who found period-doubling cascades to chaotic motion in
Figure 8. Selected Fourier coefficients for periodic motion of the arch for f = 0·01: ——, stable; – – –, unstable.
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Figure 9. Selected Fourier coefficients for periodic motion of the arch for f = 0·05: ——, stable; – – –, unstable.
similar regions of the frequency response curve. Indeed, numerical integration did not find any stable periodic solutions for that range of the excitation frequency. Doubling the forcing magnitude to a value of f = 0·1 results in a complete change in the characteristics observed in the frequency response curves. This is due to the forcing amplitude being larger than the static-buckling load. The frequency response plots for this loading condition are shown in Figure 10. Phase plane plots corresponding to specific values of the forcing frequency are shown in Figure 11. For high excitation frequencies, a stable response exists characterized by oscillations about one of the equilibrium positions. This response has a non-zero mean value as shown by branch A in Figure 10. Thus, high-frequency excitation tends to stabilize the motion of the arch about one static equilibrium position as shown by the corresponding phase–plane plot in Figure 11 (v = 1·5). Decreasing the forcing frequency results in a loss of stability due to a period doubling bifurcation at point A. This bifurcation initiates a period-doubling cascade to chaotic motion as the frequency is decreased. As shown in Figure 11, when the forcing frequency is decreased, an aperiodic response occurs about one equilibrium position (v = 1·4). This motion is unstable, and the trajectory is beginning to encompass the saddle point u − u = 0. Decreasing the frequency further results in chaotic motion encompassing all equilibrium positions. Note that for a significant range of forcing frequencies, no stable solutions are found along this branch. At point B, the 2T branch (not shown) will rejoin the main branch, which will in turn regain stability for a small range of the excitation frequency. Near the resonant frequency, both hardening and softening response curves can be seen. The hardening curve corresponds to branch C, for which the mean value of the response
Figure 10. Selected Fourier coefficients for periodic motion of the arch for f = 0·1. (a) C0 coefficient; (b) C1 coefficient. ——, stable; – – –, unstable.
364
. . .
Figure 11. u − u phase–plane plots for selected values of the forcing frequency v for f = 0·1. (a) v = 0·29; (b) v = 0·40; (c) v = 0·80; (d) v = 1·3; (e) v = 1·4; (f) v = 1·5.
is zero. Thus, the motion encompasses all three equilibrium states (v = 0·8 in Figure 11). Similar hardening responses for large amplitude motion have been reported by Plaut and Hsieh [8]. The behavior of the softening curve, branch A, was described in the previous paragraph. It is evident that branch A corresponds to the same solution branch as the curves shown in Figures 8 and 9. However, the curve no longer folds back under itself. Instead, the hardening and softening response curves intersect at point B. At this intersection, the mean value of the softening response curve (A) has gone to zero to correspond to the hardening response curve. The hardening response curve is only stable over a short range as well. A symmetry breaking bifurcation occurs at point C as indicated by the existence of a non-zero mean value branch in the C0 plot for this branch. The phase plane trajectory for this case is shown for v = 0·4 in Figure 11. Note that as in all previous cases, the asymmetric solution, branch
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D, rejoins the main symmetric, period-one solution curve. At point D on the asymmetric response branch, a period-doubling bifurcation occurs, and another cascade to chaotic motion is found. For small values of the applied frequency, f Q 0·4, only isolated regions of stable period-one motion were found. A typical example of this case is shown for v = 0·29 in Figure 11. For the unstable regions of the low frequency excitation, the motion is characterized by the arch snapping from one equilibrium state to the other with each change in direction of the applied load as shown in Figure 12. Superimposed on the large-amplitude motion are small oscillations about the equilibrium states. In order for the harmonic balance technique to accurately represent these solutions, a large number of coefficients is needed to capture the high frequency content of the response.
4. SUMMARY AND CONCLUSIONS
A technique combining harmonic balance, continuation and Floquet analysis has been applied to the study of a model of a shallow arch. This analysis technique allows for a detailed study of the solution trajectories of a non-linear dynamic system. Unlike numerical analysis techniques, both stable and unstable solution trajectories can be constructed allowing for a better understanding of the effects of the system parameters such as the magnitude and frequency of the applied load on the dynamic behavior of the system. Use of the technique in the construction phase–plane plots, time histories and bifurcation diagrams traditionally used in the analysis of non-linear systems was described. Perhaps most importantly, it was shown that simply observing the Fourier coefficients of the stable and unstable response trajectories provides the qualitative and quantiative aspects of the non-linear dynamic behavior that typcially requires the construction of phase–plane plots, time histories and bifurcation diagrams. In addition, this technique provides an understanding of the evolution of the solution as a function of the system parameters that may not be available using either numerical integration or a harmonic balance approach without a continuation scheme.
Figure 12. Displacement time history for v = 0·2 and f = 0·1.
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Using the solution techniques presented herein, it has been shown that a slight change of the values of the forcing frequency or magnitude can result in a large change in the response of the arch, both qualitatively and quantitatively. This large change in response can be of three types, each one resulting from the loss of stability along a given solution path. The first type of response change results from a jump in the magnitude of the response, while keeping qualitatively similar characteristics as seen at point A in Figure 3. The second type of transition commonly seen occurs at period-doubling bifurcations. Here, the magnitude of the response will undergo a slight change, while the character of the response will show the resulting period-doubling bifurcation. The other type of response change is the snap-buckling phenomena, where there is a large qualitative and quantitative difference in the response. Here the arch jumps from motion about one equilibrium state to motion about more than one equilibrium state, or vice versa. Thus, a large change in the magnitude and characteristic of the response is seen. Additionally, except for very low values of the applied excitation, the response plots show that there are ranges of forcing amplitude and frequency for which there exist no stable periodic solutions. In this range, chaotic motion encompassing all three equilibrium positions can be observed. For the values of the parameters studied, it is also shown that the static buckling load does not serve as an indicator of the form of the dynamic response. Comparing the results obtained by varying the forcing amplitude to those obtained by varying the forcing frequency, it is evident that much more information is available in the cases where the forcing amplitude is varied. Figures 8–10 show that even for wide ranges of the forcing frequency, the characteristics of the response do not change in a qualitative sense. The response tends to remain about the same set of equilibrium positions, making it impossible to infer any information about the snap-buckling behavior of the arch. Certainly many other qualitative and quantitative aspects of non-linear dynamic responses are typically associated with shallow arches. As with the responses shown herein, the observation of various types of non-linear dynamic response is highly dependent on the system parameters. The parameters chosen in this study resulted in several occurrences of period-doubling cascades. However, other responses such as significant second and third order superharmonic and 1/3 order subharmonic solutions were simply not observed. Investigation into the occurrence of these response characteristics will be the subject of future studies.
ACKNOWLEDGMENT
This work was supported in part by National Science Foundation Grant number MMS-9057082.
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