Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber Yushu Bian n, Zhihui Gao School of Mechanical Engineering and Automation, Beihang University, Beijing, China

a r t i c l e i n f o

abstract

Article history: Received 21 December 2016 Received in revised form 22 March 2017 Accepted 26 March 2017 Handling Editor: L.G. Tham

A semi-active vibration absorption method is put forward to attenuate nonlinear vibration of a flexible arm based on the internal resonance. To maintain the 2:1 internal resonance condition and the desirable damping characteristic, a virtual vibration absorber is suggested. It is mathematically equivalent to a vibration absorber but its frequency and damping coefficients can be readily adjusted by simple control algorithms, thereby replacing those hard-to-implement mechanical designs. Through theoretical analyses and numerical simulations, it is proven that the internal resonance can be successfully established for the flexible arm, and the vibrational energy of flexible arm can be transferred to and dissipated by the virtual vibration absorber. Finally, experimental results are presented to validate the theoretical predictions. Since the proposed method absorbs rather than suppresses vibrational energy of the primary system, it is more convenient to reduce strong vibration than conventional active vibration suppression methods based on smart material actuators with limited energy output. Furthermore, since it aims to establish an internal vibrational energy transfer channel from the primary system to the vibration absorber rather than directly respond to external excitations, it is especially applicable for attenuating nonlinear vibration excited by unpredictable excitations. & 2017 Elsevier Ltd All rights reserved.

Keywords: Flexible arm Vibration control Vibration absorber Internal resonance

1. Introduction Vibration absorption is a type of effective methods for controlling vibration of the flexible structures and mechanical systems. Generally, one or more vibrating subsystems named the Dynamic vibration absorbers (DVAs) are attached to the primary system to reduce forced vibration excited by external harmonic excitations of a specific frequency. Due to narrow effective frequency bandwidth, traditional passive DVAs lack enough flexibility and adaptability. Therefore, various measures for tuning the stiffness of DVAs have been developed to extend effective frequency bandwidth of DVAs. These measures include varying effective coil number of a mechanical spring [1], controlling the space between two spring leaves [2], adjusting the length of threaded flexible rods [3], changing effective length of a flexible cantilever beam by moving the intermediate support [4], changing the pressure of air springs [5], and adjusting the curvature of two parallel curved beams [6], etc. Recently, smart materials such as shape memory alloy [7,8], magnetorheological elastomers [9,10] and piezoelectric ceramic [11] have also been used as variable stiffness elements. Moreover, in order to replace those hardto-implement mechanical designs, some virtual passive approaches are employed to emulate the dynamics of a vibration n

Corresponding author. E-mail address: [email protected] (Y. Bian).

http://dx.doi.org/10.1016/j.jsv.2017.03.028 0022-460X/& 2017 Elsevier Ltd All rights reserved.

Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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absorber [12,13]. The stiffness, inertial and damping coefficients of the virtual vibration absorber can be readily adjusted by varying the parameters of virtual elements. In particular, the virtual spring is tuned according to the phase difference between the acceleration of primary system and the displacement of virtual mass [13]. Although there are many ways to vary effective frequency, most of aforementioned vibration absorbers are used to deal with linear vibration problem of the primary system. Compared with linear vibration, however, nonlinear vibration of the primary system is much more difficult to control. Since nonlinear terms cannot be ignored any more, many control methods based on the linear model may result in fundamental mistakes. Furthermore, these vibration absorption methods have to depend on the information of external excitations to neutralize vibration of the primary system. As a result, if the external excitations are unclear or unpredictable in such case as the outer space, these methods will deteriorate. In recent years, active control methods have made remarkable progress. Particularly, various smart material actuators have been put forward to control vibration, such as piezoelectric ceramic and shape memory alloy. However, since they rely on external energy to suppress vibration, it is difficult for them to conquer strong vibration due to limited energy output. Besides, the stability problem is crucial to active control methods. If designed unreasonably, active control forces may excite rather than suppress vibration. Therefore, when dealing with strong vibration, the semi-active vibration absorption method exhibits its own advantages. On the one hand, it absorbs rather than suppresses the vibrational energy. On the other hand, it possesses good stability as the passive methods. Based on above analyses, it is necessary to investigate an effective semi-active vibration absorption method for reducing strong nonlinear vibration excited by indeterminate external excitations. Internal resonance is a typical nonlinear phenomenon of the multi-degree-of-freedom nonlinear dynamics system, by which two vibrational modes can be coupled and the vibrational energy can be exchanged between the modes. Golnaraghi [14,15] firstly used internal resonance to reduce vibration of a flexible cantilever. A 1:2 internal resonance state was established between two vibrational modes of the beam and the slider. And vibration of the beam was dissipated by the damping of the slider. Tuer [16] and Duquette [17] conducted numerical simulations and experiments to control vibration of a similar beam using an internal resonance controller. Then Tuer [18] proposed a generalized control strategy for a cantilever beam based on the internal resonance. Afterwards, Oueini [19] built an analog electronic controller to maintain internal resonance and conducted an experiment research. Khajepour [20] examined the possibility of reducing vibration of a flexible beam using the center manifold theory. However, the flexible beam model adopted in above studies is a rigid beam connected by a torsional spring. Obviously, this assumption is not suitable for real flexible arm. Recently, the distributed flexible beam model has been researched. Pai [21] used higher order internal resonance to design a vibration absorber to reduce vibration of a cantilevered plate. Yaman [22] absorbed vibration of a cantilever beam using a pendulum attached to the tip mass. Hui [23] attenuated translational vibration of the source mass by transferring the internally resonant energy from the symmetrical to anti-symmetrical mode. Nonetheless, new research work on the flexible arm is insufficient. Firstly, since the controlled primary system in the aforementioned studies is a flexible structure without rigid motion, their research results cannot be used to deal with nonlinear vibration problem of a flexible arm with large-scale joint motion. Whether the internal resonance can be established for the flexible arm should be researched. Secondly, the controlled primary system in these studies is viewed as a linear vibration model for the simplicity. However, the flexible arm itself is a complicated nonlinear dynamics system. Therefore, unreasonable linearization may lead to fundamental mistakes. Up to now, there are few theoretical and experimental research works about nonlinear vibration control for the flexible arm based on the internal resonance. Its theoretical correctness and practical feasibility need be examined. In this paper, a semi-active vibration absorption method is put forward to attenuate nonlinear vibration of a flexible arm based on the internal resonance. To maintain the 2:1 internal resonance condition and the desirable damping characteristic, a virtual vibration absorber is suggested, which frequency and damping coefficients can be readily adjusted by simple control algorithms. Perturbation technique is utilized to study transient and steady-state response of the nonlinear dynamics equations. Through theoretical analyses and numerical simulations, it is proven that the internal resonance can be successfully established for the flexible arm. Finally, experimental results are presented to validate the theoretical predictions.

2. System description As shown in Fig. 1, a flexible arm is studied in this paper. It can rotate around the joint o0(o1), which rigid motion is denoted by θ . o0e10e20 is the fixed coordinate system and o1e11e21 is the moving coordinate system attached to the arm. The arm is simplified as a uniform Euler-Bernoulli flexible beam with the length l , rectangular cross-section of the height h and the width b. Only the lateral deformation w (x, t ) about e21 axis in the plane o1e11e21 is considered, where t is time. In this study, a virtual semi-active approach is adopted in the design of the vibration absorber. It is mathematically equivalent to a vibration absorber, and thus can emulate the dynamics of a vibration absorber. This virtual vibration absorber is implemented by a servomotor, which the stiffness and damping coefficients can be readily adjusted by simple control algorithms. In particular, its virtual spring is tuned via variable position feedback gain and its virtual damping is tuned via variable velocity feedback gain. Therefore, the frequency and damping of the virtual vibration absorber can be altered, and thus play an important role in establishing the internal resonance. As shown in Fig. 1, the servomotor is Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Fig. 1. Model of the flexible arm and the virtual vibration absorber.

attached to the flexible arm at the point A (i.e., x = xA ). A branched link with the length l2 and mass m2 is installed on the output shaft of the servomotor at an angle 90∘ with respect to the flexible arm. The coordinate system Ae12e22 is established at the point A. The branched link can rotate around the point A, which rotation is denoted by β . The servomotor and the branched link constitute a virtual vibration absorber. According to the above analysis, the effect of the virtual vibration absorber on the flexible arm is

(

)

d

(

τ3 = kV β ̇ − β ̇ + k p β d − β

)

(1)

d where τ3 is the torque of the servomotor exerted on the flexible arm, β ̇ and β d denote the desired angular velocity and displacement of the branched link respectively, kV and k p denote the velocity and position feedback gains of the servomotor respectively. d In this control model, the desired angular velocity β ̇ and displacement β d of the branched link are expected to be zero. In this case, Eq. (1) becomes

τ3 = − kVβ ̇ − k pβ

(2)

Obviously, the virtual vibration absorber introduces another degree of freedom, i.e. β . The lateral deformation w (x, t ) of the flexible arm can be expressed as n

w (x, t ) =

∑ φi(x)qi(t )

(3)

i=1

where qi(t ) is the i-th modal coordinate of the flexible arm, φi(x ) is the i-th mode shape satisfying certain geometry and force boundary conditions, n is the number of the flexural degrees of freedom of the flexible arm. In this study, only the fundamental mode of the flexible arm is taken into account due to its dominant contribution to the vibration response in most cases. By considering Eqs. (2) and (3), the flexible dynamics equations about the fundamental mode coordinates q1 and the flexural degree of freedom β of the vibration absorber are derived based on Kane's method and can be written as 2 ¨ + β 2 + βθ̇ ̇ q¨1 + 2ζ^1ω1q1̇ + ω12q1 = Q 1θ ̇ q1 + Q 2θ¨q1 − Q 3θ¨ − Q 4θ¨ + Q 5 ββ

(

(

)

(

)

(4)

)

(5)

2 2 ¨ β¨ + 2ζ^2ω2β ̇ + ω22β = − L1 θ¨q1 + 2θq̇ 1̇ + L1 q¨1β − θ ̇ q1β + L 2 −θ ̇ + θβ

(

)

where

ω12 = Q4 =

H1 G1 + m1φA21 + m2φA21 ( m1 + m2)φA1xA G1 + ( m1 + m2)φA21

, ω22 =

, Q5 =

kp m2l22

, 2ζ^1ω1 =

m2l2φA1 G1 + ( m1 + m2)φA21

f1 G1 + m1φA21 + m2φA21

, L1 =

φA1 l2

, L2 =

, 2ζ^2ω2 =

xA , l2

l

kv m2l22

, Q1 =

( m1 + m2)φA21 G1 + ( m1 + m2)φA21

, Q2 =

l

l

G1 = ∫ ρφ12dx , f1 = ∫ ρφ1xdx , H1 = ∫ EI 0 0 0

G1 G1 + ( m1 + m2)φA21

, Q3 =

f1 G1 + ( m1 + m2)φA21

,

2

( ) dx, m is the mass of the ∂φ1 ∂x

1

servomotor, m2 is the mass of the branched link, ρ is the mass per length of the flexible arm, φA1 is the value of the point A in the fundamental mode shape φ1, EI is the flexural rigidity of the flexible arm. It can be seen in Eq. (5) that the frequency ω2 of the absorber is associated with the position feedback gain k p of the servomotor and can be adjusted to establish the internal resonance with the fundamental mode of the flexible arm. On the other hand, the damping 2ζ^ ω of the absorber is associated with the velocity feedback gain k of the servomotor and can be 2 2

v

adjusted to dissipate vibrational energy of the fundamental mode of the flexible arm. Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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3. Perturbation analysis Using the following transformations

q=

q1 l

, τ = ω1t , ωs =

ω2 ω1

(6)

Eqs. (4) and (5) are nondimensionalized as

(

)

2 ¨ + β2̇ + βθ̇ ̇ q¨ + 2ζ^1q ̇ + q = N1θ ̇ q + N2θ¨q − N3θ¨ − N4θ¨ + N5 ββ

(7)

(

2 2 ¨ β¨ + 2ζ^2ω2β ̇ + ωs2β = − P1θ¨q − 2P1θq̇ ̇ + P1q¨β − P1θ ̇ qβ + P2 −θ ̇ + θβ

)

(8)

where Ni = Q i/l1 (i ¼1…5), P1 = L1l1, P2 = L2, (⋅) and (⋅⋅) represent the first and second derivatives with respect to τ . To solve Eqs. (7) and (8) using the method of multiple scales, a scaling factor 0 < ε < < 1 is introduced through the following transformations

q → εq, β → εβ, θ → εθ , ζ^1 → εζ1, ζ^2 → εζ2

(9)

Substituting Eq. (9) into Eqs. (7) and (8), one obtains

¨ + εN5β2̇ + εN5βθ̇ ̇ + o(ε2) q¨ + 2ζ1q ̇ + q = − ( N3 + N4 )θ¨ + εN2θ¨q + εN5ββ

(10)

2 ¨ + o(ε2) β¨ + 2ζ2ωsβ ̇ + ωs2β = − εP1θ¨q − 2εP1θq̇ ̇ + εP1q¨β − εP2θ ̇ + εP2θβ

(11)

To use the method of multiple scales, the following time scales are defined as

T0 = τ , T1 = ετ

(12)

According to Eq. (12), the time derivatives about τ can be rewritten in terms of T0 and T1, i.e.

dT ∂ dT ∂ d = 0⋅ + 1⋅ + O ε2 dτ dτ ∂T0 dτ ∂T1

( )

( )

= D0 + εD1 + O ε2

(13)

∂2 ∂2 d2 = + 2ε + O ε2 2 2 ∂ T ∂T0 dτ 0∂T1 = D02 + 2εD0D1

( ) + O( ε ) 2

(14)

where D0 = ∂/∂T0 , D1 = ∂/∂T1. The first order approximate solutions to Eqs. (10) and (11) take the form

( ) ( )

q( τ , ε) = u0( T0, T1) + εu1( T0, T1) + O ε2

(15)

β( τ , ε) = β0( T0, T1) + εβ1( T0, T1) + O ε2

(16)

Substituting Eqs. (13)–(16) into Eqs. (10) and (11), then equating coefficients of the same order of ε in both sides, the following differential equations are obtained. Order (ε0 ):

D02u0 + u0 = − ( N3 + N4 )D02θ

(17)

D02β0 + ωs2β0 = 0

(18)

Order (ε1):

D02u1 + u1 = − 2D0D1u0 − 2ζ1D0u0 + N2u0D02θ + N5β0D02β0 + N5 D0β0

(

2

)

(

)

+ N5 D0β0 ( D0θ )

(19) 2

(

D02β1 + ωs2β1 = − 2D0D1β0 − 2ζ2ωsD0β0 − P1u0D02θ − 2P1( D0θ )( D0u0) + P1β0D02u0 − P2( D0θ ) + P2β0 D02θ

)

(20)

Eqs. (17)–(20) can vary with the joint motion, which is different from the flexible structure. The solutions to Eqs. (17) and (18) can be expressed in the following complex form

u0 = A1eiT0 + A¯1e−iT0 − ( N3 + N4 )D02θ

(21)

β0 = A2 eiωsT0 + A¯ 2 e−iωsT0

(22)

Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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where the coefficients A1 and A2 are the functions of the slow time T1, A¯1 and A¯ 2 denote the complex conjugate of A1 and A2 respectively. Substituting Eqs. (21) and (22) into Eqs. (19) and (20), one obtains

D02u1 + u1 = − 2iA1′eiT0 − 2iζ1A1eiT0 + N2A1eiT0 D02θ − 5N5ωs2A22 e2iωsT0 + 3ωs2N5A2 A¯ 2 + iN5ωsA2 eiωsT0 D0θ 1 − N2( N3 + N4 ) D02θ 2

(

2

)

+ ζ1( N3 + N4 )D03θ + cc

(23)

D02β1 + ωs2β1 = − 2iωsA2′ eiωsT0 − 2iζ2A2 ωs2eiωsT0 − P1A1eiT0 D02θ − 2P1iA1eiT0 D0θ − P1A1A2 ei( 1 + ωs)T0

( )

−P1A1A¯ 2 ei( 1 − ωs)T0 − P1A2 ( N3 + N4 )D04 θeiωsT0 + P2A2 D02θeiωsT0 + P1( N3 + N4 ) D03θ ( D0θ ) 1 + P1( N3 + N4 ) D02θ 2

(

2

)

−

1 P2D02θ + cc 2

(24)

where, cc denotes the complex conjugate of the preceding term, and ()′ ≡ ∂() /∂T1. To obtain solutions of Eqs. (23) and (24), the solvability conditions should be determined.

4. Internal resonance analysis Whether the internal resonance can be established between the flexible arm and the vibration absorber should be researched. Since only the second order nonlinear terms exist in Eqs. (4) and (5), the 2:1 internal resonance condition is analyzed. Therefore, a detuning parameter σ is introduced, i.e.,

ωs = 0.5 + εσ

(25)

2ωsT0 and ( 1 − ωs)T0 are expressed as

2ωsT0 = T0 + 2σT1

(26)

( 1 − ωs)T0 = ωsT0 − 2σT1

(27)

Substituting Eqs. (26) and (27) into Eqs. (23) and (24), the solvability conditions are

−2iA1′eiT0 − 2iζ1A1eiT0 + N2A1eiT0 D02θ − 5N5ωs2A22 eiT0 + 2iσT1 = 0 −2iωsA2′ e

iωsT0

−

2iζ2A2 ωs2eiωsT0

−

P1A2 N3D04 θ⋅eiωsT0

−

P1A2 N4D04 θ⋅eiωsT0

(28) +

P2A2 D02θeiωsT0

− P1A1A¯ 2 ei( ωsT0 − 2σT1) = 0

(29)

For convenience, A1(T1) and A2 (T1) are expressed in polar form as follows:

A1 =

1 iθ 1 a1e 2

(30)

A2 =

1 iθ 2 a2e 2

(31)

where a1, a2, θ1, θ2 are the real functions of T1; a1 and a2 are defined as the modal amplitudes. Substituting Eqs. (30) and (31) into Eqs. (28) and (29), then separating the result into real and imaginary parts, one has

a1′ = − ζ1a1 −

5 N5ωs2a22 sin γ 4 1 a1a2 P1 sin γ 4 ωs

(33)

1 5 N1a1D02θ + N5ωs2a22 cos γ 2 4

(34)

a2′ = − ζ2a2ωs +

a1θ1′ = −

a1θ2′ =

(32)

a 1 1 a 1 aa P1( N3 + N4 ) 2 D04 θ − P2 2 D02θ + P1 1 2 cos γ 2 ωs 2 ωs 4 ωs

(35)

where

γ = 2σT1 + 2θ2 − θ1

(36)

Eliminating θ′1 and θ′2 from Eqs. (34)–(36), one obtains Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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a1γ ′ = 2σa1 + P1( N3 + N4 )

⎛ 1 a2 ⎞ a1 4 a 1 5 D0 θ − P2 1 D02θ + N1a1D02θ + ⎜⎜ P1 1 − N5ωs2a22⎟⎟cos γ ωs ωs 2 4 ⎝ 2 ωs ⎠

(37)

To investigate the energy transfer between the flexible arm and the vibration absorber, the undamped case (i.e. ζ1 = ζ2 = 0) is studied. Therefore, Eqs. (32) and (33) are written as

5 N5ωs2a22 sin γ 4 1 aa a2′ = P1 1 2 sin γ 4 ωs a1′ = −

(38) (39)

Letting

v=

5N5ωs3 P1

(40)

Multiplying Eq. (38) by a1 and Eq. (39) by va2, then adding and integrating, leads to

a12 + va22 = E

(41)

where E is a constant of integration, which is proportional to the initial energy in the system. Substituting N5 and P1 into Eq. (40), one obtains

v=

5m2l22ωs3a2

⎡ l1 2 2⎤ ⎣⎢ ∫0 ρφ1 dx + ( m1 + m2)φA1⎦⎥l1a1

(42)

Since υ > 0 in Eq. (42), it means that a1 and a2 in Eq. (41) are not only bounded but also anti-phase with each other. Because a12 and a22 denote vibrational energy of the fundamental mode of the flexible arm and the vibration mode of the absorber respectively, Eq. (41) indicates that, in the absence of the damping, the energy in the system is continuously exchanged between the fundamental mode of the flexible arm and the vibration mode of the absorber. This phenomenon has proven that the 2:1 internal resonance can be successfully established.

5. Vibration reduction based on 2:1 internal resonance Although the internal resonance provides a channel to exchange the vibrational energy between the vibration modes, the vibration response cannot be dissipated in the absence of the damping. To reduce the vibration of the flexible arm, the damping effect of the vibration absorber should be taken into account. In the presence of the damping (i.e., ζ1 > 0 and ζ2 > 0), the steady-state response is

a1′ = a2′ = γ ′ = 0

(43)

As a result, from Eqs. (32), (33) and (37), one can obtain

5 N5ωs2a22 sin γ = 0 4 1 aa −ζ2a2ωs + P1 1 2 sin γ = 0 4 ωs

−ζ1a1 −

⎛ 1 a2 ⎞ a a 1 5 2σa1 + P1( N3 + N4 ) 1 D04 θ − P2 1 D02θ + N1a1D02θ + ⎜⎜ P1 1 − N5ωs2a22⎟⎟cos γ = 0 ωs ωs 2 2 ω 4 ⎝ ⎠ s

(44) (45) (46)

It is easy to find that the system possesses an infinite number of equilibrium points defined by

a1 = 0, a2 = 0, γ ∈ R

(47)

Therefore, by evaluating the Jacobian, one can ascertain the stability of the system. The Jacobian matrix for this case is

⎡ μ 0 0⎤ ⎥ ⎢ 1 ⎢ 0 μ2 0 ⎥ ⎥ ⎢ ⎣ 0 0 0⎦

(48)

where μ1 = − ζ1, μ2 = − ωsζ2.

Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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The corresponding eigenvalues are (μ1, μ2 , 0). Since μ1 < 0 and μ2 < 0, the modal amplitudes a1 and a2 are stable, indicated by the negative eigenvalues. It means that, in the presence of the damping, the vibrational energy of the flexible arm can be transferred to the vibration absorber via the internal resonance and dissipated by the damping of the latter.

6. Essence of the proposed control scheme For a linear system, the vibration modes are orthogonal. However, for an internally resonant nonlinear system, the related vibration modes become coupled. In order to establish and maintain the internal resonance between two vibration modes, their frequencies must be commensurable or nearly commensurable, thereby providing a channel for continuously exchanging the vibrational energy between these modes. As a result, to satisfy the 2:1 internal resonance conditions in this study, the frequency of the vibration absorber should be the half of the fundamental frequency of the flexible arm. If the fundamental frequency changes, it should change accordingly. Therefore, the frequency of the vibration absorber must be adjustable. In addition, the damping of the vibration absorber is used to attenuate the nonlinear vibration of the flexible arm. Once the vibrational energy has been transferred out of the mode to be controlled, it can be dissipated by artificial damping imposed on the mode of the absorber that receives the transferred vibrational energy. However, not arbitrary damping can work effectively. Only desirable damping characteristic of the vibration absorber is acceptable. Therefore, the damping of the vibration absorber must be adjustable too. In this paper, a virtual semi-active vibration absorber is implemented based on a servomotor, which frequency and damping coefficients can be adjusted by simple control algorithms. In this control scheme, the virtual spring is tuned via variable position feedback gain and the virtual damping is tuned via variable velocity feedback gain. Once the branched link deviates from its initial position (i.e. the equilibrium position), the controller will rectify the deviation in terms of the velocity and position feedback gains, thereby emulating a vibrating system with desirable frequency and damping. If the working condition alters, the vibration absorber can still work effectively via adjustable frequency and damping. Different from conventional vibration absorption schemes in which the vibration absorber adjusts its frequency in terms of the frequency of the external excitation, the proposed vibration absorber in this study adjusts its frequency in terms of the frequency of the mode to be controlled. Therefore, the essence of the proposed control scheme is artificially establishing a channel for transferring and dissipating the vibrational energy of the mode to be controlled via the internal resonance, rather than counteracting the adverse influence caused by the external excitation. As we know, many methods control vibration according to the external excitations. However, if the types, magnitudes or positions of the external excitations are unpredictable, it is difficult for them to achieve satisfactory control results. Different from these methods, the proposed method focuses on establishing an internal channel for transferring and dissipating the vibrational energy, and thus does not necessarily rely on the information of the external excitations. Therefore, this method exhibits potential advantage in reducing vibration responses induced by unpredictable or unknown external excitations in the complicated environment, e.g. the outer space.

7. Simulations and analyses 7.1. Theoretical simulations To verify the above theoretical analysis, a flexible arm is studied, as shown in Fig. 1. Link1 is a uniform Euler-Bernoulli flexible beam with the length l , rectangular cross-section of height h and width b . The servomotor is attached to the flexible arm at the point A (i.e., x = xA ). m1 is the mass of the servomotor. A branched link with the length l2 and the mass m2 is installed on the output shaft of the servomotor at an angle 90∘ with respect to the flexible arm. These parameters are listed in Table 1. Suppose the rigid motion of the flexible arm is

θ=

⎛ π ⎞ π cos⎜ t⎟ ⎝ 18 ⎠ 6

(49)

If the flexible arm is not equipped with the vibration absorber, given the initial disturbance w (l, 0) = 100 mm , its endeffector deformation is shown in Fig. 2. Due to small damping in the flexible arm, the vibration responses of the end-effector decrease very slowly. Table 1 Parameters of the flexible arm. Components

Parameters

Flexible arm Branched link

l = 0.8 m , h ¼30 mm, b ¼5 mm, mass density 7800 kg/m3, elastic modulus 210 GPa l2 = 0.3 m , diameter: 0.008 m

Servomotor of branched link

m1 = 0.65 kg , xA = 0.6 m

Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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8

0.1 0.08

Elastic Deformation/m

0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

5

10 Time/s

15

20

Fig. 2. Responses of the end-effector (no vibration absorber).

Then a virtual vibration absorber based on a servomotor is attached to the flexible arm, which frequency is tuned to be the half of the fundamental frequency of the flexible arm. In the absence of the damping, i.e., ζ2 = 0, the modal amplitudes a1 and a2 are obtained by numerically integrating Eqs. (32), (33) and (37), as shown in Fig. 3. It can be seen that the peaks and troughs of these two modal amplitudes are out of phase. When a1 descends to the trough, a2 ascends to the peak at the same time, and vice versa. Since this case neglects the damping of both the flexible arm and vibration absorber, the system is conservative, as indicated by Eq. (41). Thus, if a1 descends and a2 ascends, it denotes that the vibrational energy is being transferred from a1 to a2, and vice versa. This phenomenon has proven that the internal resonance has been established and the vibrational energy is continuously exchanging between the two vibrational modes. To control the vibration of the flexible arm, the damping of the vibration absorber is taken into account. By numerically integrating Eqs. (32), (33) and (37) under different damping coefficients of the vibration absorber, the controlled modal amplitude a1 of the flexible arm is shown in Fig. 4. As can be seen, the response of the fundamental modal amplitude of the flexible arm can be affected by the damping of the vibration absorber. That is to say, the damping of one mode can be used to reduce the vibration of the other mode via the internal resonance. Furthermore, the effects of different damping coefficients of the vibration absorber on the modal amplitude of the flexible arm are studied, as shown in Fig. 4. If the damping coefficient is small, e.g. 0.1, although the modal amplitude of the flexible arm can be controlled, it decays very slowly. In this case, a large amount of vibrational energy is transferred back to the flexible arm and little is dissipated by insufficient damping of the vibration absorber. If increasing the damping coefficient, e.g. 0.5, desirable modal amplitude response of the flexible arm can be obtained. Due to sufficient damping of the vibration absorber, the vibration of the flexible arm can be eliminated more effectively and rapidly. However, if the damping coefficient is increased too much, e.g. 1.0, vibration response of the flexible arm cannot be effectively reduced any more. Excessive damping blocks the transfer of the vibrational energy from the flexible arm to the vibration absorber, and thus a large amount of residual vibration energy is left. Therefore, there is a reasonable range of the damping coefficient, within

0.5

Modal amplitude of the flexible arm

0.45

Modalmodal amplitude of thelink absorber Motion of branch

0.4

Amplitude

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

20

40

60 Slow Time

80

100

120

Fig. 3. Relationship of a1 and a2 .

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Fig. 4. Effects of different damping coefficient of the absorber.

which most of the vibrational energy transferred from the flexible arm can be effectively dissipated by the vibration absorber before it has the opportunity to transfer back. When ζ2 = 0.5, the modal amplitudes of the flexible arm and the vibration absorber are shown in Fig. 5. It can be seen that the transfer of energy between these two modes is occurring, verified by the coincidence of the peaks of one modal amplitude with the trough of the other. Eventually, all modal amplitudes decay to zero. On the other hand, the end-effector deformation of the flexible arm in this case is shown in Fig. 6. Compared with the uncontrolled case, the vibration response of the flexible arm is effectively reduced, as shown in Fig. 7. After 8 s, the vibration has been eliminated by 90%. It is verified that this vibration control method is effective in reducing nonlinear vibration of the flexible arm. 7.2. Virtual prototyping simulation Although above simulation results are promising, all of them come from our own theoretical model. To further examine the correctness of theoretical analysis and provide good guidance to experimental investigation, a series of virtual prototyping simulations are conducted in this section to validate proposed method using ADAMS and ANSYS software. 7.2.1. Simulation model The flexible arm is established using ANSYS software, as shown in Fig. 8. It is modeled as a uniform steel beam with the length l = 0.8 m , rectangle cross-section of height h¼30 mm and width b¼5 mm, mass density 7800 kg/m3 and elastic modulus 210 GPa. The vibrational modes of the flexible arm are extracted using the finite element analysis and exported into a modal neutral file. After importing the modal neutral file, the flexible arm model has been established in ADAMS software, as shown in Fig. 9. In addition, the main components of the vibration absorber, i.e. the servomotor (mass: 0.65 kg) and branched link (length: 0.3 m), are also built in ADAMS.

2.5 Deformation modal of flexible manipulator Modal amplitude the flexible arm Modalmodal amplitude of the absorber Motion of branch link

Amplitude

2

1.5

1

0.5

0 0

20

40

60

80

100

Slow Time Fig. 5. Modal amplitudes of the arm and the absorber.

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0.1 0.08

Elastic Deformation/m

0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

5

10 Time/s

15

20

Fig. 6. Damped response of flexible arm. 0.1 uncontrolled system controlled system

0.08

Elastic Deformation/m

0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

5

10 Time/s

15

20

Fig. 7. Comparison of vibration responses.

Fig. 8. Flexible arm in ANSYS.

The control model of the virtual vibration absorber is implemented based on the ADAMS/Control module and MATLAB/ Simulink module. As shown in Fig. 10, the ADAMS input variable comes from the MATLAB/Simulink output and the ADAMS output variable is import to the MATLAB/Simulink, so as to establish a combined simulation closed loop. In particular, the ADAMS input variable is the torque of the servomotor (VARIABLE_torque), and the ADAMS output variables are the angular position (VARIABLE_position) and angular velocity (VARIABLE_velocity) of the servomotor, as shown in Fig. 11. Then the control model of vibration absorber is established using MATLAB/Simulink, as shown in Fig. 12, where “Gain” indicates the position feedback gain module, “Gain1” indicates the velocity feedback gain module. Through adjusting “Gain” and “Gain1”, the vibration absorber with desirable frequency and damping characteristics can be obtained. Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Flexible arm

11

Vibration absorber

Branch link Servomotor Fig. 9. Flexible arm and vibration absorber in ADAMS.

Fig. 10. Combined simulation model of vibration controller.

Fig. 11. Flexible arm module in ADAMS.

Fig. 12. Control model of vibration controller.

7.2.2. Simulation analyses 7.2.2.1. Vibration response without vibration absorber. Based on above simulation models, several ADAMS and MATLAB combined simulations are conducted. Suppose the rigid motion of the flexible arm is the same as the examples of theoretical simulations. If the flexible arm is not equipped with the vibration absorber, given the initial disturbance w (l, 0) = 100 mm , the endpoint deformation is shown in Fig. 13. Due to small structural damping, it attenuates very slowly. 7.2.2.2. Internal resonance verification. To control the vibration of the flexible arm, the vibration absorber is attached to the flexible arm. In order to verify the internal resonance, the damping of the vibration absorber is not taken into account, i.e. kV = 0. In this case, the endpoint deformation of the flexible arm and the rotation angle of the vibration absorber are obtained at the 2:1 internal resonance condition (i.e. kP = 0.67), as shown in Figs. 14 and 15 respectively. In Fig. 14, the Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Fig. 13. Endpoint deformation of flexible arm without vibration absorber.

Fig. 14. Undamped response of flexible arm.

Fig. 15. Undamped response of vibration absorber.

amplitude of the flexible arm is time-varying and descends abruptly sometimes, resulting in the beat phenomenon. As we know, the vibrational energy is associated with the square of the amplitude. Therefore, the amplitude descending abruptly means the vibrational energy of the flexible arm decreasing sharply at the same time. Also, the beat phenomenon appears in the vibration response of the vibration absorber, as shown in Fig. 15. By comparing Figs. 14 and 15, it is found that when the amplitude of the flexible arm decreases, the amplitude of the vibration absorber increases at the same time, and vice versa. That is to say, when the vibrational energy of the flexible arm decreases to the minimum, the vibrational energy of the vibration absorber increases to the maximum, and vice versa. These Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Fig. 16. Response of flexible arm in frequency domain.

anti-phase beat phenomena reveal the truth that the internal resonance has been successfully established, and the vibrational energy is being exchanged between flexible arm and vibration absorber. In addition, the vibration responses of the flexible arm and vibration absorber are transformed from the time domain to the frequency domain, as shown in Figs. 16 and 17. As can be seen, main frequency ingredients of the flexible arm and vibration absorber are about 4.52 Hz and 2.23 Hz respectively, approximately satisfying the 2:1 internal resonance condition. It is verified that the 2:1 internal resonance has been established. 7.2.2.3. Vibration control based on internal resonance. Although the internal resonance has been established and can provide a channel for exchanging the vibrational energy between vibration modes, the vibration response of the flexible arm cannot be reduced in the absence of the damping. To control the vibration of the flexible arm, the damping effect of the vibration absorber should be taken into account. If keeping kP = 0.67 and letting kV ¼0.1, 0.8 and 1.2 respectively, the vibration responses of the endpoint are obtained and shown in Fig. 18. As can be seen, the damping of the vibration absorber plays an important role in dissipating the vibrational energy of the flexible arm. In whatever case, the vibration response can be reduced. However, different damping leads to different control result. Particularly, large damping does not necessarily lead to good result. In Fig. 18(a), although the vibrational energy of the flexible arm can be transferred to the vibration absorber via the internal resonance, the endpoint deformation of the flexible arm decays very slowly due to insufficient damping of the vibration absorber. If the damping of the vibration absorber is sufficient, the endpoint deformation of the flexible arm can be reduced effectively and rapidly, as shown in Fig. 18(b). However, if the damping of the vibration absorber is excessive, it will block the transfer of the vibrational energy from flexible arm to vibration absorber. As a result, the endpoint deformation of the flexible arm cannot be reduced effectively, as shown in Fig. 18(c). Therefore, there is a desired range of the damping of the vibration absorber. For the convenience of comparison, the endpoint deformation with the desired damping of the vibration absorber and that without the vibration absorber is shown in Fig. 19. It can be seen that the vibration response of the flexible arm has been effectively reduced via the internal resonance. Through above virtual prototyping simulations, it is demonstrated that nonlinear vibration of the flexible arm can be effectively decreased via the internal resonance and the virtual vibration absorber.

Fig. 17. Response of vibration absorber in frequency domain.

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Fig. 18. Endpoint response of flexible arm under different damping.

8. Experimental study Above virtual prototyping simulations have examined the correctness of theoretical analyses. In the following section, a series of experimental studies are carried out to investigate the feasibility of the proposed method.

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Fig. 19. Comparison of vibration response with and without an absorber.

8.1. Experimental device An experimental setup is developed, as shown in Figs. 20 and 21. The flexible arm possesses the same dimensional and material parameters as its counterpart in the virtual prototyping simulations. The joint is used to provide rigid motion for the flexible arm, including a motor, a reducer and a gripper. The vibration absorber is composed of a servomotor and a branched link. The branched link is installed at the output shaft of the servomotor and is set perpendicular to the flexible link as the zero reference position. The servomotor is attached to the flexible arm, which frequency and damping are tuned by variable position and velocity feedback gains respectively. The B&K accelerometer is stuck to the endpoint of the flexible arm and used to collect the acceleration signals. The dynamic signal analyzer is used to process the acceleration signals, and transfer them to the computer. 8.2. Experimental study 8.2.1. Measuring vibration response If a disturbance is exerted on the flexible arm without a vibration absorber during the motion, the acceleration signals of the endpoint are detected by the accelerometer and processed by the dynamic signal analyzer, as shown in Fig. 22. Due to small structural damping, the vibration response of the endpoint decreases very slowly. It takes about 25 s to decrease 80% of the initial amplitude. 8.2.2. Establishing internal resonance In this experiment, kP is used to adjusting 2:1 internal resonance. If kP = 3500, the frequency of the vibration absorber is measured to be 2.237 Hz in frequency domain, as shown in Fig. 23. The fundamental frequency of the flexible arm is measured to be 4.395 Hz in frequency domain, as shown in Fig. 24. In this case, the ratio of the fundamental frequency of the flexible arm to the frequency of the vibration absorber is close to 2:1. In the absence of the damping, i.e., kV = 0, the endpoint response of the flexible arm and the rotation angle of the vibration absorber are measured, as shown in Figs. 25 and 26. The internal resonance phenomena can be easily observed. For example, the endpoint vibration of the flexible arm decreases to

Branched link

Flexible arm

Accelerometer Vibration absorber

Gripper

Reducer

Encoder

Motor

Driver Encoder

PMAC Motion Controller

Driver

Servomotor

Dynamic signal analyzer

Computer

Joint Fig. 20. Scheme of experimental setup.

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PMAC motion controller

Driver

Branched link

Joint

Flexible arm

Dynamic signal analyzer Accelerometer Servomotor

Fig. 21. Photograph of experimental setup.

Fig. 22. Response of endpoint without vibration absorber.

Fig. 23. Frequency response of vibration absorber.

the minimum at about 6 s, and at the same time the amplitude of the vibration absorber increases to the maximum. It is demonstrated that the vibrational energy has been transferring between the flexible arm and vibration absorber. 8.2.3. Reducing vibration based on internal resonance In the presence of the damping, i.e., kV ≠ 0, the vibration absorber can decrease nonlinear vibration of the flexible arm. Letting kV = 500, kV = 1000 and kV = 1500 respectively, the endpoint response of the flexible arm are measured, as shown in Figs. 27–29, respectively. From above figures, it can be seen that the damping of vibration absorber is effective to reduce the vibration of the flexible arm at the internal resonance state. Unfortunately, larger damping does not always mean better control result. If kV is increased from 500 to 1000, the endpoint vibration response of flexible arm is decreased more effectively and rapidly, as shown in Figs. 27 and 28. However, if further increasing kV , i.e., kV = 1500, the endpoint vibration response of flexible arm become bad, as shown in Fig. 29. It means that excessive damping restricts the motion of vibration absorber and thus blocks the vibrational energy transfer from the flexible arm to vibration absorber. Therefore, there is a desired damping range of the vibration absorber. Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Fig. 24. Frequency response of flexible arm.

Fig. 25. Undamped response of flexible arm (kp = 3500, k v = 0 ).

Fig. 26. Undamped response of vibration absorber (kp = 3500, k v = 0 ).

Fig. 27. Endpoint response of flexible arm (kp = 3500, k v = 500 ).

Fig. 28. Endpoint response of flexible arm (kp = 3500, k v = 1000 ).

Finally, in order to compare the control effect, the endpoint vibration responses of flexible arm with and without the vibration absorber are shown in Fig. 30. If there is not the vibration absorber, it takes about 13 s to decrease 60% of the initial amplitude. If the vibration absorber is equipped, it takes about 3 s and 6 s to decrease 60% and 90% of the initial amplitude. Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i

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Fig. 29. Endpoint response of flexible arm (kp = 3500, k v = 1500 ).

Fig. 30. Comparison of endpoint response of flexible arm.

Based above experiments, it is verified that the virtual vibration absorber based on the internal resonance is effective and feasible to decrease nonlinear vibration of the flexible arm.

9. Conclusion In this paper, a semi-active vibration absorption method is put forward to attenuate nonlinear vibration of the flexible arm based on the internal resonance. To maintain the 2:1 internal resonance condition and the desirable damping characteristic, a virtual vibration absorber is suggested. It is mathematically equivalent to a vibration absorber but its frequency and damping coefficients can be readily adjusted by simple control algorithms, thereby replacing those hard-to-implement mechanical designs. The virtual vibration absorber is implemented by a servomotor, in which the virtual spring is tuned via variable position feedback gain and the virtual damping is tuned via variable velocity feedback gain. Perturbation technique is utilized to study transient and steady-state responses of the nonlinear dynamics equations. Through theoretical analyses, numerical simulations and experimental investigations, it is proven that the internal resonance can be successfully established for the flexible arm, and the vibrational energy in the flexible arm can be transferred to and dissipated by the virtual vibration absorber. Since the proposed method belongs to the semi-active vibration absorption method, it absorbs rather than suppresses vibrational energy of the primary system. Therefore, it is more convenient to reduce strong vibration than conventional active vibration suppression methods based on smart material actuators with limited energy output. Furthermore, since this method aims to establish an internal vibrational energy transfer channel from the primary system to the vibration absorber rather than directly respond to external excitations, it is especially applicable for attenuating nonlinear vibration excited by unpredictable excitations. In future studies, adaptive adjustment for the frequency and damping of the vibration absorber will be investigated.

Acknowledgement This project is supported by National Natural Science Foundation of China (No. 51675017).

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Please cite this article as: Y. Bian, Z. Gao, Nonlinear vibration absorption for a flexible arm via a virtual vibration absorber, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.03.028i