Experience and Simulation in Dynamic Systems with Discontinuities Hans I. Weber1, Sandor Divenyi2 , Marcelo A. Savi2 , and Luiz F. P. Franca3 1 PUC/Rio, Department of Mechanical Engineering hans@mec.puc-rio.br 2 UFRJ/COPPE, Department of Mechanical Engineering savi@mecanica.ufrj.br 3 CSIRO Petroleum – Drilling Mechanics Luiz.Franca@csiro.au

1 Introduction

Non-smooth nonlinearity is abundant in nature being usually related to either the friction phenomenon or the discontinuous characteristics as intermittent contacts of some system components. Non-smooth systems appear in many kinds of engineering systems and also in everyday life. Examples may be mentioned by the stick-slip oscillations of a violin string or grating brakes [1]. Some related phenomena as chatter and squeal causes serious problems in many industrial applications [2].

The mathematical modeling and numerical simulations of non-smooth systems present many difficulties, which turns their description unusually complex. Moreover, the dynamical behavior of these systems is complex, presenting a rich response. The idea that non-smooth systems can be considered as continuous in a finite number of continuous subspaces and also that the system parameters do not change in an abrupt manner, inspired some authors to try to describe non-smooth systems by a smoothed form [3,4,5,6]. Since non-smooth systems description involves many mathematical and numerical difficulties, experimental studies are of great importance. Literature presents references using experimental approaches to verify the proposed numerical methods and others discussing just the experimental point of view [7,8,9,10,11].

The objective of this research effort is the nonlinear analysis of dynamical systems with discontinuities. A single-degree of freedom system with discontinuous support is considered and, despite the deceiving simplicity of this problem, its nonlinear dynamics is very rich. Numerical and experimental approaches are employed. The mathematical model uses a smoothened switch model [3,4,6] that treats non-smooth systems defining different sets of ordinary differential equations. On the other hand, experimental approach considers an apparatus developed in order to verify numerical results. This apparatus is instrumented to obtain all system state variables, making possible the comparison of experimental results with those obtained by numerical simulations. In general, it is possible to say that numerical and experimental results are in close agreement. Besides that, the experimental apparatus is changed in order to treat different aspects related to the influence of internal impacts within the car and also support characteristics, showing how these phenomena alter the system dynamics.

2 Discontinuous Support Model

This contribution analyzes the dynamical response of a single-degree of freedom system with discontinuous support, shown in Figure 1. The oscillator is composed by a mass m, two linear springs with stiffness k. The dissipation process can be modeled by a linear viscous damping with coefficient c. Moreover the support is massless, but modeled as a linear spring with stiffness ks and, again, a dissipation process represented by a linear damping with coefficient cs. The displacement of the mass is denoted by x, relative to the equilibrium position. A gap g defines the distance between the mass and the support. Therefore, the system has two possible modes, represented by a situation where the mass presents contact with the support and another situation when there is no contact.

Fig. 1. Non-smooth system with discontinuous support.

By assuming that the relaxation time of the support is much smaller than the

time between two contact events, the support dynamics can be neglected. Under this assumption, the system governing equations may be defined by two differential equations. One for situations without contact and the other with contact:

⎩⎨⎧

=++−++=++

contact. with ,)cos()()(2contact. without ,)cos(2

txccgxkkxxmtxckxxm

ss ωρωρ

&&&

&&& (1)

The state space Γ~ of a system ),(),(

~xxuufu && == may be split into two

subspaces −Γ~ and +Γ

~ , separated by a hyper-surface Σ~ , which is defined by a scalar indicator function )(uh [3,6]. The description of the oscillator using this approach implies to perform an adequate definition of the transitions. The contact between the mass and the support occurs if the displacement becomes equal to the gap g. On the other hand, the mass loses contact with the support when the contact force vanishes, i.e., if [ ] 0)( =+−−= xcgxkf sss & . Therefore, there are two subspaces with two different transition conditions. The subspaces are defined by two indicator functions:

xkc

gxxxhgxxxhs

s &&& −−−=−= )(),(),( βα (2)

Now, the mass motion is described considering that there is no contact with the support if the state vector −∈= Γ~),( xxu & . On the other hand, there is contact

between the mass and the support if +∈= Γ~),( xxu & . The transitions are related to

the hyper-surface Σ~ , which consists of the conjunction of two hyper-surfaces αΣ~

and βΣ~ . The hyper-surface αΣ

~ defines the transition from −Γ~ to +Γ

~ , representing situations where the contact is caused when x becomes greater than g. The hyper-surface βΣ

~ defines the transition from +Γ~ to −Γ

~ as the contact is lost when the force of the support vanishes.

This non-smooth system may be smoothened redefining the subspaces and the transition hyper-surfaces, assuming that the transition has a linear variation from

−Γ~ to +Γ

~ (or from +Γ~ to −Γ

~ ) in a thin space defined by a narrow band η around the hyper-surface of discontinuity. Figure 2 shows either the subspaces related to both non-smooth and smoothened systems.

Fig. 2. Subspaces related to the system dynamics.

By assuming a transition through a narrow band with thickness η, made

considering +− +−= fqfqf )1(α with ηη 2/)( +−= gxq and

−+ +−= fqfqf )1(β with ηη 2/])/([ xkcgxq ss &−++−= (notice that 0 ≤ q ≤ 1

in both cases), the system ),( tufu =& can be written by four differential equations:

−− ∈ Γuf , , αα Σ∈uf , , ββ Σ∈uf , , ++ ∈Γuf , , where the equations are defined as follows [3,6]:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−−=− )cos(2),( tm

xmcx

mk

xtuf ωρ

&

& (3a)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++

−−−−=+ )cos()(2),( tm

xm

ccgx

mk

xmk

xtuf ss ωρ

&

& (3b)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−++−−+

+−−=

ηηηωρα

21)(

2)cos(2 gxx

mc

gxm

kt

mx

mcc

xmk

xf

asss &&

&

(3c)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+−−+−−= xm

cgx

mk

tm

xmcx

mk

xf ss &&

&

2)(

2)cos(2 ηωρβ (3d)

The subspaces and transitions hyper-surfaces are now represented by the

following sets meaning that the calculation in the transition region may be started once the value ax& is known, i.e., the velocity of the first point after the crossing of the border. With this starting, the whole transition is calculated and the mathematical process reproduces the experiments when all subspaces are crossed by the trajectory of the system. This means that following subspaces and transitions will be in the way of the trajectory: { }ηηΓ βα ≥−≤∈=− )(or )(/2 uhuhRu (4a)

{ }ηηΓ βα −≤≥∈=+ )( and )(/2 uhuhRu (4b)

{ })()( and )(/2 uhuhuhRu αβαα ηηΣ −≤+<<−∈= (4c)

})( and )()(/{ 2 ηηΣ ββαβ +<<−−>∈= uhuhuhRu (4d)

The great deal in this point is the adequate definition of η to guarantee the good results of the method. This approach allows one to deal with non-smooth systems employing a smoothened system [6].

3 Experimental Apparatus

Fig. 3. Experimental apparatus.

In order to perform an experimental analysis of non-smooth systems, it is

developed an apparatus of the one-degree of freedom oscillator discussed in the previous section (Figure 3). Basically, the apparatus is composed by an oscillator designed as a car (4), free to move on a rail (2), connected to an excitation system composed by springs (3), strings and a DC motor (1) (PASCO ME-8750 with 0-12 V e 0-0.3 A). Moreover, the apparatus has an adjustable magnetic damping device

(5). The discontinuous support (6) is constructed considering a spring with a gap related to the car position. The movement is measured with the aid of a rotary sensor (7), PASCO CI-6538, which has a precision of ±0.25 degrees, maximum velocity of 30 rev/s and maximum sampling frequency of 1000 Hz. Parameters identification is done by different procedures. For a detailed description of the procedures, see reference [12].

4 Numerical and Experimental Results

This section considers the analysis of the system with discontinuous support considering both numerical and experimental approaches. The mass of the oscillator is identified as m = 0.838kg, while the stiffness and damping are, respectively, k = 8.47 N/m and c = 0.87 Ns/m. With respect to the support, it has a stiffness ks = 1210 N/m obtained with the procedure discussed in the previous section. By analyzing an impulsive response of the system it is possible to employ classical expressions of logarithmic decrement in order to define cs = 0.60 Ns/m, or ξs = 0.0075.

The proposed mathematical formulation is integrated using the Runge-Kutta-Fehlberg method, and numerical parameters need to be properly chosen. Divenyi et al. [6] show the importance of the choice of the narrow band thickness η, which is related to the proper transition of the system response in all subspaces. On this basis, it is assumed 510−=η for all simulations.

Fig. 4. State space comparing numerical and experimental results.

The gap influence is now focused on by assuming that ω = 11.15 rad/s and ρ =

0.33. Figure 4 presents the state space for different values of the gap, g (0.0014 and 0.0018) showing, respectively, period-3 and chaotic responses. It should be pointed out the close agreement between numerical and experimental results. Strange attractors related to this chaotic response are presented in Figure 5,

considering different positions of the Poincaré section. The left side presents numerical simulations while the right side presents those results obtained by the experimental apparatus. Once again, notice the agreement between them.

Fig. 5. Strange attractors related to numerical simulations (left side) and

experimental data (rigth side), ρ = 0.33 N, ω = 11.15 rad/s and g = 0.0018 m. The forthcoming topics analyze the influence of different system

characteristics on the dynamical response. Basically, two situations are trated: car characteristics and support characteristics. With respect to the car characteristics, there is considered a further impact system within the car. Concerning the support characteristics, different properties as stiffness and damping are considered, and also the inclusion of inertia aspects, connecting a mass to the support.

5 Influence of the Car Characteristics

At this point, the influence of the car characteristics is investigated. The goal here is to consider an impact mass, which may be free to move through a guide within the car, impacting at its ends. Since this impact mass may be fixed to the car, it is possible to establish a comparison between the dynamics with or without internal impacts. Figure 6 presents the car photo, showing the device used to promote internal impacts. Different parameters configurations are carried out changing the initial position of the impact mass and also the lubrication of the gide. Under this new configuration, the car has a total mass m = 0.471 kg.

Fig. 6. Different configurations related to the initial condition of the impact mass.

A comparison of the system dynamics considering the car with and without the internal impact is presented in Figure 7. The situation without impact is generated restricting the internal mass movement. The result without impacts presents large amplitudes and the car is in the imminence to jump out of the rail. On the other hand, internal impacts tend to dissipate system energy and, therefore, small amplitudes are expected.

After this, it is analyzed the influence of the initial position of the impact mass. Two different situations are considered. Figure 6 presents these two initial configurations. In the first one, the impact mass starts at the end near the support and therefore, is associated with a smaller first impact. The second one, on the other hand, the impact mass starts the movement in the opposite end of support being related to a greater first impact.

Figure 8 presents the comparison between the responses of these two configurations for g = 62.1 mm, ρ = 0.76 N, ω = 1.03 Hz. When the system has a smaller first internal impact, the system response is related to a period-2 orbit. By considering the other situation, the first internal impact changes the orbit and the system response is related to a period-1 orbit.

Fig. 7. Effect of internal impact. g = 22.65 mm, ρ = 0.76 N, ω = 1.51 Hz.

Fig. 8. Effect of internal impact: initial position of the internal mass.

g = 62.1 mm, ρ = 0.76 N, ω = 1.03 Hz

Now, the mass guide is lubricated causing differences in the system response. Figure 9 presents a comparison between the system with and without a special lubrication assuming g = 50.4 mm, ρ = 0.76 N, ω = 1.41 Hz. Notice that the internal impacts causes perturbation in the system response, and the original orbit presents oscillations during a cycle. These oscillations are probably due to the

internal impacts force increase. Other situations may also cause the increase of the number of impacts for the same energy level.

Fig. 9. Effect of internal impact: lubricated guide. g= 50.4 mm, ρ = 0.76 N, ω =

1.41 Hz

6 Influence of the Suppport Characteristics

In order to analyze the influence of the support characteristics in the system dynamics, the experimental set up is altered. The first consideration is just the change of the support elastic element to a spring with higher stiffness and also to a rigid rubber. It is introduced a constraint in order to avoid the impact mass movement and, therefore, there are no internal impacts. The different supports employed in the analysis represents three situations: low stiffness spring, high stiffness spring and rubber. Figure 10 shows the response for each support considering an oscillator with a mass m = 0.838 kg, excitation parameters ρ = 0.75 N, ω = 0.80 Hz and two different gaps: 2.5 and 16.3 mm. For the small gap, g = 2.5 mm it is noticeable that the low stiffness spring impacts just one time in a cycle. The other two supports, high stiffness spring and rubber, on the other hand, impacts twice. Therefore, low stiffness spring presents smaller amplitudes. By changing the gap, the system presents different behavior without the second impact.

The support inertia is now focused on. This investigation is done connecting a new car to the support. Figure 11 shows the apparatus where another car is connected to the support spring. Different support masses are analyzed considering an oscillator with mass m = 0.471 kg. The first noticeable effect associated with the support mass increase is the velocity decrease when the car looses the contact. This behavior may be understood just thinking in terms of momentum conservation. It is also related to this effect the characteristics of the non-contact response, when it is increasing the velocity for the next contact with the support. By observing the state space orbit, it is perceptible the orbit inclination as the mass is increased. Besides, by increasing the support mass, the position where the car looses contact also decreases. The forthcoming results are obtained assuming g = 5.55 mm and ρ = 0.75 N. Figure 12 shows a comparison among systems with different support inertia for ω = 0.88 Hz. The increase of the

support inertia causes together with the inclination of the orbit the decrease of the second oscillation amplitude.

Fig. 10. Influence of different support characteristics.

Fig. 11. Experimental apparatus with support inertia.

Fig. 12. Comparison of the system dynamics for different values of the support

inertia.

7 Conclusions

This article presents the analysis of a non-smooth system with discontinuous support, considering both numerical and experimental approaches. A smoothened switch model is employed splitting the phase space into subspaces, defining finite regions to describe transitions among them. This procedure is useful for numerical simulations representing an effective form to integrate non-smooth equations. Moreover, an experimental apparatus is constructed in order to verify numerical

results. Numerical and experimental investigations are carried out allowing the analysis of different aspects related to the system dynamics. In general, numerical and experimental results are in close agreement. Both approaches show a very rich dynamics, presenting dynamical jumps, bifurcations and chaos. In order to analyze some variations of the original experimental apparatus, it is considered an analysis related to the effect of the car characteristics and also the support. With respect to the car characteristics, an internal impact is introduced considering a mass free to move through a guide. Concerning the support, different kinds are considered changing the stiffness and also the influence of support inertia. These aspects can change the system dynamics in a qualitative point view and also may be desirable in order to dissipate energy.

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