# A Vibration Control Method for the Flexible Arm Based on Energy Migration

Authors: Yushu Bian; Zhihui Gao; Ming Fan
Journal: Shock and Vibration
 (2015)

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## Abstract

A vibration control method based on energy migration is proposed to decrease vibration response of the flexible arm undergoing rigid motion. A type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, vibration energy of the flexible arm can be dissipated by not only the damping of the vibration absorber but also its own enhanced damping, thereby effectively decreasing vibration. Through numerical simulations and analyses, it is proven that this method is feasible in controlling nonlinear vibration of the flexible arm undergoing rigid motion.


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## Body

## 1. Introduction

Although flexible robotic arms have various important applications in space exploration, automatic assembly, undersea operation, nuclear environment, and so forth, major possible disadvantages of these arms are serious vibration response and deteriorative tracking accuracy due to large dimension, light weight, low structural damping, and small stiffness. Therefore, a great deal of research has been conducted to combat these problems.Some traditional methods have been used to control vibration, like enhancing the stiffness of links and joints, optimizing shape and dimensions [1, 2], and utilizing composite materials [3, 4]. In recent years, a number of active control methods are put forward, most of them employ such smart material actuators as the piezoelectric ceramic and shape memory alloy, and remarkable progress has been made [5, 6]. In addition to these vibration suppression methods, vibration absorption methods are especially useful to the large amplitude vibration with strong energy. However, since large amplitude can unavoidably excite nonlinear effects in the dynamic system, many methods based on linear vibration model may cause fundamental mistakes. In fact, although the nonlinearity can increase the analysis difficulty, modal interaction caused by the nonlinearity can be used to migrate and dissipate strong vibration energy of the flexible arm.Internal resonance is a typical nonlinear principle. Through modal interaction, vibration energy of one mode can be transferred to another mode which is commensurable or nearly commensurable with the former [7, 8]. Golnaraghi [9] firstly used internal resonance to decrease structural vibration of a flexible cantilever beam. By properly tuning the position gain, internal resonance was established between the beam and the slider. And the beam vibration was decreased by migrating and dissipating its vibration energy through the slider motion. Tuer et al. [10] proposed Disabled Torque Method (DTM) and Dissipated Energy Method (DEM) to control a flexible cantilever beam based on internal resonance. Afterwards, Oueini and Golnaraghi [11] finished an experimental study and built a controller with analog electronic components. Furthermore, Khajepour et al. [12] used center manifold theory to control vibration of the flexible beam. However, the flexible cantilever beam they have researched is a rigid beam connected by a linear torsional spring. This simplified model is not suitable for the long and thin arm which should be represented by a distributed flexibility model.In recent years, the above studies have been extended to control vibration of the distributed flexible beam. Pai et al. [13] used a higher order internal resonance absorber to decrease vibration of a cantilevered aluminum plate. Ashour and Nayfeh [14] designed an active vibration controller for a flexible structure based on internal resonance and saturation phenomenon. Yaman and Sen [15] studied vibration absorption problem concerning a cantilever beam with a tip mass and pendulum which is attached to the tip mass. Hui et al. [16] used the internally resonant energy transfer from the symmetrical to antisymmetrical mode to reduce the source mass vibration. But these studies only deal with the flexible cantilever beam without rigid motion, but do not involve the flexible arm undergoing large scale joint motion which exhibits much more complex dynamic behaviors.To the best of our knowledge, there is little research on controlling vibration of the flexible arm based on energy migration. In this paper, a type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Furthermore, due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, the damping of the vibration absorber can be mapped into the flexible arm, thus increasing the damping effect of the flexible arm. In this way, vibration energy of the flexible arm can not only be dissipated by the damping of the vibration absorber based on the internal resonance but also be attenuated by its own enhanced damping, thereby effectively decreasing vibration.


## 2. Oscillatory Differential Equation

In this study, a model with one flexible arm, one rigid joint, and a vibration absorber is considered, as shown in Figure1. The arm is a uniform Euler-Bernoulli beam with the length l, the rectangle cross section of height h, and width b and a tip mass mB. Only flexural deformation δ(x1,t) about y1 axis is considered, where t is time. The arm rotates around the rigid joint, and the nominal motion is denoted by q. The vibration absorber is a slider mass-spring-dashpot mechanism with mass m, stiffness k2, and damping c2 and attached to the flexible arm at x1=r. The mass displacement is denoted by s and equilibrium position is denoted by sa.Figure 1
Model of the flexible arm with a vibration absorber.Based on the assumed-modes theory, the deformation of the flexible arm can be expressed as(1)δx1,t=∑i=1nFuix1φit,where φi(t) is the ith modal coordinate describing deformation of the flexible arm, ui(x1) is the ith mode shape satisfying certain geometric and force boundary conditions, and nF is the number of flexural degrees of freedom of the flexible arm.In present study, only the fundamental mode of the arm is considered due to its most contribution to the vibration response in common cases. Equation (1) can be written as(2)δx1,t=u1x1φ1t.The angle of the tangent of the flexible arm atx1=r with respect to x1 axis is denoted by β, as shown in Figure 1. Consider(3)β=β~rφ1t,where β~r=(du1x1/dx1)x1=r. The axial displacement caused by the transverse bending of the arm is written as(4)κx1=-12∫0x1∂δϑ,t∂ϑ2dϑ=-12B1φ12,where ϑ is a dummy variable and B1=∫0x1du1/dϑ2dϑ.To use the absorber for reducing vibration response that resulted from the fundamental mode of the arm, the coupling effect between the fundamental mode coordinateφ1 of the arm and the degree of freedom s of the absorber is our concern. Based on Kane’s method, the dynamic equations concerning φ1 and s are derived and can be written as(5)m11φ¨1+m12s¨+c11φ˙1+k11+b1q¨φ1+k12q¨s=-b2φ˙1s˙-b3φ˙12-b4φ¨1φ1-b2φ¨1s-b5q¨-b6q˙2+f1φ1,φ˙1,φ¨1,s,s˙,s¨,q˙,q¨,m22s¨+m21φ¨1+c22s˙+k22s=b7φ˙12-b8q¨+b9q˙2+f2φ1,φ˙1,φ¨1,s,s˙,s¨,q˙,q¨,where m11=B2+mBu1l2+msa2β~r2+mu1r2; B2=ρ∫0lu12(x1)dx1; u1l=u1(x1)x1=l; u1r=u1(x1)x1=r; m12=mu1r; c11 is the damping of the arm; k11=EI∫0l(d4u1x1/dx14)u1(x1)dx1; b1=msa(B1r-rβ~r2); B1r=B1(x1)x1=r; k12=2msaβ~r; b2=2msaβ~r2; b3=msaβ~r(B1r-u1rβ~r); b4=2b3; b5=ρ∫0lxu1(x1)dx1+mBlu1l+msa2β~r+mru1r; b6=msa(rβ~r-u1r); m22=m; c22=c2; k22=k2; m21=m12; b7=msaβ~r2; b8=mr; b9=msa; ρ is mass per length of the flexible arm; EI is flexural rigidity; l is the length of the arm.


## 3. Nonlinear Analysis

Equations (5) are nonlinear differential equations and are solved in this section.


### 3.1. Nondimensionalization and Decoupling of the Equations of Motion

The nondimensional variablesφ∗, s∗, τ, and q∗ are defined by(6)φ∗=φ1l,s∗=sl,τ=ωφt,q∗=q,where ωφ=k11/m11 and the equations of motion (5) are nondimensionalized as follows:(7)d2φ∗dτ2+m~12d2s∗dτ2+η11dφ∗dτ+1+d1d2q∗dτ2φ∗+k~12d2q∗dτ2s∗=-d2dφ∗dτds∗dτ-d3dφ∗dτ2-d4d2φ∗dτ2φ∗-d2d2φ∗dτ2s∗-d5d2q∗dτ2-d6dq∗dτ2+f3φ∗,dφ∗dτ,d2φ∗dτ2,s∗,ds∗dτ,d2s∗dτ2,dq∗dτ,d2q∗dτ2,d2s∗dτ2+m~21d2φ∗dτ2+η22ds∗dτ+ωsφ2s∗=d7dφ∗dτ2-d8d2q∗dτ2+d9dq∗dτ2+f4φ∗,dφ∗dτ,d2φ∗dτ2,s∗,ds∗dτ,d2s∗dτ2,dq∗dτ,d2q∗dτ2,where m~12=m12/m11; η11=c11/(ωφm11); d1=b1/m11; k~12=k12/m11; d2=b2l/m11; d3=b3l/m11; d4=b4l/m11; d5=b5/(m11l); d6=b6/(m11l); m~21=m21/m22; η22=c22/(ωφm22); ωs=k22/m22; ωsφ=ωs/ωφ; d7=b7l/m22; d8=b8/(m22l); d9=b9/(m22l).To make the damping and nonlinearities appear in the same perturbation equations, letη11=εζ11, η22=εζ22, φ∗=εϕ, s∗=εw, and dq∗/dτ=ε(dχ/dτ), where ε is a small nondimensional bookkeeping parameter, 0<ε≪1. Then, (7) can be expressed as(8)d2ϕdτ2+ϕ=-m~12d2wdτ2-d5d2χdτ2+ε-ζ11dϕdτ-d1d2χdτ2ϕ-k~12d2χdτ2w-d2dϕdτdwdτ-d3dϕdτ2-d4d2ϕdτ2ϕ-d2d2ϕdτ2w-d6dχdτ2+oε,(9)d2wdτ2+ωsφ2w=-m~21d2ϕdτ2-d8d2χdτ2+ε-ζ22dwdτ+d7dϕdτ2+d9dχdτ2+oε.


### 3.2. Perturbation Analysis

The time dependenceτ is expanded in terms of multiple time scales, Ti=εiτ, (i=0,1,…), so that the first- and second-time derivatives become (10)ddτ=D0+εD1+ε2D2+⋯,d2dτ2=D02+2εD0D1+ε2D1+2ε2D0D2+⋯,where Di=∂/∂Ti, (i=0,1,…).We seek first-order approximate solutions of (8) and (9) by using the method of multiple scales in the form(11)ϕτ,ε=ϕ0T0,T1+εϕ1T0,T1,wτ,ε=w0T0,T1+εw1T0,T1.Substituting (10), (11) into (8) and (9) and equating coefficients of same powers of ε, we obtain the following:Order (ε0) is as follows:(12)D02ϕ0+ϕ0=-m~12D02w0+g1,D02w0+ωsφ2w0=-m~21D02ϕ0+g2.Order (ε1) is as follows:(13)D02ϕ1+ϕ1=-2D0D1ϕ0-ζ11D0ϕ0-m~12D02w1-2m~12D0D1w0-d1D02χϕ0-k~12D02χw0-d2D0ϕ0D0w0-d3D0ϕ02-d4D02ϕ0ϕ0-d2D02ϕ0w0-d6D0χ2,D02w1+ωsφ2w1=-2D0D1w0-m~21D02φ1-2m~21D0D1ϕ0-ζ22D0w0+d7D0ϕ02+d9D0χ2,where g1=-d5D02χ, g2=-d8D02χ.The solution of (12) can be written in the form (14)ϕ0=A1T1expjω1T0+A2T1expjω2T0+h1+cc,w0=Λ1A1T1expjω1T0+Λ2A2T1expjω2T0+h2+cc,where A1(T1) and A2(T1) are functions of slow time T1; h1=g1/2; h2=g2/(2ωsφ2); cc denotes the complex conjugate terms. According to the theory of ordinary differential equation, the ωn2 (n=1,2) are the roots of(15)1-m~12m~21ωn4-1+ωsφ2ωn2+ωsφ2=0,(16)Λn=1-ωn2ωn2m~12=m~21ωn2ωsφ2-ωn2and ωn are assumed to be distinct.In this study, because the second-order nonlinear coupling terms exist in the dynamic model, the vibration absorber is used to control vibration of the flexible arm at the 1 : 2 internal resonance condition; that is,ω2≈2ω1. It is this internal resonance condition that enables the transfer of vibration energy between the fundamental mode of the arm and the vibration mode of the absorber. In order to solve the nonlinear problem, substituting (14) into (13) yields(17)D02ϕ1+ϕ1=-2jω1A1′+ζ11jω1A1+2m~12jω1Λ1A1′+d1D02χA1+k~12D02χΛ1A1exp⁡jω1T0+-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d4ω12+ω22+d2Λ2ω12+Λ1ω22A-1A2·exp⁡jω2-ω1T0+NST+cc,D02w1+ωsφ2w1=-2jω2Λ2A2′+ζ22jω2Λ2A2+2m~21jω2A2′exp⁡jω2T0-d7ω12A12exp⁡2jω1T0+NST+cc,where ′≡∂()/∂T1 and NST denotes nonsecular terms.In the case of the 1 : 2 internal resonance, a detuning parameterσ is introduced as ω2=2ω1+εσ. To determine the solvability conditions of (17), the particular solutions are sought in the form(18)ϕ1=p11exp⁡jω1T0+p12exp⁡jω2T0,w1=p21exp⁡jω1T0+p22exp⁡jω2T0.Substituting (18) into (17) and then equating the coefficients of exp(jω1T0) and exp(jω2T0) on both sides, one obtains(19)1-ωn2p1n-m~12ωn2p2n=R1n,-m~21ωn2p1n+ωsφ2-ωn2p2n=R2n,where (20)R11=-2jω11+m~12Λ1A1′-jω1ζ11A1+-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12·A1+d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22A-1A2exp⁡jσT1,R12=-2jω21+m~12Λ2A2′-jω2ζ11A2+-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22·A2+2d2Λ1+d3+d4ω12A12exp⁡-jσT1,R21=-2jω1m~21+Λ1A1′-jω1ζ22Λ1A1+2d7ω1ω2A-1A2exp⁡jσT1,R22=-2jω2m~21+Λ2A2′-jω2ζ22Λ2A2-d7ω12A12exp⁡-jσT1.Therefore, the problem of determining the solvability conditions of (17) is reduced to that of determining the solvability condition of (19).Since the determinant of the coefficient matrix of (19) is zero according to (15), the solvability conditions are(21)R1n-m~12ωn2R2nωsϕ2-ωn2=0orR1n=-ΛnR2nm~12m~21on account of (16).Substituting (20) into (21) yields(22)-2jω11+m~12Λ1A1′-jω1ζ11A1+-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12A1+d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22A-1A2exp⁡jσT1=-Λ1·m~12m~21-2jω1m~21+Λ1A1′-jω1ζ22Λ1A1+2d7ω1ω2A-1A2exp⁡jσT1,-2jω21+m~12Λ2A2′-jω2ζ11A2+-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22A2+2d2Λ1+d3+d4ω12A12exp⁡-jσT1=-Λ2·m~12m~21-2jω2m~21+Λ2A2′-jω2ζ22Λ2A2-d7ω12A12exp⁡-jσT1.It is convenient to express the resulting modulation equations in polar form by introducing the following transformation:(23)A1=12a1exp⁡jθ1,A2=12a2exp⁡jθ2,where a1, a2, θ1, and θ2 are real functions of the slow time T1; a1 and a2 are defined as the modal amplitudes.Inserting (23) into (22), then setting the coefficients of the real and imaginary parts to zero yields the modulation equations(24)a1′=-h22h1a1+h34h1a1a2sin⁡γ,(25)a2′=-h52h4a2-h64h4a12sin⁡γ,(26)θ1′=-h72h1-h34h1a2cos⁡γ,(27)a2θ2′=-h82h4a2-h64h4a12cos⁡γ,(28)γ=θ2-2θ1+σT1,where (29)h1=1+2m~12Λ1+Λ12m~12m~21ω1;h2=ζ11+Λ12ζ22m~12m~21ω1;h3=d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22+2Λ1d7ω1ω2m~12m~21;h4=1+2m~12Λ2+Λ22m~12m~21ω2;h5=ζ11+Λ22ζ22m~12m~21ω2;h6=2d2Λ1+d3+d4ω12-Λ2d7ω12m~12m~21;h7=-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12;h8=-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22.Eliminatingθ1 and θ2 from (26) and (27) yields(30)a2γ′=σa2-h82h4a2-h64h4a12cos⁡γ+h7h1a2+h32h1a22cos⁡γ.Fromh2 and h5, it can be seen that both ζ11 and ζ22 affect a1 and a2 and thus affect the response of both flexible arm and the vibration absorber. In this way, vibration energy of the flexible manipulator can not only be dissipated by the damping of the vibration absorber based on the internal resonance but also be attenuated by its own enhanced damping, thereby obtaining better vibration control results. In this paper, it is called “the damping enhancement effect.”


### 3.3. Analysis of Secular Term

In order to better understand the dynamics of the system, the undamped case (i.e.,ζ11=ζ22=0) is studied. We have (31)a1′=h34h1a1a2sin⁡γ,(32)a2′=-h64h4a12sin⁡γ.Let(33)υ=h1h6h3h4.Then multiplying (31) by a1 and (32) by υa2 and adding them, it results in the equation(34)a1a1′+υa2a2′=0.Integrating (34), one obtains(35)a12+υa22=E0=const.,where E0 is a constant representing the total energy and is dependent on initial conditions.From (33), it is seen that υ is determined by the structural parameters of both the flexible arm and the vibration absorber. If the structural parameters of the flexible arm are given, then υ will be uniquely determined by those of the vibration absorber, that is, m, sa, and r. Due to the difficulty in determining the sign of υ in symbolic form, a numerical method is used here. Firstly, m/ml, sa/l, and r/l are viewed as three search variables, where ml is the mass of the flexible arm. Secondly, the search ranges of these variables are defined, respectively. In this example, m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. Thirdly, the search step is given. In this example, the search step is 0.001. Then, a numerical method is used to find the sign of υ in terms of the given search variables, search ranges, and search step. The final search result is υ>0 when m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. As a result, a1 and a2 in (35) are always bounded. Since the damping is neglected in this case, the system is conservative and the energy level remains constant. Therefore, if the response of the modal amplitude a1 is periodic, a2 will be periodic and out of phase from a1. Equation (35) demonstrates that, in the absence of damping, the energy in the system continues to be exchanged undamped between the fundamental mode of the flexible arm and the vibration mode of the absorber.


### 3.4. The Dissipative Energy Method

In the presence of damping (i.e.,ζ11>0 and ζ22>0), the equilibrium points for this case are defined by the solution of the set of steady state secular term equations; namely,(36)-h22h1a1+h34h1a1a2sin⁡γ=0,-h52h4a2-h64h4a12sin⁡γ=0,σa2-h82h4a2-h64h4a12cos⁡γ+h7h1a2+h32h1a22cos⁡γ=0.By inspection, it is determined that the system possesses an infinite number of equilibrium points defined by(37)a1=0,a2=0,γ∈R.Therefore, by examining the Jacobian, we can ascertain the stability of the system.The Jacobian matrix of this case is(38)μ1000μ20000,where μ1=-h2/2h1, μ2=-h5/2h4.The corresponding eigenvalues are(μ1,μ2,0). Since h2>0, the sign of μ1 is determined by h1. Since h5>0, the sign of μ2 is determined by h4. Similarly, due to the difficulty in determining the sign of h1 and h4 in symbolic form, a numerical method is used. Since h1 and h4 have relation to m, sa, and r when the structural parameters of the flexible arm are given, m/ml, sa/l, and r/l are viewed as three search variables and m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1] are defined, respectively, in this example. Also, the search step is 0.001. The search result is μ1<0 and μ2<0 when m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. Therefore, the modal amplitudes a1 and a2 are stable, as indicated by the negative eigenvalues.By numerical integrations of (24)–(28), vibration energy is transferred between the fundamental mode of the flexible arm and the vibration mode of the absorber in the presence of damping, but it is gradually dissipated. Therefore, internal resonance can be used to reduce vibration of the flexible arm.


## 3.1. Nondimensionalization and Decoupling of the Equations of Motion

The nondimensional variablesφ∗, s∗, τ, and q∗ are defined by(6)φ∗=φ1l,s∗=sl,τ=ωφt,q∗=q,where ωφ=k11/m11 and the equations of motion (5) are nondimensionalized as follows:(7)d2φ∗dτ2+m~12d2s∗dτ2+η11dφ∗dτ+1+d1d2q∗dτ2φ∗+k~12d2q∗dτ2s∗=-d2dφ∗dτds∗dτ-d3dφ∗dτ2-d4d2φ∗dτ2φ∗-d2d2φ∗dτ2s∗-d5d2q∗dτ2-d6dq∗dτ2+f3φ∗,dφ∗dτ,d2φ∗dτ2,s∗,ds∗dτ,d2s∗dτ2,dq∗dτ,d2q∗dτ2,d2s∗dτ2+m~21d2φ∗dτ2+η22ds∗dτ+ωsφ2s∗=d7dφ∗dτ2-d8d2q∗dτ2+d9dq∗dτ2+f4φ∗,dφ∗dτ,d2φ∗dτ2,s∗,ds∗dτ,d2s∗dτ2,dq∗dτ,d2q∗dτ2,where m~12=m12/m11; η11=c11/(ωφm11); d1=b1/m11; k~12=k12/m11; d2=b2l/m11; d3=b3l/m11; d4=b4l/m11; d5=b5/(m11l); d6=b6/(m11l); m~21=m21/m22; η22=c22/(ωφm22); ωs=k22/m22; ωsφ=ωs/ωφ; d7=b7l/m22; d8=b8/(m22l); d9=b9/(m22l).To make the damping and nonlinearities appear in the same perturbation equations, letη11=εζ11, η22=εζ22, φ∗=εϕ, s∗=εw, and dq∗/dτ=ε(dχ/dτ), where ε is a small nondimensional bookkeeping parameter, 0<ε≪1. Then, (7) can be expressed as(8)d2ϕdτ2+ϕ=-m~12d2wdτ2-d5d2χdτ2+ε-ζ11dϕdτ-d1d2χdτ2ϕ-k~12d2χdτ2w-d2dϕdτdwdτ-d3dϕdτ2-d4d2ϕdτ2ϕ-d2d2ϕdτ2w-d6dχdτ2+oε,(9)d2wdτ2+ωsφ2w=-m~21d2ϕdτ2-d8d2χdτ2+ε-ζ22dwdτ+d7dϕdτ2+d9dχdτ2+oε.


## 3.2. Perturbation Analysis

The time dependenceτ is expanded in terms of multiple time scales, Ti=εiτ, (i=0,1,…), so that the first- and second-time derivatives become (10)ddτ=D0+εD1+ε2D2+⋯,d2dτ2=D02+2εD0D1+ε2D1+2ε2D0D2+⋯,where Di=∂/∂Ti, (i=0,1,…).We seek first-order approximate solutions of (8) and (9) by using the method of multiple scales in the form(11)ϕτ,ε=ϕ0T0,T1+εϕ1T0,T1,wτ,ε=w0T0,T1+εw1T0,T1.Substituting (10), (11) into (8) and (9) and equating coefficients of same powers of ε, we obtain the following:Order (ε0) is as follows:(12)D02ϕ0+ϕ0=-m~12D02w0+g1,D02w0+ωsφ2w0=-m~21D02ϕ0+g2.Order (ε1) is as follows:(13)D02ϕ1+ϕ1=-2D0D1ϕ0-ζ11D0ϕ0-m~12D02w1-2m~12D0D1w0-d1D02χϕ0-k~12D02χw0-d2D0ϕ0D0w0-d3D0ϕ02-d4D02ϕ0ϕ0-d2D02ϕ0w0-d6D0χ2,D02w1+ωsφ2w1=-2D0D1w0-m~21D02φ1-2m~21D0D1ϕ0-ζ22D0w0+d7D0ϕ02+d9D0χ2,where g1=-d5D02χ, g2=-d8D02χ.The solution of (12) can be written in the form (14)ϕ0=A1T1expjω1T0+A2T1expjω2T0+h1+cc,w0=Λ1A1T1expjω1T0+Λ2A2T1expjω2T0+h2+cc,where A1(T1) and A2(T1) are functions of slow time T1; h1=g1/2; h2=g2/(2ωsφ2); cc denotes the complex conjugate terms. According to the theory of ordinary differential equation, the ωn2 (n=1,2) are the roots of(15)1-m~12m~21ωn4-1+ωsφ2ωn2+ωsφ2=0,(16)Λn=1-ωn2ωn2m~12=m~21ωn2ωsφ2-ωn2and ωn are assumed to be distinct.In this study, because the second-order nonlinear coupling terms exist in the dynamic model, the vibration absorber is used to control vibration of the flexible arm at the 1 : 2 internal resonance condition; that is,ω2≈2ω1. It is this internal resonance condition that enables the transfer of vibration energy between the fundamental mode of the arm and the vibration mode of the absorber. In order to solve the nonlinear problem, substituting (14) into (13) yields(17)D02ϕ1+ϕ1=-2jω1A1′+ζ11jω1A1+2m~12jω1Λ1A1′+d1D02χA1+k~12D02χΛ1A1exp⁡jω1T0+-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d4ω12+ω22+d2Λ2ω12+Λ1ω22A-1A2·exp⁡jω2-ω1T0+NST+cc,D02w1+ωsφ2w1=-2jω2Λ2A2′+ζ22jω2Λ2A2+2m~21jω2A2′exp⁡jω2T0-d7ω12A12exp⁡2jω1T0+NST+cc,where ′≡∂()/∂T1 and NST denotes nonsecular terms.In the case of the 1 : 2 internal resonance, a detuning parameterσ is introduced as ω2=2ω1+εσ. To determine the solvability conditions of (17), the particular solutions are sought in the form(18)ϕ1=p11exp⁡jω1T0+p12exp⁡jω2T0,w1=p21exp⁡jω1T0+p22exp⁡jω2T0.Substituting (18) into (17) and then equating the coefficients of exp(jω1T0) and exp(jω2T0) on both sides, one obtains(19)1-ωn2p1n-m~12ωn2p2n=R1n,-m~21ωn2p1n+ωsφ2-ωn2p2n=R2n,where (20)R11=-2jω11+m~12Λ1A1′-jω1ζ11A1+-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12·A1+d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22A-1A2exp⁡jσT1,R12=-2jω21+m~12Λ2A2′-jω2ζ11A2+-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22·A2+2d2Λ1+d3+d4ω12A12exp⁡-jσT1,R21=-2jω1m~21+Λ1A1′-jω1ζ22Λ1A1+2d7ω1ω2A-1A2exp⁡jσT1,R22=-2jω2m~21+Λ2A2′-jω2ζ22Λ2A2-d7ω12A12exp⁡-jσT1.Therefore, the problem of determining the solvability conditions of (17) is reduced to that of determining the solvability condition of (19).Since the determinant of the coefficient matrix of (19) is zero according to (15), the solvability conditions are(21)R1n-m~12ωn2R2nωsϕ2-ωn2=0orR1n=-ΛnR2nm~12m~21on account of (16).Substituting (20) into (21) yields(22)-2jω11+m~12Λ1A1′-jω1ζ11A1+-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12A1+d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22A-1A2exp⁡jσT1=-Λ1·m~12m~21-2jω1m~21+Λ1A1′-jω1ζ22Λ1A1+2d7ω1ω2A-1A2exp⁡jσT1,-2jω21+m~12Λ2A2′-jω2ζ11A2+-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22A2+2d2Λ1+d3+d4ω12A12exp⁡-jσT1=-Λ2·m~12m~21-2jω2m~21+Λ2A2′-jω2ζ22Λ2A2-d7ω12A12exp⁡-jσT1.It is convenient to express the resulting modulation equations in polar form by introducing the following transformation:(23)A1=12a1exp⁡jθ1,A2=12a2exp⁡jθ2,where a1, a2, θ1, and θ2 are real functions of the slow time T1; a1 and a2 are defined as the modal amplitudes.Inserting (23) into (22), then setting the coefficients of the real and imaginary parts to zero yields the modulation equations(24)a1′=-h22h1a1+h34h1a1a2sin⁡γ,(25)a2′=-h52h4a2-h64h4a12sin⁡γ,(26)θ1′=-h72h1-h34h1a2cos⁡γ,(27)a2θ2′=-h82h4a2-h64h4a12cos⁡γ,(28)γ=θ2-2θ1+σT1,where (29)h1=1+2m~12Λ1+Λ12m~12m~21ω1;h2=ζ11+Λ12ζ22m~12m~21ω1;h3=d4ω12+ω22-d2Λ1+Λ2ω1ω2-2d3ω1ω2+d2Λ2ω12+Λ1ω22+2Λ1d7ω1ω2m~12m~21;h4=1+2m~12Λ2+Λ22m~12m~21ω2;h5=ζ11+Λ22ζ22m~12m~21ω2;h6=2d2Λ1+d3+d4ω12-Λ2d7ω12m~12m~21;h7=-d1D02χ-k~12D02χΛ1+2h1d4ω12+2h2d2ω12;h8=-d1D02χ-k~12D02χΛ2+2h1d4ω22+2h2d2ω22.Eliminatingθ1 and θ2 from (26) and (27) yields(30)a2γ′=σa2-h82h4a2-h64h4a12cos⁡γ+h7h1a2+h32h1a22cos⁡γ.Fromh2 and h5, it can be seen that both ζ11 and ζ22 affect a1 and a2 and thus affect the response of both flexible arm and the vibration absorber. In this way, vibration energy of the flexible manipulator can not only be dissipated by the damping of the vibration absorber based on the internal resonance but also be attenuated by its own enhanced damping, thereby obtaining better vibration control results. In this paper, it is called “the damping enhancement effect.”


## 3.3. Analysis of Secular Term

In order to better understand the dynamics of the system, the undamped case (i.e.,ζ11=ζ22=0) is studied. We have (31)a1′=h34h1a1a2sin⁡γ,(32)a2′=-h64h4a12sin⁡γ.Let(33)υ=h1h6h3h4.Then multiplying (31) by a1 and (32) by υa2 and adding them, it results in the equation(34)a1a1′+υa2a2′=0.Integrating (34), one obtains(35)a12+υa22=E0=const.,where E0 is a constant representing the total energy and is dependent on initial conditions.From (33), it is seen that υ is determined by the structural parameters of both the flexible arm and the vibration absorber. If the structural parameters of the flexible arm are given, then υ will be uniquely determined by those of the vibration absorber, that is, m, sa, and r. Due to the difficulty in determining the sign of υ in symbolic form, a numerical method is used here. Firstly, m/ml, sa/l, and r/l are viewed as three search variables, where ml is the mass of the flexible arm. Secondly, the search ranges of these variables are defined, respectively. In this example, m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. Thirdly, the search step is given. In this example, the search step is 0.001. Then, a numerical method is used to find the sign of υ in terms of the given search variables, search ranges, and search step. The final search result is υ>0 when m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. As a result, a1 and a2 in (35) are always bounded. Since the damping is neglected in this case, the system is conservative and the energy level remains constant. Therefore, if the response of the modal amplitude a1 is periodic, a2 will be periodic and out of phase from a1. Equation (35) demonstrates that, in the absence of damping, the energy in the system continues to be exchanged undamped between the fundamental mode of the flexible arm and the vibration mode of the absorber.


## 3.4. The Dissipative Energy Method

In the presence of damping (i.e.,ζ11>0 and ζ22>0), the equilibrium points for this case are defined by the solution of the set of steady state secular term equations; namely,(36)-h22h1a1+h34h1a1a2sin⁡γ=0,-h52h4a2-h64h4a12sin⁡γ=0,σa2-h82h4a2-h64h4a12cos⁡γ+h7h1a2+h32h1a22cos⁡γ=0.By inspection, it is determined that the system possesses an infinite number of equilibrium points defined by(37)a1=0,a2=0,γ∈R.Therefore, by examining the Jacobian, we can ascertain the stability of the system.The Jacobian matrix of this case is(38)μ1000μ20000,where μ1=-h2/2h1, μ2=-h5/2h4.The corresponding eigenvalues are(μ1,μ2,0). Since h2>0, the sign of μ1 is determined by h1. Since h5>0, the sign of μ2 is determined by h4. Similarly, due to the difficulty in determining the sign of h1 and h4 in symbolic form, a numerical method is used. Since h1 and h4 have relation to m, sa, and r when the structural parameters of the flexible arm are given, m/ml, sa/l, and r/l are viewed as three search variables and m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1] are defined, respectively, in this example. Also, the search step is 0.001. The search result is μ1<0 and μ2<0 when m/ml∈(0,0.5], sa/l∈(0,0.2], and r/l∈(0,1]. Therefore, the modal amplitudes a1 and a2 are stable, as indicated by the negative eigenvalues.By numerical integrations of (24)–(28), vibration energy is transferred between the fundamental mode of the flexible arm and the vibration mode of the absorber in the presence of damping, but it is gradually dissipated. Therefore, internal resonance can be used to reduce vibration of the flexible arm.


## 4. Simulations and Analyses

To verify the above theoretical analysis, some numerical simulations are done on the conditions:l=1.0 m, h=0.05 m, b=0.003 m, mB=0.5 kg, the arm is made of aluminum, and ρ=2710 kg/m3.Supposing the flexible arm has neither rigid motion (i.e.,q=q˙=q¨=0) nor vibration absorber, it can be viewed as a flexible structure. For the initial disturbance of δ(l,0)=0.1 m, the end-effector deformation is obtained using a fourth-order Runge-Kutta numerical integration algorithm, as shown in Figure 2. For a flexible structure subjected to an initial disturbance, its end-effector response is usually a slowly attenuated vibration due to the low damping. To verify the above results, a virtual prototype simulation is conducted using ADAMS software. The dynamic model of the flexible arm with the same parameters is created (as shown in Figure 3), and the end-effector deformation is shown in Figure 4. As can be seen, the results are similar to each other.Figure 2
Response of the uncontrolled fixed flexible arm.Figure 3
A fixed flexible arm in ADAMS.Figure 4
Response of the uncontrolled fixed flexible arm in ADAMS.In order to reduce vibration of the flexible structure via modal interaction, a vibration absorber is perpendicularly attached to the flexible arm atx1=0.5 m, at which mass m=0.3 kg and equilibrium position sa=0.05 m, as shown in Figure 1. At the state of internal resonance, (24)–(28) are integrated numerically in the absence of damping. The relationship between the modal amplitudes a1 (dashed line) and a2 (solid line) is shown in Figure 5. As can be seen, the peaks and troughs of the responses are exactly 180° out of phase, verifying the fact that there is continuous exchange of the energy between modes of vibration.Figure 5
Undamped modal amplitudes in fixed flexible arm.When the damping of the vibration absorber is taken into account andζ22=0.0015, (24)–(28) are integrated numerically at the state of internal resonance. The motion of a1 (dashed line) and a2 (solid line) within the whole work process is illustrated in Figure 6. And the transfer of energy between modes can be found, verified by the coincidence of the peaks of one modal curve with the troughs of the other. In addition, the modal amplitudes decrease with time, which suggests that vibration energy of the fundamental mode of the flexible arm is absorbed by the absorber and dissipated through the damping of the absorber.Figure 6
Damped modal amplitudes in fixed flexible arm.The end-effector deformation of the flexible arm equipped with the vibration absorber is shown in Figure7. At the stage of 1 : 2 internal resonance, the initial deformation is reduced quickly within 10 seconds. After 20 seconds, the end-effector deformation is reduced by 50%. After 40 seconds, the end-effector deformation is reduced by 80%.Figure 7
Response of the controlled fixed flexible arm.Through above simulation and analysis, it is verified that this control method based on internal resonance is effective in controlling vibration of the flexible structure.Next, an example of the flexible arm undergoing rigid motion is considered. Suppose the desired joint motion of the arm is(39)q˙=0.2×sin⁡πt3+0.5×cos⁡t,0≤t≤100s.If the flexible arm is not equipped with the vibration absorber, when moving according to (39), its end-effector deformation is obtained for the initial disturbance of δ(l,0)=0.1 m, as shown in Figure 8. Compared with a flexible structure (as shown in Figure 2), rigid motion can cause more complex dynamic behaviors due to its coupling effect with flexural deformation. As can be seen, the end-effector response is no longer an attenuated vibration and its peaks even exceed the initial disturbance. Once again, the end-effector deformation is calculated using ADAMS software and shown in Figure 9. As can be seen, the results are similar to each other.Figure 8
Response of the uncontrolled moving flexible arm.Figure 9
Response of the uncontrolled moving flexible arm in ADAMS.In order to reduce vibration of the flexible arm via modal interaction, a vibration absorber is perpendicularly attached to the flexible arm and the same parameters as in the aforementioned case are used. At the state of internal resonance, (24)–(28) are integrated numerically in the absence of damping. The relationship between the modal amplitudes a1 (dashed line) and a2 (solid line) is shown in Figure 10. It can be seen that the peaks and troughs of the responses are exactly 180° out of phase, which means that there is continuous exchange of the energy between modes of vibration.Figure 10
Undamped modal amplitudes in moving flexible arm.When the damping of the vibration absorber is taken into account andζ22=0.005, (24)–(28) are integrated numerically at the state of internal resonance. The modal amplitudes a1 (dashed line) and a2 (solid line) are illustrated in Figure 11. Since each of h2 and h5 has relation to both ζ11 and ζ22, once ζ22 is added to the system, it can increase the damping of not only a2 but also a1. In this case, the vibration energy of a1 can not only be absorbed by the damping of a2 when transferred to a2, but also be dissipated by its increased damping. As demonstrated in Figure 11, the transfer of energy between modes can be found, verified by the coincidence of the peaks of one modal curve with the troughs of the other. In addition, the modal amplitudes decrease with time and approach the equilibrium value (a1=a2=0), which suggests that the vibration energy of the flexible arm has been effectively absorbed and dissipated through internal resonance.Figure 11
Damped modal amplitudes in moving flexible arm.The end-effector deformation of the flexible arm equipped with the vibration absorber is shown in Figure12. At the stage of 1 : 2 internal resonance, the end-effector deformation of the flexible arm is reduced quickly within 20 seconds. After 20 seconds, the end-effector deformation is reduced by 75%, compared with the uncontrolled case (as shown in Figure 8). Final deformation of the end-effector can be reduced by 90%. Also, an example of the initial disturbance of δ(l,0)=0.2 m and ζ22=0.005 is conducted. The end-effector deformation is shown in Figure 13. As can be seen, even though larger disturbance is imposed on the system, vibration response can still be reduced effectively.Figure 12
Response of the controlled arm (δ(l,0)=0.1 m).Figure 13
Response of the controlled arm (δ(l,0)=0.2 m).In addition, the effects of the absorber dampingζ22 are studied, as shown in Figure 14. It can be seen that if the absorber damping is very small, vibration energy flowing into the absorber cannot be effectively dissipated. However, if the absorber damping is too large, its effect becomes indistinctive. Therefore, an appropriate damping is necessary.Figure 14
Effects of the absorber damping.Furthermore, the effect of the absorber location is studied. When the absorber is attached to the flexible arm atr=0.35 m, the controlled frequency of the arm is 1.0106 Hz. Its end-effector deformation is shown in Figure 15. When the absorber is attached to the flexible arm at r=0.5 m, the controlled frequency of the arm is 0.9905 Hz. Its end-effector deformation is shown in Figure 12. When the absorber is attached to the flexible arm at r=0.75 m, the controlled frequency of the arm is 0.9237 Hz. Its end-effector deformation is shown in Figure 16. From these figures, it can be seen that the absorber location can remarkably affect the end-effector deformation, and an appropriate absorber location is necessary.Figure 15
Response of the controlled arm (r=0.35 m).Figure 16
Response of the controlled arm (r=0.75 m).Based on above simulations and analyses, it is proven that this method is effective in controlling nonlinear vibration of the flexible arm.


## 5. Conclusion

In this paper, a vibration control method based on energy migration is proposed to decrease vibration response of the flexible arm undergoing rigid motion. A type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Furthermore, due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, vibration energy of the flexible arm can be dissipated by not only the damping of the vibration absorber but also its own enhanced damping, thereby effectively decreasing vibration. Through numerical simulations and analyses, it is proven that this method is feasible in controlling nonlinear vibration of the flexible arm undergoing rigid motion.



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*Source: 382145-2015-07-15.xml*