Gianluca Gatti

Michael J Brennan

Ivana Kovacic

On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system

This paper describes the dynamic behaviour of a coupled system which includes a nonlinear hardening system driven harmonically by a shaker. The shaker is modelled as a linear single degree-of-freedom system and the nonlinear system under test is modelled as a hardening Duffing oscillator. The mass of the nonlinear system is much less than the moving mass of the shaker and thus the nonlinear system has little effect on the shaker dynamics. The nonlinearity is due to the geometric configuration consisting of a mass suspended on four springs, which incline as they are extended. Following experimental validation, the model is used to explore the dynamic behaviour of the system under a range of different conditions. Of particular interest is the situation when the linear natural frequency of the nonlinear system is less than the natural frequency of the shaker such that the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the shaker. It is found that for some values of the system parameters a complicated frequency response curve for the nonlinear system can occur; closed detached curves can appear as a part of the overall amplitude-frequency response. These detached curves can lie outside or inside the main resonance curve, and a physical explanation for their occurrence is given.

Introduction

The behaviour of forced nonlinear two degrees-of-freedom (dof) coupled oscillators has been reported extensively in the literature, mainly by researchers investigating the dynamics of vibration absorbers.

Shaw et al. [START_REF] Shaw | On the response of the non-linear vibration absorber[END_REF] considered a coupled system where the main and auxiliary springs are with a linear-plus-cubic term. By applying the method of multiple scales, they found that when the excitation frequency is halfway between their resonances, coexisting trivial and quasi-periodic steady-state solutions occur, implying that the presence of the nonlinearities can result in an amplification rather than a reduction of the vibration amplitudes. Studying a system with nonlinear damping and nonlinear springs, Zhu et al. [START_REF] Zhu | Analysis of non-linear dynamics of a twodegree-of-freedom vibration system with non-linear damping and non-linear spring[END_REF] showed that a reduction of the vibration amplitude can be obtained by properly selecting the parameters of the nonlinear dampers, nonlinear spring stiffness and excitation frequency. Resonant interactions between coupled linear and nonlinear components have been identified as the primary phenomena behind ''energy pumping'' (see, for example, [START_REF] Gendelman | Quasiperiodic energy pumping in coupled oscillators under periodic forcing[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes[END_REF][START_REF] Starosvetsky | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber[END_REF][START_REF] Starosvetsky | Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning[END_REF][START_REF] Jiang Xiaoai | Steady state passive nonlinear energy pumping in coupled oscillators: Theoretical and experimental results[END_REF]). In [START_REF] Gendelman | Quasiperiodic energy pumping in coupled oscillators under periodic forcing[END_REF], it was investigated theoretically how the system comprising a forced linear oscillator and a strongly purely cubic attachment behaves in the vicinity of a main resonance. A possible quasi-periodic response was discovered and its advantages from the viewpoint of vibration suppression were confirmed experimentally. These quasi-periodic vibrations have also been found experimentally by testing a small building, but their occurrence was detected for a certain range of the excitation amplitudes only [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF]. In the first paper in a series of two, Gendelman et al. [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes[END_REF] studied various coexisting response regimes of this system, computed their domains of existence and analysed other characteristics, such as breakdown of the resonance, phase locking, etc. A possible application of this system for vibration mitigation and absorption was described in the second paper [START_REF] Starosvetsky | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber[END_REF]. Further considering the same system [START_REF] Starosvetsky | Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning[END_REF], the authors illustrated the local bifurcations of the periodic solutions via frequency-response curves (FRCs), showing their evolution with the increase of the excitation amplitude. As a distinguishable feature, a closed detached resonance curve, which occurs above the main resonance curve, was depicted. With the increase of the excitation amplitude, it approaches the main resonance curve, merges with it and a new shape of a continuous FRC with a peak is formed. The coexistence of three distinct regimes, two of which are periodic and the third one being quasi-periodic, was also presented. In [START_REF] Jiang Xiaoai | Steady state passive nonlinear energy pumping in coupled oscillators: Theoretical and experimental results[END_REF], the co-existence of two types of the steady state solution, one of which belongs to a detached part was also shown theoretically and confirmed experimentally. It was demonstrated in [START_REF] Starosvetsky | Vibration absorption in systems with a nonlinear energy sink: Nonlinear damping[END_REF], that if this system has a piecewise-quadratic damping characteristic and is properly tuned, dangerous periodic regimes arising due to this detached curve can be completely annihilated. In a recent paper, Alexander and Schilder [START_REF] Alexander | Exploring the performance of a nonlinear tuned mass damper[END_REF] identified numerically the presence of a detached resonance curve for a system with a cubic term but with a negligible linear term.

In this paper we analyse an experimental set-up in which an electro-dynamic shaker is used to excite a geometrically nonlinear attachment which has a much smaller mass than the shaker. The system is modelled as a nonlinear hardening Duffing oscillator coupled to a linear system, representing the shaker. In the analysis of the corresponding dynamic behaviour, of particular interest is the situation where the linear resonance frequency of the nonlinear system under test is lower than the resonance frequency of the shaker. Because the system is hardening, the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the shaker. Investigating analytically and numerically the effects of the system parameters on its response, coexisting steady states are found, with some detached resonance curves appearing in the FRCs. These detached resonance curves can lie above or inside the main resonance curve. The latter is named after its shape, a ''bubble''. As far as the authors are aware, such a detached resonance curve has not been identified yet as a part of the amplitude-frequency response.

Nonlinear system excited by an electro-dynamic shaker

System description

The practical system of interest in this paper is depicted in Fig. 1. Photographs are shown in Fig. 1(a) and (b), and a schematic representation is shown in Fig. 1(c). A small mass m, is attached to a large shaker via a support frame. The four springs between the small mass and the support frame are made from fishing line, which can be modelled as a stiffness k and a damper c. The initial tension in the wires can be adjusted and has a profound effect on the stiffness of the system attached to the shaker. When the small mass vibrates in the horizontal direction, the springs stretch in tension, thus creating a geometric nonlinearity. The electro-dynamic shaker, which is used to excite the system, can be modelled as a linear system consisting of a parallel combination of a spring k s and a damper c s connected to a mass m s , which is made up of the moving mass of the shaker and the support frame, and is much larger than the mass m. If the shaker is driven with a constant current at each frequency, the excitation can be modelled as a harmonic force with constant amplitude, F cos (ωt) as shown in Fig. 1(c).

System modeling

The wires connecting the small mass to the support structure can be modelled as shown in Fig. 2(a). The distance d, is equal to the length of the spring when the system is at rest. When the mass moves in the z direction, the springs incline to accommodate the motion as shown in the figure and it is this change in their length that is the cause of the nonlinearity. The relationship between the applied static force f in Fig. 2(a), and the resulting relative displacement z is given by

f = 4kz 1 - d 0 √ z 2 + d 2 , (1)
where d 0 ≤ d is the length of the unstretched spring. Using the Taylor-series expansion to the third order for small z, Eq. ( 1) can be written as

f ≈ k 1 z + k 3 z 3 , (2)
where

k 1 = 4k 1 -d 0 d and k 3 = 2k d 0 d 3 .
The non-dimensional form of Eq. ( 1) and its approximation given by Eq. ( 2) are illustrated in Fig. 2(b), for the particular case when d 0 d = 0.9. It can be seen that for a relative displacement z, less than 40% of the length d, the percentage error between Eqs. (1) and ( 2) is less than 5%. Furthermore, this error decreases for decreasing values of

d 0 d .
Using the approximate expression for the spring restoring force, the equations of motion of the two dof system depicted in Fig. 1(c) are

m s ẍs + c s ẋs + k s x s + c (ẋ s -ẋ) + k 1 (x s -x) + k 3 (x s -x) 3 = F cos (ωt) (3a) mẍ -c (ẋ s -ẋ) -k 1 (x s -x) -k 3 (x s -x) 3 = 0, (3b)
which correspond to the simplified system depicted in Fig. 3. Eqs. (3a) and (3b) can be written in non-dimensional form as

w + y + 2ζ s w + y + (w + y)
+ 2µζ w + µω 2 0 w + µγ w 3 = cos (Ωτ ) (4a) y -2ζ w -ω 2 0 w -γ w 3 = 0, ( 4b
)
where z = x s -x, (•) = d (•) /dτ , and the following nondimensional variables are introduced

y s = x s x 0 , y = x x 0 , w = z x 0 , x 0 = F /k s , ω s = k s /m s , ω 1 = k 1 /m, τ = ω s t, Ω = ω ω s , µ = m m s , γ = k 3 µk s x 2 0 , ζ s = c s 2m s ω s , ζ = c 2mω s , ω 0 = ω 1 ω s .
It should be noted that a change in γ can be interpreted as a change in the nonlinearity or in the amplitude of excitation or in the mass ratio. It is also noted that x 0 , as defined above, represents the static displacement of the system at ω = 0. Assuming that µy y s , as in the practical situation discussed in this paper, Eqs. (4a) and (4b) can be written as

y s + 2ζ s y s + y s = cos (Ωτ ) (5a) w + 2ζ w + ω 2 0 w + γ w 3 = y s . (5b)
Eq. (5a) shows that the nonlinear system attached to the shaker has a negligible effect on the shaker vibration so that the shaker vibrates predominantly as a linear system, while Eq. (5b) describes a base-excited hardening Duffing oscillator. Approximate solutions for the equations of motion given by Eqs. (5a) and (5b) are found by the harmonic balance method (HBM), where the responses are assumed to be at the frequency of excitation and are given by

y s = Y s cos (Ωτ + ϕ s ) (6a) w = W cos (Ωτ + ϕ) . (6b)
Substituting Eqs. (6a) and (6b) into Eqs. (5a) and (5b) and applying the HBM results in

Y 2 s 1 -Ω 2 2 + 4ζ 2 s Ω 2 = 1 (7a) 9 16 γ 2 W 6 + 3 2 γ W 4 ω 2 0 -Ω 2 + W 2 Ω 4 + 4ζ 2 Ω 2 + ω 4 0 -2ω 2 0 Ω 2 -Ω 4 Y 2 s = 0. (7b)
Eqs. (7a) and (7b) form a set algebraic equations in which Eq. (7a), which describes the frequency response of the shaker, is decoupled from Eq. (7b). However Eq. ( 7b) is coupled with Eq. (7a) by the amplitude of the response of the shaker Y s . Note that Eq. ( 7b) is cubic in W 2 . The same expressions as those in Eqs. (7a) and (7b) were also found by using the method of multiple scales. The stability of the solutions were calculated following the procedure in [START_REF] Hamdan | On the steady state response and stability of non-linear oscillators using harmonic balance[END_REF][START_REF] Hassan | On the local stability analysis of the approximate harmonic balance solutions[END_REF] and the limits for stability were determined to be

Ω 1,2 = 3 2 W 2 γ -2ζ 2 + ω 2 0 ± - 3 2 W 2 γ + 2ζ 2 -ω 2 0 2 - 27 16 W 4 γ 2 -ω 4 0 -3W 2 γ ω 2 0 1 2 1 2
.

(8a, b)

The stable solutions are shown as solid lines and unstable solutions as dashed lines in the simulated FRCs shown in Figs. 678. Before these can be plotted, however, experimental work is required to determine some of the system parameters so that the predictions from the model can be compared with the experimental results.

Experimental work

The experimental setup is depicted in Fig. 4. The electrodynamic shaker was driven by a signal generator supplying a stepped-sine signal. Accelerometers (PCB type 352C22) were attached to the support structure and to the small mass as shown in Fig. 4, while a signal conditioner (PCB model 442C04) and a twochannel oscilloscope were used to observe the system response.

Before collecting data, two tests were carried out to broadly investigate the dynamic behaviour of the system. For each test, the support wires had a different initial tension. In the first test, a slow frequency sweep (where the shaker was driven by a constant voltage) was applied from zero to about 28 Hz and the response of the system was monitored using the oscilloscope. The first resonance was observed at about 19 Hz, with both masses vibrating with large amplitudes. As the frequency was increased beyond this, a second resonance occurred at about 26 Hz, in which only the vibration of the suspended mass was large. This was followed by a sudden decrease in the motion of the suspended mass (a jump-down), which can be seen in the attached movie movie 1. The frequency was then slowly swept down. A sudden increase in the amplitude was observed at a frequency of about 22 Hz, again for the suspended mass only (a jump-up), which can be seen in the attached movie movie 2. At about 19 Hz, the resonance response in which there was large motion of both the support structure and the suspended mass was observable. In the second test similar behaviour was observed, but the jump-up and jumpdown frequencies were found to occur at about 29 Hz and 34 Hz respectively.

To collect data, the shaker was then driven at discrete frequencies for the system with the wires set with low and high initial tensions, corresponding to the cases described above, respectively. The excitation frequency was increased from 10 Hz to 36 Hz, with 1 Hz increments, and then decreased to 10 Hz with the same frequency increments. As mentioned previously, the amplitude of the excitation force was maintained at a constant level for all excitation frequencies, by manually adjusting the power amplifier so that the current was 0.8 A. At each frequency, once the system was in a steady-state, five-second acceleration time histories were captured using a DataPhysics frequency analyser connected to a PC. Subsequently, these data were processed to give the displacement of the support structure and the suspended mass. The data are presented in terms of the absolute displacement x s of the support structure and the relative displacement z = x s -x between the support structure and the suspended mass. The Fourier's series coefficients were extracted from these two time histories and the amplitude of the first harmonic of each data set is plotted at the corresponding frequency. This can be seen in Fig. 5(a) and (b) for the system in which the springs have a low initial tension, and in Fig. 5(c) and(d) for the high initial tension springs, respectively. The data points in each graph are denoted by '+' for increasing frequency and '×' for decreasing frequency.

In Fig. 5(a) and (c), which depict the response of the support structure, it can be seen that, in each case, the FRC is similar, resembling the response of a single-degree-of-freedom linear system. The peak, at about 19.5 Hz corresponds broadly to the resonance frequency of the shaker and the attached mass of the support structure. It is evident, therefore, that the nonlinear system attached to the shaker has only a small effect on its response. This is because the combined mass of the moving part of the shaker and support structure is much greater than that of the suspended mass.

In the FRCs of the relative displacement Z , depicted in Fig. 5(b) and (d), in addition to the peak associated with the resonance frequency of the shaker, a jump-down and a jump-up frequency can be seen. These are due to the response of the suspended mass. The jump-down frequencies occur at approximately 26 Hz and 33 Hz for the low initial tension and high initial tension cases, respectively, and the corresponding jump-up frequencies at about 21 Hz and 31 Hz.

Parameter estimation and model validation

To compare the experimental results with the predictions from the model, the system parameters were required. One group of parameters (m s , k s , c s , m) was measured independently, and the other group (k 1 , k 3 , c) was considered to be unknown and chosen so that the FRCs were a best fit to the experimental data. The first group was estimated as follows. The combined mass of the moving part of the shaker and the support structure m s , together with the stiffness k s and damping c s of the shaker were estimated through measurements made from an impact hammer test. With the suspended mass m detached and measured directly, the Frequency Response Function (FRF) of the shaker and attached support structure was measured. The system parameters were estimated by fitting a theoretical single dof FRF to the experimental FRF.

Once these parameters were estimated, the electro-mechanical constant of the shaker, defined as the ratio of the force over the electric current (assumed to be constant), could be estimated by measuring the FRF of the same system when driven by a random signal from the DataPhysics signal generator through the power amplifier. For a given input current of 0.8 A, the force amplitude was then calculated and is given in Table 1.

The second group of parameters was then chosen to best fit the experimental data as follows. Noting that, for fairly weak nonlinearity, the damping has a negligibly small affect on the jump-up frequency and the corresponding FRC amplitude [START_REF] Brennan | On the jump-up and jumpdown frequencies of the Duffing oscillator[END_REF], a first two-parameter fit was performed to match those values and k 1 , k 3 were estimated. The remaining parameter c was then estimated by fitting the jump-down frequency value, which is affected by the degree of nonlinearity and damping. These three parameters are also listed in Table 1, but in parentheses to indicate that they were estimated this way. For completeness, Table 2 lists the equivalent system parameters for the equation of motion written in the non-dimensional form of Eqs. (4a) and (4b).

Using the parameters in Table 2 and their relation to the dimensional parameters in Table 1, the frequency response curves described by Eqs. (7a) and (7b) are plotted in Fig. 6 (a)-(d were also calculated by solving the equations of motion (Eqs. (4a) and (4b)) directly by numerical integration and the results of the first harmonic are plotted, as circles, in Fig. 6(a)-(d) as well. For the parameters of the experimental rig, the second and higher harmonics were found to be negligible compared to the first harmonic; the small differences between the analytical and numerical results around the resonances are due to the fact that the assumption that |µÿ| |ÿ s | does not hold in these frequency regions. Examining Fig. 6(a)-(d) it can be seen that there is reasonably good agreement between the approximate HBM solution and the experimental results. Thus, the analytical model qualitatively captures the behaviour of the system.

Interaction of the responses at the resonance frequencies

The model developed in the previous section is used here to qualitatively investigate the effects of the system parameters ω 0 and γ on the FRC of the relative displacement W , and in particular the interaction of the two resonances in a two dof systemone of them being linear and the other being nonlinear. For a two dof linear system, i.e. γ = 0, the interaction of the two resonances is well-known, and for convenience it is illustrated in the animation of movie 3, which shows the effect of decreasing the non-dimensional resonance frequency ω 0 of the oscillator when passing through the resonance frequency of the support structure, which is equal to unity in non-dimensional form. The damping parameters used for the simulations are ζ s = 0.046, ζ = 0.026, while ω 0 decreases linearly from 1.5 to 0.05.

If the nonlinearity increases, then the interaction between the two resonances becomes stronger, in the sense that it changes the FRC shape completely, resulting in closed detached curves which can lie outside or inside the main FRC. This is illustrated in Fig. 7 (a)-(f) and in the animation of movie 4, where the nonlinear term has been set to γ = 2 × 10 -3 . Logarithmic scales for the displacement amplitude have been used so that the behaviour for different values of ω 0 can be clearly seen. In Fig. 7(a)-(b), the resonance frequency of the shaker system is lower than the resonance frequency of the attached system, which was the case in the practical situation discussed earlier. The resonance frequency of the attached system is then further gradually decreased and its effects are illustrated in Fig. 7(c)-(e); finally in Fig. 7(f), the resonance frequency of the attached system is one tenth of that of the shaker. Depending of the values of the system parameters, if ω 0 is decreased further then a detached curve outside the main curve may form as shown in movie 5, where the damping of the linear system has been increased to ζ s = 0.06.

The effect of increasing the nonlinearity, for fixed value of the primary resonance of the Duffing oscillator, is illustrated in Fig. 8 (a)-(f) and in the animation of movie 6, where the nonlinear term has been increased from γ = 10 -4 to γ = 2 × 10 -3 in a logarithmic fashion, the primary resonance has been set to ω 0 = 0.6 and ζ s = 0.046, ζ = 0.026. It can be observed that, for very small values of γ , i.e. γ < 10 -5 , the nonlinearity is negligible and the system response is single-valued. For higher values of γ , a closed detached resonance curve appears above the main FRC, with stable and unstable parts. As γ is further increased, this detached resonance curve approaches the main FRC (Fig. 8 increase in γ causes the detached curve to join the main curve first at low frequencies (Fig. 8(c)) and then at Ω ∼ 1 to form another closed detached curve inside the main one (Fig. 8(d)) -the ''bubble''. The region of the frequency which the ''bubble'' covers becomes smaller as γ increases (Fig. 8(e)). The ''bubble'' disappears (Fig. 8 As noted above, the appearance of detached resonance curves inside or outside the main continuous one depends vitally on the values of the system parameters -damping in particular. A detailed analysis of the influence of damping on the shape of the FRCs has been performed by the authors and presented in [START_REF] Gatti | On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a non-linear quasi-zero stiffness oscillator[END_REF].

An explanation for the shape of the FRC

The appearance of detached curves that lie outside or inside the main resonance curve implies the coexistence of steady-state responses with considerably different amplitudes. Their occurrence is dependent on the initial conditions. To further investigate the shape of the FRC and, in particular, the formation of the ''bubble'' it can be observed that the attached system is effectively driven by the shaker, such that the attached system has a negligible effect on the shaker at most frequencies. Accordingly, the attached system can be considered as a baseexcited system in which the magnitude of the excitation is a function of frequency. The displacement of such a base excited system is described by Eq. (7a) and is shown in Fig. 9(a) (the green dashed line will be discussed later). A family of curves for a baseexcited system with constant amplitude of excitation is shown in Fig. 9(b): the curves are labelled i-vi and correspond to the amplitudes of excitation shown in Fig. 9(a), with the corresponding labels. Also overlaid in Fig. 9(b) is the FRC of Fig. 8(e) (the graphs have linear scales for clarity). It can be seen that the family of curves is coincident with this curve at the frequencies marked with magenta diamonds. It is evident from this graph that the FRC of the relative displacement and the ''bubble'', in particular, is formed from the multiple solutions at the specific amplitudes of excitation and corresponding frequencies in the family of curves.

It is possible to take this analysis one step further by noting that the frequencies at which a jump-up in the response occurs is the same in the base-excited systems as well as the two dof system. However, in the two dof system, there are three jumpup frequencies; two of these occur in the ''bubble'' -one at each end, and the other occurs in the main FRC. A general expression for the amplitude, Y u at the jump-up frequency Ω u for a base-excited Duffing oscillator is given in [START_REF] Milovanovic | On the displacement transmissibility of a base excited viscously damped non-linear vibration isolator[END_REF], which can be rewritten in terms of the variables used in this paper as This is plotted as a green dashed line in Fig. 9(a). Its intersection with the graph for Y s gives the frequencies at which the response jumps up; the intersections marked 1 and 2 correspond to the frequencies at the left and right-hand of the ''bubble'' respectively. The intersection marked 3 corresponds to the jump-up frequency in the main FRC.

Y 2 u = 8 81Ω 4 u γ Ω 2 u -ω 2 0 3 + 36ζ 2 Ω 2 u Ω 2 u -ω 2 0 + Ω 2 u -ω 2 0 2 -12ζ 2 Ω 2 u 3 2 . ( 9

Conclusions

In this paper, an investigation into the dynamics of a nonlinear system attached to a shaker which was driven harmonically, has been presented. The mass of the nonlinear system was much less than the support structure connecting it to the shaker, consequently the nonlinear system had a negligible effect on the response of the shaker for most frequencies. The nonlinearity of the attached system was due to the particular geometrical configuration of the springs. The system was modelled using the harmonic balance method to determine the primary frequency response equations and the stability conditions which define the stable and unstable steady-state solutions. The system of equations was decoupled and solved in closed form. Good agreement was found between the experimental results and the analytical and numerical solutions.

The effect of such nonlinearity was then further investigated through simulations and it was found that the frequency response curves for the two dof system can have unusual shapes. In particular, closed detached resonance curves can appear. They can lie outside or inside the main resonance curve and have stable and unstable parts. The latter has a shape of a ''bubble'' and the reason for this phenomenon was investigated using a family of base-excited Duffing oscillators with different excitation levels.

Fig. 1 .

 1 Fig. 1. Practical system under consideration, consisting of a nonlinear system attached to an electro-dynamic shaker. (a) Photograph of the system, (b) Photograph showing the details of the nonlinear system attached to the shaker, (c) Schematic view.

Fig. 2 .

 2 Fig. 2. The nonlinear system attached to the shaker. (a) Schematic view, (b) Non-dimensional reaction force as a function of the non-dimensional relative displacement of the mass for d o /d = 0.9; exact expression given by Eq. (1) (blue solid line), approximate expression defined by Eq. (2) (red dashed line).

Fig. 3 .

 3 Fig. 3. Simplified model of the nonlinear system attached to the shaker.

Fig. 4 .

 4 Fig. 4. Experimental setup.

Fig. 5 .

 5 Fig. 5. Experimental results for a stepped-sine input to the shaker with a constant force amplitude. Wires with a low initial tension: (a) Absolute displacement of the support structure, (b) Relative displacement between the suspended mass and support structure. Wires with a high initial tension: (c) Absolute displacement of the support structure, (d) Relative displacement between the suspended mass and support structure. Increasing frequency (magenta '+'), decreasing frequency (green '×').

Fig. 6 .

 6 Fig. 6. Analytical, simulated and experimental results for a stepped-sine input to the shaker with a constant force amplitude. Wires with a low initial tension: (a) Absolute displacement of the support structure, (b) Relative displacement between the suspended mass and support structure. Wires with a high initial tension: (c) Absolute displacement of the support structure, (d) Relative displacement between the suspended mass and support structure. HBM solution: stable solution (blue solid line), unstable solution (red dashed line). Numerical solution by integrating Eqs. (4a) and (4b) (black '•'). Experiment: increasing frequency (magenta '+'), decreasing frequency (green '×').

Fig. 7 .

 7 Fig. 7. FRCs of the normalised relative displacement W as a function of the normalised frequency, Ω, for γ = 2 × 10 -3 , ζ s = 0.046, ζ = 0.026 and for different values of the normalised primary resonance of the oscillator: (a) ω 0 = 1.4, (b) ω 0 = 1.1, (c) ω 0 = 0.7, (d) ω 0 = 0.5, (e) ω 0 = 0.3, (f) ω 0 = 0.1. Stable solution (blue solid line), unstable solution (red dashed line). Numerical solution by integrating Eqs. (4a) and (4b) for µ = 0.001 (black '•').

  (f)) for values of γ greater than approximately 1.5 × 10 -3 .Depending of the values of the system parameters, the detached curve may also join the main curve first at Ω ∼ 1 and then at lower frequencies, as shown in the animation of movie 7, where the damping of the linear system has been decreased to ζ s = 0.03. Note that in Figs.7 and 8(a)-(f) unstable parts of the FRCs are indicated by dashed lines by applying the conditions for instability given by (8a, b). In addition, by numerical integration of the equations of motion, Eqs. (4a) and (4b), with the mass ratio fixed to µ = 0.001 to satisfy the assumption |µÿ| |ÿ s |, it is verified that the response of the system is periodic at the frequency of excitation, and higher harmonics are negligible. In Figs.7(a)-(f) and 8(a)-(f), numerical solutions are also shown as circles as validation of the analytical results.

Fig. 8 .

 8 Fig. 8. FRCs of the normalised relative displacement W as a function of the normalised frequency Ω for ω 0 = 0.6, ζ s = 0.046, ζ = 0.026 and for different values of the nonlinear parameter: (a) γ = 10 -4 ; (b) γ = 2 × 10 -4 ; (c) γ = 3 × 10 -4 ; (d) γ = 5 × 10 -4 ; (e) γ = 1 × 10 -3 ; (f) γ = 2 × 10 -3 . Stable solution (blue solid line), unstable solution (red dashed line). Numerical solution by integrating Eqs. (4a) and (4b) for µ = 0.001 (black '•').

) a b Fig. 9 .

 b9 Fig. 9. Equivalence to a base-excited system. (a) Normalised absolute response of the support structure of the two dof system (blue solid curve), where dashed black lines i-vi indicate the input displacement amplitudes used to generate the family of curves i-vi in (b) and magenta '♦' indicate their intersections to the solid blue curve; the amplitude of Y s at the jump-up frequency of the equivalent base-excited system (green dashed curve), where red ' ' indicate the jump-up frequencies for the FRC of Fig. 8(e) overlaid in (b). (b) The family of curves for the amplitude response of the equivalent base-excited system for the different levels of input displacement i-iv in (a) (black solid and dashed curves), where magenta '♦' indicates the amplitudes of the response at the frequencies corresponding to the '♦' in (a); FRC of the normalised relative displacement W in Fig. 8(e) (blue solid and red dashed curves).

Table 1

 1 The system parameters in Eqs. (3a) and (3b) for the two different experimental tests.

			m s (kg)	c s (N s/m)	k s (N/m)	m (kg)	c (N s/m)	k 1 (N/m)	k 3 (N/m 3 )	F (N)
	Wires with low initial tension	1.72		19.12		2.51 × 10 4	19.4 × 10 -3	(0.234)	(1.3 × 10 2 )	(1.1 × 10 7 )	8.59
	Wires with high initial tension	1.72		19.12		2.51 × 10 4	19.4 × 10 -3	(0.122)	(5.5 × 10 2 )	(1.3 × 10 7 )	8.59
	Numerical values in parentheses are estimated by fitting the FRC to the experimental data.
	Table 2
	Equivalent non-dimensional system parameters in Eqs. (4a) and (4b).
		µ	ω 0	γ		ζ s	ζ
	Wires with low	0.011	0.677	4.6 × 10 -3	0.046	0.050
	initial tension
	Wires with high	0.011	1.394	5.4 × 10 -3	0.046	0.026
	initial tension

Acknowledgements

The first author would like to acknowledge the financial support of Regione Calabria (FSE-POR CALABRIA 2000-2006 Mis.3.7 Az.3.7B Cap.3421107) for covering travel and accommodation expenses at the Institute of Sound and Vibration Research (UK). Professors Brennan and Kovacic would like to acknowledge the support received from the Royal Society, UK, International Joint Project' 'Using nonlinearity to improve the performance of vibrating systems'.

The authors would also like to acknowledge the helpful discussions with Dr Dane Quinn from the University of Akron, concerning some of the features of the response curves.

Appendix. Supplementary data

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.physd.2010.01.006.


B Vujanovic

Novi Sad

A Study of Conservation Laws of Dynamical Systems by Means of the Differential Variational Principles of Jourdain and Gauss*

In this report we consider the possibility of using the differential variational principles of Jourdain and Gauss as a starting point for the study of conservation laws of holonomic conservative and nonconservative dynamical systems with a finite number of degrees of freedom. We demonstrate that this approach has the same status as the method based on the D'Alembert's differential variational principle developed in a previous paper.

Introduction

It is generally accepted that the conservation laws (or first integrals) of the classical dynamical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principle.

In general, there are a variety of approaches to take in finding conservation laws. For example, in the early times of the development of :nalytical mechanics the attention was paid mainly to the conservation laws of the total energy, momentum and moment of momentum which can be easily derived directly from the differential equations of motion. It is also interesting to note that some conservation laws were discovered much before the appearance of the differential calculus and the Newton's laws of motion [START_REF] Wigner | Symmetries and reflections-scientific essays[END_REF]. For example Jordanus de Nemore (about 1300 A.D.) was acquainted with a primitive form of the law of conservation of energy and the Kepler's second and third laws are also the classical conservation laws for the two body problem.

Probably the best known and mostly used modern method for finding conservation laws is based on the study of the invariant (or gauge-invariant) prop-B. Vujanovic: erties of the Hamilton's action with respect to the infinitesimal transformations of generalized coordinates and time. Namely as shown by E. Noether [START_REF] Noether | Invariante Variationsprobleme[END_REF], for every. infinitesimal transformation which leaves the Hamilton's action absolute or gauge invariant, there exists a conservation law of the dynamical system.

However, the Noether's theory can be successfully applied to so called monogenic or Lagrangian dynamical systems i.e. systems which can be completely described by means of the Lagrangian function. Naturally, for purely nonconservative dynamical systems which are subject to nonconservative (nonpotential or polygenic) forces, the Noether's theory is not applicable since the action does not exists for this class of systems.

In order to study the conservation laws of nonconservative dynamical systems several attempts have been made by employing a version of the Hamilton's variational principle suitable for the study of nonconservative systems and also the D' Alembert's differential variational principle [START_REF] Djukic | Noethers's theory in classical nonconservative mechanics[END_REF], [START_REF] Vujanovic | Conservation laws of dynamical systems via D'Alembert's principle[END_REF]. Recently, an integrating multipliers method for the study of conservation laws of nonconservative systems have been established in [START_REF] Jourdain | Note on a analogue of Gauss' principle[END_REF].

In this work, we consider the possibility of using the differential variational principles of Jourdain and Gauss as a starting point for the study of conservation laws of conservative and nonconservative dynamical systems. We demonstrate that this approach has the same status as methods described in [START_REF] Djukic | Noethers's theory in classical nonconservative mechanics[END_REF] and [START_REF] Vujanovic | Conservation laws of dynamical systems via D'Alembert's principle[END_REF]. In addition, for the case of purely conservative systems, the general feature of our theory is the same as the Noether's theory.

A Short Outline of the Differential Variational Principles

1. Let us consider a holonomic dynamical system consisting of .Zvl particles. We denote coordinates of the particles, refered to fixed orthogonal axes, by , YN where N = 3M. The x, y and z coordinates of a particle enumerated by the integer f3 ar~ y 3 p_ 2 , YaP-1' Yap-The mass of this particle is denoted indifferently by m 3 P-l' m 3 p_ 2 , map• The system is subject to the applied forces Ya and constraint forces R., (<X = 1, 2, ... , N). Note that this classification is performed in such a way that R. are so called ideal reaction faces i.e. their virtual work is zero (2.1) where C!y., (<X= 1, 2, ... , n) stands for the virtual displacement vector or simultaneous (Lagrangian) variation. Note also, that the nonideal components of the reaction forces are usually classified as the nonconservative applied forces for which, as a rule, we have to know in advance the character of the physical interaction between particles and constraints, as for example: dry friction forces, laminar friction forces depending on the velocities of particles etc. Denoting by fj. the acceleration vector, we define the D' Alembert's principle of the virtual work in the following invariant form

y 1 ,
N }; (Ya-m.fj.) ~y. = 0. (2.2)
It is of interest to underline, that the virtual displacement vector ~y. refers to an arbitrary and infinitesimal change of the configuration of the system consistent with the imposed constraints at the given instant of time t. In other words, the virtual displacement is of purely kinematic character and the operator~ does not affects the time ~t = 0. This fact can be used for the proof that the symbols djdt and~ are commutative i.e. d~bd = 0, which is one of the basic supposition of the classical variational calculus.

Let us introduce the independent generalized coordinates xi(t), (i = 1, ... , n) whose values at timet determine the configuration of the system. We suppose that all Cartesian coordinates can be uniquely represented in terms of xi and t

From this relation we have oy. . ~y. = -. ~x', ox'

(01-= 1, ... , N).

(i = 1, ... , n, IX= 1, ... , N) (2.3) (2.4)
where the summation convention with respect to the Latin repeated indeces is assumed. Note, that this convention will be used throughout the subsequent text. The time derivative of (2.4) is

-" • = ( ozy. (01-= 1, ... , N, i, j = 1, ... , n)

2. Next, we introduce the Jordain's variational principle by the scalar product [START_REF] Jourdain | Note on a analogue of Gauss' principle[END_REF] N }; ( Y.m.fj.) ~i'[J. = 0 (2.8) where ol'[j. denotes the Jourdain's variation which affects only the infinitesimal arbitrary changes of the velocity vector but without time and space deformations: Since the J ourdainian type of variations are introduced in full accordance with the holonomic constraints, we have by varying (in the sense of Jourdain) the Eq. (2.6), or by putting oxi = 0 in (2.5):

The Jourdainian variation of the acceleration vector ( Note that in contrast to the D' Alembert's and Jourdain's variational principles, the Gauss' variational principle can be formulated as a minimum problem with respect to the true components of the acceleration vector (see, for example [START_REF] Pars | A treatise on analytical dynamics[END_REF]). Namely, let us consider the constraint in the sense of Gauss:

N 1 Z = } ; -(Y.-m.jj.) 2 • 1 1n.
(2.15) Suppose that jj. is the actual acceleration and let y. denotes any other possible acceleration which is compatible with the constraints acting on the system and which differs infinitesimally from jj. i.e. (2.16) where e is a small positive dimensionless parameter and K, is a continuous and differentiable function which can depend on time, configuration and velocities of the system. The change in the value of the Gauss' constraint Z subject to the small changes of jj. is found to be

N N Z -Z = -2 .E ( Y. -m.jj.) b2jj. + .E m.(b2jj.) 2 •
(2.17)

Since by the supposition (2.12) the first term on the right-hand side vanishes for the actual acceleration, we have Z-Z > 0.

However, it is of importance to note, that if we interpret the Eq. ( 2.16) as a characteristic infinitesimal transformation of the acceleration vector, then the expression (2.12) can be contemplated as the condition of invariance (up to ordere)of the Gauss constraint Z. This fact will be used in the subsequent text.

Lagrangian, Jourdainian and Gaussian Generalized (Nonsimultaneous) Variations

In order to enlarge the class of conservation laws of dynamical systems, it is necessary to introduce the deformation (variation) of the time into the Lagrangian, Jourdainian and Gaussian variations.

1.

For the sake of clarity, consider first the classical nonsimultaneous variations which are usually arising in the variable end-points problems of the classical variational calculus.

As defined in the previous section, the simultaneous variation bxi of a generalized coordinate xi is an infinitesimally small increment of this coordinate without the time change

xi(t) = x;(t) + ox; t = t (3.1)
where an overbar denotes the varied path. Obviously, we can look at (3.1) as an infinitesimal transformation correlating the point (x 1 , ... , x") on the actual path to a contemporaneous point (x 1 + oxl, ... , x" + ox") on the varied trajectory.

Next, let us introduce the varied path xi(t + Lit) which is infinitesimally close to the actual path xi(t) but suffers a small continuous variation of time Lit. Developing and retaining the first order terms, we have

(3.2)
Inserting (3.1) into this relation and defining so called generalized or nonsimultaneous (noncontempotaneous) variation Llxi as

(3.3)
we have (3.4) As in the case of the simultaneous variations, we can consider the generalized variations as an infinitesimal transformation of the generalized coordinates and time correlating the point xl, ... , xn on the actual trajectory at the time t with a point x 1 + Llxl, ... , xn + Llxn on the varied path at the timet+ Lit, namely

(3.5) t = t
Lit.

The relation (3.4) which relates the generalized variations Ll and classical variations <5 can serve as a pattern for the variation of any scalar, vector or tensor quantity.

In other words, we can establish the relation

(3.6)
For example, applying (3.6) to the generalized velocity vector one finds (3.9) therefore, the symbols Ll and djdt are not commutative.

Repeating the same process we find the following relations for the generalized variations of the acceleration vector Differentiating (3.9) we also have At this point it should be noted that the structure of the generalized variations LJxi and Lit introduced by (3.5) is of a great importance in the study of conservation laws. We will suppose that this structure is of the form (see also [START_REF] Vujanovic | Conservation laws of dynamical systems via D'Alembert's principle[END_REF]) (3.14) We usually call Fi and f the space and time generators of the infinitesimal transformation (3.5).

(Lix;)" = (Lixir• -x;(Lit)" -x;(Lit)" ..

Let us define the

(LI1x;)" = olx; + x;(LI1t)", Ll1x; = (LI1xi)" -x;(LI1t)". (3.16) (3.17) (3.18)
The following infinitesimal transformations of the generalized coordinates, time and velocity can be of use for better understanding the nature of the Jourdainian variations :

t = t, (3.19)
We will consider the infinitesimal quantities (LI 1 xi)" and (Lit)" as the constitutive elements or primitives of the Jourdain's infinitesimal transformation, and we introduce the following space and time Jourdainian generators of the transformation where X= {xl, ... , xn} and X = {xl, ... , xn}.

3.

Finally, let us introduce the Gauss' nonsimultaneous variations: Ll 2 xi, (LI 2 xi)"•, (LI 2 xi)" and (LI 2 t)"" together with the following requirements and also We can now establish the following infinitesimal transformations of the acceleration vector m the sense of Gauss' which include the variations of the time elements:

t = t, dxi dxi dxi dxi ---= ---= 0 dt dt dt dt ' (3.29)
Let us introduce the space and time Gauss' generators F 2 ; andi 2 of the infinitesimal transformation of the generalized acceleration vector (3.31)

Condition of Invariance of the Gauss Constraint

In this section we study the condition of the invariance of Gauss' constraint, given by (2.15). Namely we make the following transformation of the Gauss' principle (2.12) into an expression involving the Gauss' nonsimultaneous variations of the generalized acceleration vector and time elements. Substituting (3.31) into (2.14) and entering with it into (2.12), we have ~ (y oya .. oya) (F; •it) 0 e.::..., a -. -maYa -.

( oy~)• oiJ. ox' = oxi (4.3) (4.4)
and introducing the kinetic energy of the system T = (lj2)l)m.y.

-F -- -X -x ' 2
2)

+ oxi 2 + oxi 2 ot + oxi + ()xi t2 U U .. d[U . ]} - --x ' ---F -x ' =0 + ot
•) oxi F2' + L-oxi x' /2 -P(t, x, x) = 0 = const. 8L
Note that the condition for the existence of the conserved quantity (4.8) and corresponding conservation law are identical with those obtained by the study of the invariant properties of a special form of the Hamilton's principle [START_REF] Djukic | Noethers's theory in classical nonconservative mechanics[END_REF] and also from the D'Alembert's differential variational principle [START_REF] Vujanovic | Conservation laws of dynamical systems via D'Alembert's principle[END_REF]. For the case of a conservative dynamical system, i.e. Q; = 0, the Eqs. (4.8) and (4.9) constitute the classical form of the Noether's theorem (see [START_REF] Vujanovic | A group-variational procedure for finding first integrals of dynamical systems[END_REF] and [START_REF] Sarlet | Generalization of Noether's theorem in classical mechanics[END_REF]), except, the meaning of the generators F 2 i and / 2 introduced here, is completely different than used in the cited references.

It is interesting to note that the condition of the invariance of the Gauss' variational principle (4.7) is producing simultaneously the condition for the existence of a conservation law (4.8) and the conservation law itself (4.9). The situation in the classical Noetherian approach is different. The condition of the invariance of the Hamilton's action (with respect to the infinitesimal transformations of the generalized coordinates and time) is generating first the single relation (4.8) and after an additional transformation of this equation, we arrive to the conservation law (4.9) (see [START_REF] Vujanovic | A group-variational procedure for finding first integrals of dynamical systems[END_REF] and [START_REF] Sarlet | Generalization of Noether's theorem in classical mechanics[END_REF]). Note that this fact is not surprising, since the Gauss differential variational principle is at the higher differential level than the Hamilton's integral variational principle whose structure does not contain the acceleration in itself. However, despite these differences it is clear that the Gauss' principle can be used for finding conservation laws of conservative and nonconservative dynamical systems, whose behaviour is described by the monogenic function i.e. the Gauss' constraint Z.

It is also of interest to note, that the derivation of the condition of the invariance (4.7) can commence from the Z which is preliminary expressed in terms of the independent generalized coordinates xi as (see [9]) (i, j = 1, ... , n). (4.10) As in ordinary tensor dynamics, the following notation in the equation (4.10) is introduced: gii and gii are covariant and contravariant forms of the fundamental metric tensors respectively and the symbol Di}JDt denotes the intrinsic derivative of i} with respect to time, namely (4.11) where {/k} stands for the Christoffel symbol of the second kind, formed with respect to the metric tensor. Finally, Q;* denotes the generalized force which includes conservative and nonconservative parts.

Applying the symbol of the Gauss' variation () We conclude this section with the following theorem: For every set of Gauss' generators F 2 ;(t, x, x), / 2 (t, x, x) and the gauge function P = P(t, x, x) which satisfy the functional relation (4.8), there exists a conserved quantity of the dynamical system in the form of (4.9).

An Equivalent Transformation of the Jourdain's Principle

In this section we will demonstrate that the Jourdain's variational principle (2.8) can be transformed into a relation equivalent to (4.7). Thus, this variational principle can be also used as a basis for obtaining of conservation laws.

The proof is simple. Substituting (3.21) into (2.10) and inserting this into (2.8), we find ~ (Y ay" .. ay")

(F i •it ) o ..:... a -i -maYa -i 1 - X 1 = ' "=1
ox ox (i=1, ... ,n) (5.1) which is structurally identical to the Eq. (4.1). Therefore, repeating the same procedure as in the previous section, we arrive at the relation equivalent to (4.7), which confirms our statement.

Brief Comments on the Relation (4.8). Generation of Conservation Laws

Here we shortly discuss the question: Can the theorem established in the section 4, be used as a working tool for obtaining conservation laws? Actually, the problem of generating constant quantities of a dynamical system depends on the possibility of finding a set of n + 2 functions

P = P(t, x, x) (6.1)
from the single relation (4.8), which we usually call the basic Noether's identity. Generally it is a difficult problem and it is not easy to trace any universal approach for the systematic selection of these functions. Sometimes, with experience as a guide, the set (6.1) is readily evident directly from (4.8). But generally, for nonconservative systems with more than one degree of freedom it is not an easy task.

As proposed in the ref. [START_REF] Vujanovic | A group-variational procedure for finding first integrals of dynamical systems[END_REF] and accepted by many authors, the relation (4.8) can be frequently decomposed into a system of partial differential equations of the first order with respect to F 2 i, / 2 and P. We call this system the generalized Killing's equations. Naturally, this decomposition is based on the premise that it is easier to find a solution of the generalized Killing's equations than to analize the basic Noether's identity (4.8) directly. As demonstrated in numerous practical situations solutions of these equations are frequently obtainable. A detailed study of the generalized Killing's equations is out of scope of this paper. For more details see [START_REF] Djukic | Noethers's theory in classical nonconservative mechanics[END_REF], [START_REF] Vujanovic | Conservation laws of dynamical systems via D'Alembert's principle[END_REF], [START_REF] Vujanovic | A group-variational procedure for finding first integrals of dynamical systems[END_REF], [START_REF] Sarlet | Generalization of Noether's theorem in classical mechanics[END_REF], [10] and [START_REF] Vujanovic | A geometrical approach to the conservation laws of non-conservative dynamical systems[END_REF].

It is of interest to note that the structure of the generators F 2 i, / 2 and the gauge function P, especially their velocity dependence, as indicated by (6.1), has been the subject of a contraversial discussion from the theoretical point of view. A rather exhaustive account concerning this question is given in the ref. [START_REF] Sarlet | Generalization of Noether's theorem in classical mechanics[END_REF]. Note, that the velocity and even acceleration dependent infinitesimal transformations were introduced by E. Noether in her pioneering work [START_REF] Noether | Invariante Variationsprobleme[END_REF]. It seems, appart from the various subtle theoretical analysis, that the structure of infinitesimal transformations is mainly dictated by the form of a conservation law we wish to obtain. Namely, it is easy to demonstrate that many important conservation laws are not derivable by means of the Noether-type theory without the supposition of the velocity dependent infinitesimal transformations. To show this, we turn to a simple concrete example.

Let us consider the differential equation which arise in gas dynamics

(a = const.). (6.2)
The Lagrangian function is

L = (1/2) x 2 + 2axll2. (6.3)
Therefore, the basic Noether's identity (4.8) becomes

ax-1 1 2 F 2 + xF 2 -i2 c; -2ax 1 1 2 ) -P(t, x, x) = o. (6.4)
Let us try to find a conservation law of (6.2) by supposing that the structure of the space and time generators is

F2 = F(x), /2 = f(x). ( 6.5)
Substituting this into (6.4) and employing (6.2), we find

p = i;2 _ _ _ x-112 -.-+ ax-112F(x) + 2a2 --. . . [ dF a df(x)] df(x) ~ 2 ~ ~ (6.6)
By taking dj(x)jdx = const. = 1, i.e. (

the expression in brackets is independent of x. Equating this to zero, one obtains dFjdx = (a/2) x-1 1 2 , namely F = ax112.

(

The Eq. (6.6) becomes P = 3a 2 , namely (6.9) Therefore, with (6.7), (6.8) and (6.9) the expression (4.9) generates the following cubic conservation law of the Eq. (6.2) which was found by Parsons [12] using a different approach (6.10)

It is clear that this conservation law cannot be obtained by using the Noether's approach without the supposition (6.7).

An Example. Energy Like Conservation Laws of a Linear Nonconservative Dynamical System

As an illustration of the foregoing theory, we consider the problem of finding quadratic conservation laws of the linear dynamical system whose differential equations of motion are (i, j = 1, ... , n) (7.1) where k and a;; are given constant parameters. It should be noted, that the position forces X; = -a;;xi are nonconservative, namely 8X;j8xi =!= 8Xif8x;.

To analize the system (7.1) we introduce the Lagrangian function (7.2) and nonconservative generalized forces

(7.3)
It is easy to verify that the Lagrangian equations of the second kind: (oLjori)"

-(oLjoxi) = Qx,
• are producing the differential equations of motion (7.1).

Denoting the space and time generators by F;, (i = 1, ... , n) and f, we write the basic Noether's identity (4.8) as

• • 2kt • [x;F; + (L'x; 2 ) (kf-f/2)-(F,-x;f) a;
;] e -P = 0.

(7.4)

The following considerations will be simplified if we suppose that the infinitesimal transformation does not contain the time transformations i.e. f = 0. We select the space generators in the form Let us select the space generator in the form (7.12)

where A and Bare constants. Substituting (7.12) into (7.10), we have

[A(OO-Q 2 80) + B(0 2 -Q 2 8 2
)J e 2 k t -P = o.

(7.13)

For A = 1 and B = k, (7.13) can be integrated. Therefore, the gauge function is (7.14) Inserting f = 0, (7.5), (7.12), (7.14) and (7.2) into (7.9), we have by employing (7.11) It should be noted that we have obtained n independent quadratic conservation laws of the dynamical system (7.1) without any restrictions on the coefficients a; 1 .

( ~ 1 ) 2 + 2 !J 2 () 2 + k()O e 2 kt = D = const. ( 7
To show this, we turn to the algebraic problem of finding A.. (,x = 2, 3, ... , n) and Q 2 from n relations (7.8) and (7.9). Actually, using (7.8), we write (7.9) in the form A2(a22-Q 2 ) + Asaa2 + ... + J.,.a,.2 = -au A2a2s + As(ass -!J 2 ) + ... + ?.,.ana = -ala (7.17) Solving this system of linear algebraic equations with respect to A,, (,x = 2, 3, ... , n) and substituting the solution into (7.8), we conclude after simple calculation, that Q 2 can be obtained as a root of the determinantal equation =0 (7.18) a,. 1 , a,.2, ... , a,.,. -Q 2 which is an equation of the n-th order with respect to [J2. Solving (7.18) we find f2 1 2 , !:22 2 , ••• , f2n 2 • Entering with this into (7.17) we find also A•<Il' ••• , A•<n)' which completes our proof. Note, that the case of the multiple roots of the Eq. ( 7.18) will be not considered here.

1. Since the dissipative parameter k is not involved in the algebraic problem (7.18)-(7.17) our analysis is valid for the special case k = 0. Thus, we have the following result: Linear dynamical nonconserva ti ve system X; = -a;ixi, ( i, j = 1, ... , n) has n idependent quadratic conservation laws of the form ()~i)/2 + Q~;/)~il/2 = D(i) = const. (the summation convention with respect to (i) is not applied), where e(i) = XI + A2(i)x2 + ... + An(i)Xn• The coefficients Q~i)' A•(i)' (r:x = 2, 3, ... , n, (i) = 1, ... , n) can be determined from the algebraic problem (7.18)-(7.17).

2. The analysis given in the previous point is valid also for the case of conservative dynamical systems i.e. k = 0, a;i = aii• Naturally, the conservation law of the total mechanical energy can be represented as linear combination of the conservation laws D<il mentioned above.

3.

Note that for conservative and oscillatory dynamical systems the Eq. (7.11) is identical with the principal coordinates of the vibrating system. The conservation laws D(i) thus obtained can be interpreted as the energy associated to each normal mode of vibration. As in the previous point, the total mechanical energy of the vibrating system can be expressed as a linear combination of the conservation law D<il' (i = 1, ... , n). As a simple concrete example, consider the nonconservative dynamical system analized in [START_REF] Djukic | Noethers's theory in classical nonconservative mechanics[END_REF] and [13]:

(7.19) From (7.18) and (7.17) we find Q~l) =a+ bi, Q~2l =a-bi, A. 2 (Il = i, A. 2 < 2 > = -i, where i = ( -1) 1 1 2 • Therefore, using (7.16) we have following two conservation laws The total energy integral of (

[ ~ (x1 + ixz) 2 + ~ (a+ bi) (x1 + ix2) 2 + k(x1 + ix2) (i1 + ix2)]

Conclusions

The main purpose of this report is to show that the differential variational principles of Jourdain and Gauss can be employed as a starting point for the study of conservation laws of conservative and nonconservative systems with a finite number of degrees of freedom. It is demonstrated that the introduction of the generalized (nonsimultaneous) variations in the sense of Jourdain and Gauss, a Noether's-type theory for finding conservation laws can be established. For the case of purely conservative dynamical systems, the outcome is identical with the classical Noether's theory obtained by means of the Hamilton's principle. As a characteristic feature of the theory, the condition of the invariance of the Gauss' constraint with respect to the infinitesimal transformations of the acceleration vector (which includes the time deformations), produces simultaneously the condition for the existance of a conserved quantity (the basic Noether's identity) and corresponding conservation law.

In recent years a lot of publications have been devoted to the quadratic conservation laws of linear dynamical systems. A great deal of them are not based on the Noetherian approach. Contrary to the opinion expressed in [13] that the considerations builded on the basic Noether's identity (4.8) "restrict the range of possibilities" we demonstrate by means of a general example that the linear conservative and nonconservative systems can be advantageously studied in the light of generalized Noether's theory described in this paper. The author believes that the form of the conservation laws given by (7.16) (including the special cases discussed therein) did not appear previously.

(3. 7 )

 7 Differentiating (3.4) we have (Lix;)" = <5x; + x;LI ;+ x;(Lit)". (3.8) Combining (3. 7) and (3.8) we find Llx; = (Lix;)" -x;(Lit)"

Finally

  , from (3.10) and (3.11) one finds Llx; = (LJxir• -x;(Lit)". -2x;(Lit)".

  Jourdain's generalized (nonsimultaneous) variations L1 1 xi, (LI 1 x;)" and (LI 1 t)" together with the requirement: (3.15) By changing the symbols Ll and o into Ll 1 and o 1 and applying (3.15), we obtain from (3.7)-(3.9) respectively:

  (3.20) From the relation (3.16)-(3.18) we express the simultaneous Jourdainian variation in terms of these genera tors

  (3.23) By changing the symbol L1 and o into Ll 2 and o 2 and applying (3.22), we obtain from (3.10)-(3.13) the following relations A simple comparison of these equations shows that we also have (

(3. 30 )

 30 By using the relation (3.24), we express the Gauss' simultaneous variation in terms of the generators as o2x; = e[F/(t, x, x)-x;i2(t, x, x)].

( 7

 7 2 , A. 3 , •.. , A.,. are constant factors to be determined and F is an unknown function of time, position and velocity. Equation (7.5) becomes {F(x 1 + A..x.) -F[(au + A..a.l) x1 + (a12 + A..a.2) x2 + ... or Denoting by and selecting A 2 , A. 3 , ... , A.,. as we write (7.7) in the form where ala+ A..a.a = Aa, ... , au+ J..a.l (Fe-Q 2 FO) e 2 k t -P = o

e 2 " 1 = 2 " 1 X 1

 1211 D<r>• (7.20) r 1 ( . . . )2 1 ( b .) ( . )2 + k( . ) ( . . . )] 2kt D l2 X1 -~X2 + 2 a -~ X1 -~Xz xi -~X2 xi -~Xz e = <2> •Separating the real and imaginary parts, one has[ ~ (xi 2 -:i2 2 ) + ~ a(x1 2 -Xz 2 ) -bxix2 + k(x1x1 -x2iz) J e Xz + axixz 1 2 b(xi -X2 ) -r k(xix2 -xixz) e = G2.Next, consider the conservative linear system (7.22} Repeating the same procedure as in the previous case we arrive to following two independent conservation laws (x1 -x2) 2 + (ab) (xl -x2) 2 = 2Ez (7.23}

  2

  •j + o2ya ) -" i + oy. -" •i

	uy.	. ox' ox1	X	. ox' ot	ux	_ uX ox'	(2.5)
	where an overdot denotes the time derivative. Naturally, the same relation can
	be derived by variating the velocity vector
		oy. •i oy. v.=--. x +-. ox' ot		(2.6)
	An expression similar to (2.5) can be obtained for the Lagrangian variation of the
	acceleration vector which is given by the relation
	.. -oy. ••i + o2y. •i •j + 2 o2y. •i + o2y. Ya-. X • . X X . X ox' ox' oxJ ox' ot ot 2	(2.7)

  2.7) is found to be o .. = oy. o xi + 2 ( 0 . 2 y • . xi + 02 ~• ) o xi.

					(2.10)
	1 y.	ox' 1	ox' ox'	ox' ot 1	(2.11)
	3. Let us introduce the Gauss' differential variational principle in the form
		N
		}; (Y.-m.jj.) o2jj. = 0,	(2.12)
	where o 2 jj. denotes the Gauss' infinitesimal variation of the acceleration vector by
	supposing that the configuration y., velocities y. and the timet are given i.e.
		o2y. = (o2y.) .. =1= o, o2y. = (o2y.)"'• =1= o, ...
					(2.13)
	By varying the expression (2. 7) in the sense of Gauss, we find
		~ .. U2Y• =	oy. ~ .. ; ox' u2X		(2.14)
	It is obvious, that the last expression can be obtained from (2.11) by changing the
	symbol 01 into the symbol 02 and applying the rule o2y. = 0.

  Let us suppose that each particle of the dynamical system is subject to the conservative force -8Ilj8xi and a purely nonconservative force Q; = Q;(t, x, x), where II= Il(x, t) is the potential function. Thus we have

	1	ox'	ox'	2 -	X 2 =	(i=1, ... ,n)	(4.1)
	where (as we already mentioned) the summation convention with respect to the
	Latin repeated indices is assumed.
			N I; Y. -. = --+Q;(t,x,x). oy. oil . •=1 ox' ox!		(4.2)
	Performing a simple transformation of the second group of terms in (4.1) we write
	this equation as

N (oy.)• . . N oy. . . } + E m.iJ. -; (Fz'-x'f2l + E m.y. -; (F2'-x'tzl' = o.

  2 , the Eq.(4.3) Introducing the Lagrangian function L = T-II and noting that oTfoxi = oLjoxi, the last equation, after adding and subtracting the term (oLfot) t 2 ,

	becomes
	{(	oil) ox'	.	.	d [ oT dt ox'	.	. ]	oT ox'	.	.
										(4.5)
	oT + -. ox'	.	.	oT . . ox'	.	.. }

e Q; --. (F2' -x't2l ---.. (F2' -x'/2) + -. (F/ -x't2l (F2'-x't2l + -. (Fz'-x'/2x'/2) = o. reads {Q (F ; •it oL F ; oL . ; (oL oL .; oL .. ;) e • -X -

  7.22) is 2E = E 1 + E 2 i.e. the linear combination of the conservation laws E 1 and E 2 • Note, that the conservation laws (7.21) were obtained in [13] by means of a different approach.

* This research was supported by U.S. NSF Grant JPP 524 1


R Kalaba

H Natsuyama

S Ueno

R Xu

The Bellman-Gauss Principle for Constrained Motion

Keywords: Principle of optimality, Dynamic programming, Gauss' principle of least constraint, Constrained motion

Gauss' principle of least constraint is solved in a sequential fashion via dynamic programming in this paper. The solution itself constitutes a new principle for constrained motion, which we may name the Bellman-Gauss principle for constrained motion.

INTRODUCTION

In 1829, C. F. Gauss stated his celebrated principle of least constraint, which subsumes all of analytical mechanics [1 J. The principle takes the form of a constrained quadratic optimization problem. In this paper, Bellman's principle of optimality is used to solve Gauss' problem in a sequential fashion. The solution itself constitutes a new principle of analytical mechanics, which we may name the Bellman-Gauss principle for constrained motion.

GAUSS' PRINCIPLE OF LEAST CONSTRAINT

Consider a system of p particles, the ith particle of which has mass mi, displacement vector xi in inertial Cartesian coordinates, velocity vector xi, and acceleration vector Xi• The external force on the ith particle in rectangular coordinates in the inertial frame of reference is fi. If there were no constraints present, the free motion acceleration of the ith particle would be 8i = f/mi. Assume, though, that the particles are subject to equality constraints of both holonomic and nonholonomic type. A task of theoretical mechanics is to determine, at any timet, the actual accelerations x, x 2 , ... , :Xp resulting from both the impressed forces and the constraint forces. Let x be the n x 1 actual acceleration vector obtained by stacking the three-dimensional ac- celeration vectors X1, x2, ... , Xp in the usual fashion, n = 3p, and a be the n x 1 free motion acceleration vector obtained by stacking the'three-dimensional free motion acceleration vectors a 1 , a 2 , ... , ap. Then, Gauss' principle of least constraint states that the actual acceleration vee-tor x is the one that minimizes G, where [START_REF] Gauss | Uber ein neues allgemeines Grundgesetz der Mechanik[END_REF] subject to whatever the constraints might be on the accelerations. As usual, superscript T denotes transposition, and M is the n x n diagonal mass matrix.

For all the bilateral constraints treated iD Lagrangian mechanics, upon differentiating the holonomic constraints twice and the nonholonomic constraints once with respect to the time t, one is left with a consistent linear system of constraint equations for the acceleration vector x Ax=b,

where A is an m x n matrix and b is an m x 1 vector in which m is the number of constraints.

The matrix A need not be of full rank. Given initial conditions on x and x, such differentiations of the constraints do not produce any loss in generality. In this case, Gauss' principle actually takes the form of minimizing G subject to the linear constraints [START_REF] Lawson | Solving Least Squares Problems[END_REF]. The matrix A and the vector b may be functions oft, x, and x.

To recast Gauss' principle, let

so that

x = M-1 1 2 y +a. (4)
Consequently, equations ( 1) and ( 2) are equivalent to [START_REF] Kalaba | Algorithms for generalized inverses[END_REF] and

AM-1 1 2 y = b -Aa. (6)
Gauss' principle is then reduced to the problem of finding the shortest length vector y such that the consistent equation ( 6) is satisfied. As is known [START_REF] Lawson | Solving Least Squares Problems[END_REF], the solution to this problem is [START_REF] Bellman | Numerical Inversion of Laplace Transforms[END_REF] or equivalently [START_REF] Kalaba | On constrained motion[END_REF][START_REF] Udwadia | On motion[END_REF],

x =a+ M-1 1 2 ( AM-1 1 2 ) +(b-Aa), (8)
where (AM-1 1 2 )+ is the Moore-Penrose generalized inverse [START_REF] Kalaba | Algorithms for generalized inverses[END_REF] of the matrix AM-1 1 2 .

In the following, we present an alternative approach to solving this shortest length problem via Bellman's principle of optimality [START_REF] Bellman | Dynamic Programming[END_REF][START_REF] Bellman | Numerical Inversion of Laplace Transforms[END_REF].

SOLUTION VIA BELLMAN'S PRINCIPLE OF OPTIMALITY

Let C = AM-1 1 2 and d = b-Aa. Apparently, C is an m x n matrix and d is an m x 1 vector.

Suppose that the rank of the matrix C is r, and its first r columns, c1. c 2 , ..• , Cr, are linearly independent (if C is not in this form, it can be transformed into this form in various ways [START_REF] Kalaba | Solving shortest length least squares problems via dynamic programming[END_REF]). Let ck be the kth column of the matrix C, k = 1, 2, ... , n. Gauss' principle is then equivalent to the problem of finding the set of scalars Y1, Y2, ... , Yn such that y~ + y~ + • • • + y~ is minimum, and concurrently the consistent set of linear equations C1Y1 + C2Y2 + • • • + CnYn = d is satisfied.

The imbedding procedure of dynamic programming suggests the following. Let gk(dk) be the value of y~ + y~ + • • • +y~ when using an optimal set of scalars Y1, y2, ... , Yk, where c1Y1 + C2Y2 + • • • + CkYk = dk is a consistent set of linear algebraic equations, dk is an m x 1 vector, and k = r, r + 1, ... , n. Optimal is in the sense that this particular set of scalars Ylt y2, ... , Yk makes the value of Y? + y~ + • • • + y~ be minimum. Then, Bellman's principle of optimality1 leads to the basic recurrence relationship k = r + 1, r + 2, ... , n. [START_REF] Kagiwada | Numerical Derivatives and Nonlinear Analysis[END_REF] . ~ This is because once Yk is chosen, there is an immediate cost of y~, and Y1. Y2, ... , Yk-1 have to be chosen so that Y?

+ y~ + • • • + yL 1 is minimum subject to the consistent constraint (10)
But, by definition, this minimum is Yk-l(dk-CkYk). Therefore, Yk must be chosen to minimize the sum in equation [START_REF] Kagiwada | Numerical Derivatives and Nonlinear Analysis[END_REF]. The reason k starts from r + 1 is that there is one and only one set of scalars Y1, Y2, ... , Yr that fulfills the constraint C1Y1 + c2Y2 + • • • + CrYr = dr, and hence there is no freedom in choosing the scalars Yl. Y2, ... , Yk, when k ~ r, in view of the independence assumption.

Denote Cr to be the m x r matrix whose columns are c1, c2, ... , Cr and y(r) to be the r x 1 vector whose elements are y1, Y2, ... , Yr• Since c1, c2, ... , Cr are assumed to be independent, so that the matrix C~Cr is of dimension r x r with rank r, it follows that the inverse (C~Cr)-1 exists. Thus, the unique solution to the consistent set of linear algebraic equations Cry(r) = dr, in vector form, is

Y <rl -(eTc )-1 cT d - r r r r• (11)
Consequently, by definition,

where R,. is an m x m positive semidefinite symmetric matrix. Assuming Yk-1(dk-1) = dL 1 Rk-1dk-1, where Rk-1 is an m x m positive semidefinite symmetric matrix, we now prove that Yk(dk) has the form Yk(dk) = dJRkdk, where Rk is an m x m positive semidefinite symmetric matrix and k = r + 1, r + 2, ... , n.

From the recurrence relation ( 9), we see that fork= r + 1, r + 2, ... , n,

Yk(dk) =min {y~ + Yk-l(dk-CkYk)} =min {y~ + (dk-CkYk) TRk-l(dk-CkYk)} Yr. Yr.
=min {y~ + dJRk-1dk-2dJRk-1CkYk + cJRk-1CkYn

Yr.

(15) =min { (1 + cJRk-1ck) y~ + dJRk-ldk-2dJRk-1CkYk}.

Yr.

Since the first order condition for the minimizing value of Yk is (25)

k = k-1- TR ' 1+ck k-1Ck k = r + 1,r + 2, ... ,n.
Therefore, the sequence of Rn Rr+b ... , Rn is now obtainable through formulas ( 13) and ( 22).

The optimal values of y(r), Yr+1,Yr+2, ... ,JJn can be calculated via equation ( 11) and equation (23) or (24). In brief, to find the shortest length solution to the consistent set of linear algebraic equations Cy = d, where C is a matrix with rank r whose first r columns are linearly independent, we may use the following {3 -R algorithm. We carry out the following sequence of calculations:

(28)

f3n = Rn-lCn, D D f3nf3J ... "n = .L"n-l -1 + cJ f3n • (29)
Only f3r+l, f3r+2, ... , f3n need to be stored. The optimal set of scalars Yn, Yn-1> ••• , Yl is then obtained by Once y is determined, xis obtained from equation (4).

THE BELLMAN-GAUSS PRINCIPLE FOR CONSTRAINED MOTION

Gauss' principle describes the relationship between the actual accelerations and the free motion accelerations. The algorithm derived from Bellman's principle of optimality gives an algorithm for converting a set of free motion accelerations to a set of actual accelerations. It is thus seen that equations (26)-(33) plus equation (4) constitute a free standing principle of analytical mechanics, which we may call the Bellman-Gauss principle for constrained motion.

AN APPLICATION

Consider a simple pendulum problem in mechanics, in which we want to find the equations of motion for a material point with mass m and• coordinates ( X1, x2). See Figure 1.

According to Gauss, the actual acceleration vector x for this pendulum is the one that minimizes

-C2Y2 = m1/2x11 -(±~ + ±~) + gx2-m 2x2m2 X2 [gx2-(±~ + ±~)] { -1/2 1/2 } x 1 + x 2 (44) m 1 / 2 x1 [gx2-(±i + ±~)] = x2 + x2 1 2
From equation ( 4), it is seen that the equations of motion for the pendulum are .

. -1/2 opt YX1X2 -X1 (±i + ±~) x1 = m Y1 = 2 2 ' x1 +x2 X -gx21 -X2 (±21 + j;22) .. --1/2 opt - 2 [gx2 -(±21 + ±22)] -g = _ __;;_---::,..--'-:----"'-'- X2 -m Y2 -g -xi + X~ xi + X~ (45)
which are seen to be correct by standard methods. These equations were derived for x1 =/:-0. However, they hold even when X1 = 0.

DISCUSSION

The Bellman-Gauss principle introduced in this paper provides new insights into the nature of constrained motion. Not only is the relationship between the actual accelerations and free motion accelerations specified in a new way, but also the interrelationship among the actual accelerations of all the particles is prescribed. Since Gauss' principle holds even when the system is described in generalized coordinates [START_REF] Kalaba | Solving shortest length least squares problems via dynamic programming[END_REF], the possibilities are far wider than is indicated above. In practical applications, the FEED (Fast and Efficient Evaluation of Derivatives [START_REF] Kagiwada | Numerical Derivatives and Nonlinear Analysis[END_REF]) procedure would be employed to automatically calculate all the needed derivatives for the holonomic and nonholonomic constraints. The potential utility of the Bellman-Gauss principle, both conceptually and computationally, remains to be explored further.

  denominator 1 +clRk-1Ck =1-0 always holds, since Rk-1 is positive semidefinite. Let k = r + 1, r + 2, ... , n.

k

  (20) is equivalent to k = r + 1, r + 2, ... , n. (22) Substituting equation (22) into equations (17) and (18= k+1 -Ck+1Yk+l' k = n -1, n -2, ... , r.

GFigure 1 . 2 m~/ 2 ]

 122 Figure 1. The motion of a pendulum.

Bellman's principle of optimality states: "An optimal policy has the property that whatever the initial decision is, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision."