Using of a snap-through truss absorber in the attenuation of the sommerfeld effect

This work, considers a vibrating system, which consists of a snap-through truss absorber (STTA) coupled to an oscillator, under excitation of an DC motor, with an eccentricity and limited power, characterizing a non-ideal oscillator (NIO). It is aimed to use the absorber STTA, to establish the conditions, that we have the maxim attenuation of the jumpphenomenon (Sommerfeld Effect). Here, weare interestedin determining the conditions of the vibrating system, in which there arereduced amplitudes of the oscillator, when it passes through the region of resonance.


Introduction
This work, considers a non-ideal vibrating system, which is characterized by the mutual interaction between the system's response and the excitation.The response influences system's excitation, in opposite to the traditional one, called ideal systems.The non-ideal systems theories can be seenin details in: [1] and [2], undeserving of others authors.
Here, we used a vibrating system, formed by an oscillating block and a DC motor, with limited power, which works as an excitation source.This situation, characterizes a non-ideal system.The passage through resonance reveals interesting behaviors, once the motor is able to transfer a big part of its energy to carry out system oscillations, generating large movement's amplitude.Thereby, we used a DC motor, as a model of linear torque.This torque was used as a control parameter, in order to obtain the passage through resonance and control motor's frequency.
The main phenomenon observed in non-ideal systems is the called Sommerfeld effect, in which the oscillator presents unstable movements at resonance regions.A jump can be seen in the frequency-response curve, revealing the conditions where there is no permanent state.
In this work, we used the snap-through trussSTTA, in the absorption of the longitudinal vibrations of a nonideal oscillating system.Here, the oscillation energy of the main system is transferred to STTA, which fluctuates around an equilibrium point.The analysis ofthe free oscillation of this system was recently studied by [3].
Later, the forced system coupled STTA, was studied by [4].Also, in recent work [5]analyzed the interaction between STTA and an elastic system.Also, two previous works, [6,7] analysed the use of STTA in attenuation of the jump phenomenon.
Thegoal of this paper is to analyse the phenomenon of the mutual interaction, between the non-ideal oscillator and the STTA, so that the vibration amplitudes are reduced in the passage through resonance and the jump phenomenon (Sommerfeld effect) is attenuated.
This paper is organized as the follows:Section 2 presents the adopted mathematical model, whichrepresents the non-ideal system connected to a STTA.In Section 3, the stability analysis is exhibited.In Section 4, results and numerical simulations are showed and in Section 5, conclusions will be presented.Finally, we list the bibliographic references

Mathematical model
The system considered here, is based onan extension of the previous works, from [3 -6], and may be described by figure 1.In this research, we will aim to show: How the coupling of a small snap-through truss mass, can absorb part of the non-ideal system vibrations, considering especially the passage through resonance and the Sommerfeld effect [1].The governing equations of motion, which the mathematical model of the non coupled to snap-through truss absorber, are shown below: u 1 is related to the voltage applied to the motor as a control parameter for our problem, and for each type of motor.

Dimensionless system
Using the following dimensionless parameters: We can write the governing equations attachment, coupled to snap- , which represent non-ideal oscillator , are shown below: ( cos ) 2 ( sin cos ) ) are the generalized coordinates of NIO, ) is the angular velocity rotation angle of the DC motor ) are theNIO mass and STTA mass, are the eccentricity and unbalanced I is the moment of ) are the linear stiffness of the ) are NIO and STTA linear damping.
is the angle that By using a linear the voltage applied to the motor and acts as a control parameter for our problem, and u 2 is constant Using the following dimensionless parameters: equations of motion,as:

Equilibrium points and
Defining new variables, we the space state.

J q x
x q a bx x g q a bx x q q (1 2( ) ) where : Now, using the numerical values to the parameters exhibited on table 1, we evaluated the eigenvalues at each equilibrium point and then we will find the stable points of the system (the corresponding eigenvalues must have negative real parts to ensure stability).Thus, for system (table1), we will obtain: System: x 1 = 0, x 2 = 0, x 3 = 0, x 4 = 0, x 5 = 0, x 6 = 0;=> STABLE , x 2 = 0, x 3 = -s, x 4 = 0, x 5 = 0, x 6 = 0;=> STABLE Table 1.Dimensionless parameters of the non-ideal system [6].

Numerical simulations results and discussionsof the obtained results
In this paper, numerical simulations were carried out by using Matlab®, with the numerical integrator ode113, Adams-Bashforth-Moulton PECE solver algorithm with variable step-length.Values s=0.39 and c=0.92 were obtained from φ.
We will consider the non-ideal oscillator (NIO) coupling to STTA.In this case, we will investigate the dynamicbehaviour of the considered system, when the STTA is coupled to it and also we will verify the reduction of amplitudesof the motion, compared to the system without STTA (we are going to analyse how the system works before, during and after passing through the resonance region).

Non-ideal vibrating system (NIO)
Now, considering the case of the non-ideal oscillator (NIO), we will determine through the figure2: what is the input value of the torque of the DC motor (a), in which the jump phenomenon (Sommerfeld effect) occurs, without coupling the STTA to the NIO (black point).
Figure 2, displays the jump phenomenon present in the NIO, with and without coupling, with STTA.In the coupled system, different values for the stiffness parameter (γ) were used, in order to analyze the influence of this parameter on the system (considering the other values in table 1).In this figure, we used γ = 0.05 (gray point), γ 1 = 0.35 (red point), γ 2 = 0.50 (blue point), γ 3 = 0.70 (green point), γ 4 = 1.0 (magenta point).For all cases analyzed, the initial conditions are taken nulls.
Figure 2a, shows the maximal amplitudefor each "averaging'' value of the angular velocity of considered DC motor.Figure 2b, shows the maximal amplitudes versus the control parameter (a).We notice that ∆a = We can see in both figures 2a and 2b, that the maximum amplitude and the jump of the frequency of the NIO without coupling occur, when the (a) is close to 1.8 and (ϕ') to 1.0.What we can also confirm when NIO is coupled to STTA, is that while the value of ( increased, the maximum amplitudes of the oscillator tend to decrease to (a), value close to 1.8, region of the jump of NIO.Another aspect tobe verifiedin this figureis the effectthat the change invalue(γ) causes to shown.It iseasy to see thatas we increasethe value of ( the curves shift to the right ,indicating occurs for values of (a) above1.8.For angular velocity (ϕ') the same happens, so up being larger than1.0 for the jump.After determining the points, where phenomenon occurs, the behavior of the without the STTA is then analyzed.To of the NIO ,in the passage through resonance, observe the displacement and angular velocity of the motor.
The displacement of the NIO and angular velocity the motor are shown in figure3, in three situations: (a = 1.4) before the resonance region of resonance and (a = 2.2) after the region parameters used are the same as in table note that when the NIO is coupled to STTA, the velocity tends to fluctuate less and maintai value.Also, for the displacement of the NIO the range of motion is reduced when we have the , in the parameter control.b, that the maximum amplitude and the jump of the frequency of the NIO ) is close to 1.8 and What we can also confirm when NIO is coupled to STTA, is that while the value of (γ) is increased, the maximum amplitudes of the oscillator tend egion of the jump Another aspect tobe verifiedin this figureis the γ) causes to thecurves shown.It iseasy to see thatas we increasethe value of (γ), ,indicating that the jump the frequency or , so the valuesend where the jump the behavior of the NIO with and To check the action resonance, we will angular velocity of the DC angular velocity of in three different resonance (a = 1.8) in the after the region.The le 1.In figure3, we STTA, the angular maintain a constant NIO, we find that when we have the STTA active in the system.
In figure 3, we note that resonance region (figure 3a) (figure 3b), the angular velocity less and maintain a constant value system with the STTA.
The comparisonbetween the displacementsalso revealsthat theoscillations can bedramaticallydecreased whenthe absorberis attached to t of the resonance region (figure coupling oscillates less, while own region of resonance, with reflected in the figure2.2b 3a 3b 2a that: before entering of the a) and within this region the angular velocity tends to fluctuate much constant value when we have the The comparisonbetween the displacementsalso revealsthat theoscillations can bedramaticallydecreased, attached to theNIO.Since the output ure3c), the system without the coupled system is in his with large fluctuations, as 3a MATEC Web of Conferences
,y,ϕ) are the generalized coordinates of NIO, STTA and rotor, respectively.( ) is the angular velocity of the rotor and (ϕ) is the rotation angle of the DC motor shaft.(M, m) are theNIO mass and STTA mass, respectively.(r, m 0 ) are the eccentricity and u mass of the electric motor considered.I inertia of the rotor.(k, k 1 ) are the linear stiffness of the springs.(c 1 , c 2 ) are NIO and STTA linear Finally, l is the spring length and φ defines STTA equilibrium position.By using a linear torque model, we consider that ( ) u u , refers to the step length, in the parameter control.