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Abstract

The main task of this paper is to put forward a vibration absorption method for attenuating nonlinear vibration of the flexible manipulator based on modal interaction. A vibration absorber is suggested to establish the 1:1 internal resonance state with the flexible manipulator, thereby transferring the vibration energy from the flexible manipulator to the vibration absorber. In the presence of damping, the vibration energy of the flexible manipulator can be effectively dissipated by the vibration absorber. Since this method puts an emphasis on constructing an internal energy transfer channel between the flexible manipulator and the vibration absorber rather than directly responding to external excitations, it is particularly convenient to reduce nonlinear vibration induced by unknown external excitations. Numerical simulations and virtual prototyping simulations have verified this method’s feasibility.

Keywords

Flexible manipulator nonlinear vibration internal resonance

Introduction

In the past years, mainly because of space applications, robotic manipulators with flexible members have attracted great attention. Typically, these flexible manipulators possess low structural damping and small stiffness, and thus any disturbances may cause serious control issues or structural damage. Therefore, a great deal of research has been conducted on designing vibration control schemes to deal with these problems.

Generally, the vibration response can be reduced by enhancing the structural stiffness or increasing the fundamental frequencies. Another useful way is to tailor the frequency response of a vibrating structure by attaching additional masses or oscillators. Through optimizing the distributions of the dynamic properties in an array of additional masses, ribs or subordinate oscillators, the desired frequency response can be precisely obtained.^1–3

In addition to passive control methods, a large number of active control methods have been developed to damp out transient vibration, including: PID control, computed torque control, adaptive control, robust control, neural network control, input shaping, sliding mode control, variable structure control, fuzzy control, and so on.^4–9 Moreover, various actuators based on smart materials, like piezoelectric ceramic and shape memory alloy, have also been investigated.^10–12 Nevertheless, they are probably faced with severe challenges when dealing with vibration problems with strong energy. Since these methods rely on external energy to suppress vibration, it is difficult for most of the smart material actuators to conquer strong vibration due to limited energy output. Besides, since strong energy always excites large amplitude, a number of nonlinear terms become significant and cannot be ignored any more. Therefore, any control methods based on linearized models will become invalid.

In addition to vibration suppression methods, vibration absorption methods are other useful ways to control vibration. Rather than suppressing vibration with the help of external energy, they absorb vibration of the primary system via vibration absorbers. Since these methods do not need much energy, and thus are suitable for reducing strong vibration with large amplitude. However, only after detailed information of external excitations has been obtained, can most of them effectively neutralize vibration responses of the primary system. As a result, if external excitations are unclear or unpredictable, such as in the outer space, these methods will deteriorate.

In fact, it is a valuable way to resort to appropriate nonlinear theory for nonlinear vibration control. For a multi-degree-of-freedom nonlinear vibrating system, it is found that vibration energy of one mode can be transferred via internal resonance to another mode which is commensurable with the former.¹³ Golnaraghi¹⁴ and Golnaraghi et al.¹⁵ firstly attempt to use internal resonance to control structural vibration of a flexible cantilever beam. Afterwards, Tuer et al.^16,17 and Oueini et al.¹⁸ and Oueini and Golnaraghi ¹⁹ have conducted theoretical and experimental studies and regulated the motion of a similar beam. However, their flexible cantilever beam model was a rigid beam connected by a torsional spring. Obviously, this simplified model is not feasible to the flexible manipulator. Recently, the above studies have been extended to the distributed flexible beam.^20,21 But their models belonged to the flexible structure without rigid motion and did not involve the flexible manipulator undergoing large scale joint motion. Furthermore, their flexible structure is assumed as a linear vibration model for the simplicity, while the flexible manipulator itself is a nonlinear dynamics system. Obviously, the latter is much more complicated than the former.

The main task of this paper is to put forward a vibration absorption method for attenuating nonlinear vibration of the flexible manipulator based on the internal resonance. The paper is composed of six sections. Following the introduction, the next section suggests a vibration absorber and creates a nonlinear dynamics model for the flexible manipulator based on Kane’s method. Then, the method of multiple scales is used to investigate vibration response of the nonlinear dynamics equations. In the subsequent section, a 1:1 internal resonance method is analyzed for two cases: undamped and damped. It is proven that the internal resonance can be successfully established between the flexible manipulator and the vibration absorber, and the vibration energy of the flexible manipulator can be transferred to the vibration absorber and then dissipated by the damping of the latter. Numerical simulations and virtual prototyping simulations are then conducted to verify theoretical analyses. Finally, the last section provides the conclusions.

Formulation of nonlinear dynamics model

In this paper, a three-link manipulator is researched, in which the first two links (Link 1 and Link 2) are rigid and the third link (Link 3) is flexible, but three joints (Joint 1, Joint 2 and Joint 3) are rigid, as shown in Figure 1. Link 3 is viewed as a uniform Bernoulli beam with the rectangle cross-section of height $h$ and width $b$ . Only flexural deformation about $y_{3}$ axis, i.e. $δ (x_{3})$ , is considered, as shown in Figure 2.

Figure 1.

Model of a flexible manipulator with a vibration absorber.

Figure 2.

Description of deformation.

To reduce vibration of the flexible manipulator, another vibrating system, used as a vibration absorber, is attached to the flexible manipulator at $x_{3} = r$ , as shown in Figure 1. It consists of a rigid link (Link 4), a flexible joint (Joint 4) and a damper. Its flexibility and damping are provided by a torsional spring and a damper. The Link 4 is connected through the Joint 4 and damper, to the Link 3 at a fixed angle $q_{B}$ to $x_{3}$ axis. When the Link 3 deforms, as shown in Figure 2, the rotation angle $χ$ of the Link 4 with respect to $x_{3}$ axis is

χ = q_{B} + θ + β

(1)

where

θ

is the rotation angle of the Joint 4 due to its flexibility,

β

is the angle of the tangent at the Joint 4 with respect to

x_{3}

axis, i.e.

β = {\frac{\partial δ (x_{3}, t)}{\partial x_{3}} |}_{x_{3} = r}

(2)

where

t

is the time.

Based on Kane’s method, the dynamics equation of the flexible manipulator and vibration absorber is derived and can be written as²²

M_{T} \ddot{ψ} + C_{T} \dot{ψ} + K_{T} ψ = Q_{T}

(3)

where

M_{T} \in R^{n \times n}

is the system mass matrix;

C_{T} \in R^{n \times n}

is the system damping matrix;

K_{T} \in R^{n \times n}

is the system stiffness matrix;

Q_{T} \in R^{n}

is the sum of coriolis, gravitational, centripetal, and control torques;

ψ \in R^{n}

is the set of rigid and flexural degrees of freedom,

ψ = {[\begin{matrix} q^{T} & ϕ^{T} \end{matrix}]}^{T}

;

q \in R^{n_{R}}

is the set of joint angles;

n_{R}

is the number of joints.

ϕ \in R^{n_{F}}

is the set of flexural degrees of freedom which can be expressed by a series of modal coordinates

ϕ_{i} (t)

(i = 1 to

n_{F}

) to describe the flexural displacement of links;

n_{F}

is the number of flexural degrees of freedom in the flexible links;

n

is the total number of degrees of freedom

n = n_{R} + n_{F}

Equation (3) can be separated into two equations describing the dynamics of $q$ and $ϕ$

D \ddot{q} + U \ddot{ϕ} = τ + τ_{R}

(4)

G \ddot{q} + M \ddot{ϕ} + C \dot{ϕ} + K ϕ = f

(5)

where

D \in R^{n_{R} \times n_{R}}

U \in R^{n_{R} \times n_{F}}

G \in R^{n_{F} \times n_{R}}

M \in R^{n_{F} \times n_{F}}

are the block matrices of

M_{T}

;

τ \in R^{n_{R}}

is the set of control torques applied to the joints;

τ_{R} \in R^{n_{R}}

is the rigid component of the nonlinear torque;

C \in R^{n_{F} \times n_{F}}

;

K \in R^{n_{F} \times n_{F}}

;

f \in R^{n_{F}}

is the flexible component of the nonlinear torque.

Equation (5) exhibits flexible dynamic behavior of both the flexible manipulator and vibration absorber, playing an important role in vibration control. By exploring nonlinear coupling between the flexible manipulator and vibration absorber in this equation, the internal resonance is expected to be established and used to reduce vibration.

According to the vibration theory, the transverse displacement of Link 3 can be expressed in terms of the mode shapes

δ (x_{3}, t) = \sum_{i = 1}^{n_{F}} u_{i} (x_{3}) ϕ_{i} (t)

(6)

where

ϕ_{i} (t)

is the i-th modal coordinate describing the deformation of the Link 3,

u_{i} (x_{3})

is the i-th mode shape satisfying certain geometry and force boundary conditions.

In this study, only the fundamental mode of the Link 3 is considered due to its dominant contribution to the vibration response in common cases. By considering equations (1), (2) and (6), the flexible dynamics equations about the fundamental mode coordinates $ϕ_{1}$ of the Link3 and the flexural degree of freedom $θ$ of the vibration absorber are derived from equation (5) and can be written as

{\ddot{ϕ}}_{1} + (c_{1} + f_{c 1}) {\dot{ϕ}}_{1} + (ω_{10}^{2} + f_{k}) ϕ_{1} = f_{1} \dot{θ} + b_{1} {\dot{θ}}^{2} + b_{2} θ + f_{2}

(7)

\ddot{θ} + (c_{2} + f_{c 2}) \dot{θ} + ω_{20}^{2} θ = (b_{3} + f_{3}) ϕ_{1} + f_{4} {\dot{ϕ}}_{1} + b_{4} {\dot{θ}}^{2} + f_{5}

(8)

And the detailed descriptions of above parameters are listed in Appendix 1.

In equations (7) and (8), it is seen that these two modal coordinates $ϕ_{1}$ and $θ$ are nonlinear coupled. The incorporation of the vibration absorber into the flexible manipulator introduces dynamic nonlinearities to the system, thereby yielding modal coupling effects between them. In addition, several parameters, like: $f_{c 1}$ , $f_{c 2}$ , $f_{k}$ , $f_{1}$ , $f_{2}$ , $f_{3}$ , $f_{4}$ and $f_{5}$ , are time-varying and vary with the joint motion. This exhibits the influences of rigid motion on flexural vibration, which is one of the main differences between a flexible manipulator and a flexible structure.

In the following sections, modal interaction effects between the fundamental mode of the flexible manipulator and the vibration mode of the vibration absorber will be investigated.

Nonlinear analysis

Perturbation equation

Since there are no closed-form solutions to equations (7) and (8), an approximate analytical solution is generated using perturbation analysis. A small non-dimensional bookkeeping parameter $ε, 0 < ε < 1$ is introduced to represent the order of nonlinear terms and coupling. In this way, these two equations are transformed into the following form

{\ddot{ϕ}}_{1} + ω_{10}^{2} ϕ_{1} = ε [- (c_{1} + f_{c 1}) {\dot{ϕ}}_{1} - f_{k} ϕ_{1} + f_{1} \dot{θ} + b_{1} {\dot{θ}}^{2} + b_{2} θ] + f_{2}

(9)

\ddot{θ} + ω_{20}^{2} θ = ε [- (c_{2} + f_{c 2}) \dot{θ} + (b_{3} + f_{3}) ϕ_{1} + f_{4} {\dot{ϕ}}_{1} + b_{4} {\dot{θ}}^{2}] + f_{5}

(10)

The method of multiple scales is used to determine a first-order approximation to the solutions of equations (9) and (10) in the form

ϕ_{1} (t, ε) = ϕ_{10} (T_{0}, T_{1}) + ε ϕ_{11} (T_{0}, T_{1})

(11)

θ (t, ε) = θ_{0} (T_{0}, T_{1}) + ε θ_{1} (T_{0}, T_{1})

(12)

where independent time variables

T_{m}

are introduced according to

T_{m} = ε^{m} t (m = 0, 1)

(13)

In terms of $T_{0}$ and $T_{1}$ , the first and second time derivatives are transformed into

\frac{d}{d t} = D_{0} + ε D_{1}

(14)

\frac{d^{2}}{d t^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1}

(15)

where

D_{m} = \partial / \partial T_{m}

Substituting equations (11), (12), (14), (15) into equations (9) and (10) and equating coefficients of the same order of $ε$ in both sides, we obtain the following set of differential equations

Order ( $ε^{0}$ )

D_{0}^{2} ϕ_{10} + ω_{10}^{2} ϕ_{10} = f_{2}

(16)

D_{0}^{2} θ_{0} + ω_{20}^{2} θ_{0} = f_{5}

(17)

Order ( $ε^{1}$ )

D_{0}^{2} ϕ_{11} + ω_{10}^{2} ϕ_{11} = - 2 D_{0} D_{1} ϕ_{10} - (c_{1} + f_{c 1}) D_{0} ϕ_{10} - f_{k} ϕ_{10} + f_{1} D_{0} θ_{0} + b_{1} {(D_{0} θ_{0})}^{2} + b_{2} θ_{0}

(18)

D_{0}^{2} θ_{1} + ω_{20}^{2} θ_{1} = - 2 D_{0} D_{1} θ_{0} - (c_{2} + f_{c 2}) D_{0} θ_{0} + (b_{3} + f_{3}) ϕ_{10} + f_{4} D_{0} ϕ_{10} + b_{4} {(D_{0} θ_{0})}^{2}

(19)

Equations (16) to (19) are time-varying, because $f_{c 1}$ , $f_{c 2}$ , $f_{k}$ , $f_{1}$ , $f_{2}$ , $f_{3}$ , $f_{4}$ and $f_{5}$ vary with the joint motion. To solve these equations, the time period $t$ of whole work process is divided into $n_{h}$ equal time subintervals, and the time step corresponding to each subinterval is termed $Δ t$ , i.e. $Δ t = t / n_{h}$ . Afterwards, within each subinterval, the joint motion is substituted into the equations, and $f_{c 1}$ , $f_{c 2}$ , $f_{k}$ , $f_{1}$ , $f_{2}$ , $f_{3}$ , $f_{4}$ _, and $f_{5}$ are considered as constant for the particular subinterval. Finally, equations (16) to (19) are solved for each subinterval using the method of multiple scales. By repeating the above procedures, total vibration responses can be obtained for whole work process.

Perturbation solution

The linear problem is governed by equations (16) and (17). As mentioned above, these two equations are solved within the k-th time subinterval $Δ t_{k}$ , ( $k = 1, 2, \dots, n_{h}$ ). In this way, the solutions to these two equations can be written as

ϕ_{10, k} = \frac{f_{2, k}}{ω_{10}^{2}} [1 - \cos (ω_{10} T_{0})]

(20)

θ_{0, k} = \frac{f_{5, k}}{ω_{20}^{2}} [1 - \cos (ω_{20} T_{0})]

(21)

The nonlinear problem is governed by equations (18) and (19). Similarly, these two equations are solved within the k-th time subinterval $Δ t_{k}$ , ( $k = 1, 2, \dots, n_{h}$ ). In order to solve the nonlinear problem, the general solution (20) and (21) can be expressed in the form of complex function for the convenience of computation, i.e.

ϕ_{10, k} = \frac{1}{2} a_{1, k} {1 - \exp [j (ω_{10} T_{0} + φ_{1, k})]} + c c

(22)

θ_{0, k} = \frac{1}{2} a_{2, k} {1 - \exp [j (ω_{20} T_{0} + φ_{2, k})]} + c c

(23)

where

a_{1, k}

φ_{1, k}

a_{2, k}

φ_{2, k}

are real functions of

T_{1}

a_{1, k}

and

a_{2, k}

are defined as the modal amplitudes,

c c

denotes the complex conjugate of the preceding term.

Substituting equations (22) and (23) into equations (18) and (19), the following equations in $Δ t_{k}$ can be obtained

\begin{matrix} D_{0}^{2} ϕ_{11, k} + ω_{10}^{2} ϕ_{11, k} = [{a^{'}}_{1, k} j ω_{10} - a_{1, k} {φ^{'}}_{1, k} ω_{10} + \frac{1}{2} (c_{1} + f_{c 1, k}) a_{1, k} j ω_{10} + \frac{1}{2} a_{1, k} f_{k, k}] \exp [j (ω_{10} T_{0} + φ_{1, k})] \\ - \frac{a_{2, k}}{2} (f_{1, k} j ω_{20} + b_{2}) \exp [j (ω_{20} T_{0} + φ_{2, k})] + \frac{1}{4} b_{1} a_{2, k}^{2} ω_{20}^{2} {1 - \exp [2 j (ω_{20} T_{0} + φ_{2, k})]} + c c \end{matrix}

(24)

\begin{matrix} D_{0}^{2} θ_{1, k} + ω_{20}^{2} θ_{1, k} = [{a^{'}}_{2, k} j ω_{20} - a_{2, k} {φ^{'}}_{2, k} ω_{20} + \frac{1}{2} (c_{2} + f_{c 2, k}) a_{2, k} j ω_{20}] \exp [j (ω_{20} T_{0} + φ_{2, k})] \\ - \frac{a_{1, k}}{2} (b_{3} + f_{3, k} + f_{4, k} j ω_{10}) \exp [j (ω_{10} T_{0} + φ_{1, k})] + c c \end{matrix}

(25)

To obtain the solutions of equations (24) and (25), it is necessary to determine the solvability conditions by eliminating the secular terms.

Analysis in internal resonance

In this section, we will study the responses of the system at the 1:1 internal resonance condition, i.e. $ω_{10} \approx ω_{20}$ . It is this internal resonance condition that enables the transfer of the vibration energy between the fundamental mode of the flexible manipulator and the vibration mode of the absorber. The analysis of the modal amplitudes will verify that the transfer of vibration energy can be realized. To this end, two cases will be studied, i.e. undamped and damped.

Resonant case, without damping

In the case of the 1:1 internal resonance, a detuning parameter $σ$ is introduced as follows

ω_{20} = ω_{10} + ε σ

(26)

In the absence of damping (i.e. $c_{1} = c_{2} = f_{c 1, k} = f_{c 2, k} = 0$ ), substituting equation (26) into equations (24) and (25) yields the following conditions for the elimination of secular terms

({a^{'}}_{1, k} j ω_{10} - a_{1, k} {φ^{'}}_{1, k} ω_{10} + \frac{1}{2} a_{1, k} f_{k, k}) \exp (j φ_{1, k}) - \frac{a_{2, k}}{2} (f_{1, k} j ω_{20} + b_{2}) \exp [j (ε σ T_{0} + φ_{2, k})] = 0

(27)

({a^{'}}_{2, k} j ω_{20} - a_{2, k} {φ^{'}}_{2, k} ω_{20}) \exp (j φ_{2, k}) - \frac{a_{1, k}}{2} (b_{3} + f_{3, k} + f_{4, k} j ω_{10}) \exp [j (φ_{1, k} - ε σ T_{0})] = 0

(28)

Separating real and imaginary parts, one obtains

{a^{'}}_{1, k} = \frac{a_{2, k}}{2 ω_{10}} \sqrt{{(f_{1, k} ω_{20})}^{2} + b_{2}^{2}} \sin (γ_{k} + Ω_{1, k})

(29)

{a^{'}}_{2, k} = - \frac{a_{1, k}}{2 ω_{20}} \sqrt{{(f_{4, k} ω_{10})}^{2} + {(b_{3} + f_{3, k})}^{2}} \sin (γ_{k} + Ω_{2, k})

(30)

{φ_{1, k}}^{'} = \frac{f_{k, k}}{2 ω_{10}} - \frac{a_{2, k}}{2 a_{1, k} ω_{10}} \sqrt{{(f_{1, k} ω_{20})}^{2} + b_{2}^{2}} \cos (γ_{k} + Ω_{1, k})

(31)

{φ_{2, k}}^{'} = - \frac{a_{1, k}}{2 a_{2, k} ω_{20}} \sqrt{{(f_{4, k} ω_{10})}^{2} + {(b_{3} + f_{3, k})}^{2}} \cos (γ_{k} + Ω_{2, k})

(32)

{γ^{'}}_{k} = {φ^{'}}_{2, k} - {φ^{'}}_{1, k} + σ

(33)

where

γ_{k} = φ_{2, k} - φ_{1, k} + σ T_{1}

Ω_{1, k} = \arctan (f_{1, k} ω_{20} / b_{2})

Ω_{2, k} = \arctan [- f_{4, k} ω_{10} / (b_{3} + f_{3, k})]

Multiplying equation (29) by $a_{1, k}$ and equation (30) by $υ a_{2, k}$ , where

υ = \frac{ω_{20}}{ω_{10}} \sqrt{\frac{{(f_{1, k} ω_{20})}^{2} + b_{2}^{2}}{{(f_{4, k} ω_{10})}^{2} + {(b_{3} + f_{3, k})}^{2}}}

(34)

Adding the resulting equations and integrating, one obtains

a_{1, k}^{2} + υ a_{2, k}^{2} = E_{k}

(35)

where

E_{k}

is a constant of integration proportional to the initial energy of the system in

Δ t_{k}

Since the effect of damping is not taken into account, the system is conservative and the energy level remains constant. Because $υ > 0$ , $a_{1, k}$ and $a_{2, k}$ are bounded and anti-phase with each other. Equation (35) demonstrates that, in the absence of damping, the vibration energy in the system exchanges undamped between these two modes, verifying that the 1:1 internal resonance has been successfully established.

Resonant case, with damping

In the presence of the damping (i.e. $c_{1} \neq 0$ , $c_{2} \neq 0$ , $f_{c 1, k} \neq 0$ , $f_{c 2, k} \neq 0$ ), substituting equation (26) into equations (24) and (25) yields the following solvability conditions

\begin{matrix} [{a^{'}}_{1, k} j ω_{10} - a_{1, k} {φ^{'}}_{1, k} ω_{10} + \frac{1}{2} (c_{1} + f_{c 1, k}) a_{1, k} j ω_{10} + \frac{1}{2} a_{1, k} f_{k, k}] \exp (j φ_{1, k}) \\ - \frac{a_{2, k}}{2} (f_{1, k} j ω_{20} + b_{2}) \exp [j (ε σ T_{0} + φ_{2, k})] = 0 \end{matrix}

(36)

\begin{matrix} [{a^{'}}_{2, k} j ω_{20} - a_{2, k} {φ^{'}}_{2, k} ω_{20} + \frac{1}{2} (c_{2} + f_{c 2, k}) a_{2, k} j ω_{20}] \exp (j φ_{2, k}) \\ - \frac{a_{1, k}}{2} (b_{3} + f_{3, k} + f_{4, k} j ω_{10}) \exp [j (φ_{1, k} - ε σ T_{0})] = 0 \end{matrix}

(37)

Separating real and imaginary parts, one obtains

{a^{'}}_{1, k} = - \frac{1}{2} (c_{1} + f_{c 1, k}) a_{1, k} + \frac{a_{2, k}}{2 ω_{10}} \sqrt{{(f_{1, k} ω_{20})}^{2} + b_{2}^{2}} \sin (γ_{k} + Ω_{1, k})

(38)

{a^{'}}_{2, k} = - \frac{1}{2} (c_{2} + f_{c 2, k}) a_{2, k} - \frac{a_{1, k}}{2 ω_{20}} \sqrt{{(f_{4, k} ω_{10})}^{2} + {(b_{3} + f_{3, k})}^{2}} \sin (γ_{k} + Ω_{2, k})

(39)

By numerical integrations of equations (31) to (33) and (38) and (39), it is shown that, in the presence of damping, the vibration energy is continually exchanged between the flexible manipulator and the vibration absorber, and dissipated by the damping of the vibration absorber.

Discussions about the proposed internal resonant absorber

Internal resonance is an important nonlinear principle, by which two coupled vibration modes can exchange the vibration energy with each other. Therefore, an internal resonant absorber is proposed in this paper to provide a vibration mode used to establish the internal resonance with the controlled mode of the flexible manipulator. Once the internal resonance has been established, the vibration energy can be transferred from the controlled mode to the vibration mode provided by the vibration absorber, then dissipated by the damping of the vibration absorber.

To this end, two conditions should be satisfied. The first condition is the frequency relationship between the vibration absorber and the flexible manipulator. To establish the internal resonance between two vibration modes, their frequency relationship must be commensurable or nearly commensurable. Therefore, the frequency of the vibration absorber in this study should be equal to the controlled mode frequency of the flexible manipulator. The second condition is $υ > 0$ in equation (34). Only $υ > 0$ , can the vibration energy of the flexible manipulator be exchanged between these two modes. When both conditions are satisfied, the internal resonance can be successfully established.

In addition, appropriate damping of the vibration absorber is an important parameter for energy absorption. If the damping is excessive small, the vibration energy cannot be sufficiently absorbed by the vibration absorber. However, if the damping is excessive large, it will block the transfer of the vibration energy from the flexible manipulator to the vibration absorber. Therefore, appropriate damping of the vibration absorber should be obtained with the help of the relevant experiments. Besides, the position of the vibration absorber is an important parameter for enhancing energy absorption. Since the internal resonance relies on the nonlinear coupling, the vibration absorber should be mounted on the appropriate position where nonlinear coupling terms are strong. This can be implemented by optimizing the position of the vibration absorber.

Simulation and analysis

To verify the above theoretical study, a three-link manipulator with last link flexible is researched in numerical simulations, as shown in Figures 1 and 2. The parameters of the links are given as follows: each of the first two links is made of steel, which length is 0.4 m, the square cross-section of side length is 0.05 m. The third link is made of aluminum with elastic modulus 71 GPa and mass density 2710 kg/m³. Its length is 0.8 m, the rectangle cross-section of height $h$ is 0.05 m and width $b$ is 0.002 m. Only the flexural deformation about $y_{3}$ axis is considered in numerical simulations.

Suppose the desired joint motion of the manipulator is

{\begin{array}{l} q_{1} = \frac{π}{50} (1 - \cos \frac{π t}{3}) \\ q_{2} = \frac{2 π}{3} + 0.07 \sin 2 π t \\ q_{3} = \frac{1}{3} \sin \frac{π t}{5} \end{array}

(40)

If the flexible manipulator is not equipped with the vibration absorber, its end-effector response is obtained when moving according to equation (40), as shown in Figure 3. Given the initial disturbance of 20 mm, its vibration response attenuates very slowly due to small structural damping.

Figure 3.

Vibration response of the uncontrolled manipulator.

In order to reduce vibration of the flexible manipulator via the internal resonance, a vibration absorber is attached to the manipulator, as shown in Figure 1. The distance r measured from the Joint 4 to the Joint 3 is 0.4 m. The distance $r_{B}$ measured from the centroid of the Link 4 to the Joint 4 is 0.15 m. The mass $m_{B}$ of the Link 4 is 0.2 kg. At the state of internal resonance, equations (31) to (33) and (38) and (39) are integrated numerically in the absence of damping. The modal amplitude $a_{1}$ and $a_{2}$ is obtained and shown in Figure 4. It is found that if $a_{1}$ is descending then $a_{2}$ is ascending, and vice versa. When $a_{1}$ descends to the trough, $a_{2}$ ascends to the peak at the same time, and vice versa. Since the vibration energy is proportional to the square of the amplitude, it means that the vibration energy is being exchanged between $a_{1}$ and $a_{2}$ as predicted. This phenomenon demonstrates that the internal resonance has been established between the flexible manipulator and the vibration absorber.

Figure 4.

Undamped modal amplitudes.

When the damping of the vibration absorber is considered, e.g. $c_{2} = 0.01$ , the modal amplitudes $a_{1}$ and $a_{2}$ are not only anti-phase but also decreased gradually, as shown in Figure 5. It is proven that the vibration energy of the flexible manipulator can be transferred to the vibration absorber via the internal resonance and dissipated by the damping of the latter. However, since the damping of the vibration absorber is insufficient, there is still small amount of vibration energy which cannot be effectively reduced. In this case, the end-effector deformation of the flexible manipulator decays slowly and the residual vibration cannot be effectively eliminated, as shown in Figure 6.

Figure 5.

Response of modal amplitudes.

Figure 6.

Response of flexible manipulator and vibration absorber.

If the damping of the vibration absorber is increased, e.g. $c_{2} = 0.05$ , the modal amplitudes $a_{1}$ and $a_{2}$ are obtained and are shown in Figure 7. Compared with Figure 5, the vibration energy of the flexible manipulator is reduced more effectively and rapidly. As shown in Figure 8, the end-effector deformation of the flexible manipulator is also effectively controlled. After 10 s, the deformation has been eliminated by 80%. Also, the residual vibration response becomes very small.

Figure 7.

Response of modal amplitudes.

Figure 8.

Response of flexible manipulator and vibration absorber.

However, if the damping of the vibration absorber is excessive, e.g. $c_{2} = 0.1$ , the vibration energy of the flexible manipulator cannot be effectively dissipated, as shown in Figure 9. It implies that, excessive damping will block the exchange of the vibration energy, and thus a large amount of residual vibration energy is left. As a result, the residual vibration response is large, as shown in Figure 10.

Figure 9.

Response of modal amplitudes.

Figure 10.

Response of flexible manipulator and vibration absorber.

To further verify above theoretical study, several virtual prototyping simulations are conducted using well-known ADAMS software (Automatic Dynamic Analysis of Mechanical System, MSC Software Corp.). Since dynamics analysis and vibration solution for the flexible link and the vibration absorber are all implemented by ADAMS software rather than by our own numerical simulation program, more convincing results can be obtained.

Dynamics models of the flexible link and the vibration absorber are established using ADAMS software, as shown in Figure 11. If the damping of the vibration absorber is not taken into account, given the initial disturbance 20 mm, the vibration responses of the flexible manipulator and the vibration absorber are obtained under the internal resonance condition and shown in Figure 12. The “beat” phenomenon can be observed, that is, when the amplitude of the flexible manipulator is decreasing to the minimum, the amplitude of the vibration absorber is increasing to the maximum, and vice versa. It means that the internal resonance has been successfully established and the vibration energy is exchanging between the flexible manipulator and the vibration absorber. If the damping of the vibration absorber is taken into account, e.g. 0.089 in ADAMS, the vibration responses are shown in Figure 13. It takes only 10 s to decrease about 80% of the initial vibration amplitude.

Figure 11.

Models of flexible link and vibration absorber in ADAMS.

Figure 12.

Vibration responses without the damping of vibration absorber (in ADAMS).

Figure 13.

Vibration responses with the damping of vibration absorber (in ADAMS).

Through above simulation and analyses, it is verified that the internal resonance can be used as a feasible tool to reduce nonlinear vibration of the flexible manipulator.

Conclusion

In this paper, a vibration absorption method is suggested to damp out nonlinear vibration of the flexible manipulator based on modal interaction. It is implemented based on a nonlinear dynamics phenomenon known as the internal resonance. The method of multiple scales is used to investigate modal interaction behaviors between the flexible manipulator and vibration absorber. Through theoretical analyses, it is proven that the internal resonance can be successfully established via the modal interaction for dissipating the vibration energy of the flexible manipulator using the damping of the vibration absorber. Since the proposed method absorbs rather than suppresses the vibration energy of the flexible manipulator, it is more suitable for attenuating strong vibration than conventional active control methods based on smart material actuators with limited energy output. In addition, since this method puts an emphasis on constructing an internal energy transfer channel from the flexible manipulator to the vibration absorber rather than directly responding to external excitations, it is particularly convenient to reduce nonlinear vibration induced by unknown or unpredictable external excitations.

Supplemental Material

Supplemental material for Nonlinear vibration control for flexible manipulator using 1: 1 internal resonance absorber

Supplemental material for Nonlinear vibration control for flexible manipulator using 1: 1 internal resonance absorber by Yushu Bian and Zhihui Gao in Journal of Low Frequency Noise, Vibration and Active Control

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by National Natural Science Foundation of China (grant no. 51675017) and Civil Astronautics Pre-Research Project during the “13th Five-Year Plan” (grant no. D020205). In addition, the authors want to thank Dr. Jie Li and Dr. Sicheng Yi for their support.

Appendix 1

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