DYNAMIC EFFECT OF THE PARAMETRIC EXCITATION FORCE ON AN AUTOPARAMETRIC VIBRATION ABSORBER Текст научной статьи по специальности «Медицинские технологии»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 1, pp. 137-157. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220109

NONLINEAR ENGINEERING AND ROBOTICS

MSC 2010: 70E20, 70E55, 70E99, 93D05, 93C73, 93C10

Dynamic Effect of the Parametric Excitation Force on an Autoparametric Vibration Absorber

L. Atepor, R. N.A.Akoto

Autoparametric vibration absorber is a machine invented to suppress vibration and has been widely employed in many fields of engineering. Previous works reported by various researchers have shown that dangerous motions, like the full rotation of the pendulum subsystem or chaotic motion, can emerge due to small perturbations of initial conditions or system parameters. To tackle this problem, a new model of the autoparametric vibration absorber with an attached piezoelectric actuator exciter is proposed in this paper. Under the effects of parametric excitation forces produced by the exciter, the vibration absorber will absorb more vibration energy. The dynamic response of the new system is studied analytically using the method of multiple scales and the results validated numerically using the continuation method and detailed bifurcation analysis. The results show that the vibration amplitudes of the subsystems are reduced, the region over which the absorption takes place gets widened and chaotic regions are removed with the introduction of parametric excitation forces in contrast to that of the original model of the autoparametric vibration absorber.

Keywords: autoparametric vibration absorber, bifurcation, broaden, chaos, parametric excitation forces

1. Introduction

In this paper an account is given of a recent study of the autoparametric vibration absorber by means of nonlinear coupling between the primary system (the system under absorption) and a secondary subsystem (absorber element), and a method is proposed for proper tuning by introducing parametric excitations through a piezoelectric exciter in order to enhance performance.

Received July 16, 2021 Accepted October 28, 2021

Lawrence Atepor lawrence.atepor@cctu.edu.gh

Department of Mechanical Engineering, Cape Coast Technical University Cape Coast Poly Rd., P. O. Box AD 50, Cape Coast, Ghana

Richard Nii Ayitey Akoto nii.ayitey-akoto@upsamail.edu.gh

School of Graduate Studies, University of Professional Studies, Accra New Rd, Madina, P. O. Box LG 149, Accra, Ghana

Since its invention, the autoparametric system has been intensively studied for many years [1]. The autoparametric absorber system arrangement was primarily meant for the secondary subsystem to absorb energy from the primary subsystem and modify its response. However, published results show that the nonlinear coupling between systems under absorption and the absorber can lead to very interesting and unexpected results [26], which can lead to some undesired resonances, and for some conditions, the system can transit from periodic to chaotic response behaviors [2-5]. This kind of behavior occurs when the external resonance and the internal resonance meet and these are caused by the presence of coupling terms [6]. Also, small changes in parametric excitations and or small changes of initial conditions of the nonlinear coupled system most often produce large responses when the excitation frequency gets close to one of the natural frequencies of the system [7, 24].

Some of the undesired responses mentioned above can be reduced by energy transfer and dissipation [8]. However, the effectiveness of this reduction method is minimal [9]. An interesting mathematical method known as the method of fractional steps may also be used if the problem is appropriately formulated. An application of this method has been used to describe plasma-physical processes [25]. In trying to enhance the performance of an autoparametric vibration absorber, [4, 10] presented a nonlinear frequency analysis using the multiple scales method. To improve the vibration suppression performance of the nonlinear autoparametric vibration absorber, some researchers proposed other configurations of pendulum absorbers. Sado [8] proposed the use of magnetorheological (MR) damper to change the dynamic behavior of the system giving reliable semiactive control possibilities. However, the factor by which this damper can shift the chaotic regions has not been proven. Kecik et al. [1] also proposed the use of magnetorhe-ological damper on an autoparametric pendulum system attached to a nonlinear spring. The researchers investigated the effect of the MR damper on the vibration suppression performance. Their results show that vibration mitigation can be obtained using the MR damper, however, the researchers could not show by what factor the absorption effect was altered. In another method, [11] proposed the use of magnetic forces on an autoparametric pendulum absorber and their results show a good reduction in vibration. Although the vibration reduction system mentioned above recorded good performance vibration absorption, the paper did not consider higher amplitudes of the pendulum oscillations and the emergence of unstable regions. To achieve better performance, absorber designs need to be studied taking into consideration higher amplitudes of oscillation and possible emergence of unstable regions and the effect of the proposed control methods on the absorption effect regions.

In this paper, we couple a piezoelectric exciter to the coupled nonlinear autoparametric absorber system and propose a novel autoparametric pendulum absorber. The exciter is attached to the oscillator introducing parametric excitation force into the absorber system. The modified equations of motion for the coupled system are solved using the method of multiple scales [12, 13]. The dynamics of this new system is investigated theoretically and the influence of the introduced parametric forces are analyzed analytically and numerically. The results show that effective damping and shifting of chaotic regions can be achieved without reducing the vibration effect regions significantly.

This paper is structured as follows: Section 2 presents the model of the novel autoparametric vibration absorber. In Section 3, the piezoelectric actuator exciter model is discussed. Section 4 is concerned with the solution of the equations of motion and the nondimensionalization of the system's dynamics is briefly presented in Section 5. In Section 6, the analytical results and their validation using two numerical methods are discussed. The conclusions are presented in Section 7.

2. The model of the new autoparametric absorber

The new autoparametric absorber model, presented in Fig. 1, is made up of two subsystems: a nonlinear oscillator (primary system), a pendulum (secondary subsystem) and a piezoelectric actuator exciter (piezoexciter). The pendulum is made up of two masses m2 (for the ball) and m3 (for the rod) and is attached in a bearing to the mass m1. The piezoexciter is fixed at one end to the mass and its other end to an unmovable foundation. The oscillator is forced by an external harmonic force and is supported by a nonlinear spring having Duffing's type k1 x + k2x3 characteristic, and a linear viscous damper with damping coefficient c1 as found in [14]. We modified the equations of motion for the degrees-of-freedom autoparametric vibration absorber which are explained in [14, 15]:

(m^ + m2 + m3)x + c1.t + kxx + k2x3 + I (j) sin 9 + 92 cos ^'m2 + =

= Q cosQt + xFa cos q2t, (2.1)

m2 + ^ + + (m2 + + sin ° = °> (2-2)

where x denotes the displacement of the oscillator in the horizontal direction, 9 is the angular displacement of the pendulum, g is acceleration due to gravity, l is the length of the pendulum, c2 is the coefficient of viscous damping of the pendulum, Q is the amplitude of the external harmonic force, Fa is the parametric excitation force, q2 is the parametric excitation frequency, and Fa cos(Q2t)x is the parametric force term. The nonlinear oscillator's spring stiffness is

described by the parameters k1 and k2. The system's parameters are defined by

2 k1 2 k3 1 2

'"i ' i" > ■ '":■.• = —, uj2 = —, N = -ni2 + m3 I ,

M 2 N V3

ef = eFa = 77' ea = 77' £Cl = 7-T77 '

M a M M 2u1M 2u2n

u 1 (1 \ , 1(1

eh = Jj [ 2m2 + ms J I, = jj ( 2m2 + ms

c

2

Fig. 1. Schematic diagram of piezoexciter inertia-coupled system RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(1), 137 157_

By defining representative parameters and assuming small angle oscillations, Eqs. (2.1) and (2.2) can be transformed in order to get an approximate analytical solution for the nonlinear frequency response. This procedure results in the following two nonlinear coupled differential equations:

x + 2e(1 u1x + ufx + eax3 + eh (j)9 + 92^ — eFax cos q2t = ef cos (2.3)

() + 2eC2u20 + + e^xd + e^gd = 0. (2.4)

3. The model of the piezoelectric actuator exciter

The piezoelectric actuator exciter provides a modern and novel solution for active control in many different applications, with several advantages, namely, compactness of structure, noise-free operation, low power demands, small displacements and large force outputs.

F ■

smin

Fig. 2. (a) Piston-actuator assembly when the system is not excited. (b) Piston-actuator assembly when the system is excited at maximum amplitude. (c) Free length of the spring

To get a suitable parametric excitation force there is the need to select a corresponding actuator, and there is also the need to determine the likely force levels needed to excite the au-toparametric system parametrically through the primary subsystem, and suitable exciters which can provide this, at approximate levels of displacement. As the primary subsystem oscillates, there will be a millimeter level of gap, which will have to be taken up by the piston of the exciter system. The actuator only displaces by micrometers so there will be a potential gap between the actuator-piston connection and the point of attachment of the primary subsystem when the system is excited. The actuator-piston connection will therefore have to follow the primary subsystem as it gets displaced, but because the other end of the actuator has to react against something, a spring is needed to provide sufficient reaction, and to take up the space left as the primary subsystem moves to and fro. The maximum spring force available is given as

Fsmax = ks $2, (3.1)

where Fsmax is the maximum spring force, ks is the spring constant, and 62 is the maximum spring compression. Figure 2a depicts the piezoelectric excitor piston-actuator assembly when the system is not excited. Figure 2b shows the piston-primary subsystem having displaced as a consequence of excitation. The spring has extended to fill the gap, A, and the remaining spring compression is ¿1. This is a precompression and is set up via Eq. (3.2) such that it satisfies the

need for the minimum spring force Fsmin offered by the spring to equal at least the maximum force which the actuator is capable of (Fa(max)), meaning

Fsmin = Fa(max) = ks ¿1, (3-2)

where ¿1 is the "preload" precompression. As the minimum spring force available must be enough to resist the maximum force generated by the actuator, the actuator then can transmit its force to the piston-primary subsystem, even when the subsystem has travelled by its maximum displacement. Figure 2c is the free length of the spring shown in Figs. 2a and 2b. It can easily be seen that the relationship between the precompression ¿1, the maximum compression ¿2, and the maximum displacement, A, is given by

S1 = ¿2 - A. (3.3)

This means that the maximum spring force can be written as

Fsmax = ^(¿1 + A). (3.4)

Eqs. (2.3) and (2.4) were used to find the parametric excitation force that is actually needed for the system, in order to get parametric resonances and the displacement due to the movement of the mass. The ND Solve integrator within MathematicaTM code was employed to solve the differential equations. All other parameters were fixed and the value of the parametric excitation force term was varied until a parametric excitation plot was obtained as shown in Fig. 3 and the value at which the response is predicted was taken as a threshold value for the parametric excitation force [16].

0.25 0.20 0.15 0.10 0.05 x 0.00 -0.05 -0.10 -0.15 -0.20 -0.25

Fig. 3. Parametric plot

From the above analysis, the maximum actuator force Fsmax is found to be 100 N. Calculation of the maximum spring force is done using data from an autoparametric absorber test rig being set up at Cape Coast Technical University laboratory and it is calculated using Eqs. (3.1) to (3.4) and found to be 100.1 N. A spring was chosen based on the maximum required spring force leading to the calculation of the actuator force and hence the selection of the appropriate actuator.

Figure 4 is a schematic diagram of the piezoelectric exciter (shown as a symbol in Fig. 1) designed purposely for this investigation and the voltage applied through a piezoelectric voltage amplifier to the actuator, which in turn develops the parametric excitation force set at frequency twice the natural frequency of the primary subsystem.

t (sec)

Sliding Bearing

Compressor Spring

Spring Compressor

Drive current Thin Plate

Piezoelectric Actuator

Brass Plate

Fig. 4. Schematic of the piezoelectric exciter

4. Solution of equations of motion

The method of multiple scales [4, 13, 17] is used to solve Eqs. (2.3) and (2.4), and as required by the method, the coordinates x and 9 are stated in power series form, as are the total derivatives with respect to time:

X — Xq + SX'i

0 — Qq + £el +... + en 0n, d_ dt d2

— Dq + eD1 + ...,

dt2

— D2 + 2eD0 D1 + ....

(4.1)

(4.2)

(4.3)

(4.4)

For determination of the first-order approximation, we present two-time scales, where Tn = = ent and the derivatives D = -Mr- (n = 0, 1). Applying multiple scales by substituting

n

Eqs. (4.1)-(4.4) into Eqs. (2.3) and (2.4), and grouping terms of order of £ leads to

D^XQ + ufx 0 = 0, DQOQ + U229O = 0, D0>x1 + wfo = -2D!(DqXQ) - 2£i w(DqXQ) - axQ - h(Do6Q)2-

-h(D0290)90 + Fa cos(02t)x0 + f cos(Qt), D261 + w26i = -Di(DQOQ) - w2(DQ6Q) - »(DjjXQ)6Q -

General solutions to Eqs. (4.5) and (4.6) are stated in convenient polar forms as

x0 = A(Tl)eiUJlT° + A(T1)e~tu i'Jo,

—iu-, Tr

0q = B(Ti)eiuj'2T° + B{T{)e~w^

—iu0Tr

(4.5)

(4.6)

(4.7)

(4.8)

(4.9) (4.10)

The autoparametric interaction between two modes is typified by an internal resonance of the simplest form of

(4.11)

a statement of principal parametric resonance internal to the system. According to [4], this can be expressed more conveniently by using a detuning parameter en such that

w,

w2 = — + er].

(4.12)

The primary subsystem will be excited at Q, which is occurring at the same time as w1, then we have

Q = u1 + ea. (4.13)

For the case in which the parametric excitation frequency Q2 is twice the external frequency Q, so that the detuning of one frequency automatically decides that for the other we have

Q2 = 2w1 + eY. (4.14)

en, ea and eY are defined as the internal, external and parametric detuning parameters, respectively.

Therefore, substituting Eqs. (4.9) and (4.10) into (4.7) and (4.8) gives

D02X, + wfx, = To

d02 e, + wie, = é^ To

-i2w1D1 A - i2(1w21A - aA3e2iwiTo --3aA2A - 2hwiB2ei(2u2)To + +^Ae2inTo + + cc.

-i%j2DxB - i'2i2w2B - ßcofABe^i-^o--WABei(ui +w2)To - ßgB + cc.

(4.15)

(4.16)

where cc denotes the complex conjugate of the preceding terms. The complex amplitudes A and B can be expressed in polar form as

A = Ie*,

B = -etK. 2

(4.17)

(4.18)

Substituting Eqs. (4.17) and (4.18) into the secular terms of Eqs. (4.15) and (4.16) leads to

3 1 F f

—iui-^ci! + uj^iS' - i^ujja - -act3 - Thw\b2ë^ + -faé^ + f e^ = 0

8

2

1

4

2

-iw2b' + u}2bn' - ii2tjJ22b - lujjabe- -y 6 = 0,

(4.19)

(4.20)

where

01 = vti - 5, <2 = 2k - 5 - 2nT,, <3 = 2aT, - 25.

(4.21)

(4.22)

(4.23)

Separating the real and imaginary terms results in the following four equations:

3 1 f F

uiaô' — -aa,3 — -hw2b2 cos <f>l + — cos 02 + —fa, cos <f>3 = 0, (4.24)

8 2 2 4

1 f F

-u^a' - thuja - -hull)2 sin cj)l + — sin 02 + sin = 0, (4.25)

uj2 bu1 — ^iojfabcos<p1 — ^-b = 0, (4.26)

—u)2b' — £,20J2b + ab sin (f)1 = 0. (4.27)

The form of Eqs. (4.24)-(4.27) renders the system autonomous. For the steady state, the condition is required whereby

a = b' = 01 = 02 = 0'3 = 0, (4.28)

and so differentiating Eqs. (4.21)-(4.23) with respect to T1 leads to

01 = a - ô' = 0 ^ a = ô', (4.29)

2 k' - 6' - 2r? = 0 k' = (4.30)

Substituting Eqs. (4.28)-(4.30) into Eqs. (4.24)-(4.27) leads to

3 1 f F

w1acr--aa,3 — -hu)2b2 cos <f>l + — cos 02 + —fa. cos <f>3 = 0, (4.31)

8 2 2 4

1 f F

-CiW?a, - -huo'lb2 sin ^ + - sin <f)2 + sin <p3 = 0, (4.32)

^b

a

2-1.

- ^iuj jab cos (¡)1 - Yb = 0> (4-33)

~(2uj'2b + ^i(jj\ab sin cf)1 = 0. (4.34)

Squaring and adding Eqs. (4.33) and (4.34) isolates sin and cos multiplying the numerator and the denominator by e allows returning to the originally defined parameters in the equations of motion leading to an expression for the amplitude response for the primary subsystem:

4^2

» -lY {en){eg) ( ii \ (e^)2(eg)2 2

(S^uj2 \2UJ2 J UJ2 \2UJ2 J 4 ujîujf

1/2

. (4.35)

Applying a similar procedure to Eqs. (4.31) and (4.32) leads to a biquadratic equation in b2 for the amplitude response for the secondary subsystem, Eq. (4.36). Note that Eqs. (4.35) and (4.36) describe the nonlinear response of the system with the parametric excitation. It is worth noting that the primary response, Eq. (4.35), does not depend on the cubic stiffness, the external force into the Duffing system [15] and the parametric excitation force. However, the secondary subsystem with response, Eq. (4.36), is evidently influenced by the cubic nonlinearity ea and the parametric excitation force eFa, meaning that the cubic term and more significantly the parametric force term have been transferred from the primary subsystem and exciter to the secondary subsystem to manipulate or enhance the performance of the pendulum absorber:

b4 +

12uj2(ea)(Q-2uj2)3 _ 8(0-1^X0-2^) + |

(eh) (t/x)3 (t gfu)\ (eh) (e /x) (eg)w1 uj2 (eh) (t/x) (eg)

a

+

48w23(t-a)(t-e2)2(Q - 2u>2) 2uj3(eh)(eFa)

(e:/n)(eg)ujf

(sh)(eß)3 (eg)3 w6 (c^^w,

'8w2(eFa)[1 - (e^i)w,](Q - 2w,)

b2+

+

+

+

8io2(eFa)(n-2co1f (e:/n)2(sg)2ujf

36w2 (ea)2

(eß)2(eg)2 w3

n

2

- M + (e&)2

(eh)2 (e^)4(eg)4 wf

+

7 Q \2 ^3/2

64[(Q-a;1)2+a;2(fc-ei)2] (eh)2(eß)2(eg)2uj2

+

48u)2(ea)(eFa)

(eriHegyul

dr1)'*«"*

768u)42(ea)(Q -ujJ

+

» 2w

2

- 1 +(e^2)2

+

+

(■ef)2

(eh)2 w4

= 0. (4.36)

4.1. System dynamics

In addition to relying on analytical solutions for the new autoparametric absorber system, emphasis can also be placed on its qualitative behavior. The analysis method employed in this study includes dynamic trajectories of the absorber system, displacement-time plots, Poincare maps and bifurcation diagrams. Nonlinear dynamic analyses of the system are carried out using the Dynamics 2 software [21] and the numerical continuation software of Auto 07p [18]. The equations of the model contain dimensional parameters that are nondimensionalized by scaling to reduce the order of the design space. This does not change the dynamics of the system.

The models of the new autoparametric absorber system are used after some modifications for analyzing the behavior. We therefore write the equations in the following form:

x + Cxx + C2x + C3x3 + CA60 + CA62 - C5x cos ü2t = pcos Q, ë + c6è + c7e + c8xe = o,

(4.37)

(4.38)

where C1 = 2e^lujl, C2 = oj2, C3 = ea, C4 = eh, C5 = eFa, C6 = 2e(2uj2, C7 = uj2, p = ef and C8 = t¿t. The time t is nondimensionalized by using the system natural frequencies of u1 and w2. Nondimensionalization of the timescales in Eqs. (4.37) and (4.38) is introduced by t1 = = v/w1i and r2 = s/uJ2t, leading to first-order ordinary differential equations in the following form:

u,

u' = — cos (<fit) — CiU — C2x — C3x3 — C^v'O — C4v4 + C5x cos^i),

w

e' = v,

v = -c6v - c7e - cfu'e,

(4.39)

(4.40)

(4.41)

(4.42)

wlnor-o n __n __(~1 _ C3 (~1 _ C4 (~1 _ C5 (~1 __(~1 __f-

wùeie w - 75r> °2 - - - - - o7 - -j=, -

_ ef

and Cf =

G*

presented in Tables 1 and 2, respectively.

The dimensional and nondimensional parameters used for the study are

2

2

2

X

1

Table 1. Dimensional parameters

m¡ = 5.0 kg k1 = 474.92 N/m c2 = 0.01 Nm/ (rad/s) Kot = 100 N u1 = 8.2368 rad/s Q2 = 17.56 rad/s e^2 = 0.00433 e = 0.55 kg • m

m2 = 1.0 kg

k2 = 77.0 N/m3

Qo = 10 N

eFa = 14.2857 m/s2

w2 = 3.4658 rad/s

ea = 74.25 (N/m3)/kg

eh = 0.1071

eg = 5.3955 kg • m2/s2

m3 = 1.0 kg c1 = 1.0 N/(m/s) l = 0.5 m ef = 1.4285 m/s Q = 8.76 rad/s e^1 = 0.00867 en = 2.2502

Table 2. Nondimensional parameters C\ = 0.05 C2 = 23.64 C3 = 25.87 C4 = 0.0575 C5 = 4.96 C6 = 0.016 C7 = 6.45 C8 = 1.21 4> = 8.76 (/>2 = 17.56 ^=0.17

5. Results and discussions

The analytical response curves are generated from Eqs. (4.35) and (4.36). Numerical values (with the exception of the parametric excitation force value), taken from [15], are substituted into these equations to arrive at the response curves for the system.

A computer program is written using MathematicaTM software [23] to perform the calculations, and the results are those shown in Figs. 5-8. Some response curves are illustrated for variations of the parametric excitation force, keeping the external harmonic force constant, for the exact tuning.

ea

Fig. 5. Frequency response for the nonlinear primary subsystem when F0 = 10 N, Fa =0 and the secondary subsystem is not swinging

The amplitude frequency response for the nonlinear primary subsystem is shown in Fig. 5, where a denotes the response amplitude of the nonlinear primary subsystem and ea is the external resonance condition. This results when the external harmonic force with amplitude F0 = 10 N, there is no autoparametric interaction with the secondary subsystem absorber and there is no

parametric excitation force acting on the system. This response describes the classical bent response due to the cubic stiffness associated to a hardening spring and when no energy pumping occurs [15].

The frequency responses for both the primary and secondary subsystems for the autopara-metric vibration absorber system under external harmonic force with amplitude F0 = 10 N are shown in Fig. 6. In Fig. 6a, the nonlinear amplitude decreases when ea is close to and increases after crossing the zero point. ea is termed the exact resonance condition. At the exact resonance condition, amplitudes of 0.026 m and 1.1 rad are observed for the primary and secondary subsystems, respectively, in Figs. 6a and 6b when F0 = 10 N and for no parametric excitation force, that is, when Fa = 0.

0.000

-0.6

Fig. 6. Frequency response for (a) primary subsystem (b) secondary subsystem, when F0 and F„ =0

10 N

In Fig. 7, introducing the parametric excitation force of Fa = 50 N, the primary subsystem's response amplitude reduces to 0.013 m in Fig. 7a from 0.026 m in Fig. 6a and that of the secondary subsystem reduces to 0.7 rad in Fig. 7b from 1.1 rad in Fig. 6b.

r 1

: \ i i 1 i i 1 i \---/ 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.000n ß —0.6

<N

t—I

I

Ci

o I

to Ö I

CO o Ö

CO Ö

tO Ol Ö Ö

(a)

ea (b)

Fig. 7. Frequency response for (a) primary subsystem (b) secondary subsystem, when F0 = 50 N

10 N and Fn

It is also observed that the original V-shaped response characteristic of the absorbed primary subsystem is altered by the introduction of the parametric excitation force. The V-curve widens out to become a flat-bottomed U-shape. Introducing the parametric excitation force of Fa = 100 N into the autoparametric absorber system, the response amplitude of the primary subsystem further reduces to 0.004 m in Fig. 8a and the response amplitude of the secondary subsystem reduces to 0.44 rad in Fig. 8b. Increasing the parametric excitation force from 50 N to 100 N causes further widening of the V-shaped curve, making it more flat-bottomed, response curves as depicted in Fig. 8a.

■ ' I ■ i I ■ ■ I ■

in i—i I

<31

o I

(a)

■ I ■ ■ I ' i I ■ i

Fig. 8. Frequency response for (a) primary subsystem (b) secondary subsystem, when F0 = 10 N and Fa = = 100 N

Fig. 9. Frequency response for (a) primary subsystem (b) secondary subsystem, for Fa =0 N, obtained by the numerical method

In order to verify the analytical results, the numerical methods of Auto 07p and Dynamics 2 softwares are used to solve the nondimensionalized equations of Eqs. (4.39)-(4.42) with different initial conditions. In Figs. 9-11, the response curves for the primary and secondary subsystems are obtained using the numerical continuation method software of Auto 07p. Doedel et al. [18]

e 0.4

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

fl n

(a) (b)

Fig. 10. Frequency response for (a) primary subsystem (b) secondary subsystem, when Fa = 50 N, obtained by numerical method

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Q, (a)

e o.4

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 f2 (b)

Fig. 11. Frequency response for (a) primary subsystem (b) secondary subsystem, when Fa = 100 N, obtained by numerical method

used a similar software of Auto 97 for numerical continuation and bifurcation analysis. The Auto 07p software has also been used for systems with pendulum by [1, 19].

The responses in Figs. 9-11 are sets of system responses in which the linear and nonlinear response curves have been superimposed. At around resonance of 0.76, the response curves for the subsystems are presented in Fig. 9. The case of response for the primary subsystem (where the secondary subsystem is not oscillating or swinging) is marked by the black line, while the red line corresponds to the case where the secondary subsystem swings or oscillates. The black lines in Figs. 9b, 10b and 11b show the linear (zero) amplitudes of the secondary subsystem. Dash-dotted lines represent the unstable solutions and the solid lines represent stable solutions. The dynamical elimination of the primary subsystem's vibration caused by the secondary subsystem's

swing is observed in the region of Q = 0.76. Note that the primary subsystem vibrates with the same amplitudes for both positive and negative shifts of the secondary subsystem.

Introducing the parametric excitation force of Fa = 50 N, we reduced the amplitude of the primary subsystem from 2.24 in Fig. 9a to 1.70 in Fig. 10a and that of the secondary subsystem from 0.36 in Fig. 9b to 0.34 in Fig. 10b. It is observed that the absorption effect still exists and that in Fig. 10a the V-shaped response characteristic of the absorbed primary system is altered by the introduction of the parametric excitation force. Again, by introducing the parametric excitation force of Fa = 100 N, the amplitude of the primary subsystem reduces further to 0.90 in Fig. 11a and that of the secondary subsystem to 0.28 in Fig. 11b. The V-shaped response characteristic of the absorbed primary system is further altered, becoming more flattened.

The dynamic behavior here is quite consistent with the frequency responses predicted in the analytical responses of the system in Figs. 6-8. The slight difference between the analytical solutions and the numerical solutions is due to the approximations made and ignoring of the higher-order harmonic terms of the Maclaurin series in the analytical study. It is evident that the general V-shaped response characteristic of the absorbed primary subsystem is altered due to the introduction of the parametric excitation force at parametric resonance, and this means an obvious improvement in performance. The significant effect is that the operating region broadens and the sharp V-curve widens out to become a flat-bottomed, U-shaped response curve corresponding to that which was observed by [4]. This is a very important performance effect from a practical point of view, because this exciter piece can be used to enhance the performance of the dynamics of the vibration absorber without decreasing its efficiency.

Unstable vibrations leading to chaotic motions may occur near parametric resonances considering certain parameters [20] in the exciter-autoparametric absorber system. The global and local dynamics of the system's equations (4.39)-(4.42) are studied numerically using Dynamics 2 software of [21]. The bifurcation diagram summarizes the essential dynamics of systems and is considered a very useful way of observing nonlinear dynamic behavior [22]. The bifurcation diagrams are calculated to investigate the effects of the influence of the introduced parametric excitation forces on the dynamics of the system for a typical resonance range. An increase in the amplitude of external excitation causes the system to show possible bifurcation to chaos. Bifurcation diagrams, Poincare maps and time plots are studied for the system with and without parametric excitation forces. Figure 12 shows the bifurcation of the primary subsystem as controlled by the normalized excitation. It appears that, for the normalized excitation of 0.5 to 0.65, the iterates settle down to a fixed point. This is considered period one motion. At the point 0.65, a jump occurs from an amplitude of -0.26 to 0.00. From this point, there remains a gradual increase in amplitude up to the level of 0.02 at Q = 0.75. Beyond the 0.75 point, the amplitude decreases to the end. A stable period one motion is observed.

Figure 13 shows the bifurcation of the secondary subsystem as controlled by the normalized excitation. In Fig. 13a one finds that three kinds of system motions exist over the range of excitation frequency values. These are stable periodic motions, stable period two motions and chaotic motions. In Fig. 13b the positive Lyapunov exponents show a clear indication of chaos. Figure 14 shows the bifurcation of the primary subsystem as controlled by the normalized excitation when a parametric excitation force is applied at a parametric frequency twice the first mode resonance frequency. Upon introducing the parametric excitation force into the system, the amplitude level reduces from 0.02 in Fig. 12 to -0.06 in Fig. 14a. A stable period one motion is observed throughout. It is also observed that the system's motions shown in Fig. 13 turn into stable period one and period two motions in Fig. 14 when the parametric excitation force is introduced into the autoparametric vibration absorber system.

0.50.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 ii

Fig. 12. Bifurcation diagram of the primary subsystem's displacement versus excitation frequency at Fa =0

e -o.i -

-0.3 -

- a -o.i -

-0.5

0.5 0.7 0.9 1.1 1.3 1.5 0.5 0.7 0.9 1.1 1.3 1.5 Q fi

(a)

(b)

Fig. 13. (a) Bifurcation diagram of the secondary subsystem's angular velocity versus excitation frequency when Fa =0 and (b) corresponding Lyapunov exponent diagram

Discrete excitation frequency points in Figs. 12-14 are selected to plot Poincare maps and time plots for a more detailed understanding of the dynamics of the new autoparametric vibration absorber system.

A Poincare map is a qualitative topological approach widely applied to the predictions of chaos and the study of stability in the phase case through exploring the geometric features of the sequence of points on a Poincare section. For a periodically forced, second-order nonlinear oscillator, a Poincare map can be obtained by stroboscopically observing the position and velocity at a particular phase of the forcing functions. For quasi-periodic motion, the return points in the Poincare map form a closed curve. For a system undergoing chaotic motion, its associated Poincare map shows specific shapes or many irregular points and features indicating the state and extent of bifurcation [16].

1 1 1 0. 1 1 9 1. \ i 2 1. \ i i i i

1 1 1 / i i 1 i

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5 ÍÍ

(a)

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5 ÍÍ (b)

Fig. 14. Bifurcation diagrams of the (a) primary subsystem's displacement versus excitation frequency and (b) secondary subsystem's angular velocity versus excitation frequency when Fa = 100 N

A more detailed analysis of Figs. 12-14 is extended to Poincare maps and time plots, that is, Figs. 15-21. The Poincare maps are plotted from the transient time, that is, t = = 0 to 1000 seconds, while their time plots are plotted for assumed steady-state conditions, taken to be during the interval t = 800-1000 seconds. A breakdown of the observations of the autoparametric absorber system's dynamics recorded in Figs. 12-14 with and without parametric excitation force is as follows:

At normalized excitation frequency of 0.75 (Fig. 15):

• The bifurcation diagram shows periodic and stable motion as depicted in Fig. 12.

• The Poincare map converges into a circle of points with a darker single point at the center, which implies periodic motion, it indicates a period one motion with a stable attractor.

• The time plot shows evidence of a periodic response. At normalized excitation frequency of 0.9 (Fig. 16):

• The bifurcation diagram shows chaotic motion with positive Lyapunov exponents as depicted in Figs. 13a and 13b.

• The Poincare map shows an irregular shape and is that of chaotic motion.

• The time plot is nonperiodic, the oscillations do not repeat, indicating chaotic motion. At normalized excitation frequency of 1.2 (Fig. 17):

• The bifurcation diagram shows period two motions with negative Lyapunov exponents as depicted in Figs. 13a and 13b.

• The Poincare map shows two darker distinct points circled with dotted points.

• The time shows stable motion.

a o

-0.1 -0.06 -0.02 0.02 0.06 0.1 a

(a)

950 1000

Fig. 15. Dynamical analysis of response in Fig. 12 at normalized frequency of 0.75. (a) Poincare map and (b) time plot

950 1000

-0.1 -0.06 -0.02 0.02 0.06 0.1 e

(a)

Fig. 16. Dynamical analysis of response in Fig. 13 at normalized frequency of 0.90. (a) Poincare map and (b) time plot

6 0

-0.1 -0.06 -0.02 0.02 0.06 0.1 6

(a)

-1.5

950 1000

Fig. 17. Dynamical analysis of response in Fig. 13 at normalized frequency of 1.2. (a) Poincare map and (b) time plot

à 0

-0.1 -0.06 -0.02 0.02 0.06 0.1

950 1000

Fig. 18. Dynamical analysis of response in Fig. 14a at normalized frequency of 0.85. (a) Poincare map and (b) time plot

0 0

-0.1 -0.06 -0.02 0.02 0.06 0.1 e

(a)

950 1000

Fig. 19. Dynamical analysis of response in Fig. 14b at normalized frequency of 0.90. (a) Poincaré map and (b) time plot

At normalized excitation frequency of 0.85 with excitation force of Fa = 100 N (Fig. 18):

• The bifurcation diagram in Fig. 14a shows a jump but period one motion.

• The Poincaré map shows a dark distinct point surrounded by small-scattered dotted points indicating stable periodic motion.

• The time shows repeated oscillations, indicating periodic motion.

At normalized excitation frequency of 0.90 with excitation force of Fa = 100 N (Fig. 19):

• The bifurcation diagram in Fig. 14b shows a period one and stable motion.

• The Poincaré map shows a dark distinct point circled by smaller dotted points indicating stable periodic motion.

• The time shows a close periodic motion.

0 o

-0.1 -0.06 -0.02 0.02 0.06 0.1 800 850 900 950 1000

e

(a)

Fig. 20. Dynamical analysis of response in Fig. 14b at normalized frequency of 1.2. (a) Poincare map and (b) time plot

è o

-0.1 -0.06 -0.02 0.02 0.06 0.1 e

(a)

950 1000

Fig. 21. Dynamical analysis of response in Fig. 14b at normalized frequency of 1.4. (a) Poincaré map and (b) time plot

At normalized excitation frequency of 1.2 with excitation force of Fa = 100 N (Fig. 20):

• The bifurcation diagram in Fig. 14b shows a period two motion.

• The Poincaré map shows two dark distinct points circled by smaller dotted points indicating stable periodic motion.

• The time displays two types of repeated oscillations, which is still an indication of stable periodic motion.

At normalized excitation frequency of 1.4 with excitation force of Fa = 100 N (Fig. 21):

• The bifurcation diagram in Fig. 14b shows a stable period one motion.

• The Poincaré map shows a dark distinct point circled by smaller dotted points indicating stable periodic motion.

• The time shows repeated oscillations indicating stable periodic motion.

The numerical analysis using Dynamics 2 here indicates that chaos is evident in the secondary subsystem's motion of the autoparametric vibration absorber system and that with the introduction of parametric excitation forces the system's motions become periodic and stable.

6. Conclusions

In this paper, the dynamics of a new type of autoparametric vibration absorber system making practical and effective use of the effect of parametric excitation forces is investigated. The paper shows that it is possible to improve the response performance of an autoparametric vibration absorber system by introducing a form of intelligent control. First, the source of the parametric excitation forces is modeled. Then the responses of the primary and secondary subsystems that are coupled to a piezoelectric actuator exciter are obtained analytically using the method of multiple scales. The results are then validated numerically using continuation and bifurcation methods. It is shown that the proposed autoparametric vibration absorber system exhibits better performance in contrast to the existing ones. The vibration amplitudes of both the primary and secondary subsystems get reduced with the introduction of parametric excitation forces. Numerical studies have shown that chaotic regions that are evident in the system's motion when the amplitudes of external excitation are increased are removed upon introduction of parametric excitation forces. According to the results, the parametric excitation forces broaden the response characteristics of the absorber so that it efficiently removes vibrational energy over more of the region of autoparametric interaction. In the next step, the results will be verified experimentally.

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