Theoretical and experimental study on vibration control of flexible manipulator based on internal resonance

Abstract

Theoretical and experimental studies are conducted to control nonlinear vibration of a two-link flexible manipulator via internal resonance. A vibration control method is proposed and an effective vibration absorber is implemented based on a servomotor to establish a 2:1 internal resonance relationship with the flexible manipulator. By way of perturbation analysis, it is proven that internal resonance can be successfully established for the flexible manipulator undergoing rigid motion. In the presence of damping, the vibration energy of the flexible manipulator can be transferred to and dissipated by the vibration absorber via internal resonance. Numerical simulations and experimental investigation have verified the effectiveness and feasibility of the proposed method.

Keywords

Flexible manipulator nonlinear vibration internal resonance modal interaction vibration control

1. Introduction

With the development of robotic manipulators for use at high speed, with heavy loads, precise motion and large work spaces, various vibration problems have become the main reasons for the deterioration of their dynamic performance. Consequently, how to alleviate the influence of adverse vibration of flexible manipulators is an important issue (Heidari et al., 2011, 2015).

Due to the small amount of damping, low stiffness, and strong nonlinearity, vibration control of flexible manipulators becomes extremely difficult, and thus has attracted much attention in the past two decades. Several passive control methods have been put forward, including the optimization of shapes and dimensions, adopting composite materials, etc. (Krishnamurthy et al., 1990; Gordaninejadd et al.,1992; Ghazavi et al., 1993; Cui and Xiao, 2003; Dixit et al., 2006). Furthermore, a number of active control schemes have been developed, including PID control, adaptive control, robust control, neural network control, input shaping, sliding mode control, variable structure control, and so on (Theodore and Ghosal, 2003; Sasaki et al., 2009; Abiko and Yoshida, 2010; Ahmad et al., 2010; Mahamood and Pedro, 2011; Mojtaba et al., 2011; Orszulik and Shan, 2011).

In recent years, a novel method for vibration control based on internal resonance has been suggested. It is different from other methods in that it does not purposely avoid but utilizes sufficient nonlinear coupling between the modes to dissipate vibration energy, exhibiting a potential advantage in combating strong nonlinear vibration. When internal resonance occurs, the vibration energy of one mode may be transferred back and forth between the related modes. In the presence of damping, vibration can be alleviated through the damping of the related modes.

Golnaraghi (1991) and Golnaraghi et al. (1994) were the first to try to reduce large amplitude oscillations of a flexible cantilever beam using internal resonance. A 1:2 internal resonance relationship was established between the beam and the slider. Vibration of the beam was absorbed by the slider and dissipated through the slider damping. Tuer et al. (1993) and Duquette et al. (1993) controlled the vibration of a similar beam using a rotational internal resonance controller at the tip of the beam. In addition, Tuer et al. (1994) proposed two methods based on internal resonance, i.e. the disabled torque and dissipated energy methods, to control a flexible cantilever beam. Oueini and Golnaraghi (1996) built and experimented upon an analog electronic controller. Khajepour et al. (1997) used the center manifold theory to control the vibration of a flexible beam. However, these flexible beam models are treated as a rigid beam connected by a linear torsional spring. Obviously, this assumption is not suitable for the analysis of a real flexible arm. The studies mentioned above have recently been extended to the distributed flexible beam. Pai et al. (2000) designed a higher order internal resonance absorber to decrease the vibration of a cantilevered plate. Ashour and Nayfeh (2002) used an adaptive controller to control a flexible structure via internal resonance and saturation. Yaman and Sen (2007) absorbed the vibration of a cantilever beam using a tip mass and a pendulum attached to the tip mass. Hui et al. (2011) reduced the source mass translation vibration by transferring the internally resonant energy from a symmetrical to an anti-symmetrical mode.

Although remarkable progress has been made, research upon the flexible manipulator is still needed. Firstly, all of the abovementioned studies only concerned the flexible beam and did not involve the flexible manipulator. Unlike the former, the flexible manipulator possesses large-scale joint motion, and thus exhibits much more complex behaviors resulting from the dynamic coupling between joint motion and flexural vibration. Secondly, for simplicity the main system, i.e. the flexible beam, is usually viewed as a linear vibration model. However, the flexible manipulator itself is a complex nonlinear dynamic system. As a result, unreasonable linearization may lead to fundamental mistakes. Thirdly, some authors artificially introduced certain nonlinear coupling terms to improve control results. But the corresponding implementation schemes and experiments were not described in detail and their control effects were not verified.

To the best of our knowledge, little research work has been conducted on controlling large amplitude nonlinear vibration for the flexible manipulator via internal resonance, and no related experimental investigations have been reported either. Therefore, whether internal resonance can be successfully established for the flexible manipulator must be researched. Moreover, the theoretical and practical feasibility of decreasing nonlinear vibration for the flexible manipulator should be examined.

In this paper, a vibration control method based on internal resonance is put forward for the two-link flexible manipulator. Internal resonance is used as an important bridge to transfer and dissipate the nonlinear vibration energy of the main system. An effective vibration controller based on a servomotor is designed to establish a 2:1 internal resonance relationship with the flexible manipulator. In the presence of damping, the vibration energy of the flexible manipulator can be transferred to and dissipated by the vibration absorber through internal resonance. By means of numerical simulations and experimental investigation, the feasibility and effectiveness of this control method are successfully verified.

2. Mathematical model

This paper studies a two-link manipulator with final flexible link, as shown in Figure 1. Link 1 is a hollow uniform rigid beam with length l₁, thickness δ, with a rectangular cross-section of width s₁ and height s₂. Link 2 is a uniform thin Euler–Bernoulli flexible beam with length l₂, with a rectangular cross-section of height h and width b. Only the flexible deformation w(x₂,t) about the y₂ axis in the plane o₂x₂y₂ is considered, where t is the time. The joint o₁ and o₂ provide rigid motion of the links around the z₁ and z₂ axis, denoted by q₁ and q₂, respectively.

Figure 1.

Model of the two-link flexible manipulator with a vibration absorber.

To control the vibration of the flexible manipulator, a vibration absorber consisting of a servomotor and a branch link (i.e. a rigid link) is attached to the flexible link at x₂ = r. The branch link with the length r_c₃ and mass m₃ is connected to the servomotor at an initial angle of 90° to the flexible link. It can rotate around the z₃ axis in the plane o₂x₂y₂. Through adjusting the velocity feedback gain k_d and position feedback gain k_p, the servomotor control system can serve as the damping and elasticity components of a vibration absorber. If the branch link deviates from its initial position, the controller will rectify the deviation angle β in terms of PD control algorithm with appropriate k_d and k_p.

Based on Kane’s method, dynamics equations for the flexible manipulator and the vibration absorber are derived and can be written as (Bian et al., 2011)

M s \overset{··}{χ} + C s \overset{\cdot}{χ} + K s χ = Q

(1)

where

M s \in R n \times n

is the system mass matrix;

C s \in R n \times n

is the system damping matrix;

K s \in R n \times n

is the system stiffness matrix;

Q \in R n

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

χ \in R n

is the set of rigid and flexural degrees of freedom,

χ T

[χ_{1}^{T} χ_{2}^{T}] T

;

χ 1 \in R n R

is the vector describing the rigid joint angles;

χ 2 \in R n F

is the set of flexural displacement of the flexible links; n is the total number of degrees of freedom, n = n_R + n_F; n_R is the number of joints; and n_F is the number of flexural degrees of freedom.

Equation (1) can be separated into two equations describing the dynamics of $χ 1$ and $χ 2$

D \overset{··}{χ} 1 + U \overset{··}{χ} 2 = τ + τ R

(2)

G \overset{··}{χ} 1 + M \overset{··}{χ} 2 + C \overset{\cdot}{χ} 2 + K χ 2 = f

(3)

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

, and

M \in R n F \times n F

are block matrices that form

M s

;

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

; and

f \in R n F

is the flexural component of the nonlinear torque.

Equation (3) describes the flexible dynamics of the flexible manipulator and vibration absorber. It plays an important role in vibration control. By analyzing nonlinear coupling between the flexible manipulator and vibration absorber in equation (3), internal resonance is expected to be established and used to reduce vibration. Since our method aims to control the vibration of the flexible manipulator, and its implementation mainly relies on the vibration equation, only equation (3) is considered here.

The transverse deformation $w (x 2, t)$ of link 2 can be expressed in terms of the mode shapes

w (x 2, t) = \sum_{i = 1}^{n L} u i (x 2) ϕ i (t)

(4)

where

ϕ i (t)

is the ith modal coordinate of link 2,

u i (x 2)

is the ith mode shape satisfying certain geometry and force boundary conditions, and

n L

is the number of flexural degrees of freedom of link 2.

In this paper, only the fundamental mode of the flexible link is studied due to its dominant contribution to the vibration response in most cases. Therefore, the flexible dynamics equations about the fundamental mode coordinates u₁ and the flexural degree of freedom β of vibration absorber are derived from equation (3) and can be written as

m 11 \overset{··}{ϕ} 1 + c 11 \overset{\cdot}{ϕ} 1 + k 11 ϕ 1 = b 1 (\overset{··}{q} 1 + \overset{··}{q} 2) β + b 1 \overset{\cdot}{β} 2 + b 1 \overset{··}{β} β - b 2 \overset{··}{q} 1 - b 3 \overset{··}{q} 2 + b 1 (\overset{\cdot}{q} 1 + \overset{\cdot}{q} 2) 2 - b 4 {\overset{\cdot}{q}}_{1}^{2} + f 1 (ϕ 1, \overset{\cdot}{ϕ} 1, \overset{··}{ϕ} 1, β, \overset{\cdot}{β}, \overset{··}{β}, \overset{\cdot}{q} 1, \overset{··}{q} 1, \overset{\cdot}{q} 2, \overset{··}{q} 2)

(5)

m 22 \overset{··}{β} + c 22 \overset{\cdot}{β} + k 22 β = - b 1 (\overset{··}{q} 1 + \overset{··}{q} 2) ϕ 1 + b 1 \overset{··}{ϕ} 1 β + (b 5 \overset{··}{q} 1 + b 6 \overset{··}{q} 2) β - b 7 \overset{··}{q} 1 - b 8 \overset{··}{q} 2 - b 9 {\overset{\cdot}{q}}_{1}^{2} - b 6 (\overset{\cdot}{q} 1 + \overset{\cdot}{q} 2) 2 + f 2 (ϕ 1, \overset{\cdot}{ϕ} 1, \overset{··}{ϕ} 1, β, \overset{\cdot}{β}, \overset{··}{β}, \overset{\cdot}{q} 1, \overset{··}{q} 1, \overset{\cdot}{q} 2, \overset{··}{q} 2)

(6)

where

m 11 = (m 3 + m B) u_{1 r}^{2} + ρ \int_{0}^{l 2} u_{1}^{2} (x 2) d x 2

;

m B

is the mass of the vibration absorber;

u 1 r = u 1 (x 2) | x 2 = r

;

c 11

is the damping of the flexible link;

k 11 = EI \int_{0}^{l 2} \frac{d 4 u 1 (x 2)}{d x_{2}^{4}} u 1 (x 2) d x 2

;

b 1 = m 3 r c 3 u 1 r

; E is the Young’s modulus of the flexible link; EI is the flexural rigidity of the flexible link;

b 2 = (m 3 + m B) u 1 r (r + l 1 \cos q 2) + I_{zz}^{1} + l 1 ρ \int_{0}^{l 2} u 1 (x 2) d x 2 \cos q 2 + ρ \int_{0}^{l 2} u 1 (x 2) x 2 d x 2

;

I_{zz}^{1}

is the moment of inertia of link1 with respect to its centroid axis; ρ is the mass per length of link2;

b 3 = (m 3 + m B) u 1 r r + ρ \int_{0}^{l 2} u 1 (x 2) x 2 d x 2

;

b 4 = m 3 u 1 r l 1 \sin q 2

;

m 22 = I_{zz}^{3}

;

I_{zz}^{3}

is the moment of inertia of the rigid link of the absorber with respect to its centroid axis;

c 22 = k d

, which is related to the damping of vibration absorber and adjustable in an active form;

k 22 = k p

, which is related to the stiffness of vibration absorber and adjustable in an active form;

b 5 = m 3 r c 3 (l 1 \cos q 2 + r)

;

b 6 = m 3 r c 3 r

;

b 7 = - m 3 r c 3 l 1 sin q 2 + I_{zz}^{3}

;

b 8 = I_{zz}^{3}

;

b 9 = m 3 r c 3 l 1 \cos q 2

; and f₁ and f₂ are the quadratic nonlinearity terms of

ϕ 1

and β.

From equation (6), it is seen that $k d$ and $k p$ can be used to establish the internal resonance conditions.

Using the transformations

ϕ * = \frac{ϕ 1}{l 2}, β * = β, τ = ω β t, q_{1}^{*} = q 1, q_{2}^{*} = q 2

(7)

where

ω β = \sqrt{k 22 / m 22}

, equations (5) and (6) are nondimensionalized as

\overset{··}{ϕ} * + η 11 \overset{\cdot}{ϕ} * + ω_{ϕ β}^{2} ϕ * = d 1 ({\overset{··}{q}}_{1}^{*} + {\overset{··}{q}}_{2}^{*}) β * + d 1 \overset{\cdot}{β} 2 + d 1 \overset{··}{β} * β * - d 2 {\overset{··}{q}}_{1}^{*} - d 3 {\overset{··}{q}}_{2}^{*} + d 1 ({\overset{\cdot}{q}}_{1}^{*} + {\overset{\cdot}{q}}_{2}^{*}) 2 - d 4 {\overset{\cdot}{q}}_{1}^{*} 2 + f 3 (ϕ *, \overset{\cdot}{ϕ} *, \overset{··}{ϕ} *, β *, \overset{\cdot}{β} *, \overset{··}{β} *, {\overset{\cdot}{q}}_{1}^{*}, {\overset{··}{q}}_{1}^{*}, {\overset{\cdot}{q}}_{2}^{*}, {\overset{··}{q}}_{2}^{*})

(8)

\overset{··}{β} * + η 22 \overset{\cdot}{β} * + β * = - d 10 ({\overset{··}{q}}_{1}^{*} + {\overset{··}{q}}_{2}^{*}) ϕ * + d 10 \overset{··}{ϕ} * β * + (d 5 {\overset{··}{q}}_{1}^{*} + d 6 {\overset{··}{q}}_{2}^{*}) β * - d 7 {\overset{··}{q}}_{1}^{*} - d 8 {\overset{··}{q}}_{2}^{*} - d 9 {\overset{\cdot}{q}}_{1}^{*} 2 - d 6 ({\overset{\cdot}{q}}_{1}^{*} + {\overset{\cdot}{q}}_{2}^{*}) 2 + f 4 (ϕ *, \overset{\cdot}{ϕ} *, \overset{··}{ϕ} *, β *, \overset{\cdot}{β} *, \overset{··}{β} *, {\overset{\cdot}{q}}_{1}^{*}, {\overset{··}{q}}_{1}^{*}, {\overset{\cdot}{q}}_{2}^{*}, {\overset{··}{q}}_{2}^{*})

(9)

where

η 11 = c 11 / m 11 ω β

;

ω ϕ β = ω ϕ / ω β

;

ω_{ϕ}^{2} = k 11 / m 11

;

d 1 = b 1 / m 11 l 2

;

d 2 = b 2 / m 11 l 2

;

d 3 = b 3 / m 11 l 2

;

d 4 = b 4 / m 11 l 2

;

η 22 = c 22 / (m 22 ω β)

;

d 5 = b 5 / m 22

;

d 6 = b 6 / m 22

;

d 7 = b 7 / m 22

;

d 8 = b 8 / m 22

;

d 9 = b 9 / m 22

; and

d 10 = l 2 b 1 / m 22

;

(\cdot)

represents the derivatives with respect to τ.

The method of multiple scales is used to solve equations (8) and (9). To this end, a scaling factor $ɛ << 1$ is introduced to the equations through the transformations

ϕ * = ɛ φ, β * = ɛ ψ, {\overset{\cdot}{q}}_{1}^{*} = ɛ \overset{\cdot}{Ω} 1, {\overset{\cdot}{q}}_{2}^{*} = ɛ \overset{\cdot}{Ω} 2, η 11 = ɛ ζ 11, η 22 = ɛ ζ 22

(10)

Substituting equation (10) into equations (8) and (9) gives

\overset{··}{φ} + ω_{ϕ β}^{2} φ = - d 2 \overset{··}{Ω} 1 - d 3 \overset{··}{Ω} 2 + ɛ [- ζ 11 \overset{\cdot}{φ} + d 1 (\overset{··}{Ω} 1 + \overset{··}{Ω} 2) ψ + d 1 \overset{\cdot}{ψ} 2 + d 1 \overset{··}{ψ} ψ + d 1 (\overset{\cdot}{Ω} 1 + \overset{\cdot}{Ω} 2) 2 - d 4 {\overset{\cdot}{Ω}}_{1}^{2}] + o (ɛ 2)

(11)

\overset{··}{ψ} + ψ = - d 7 \overset{··}{Ω} 1 - d 8 \overset{··}{Ω} 2 + ɛ [- ζ 22 \overset{\cdot}{ψ} + (d 5 \overset{··}{Ω} 1 + d 6 \overset{··}{Ω} 2) ψ - d 10 (\overset{··}{Ω} 1 + \overset{··}{Ω} 2) φ + d 10 \overset{··}{φ} ψ - d 9 {\overset{\cdot}{Ω}}_{1}^{2} - d 6 (\overset{\cdot}{Ω} 1 + \overset{\cdot}{Ω} 2) 2] + o (ɛ 2)

(12)

3. Perturbation solution

Using the method of multiple scales, the timescales are defined as

T i = ɛ i τ (i = 0, 1)

(13)

Then, the first-order approximate solutions to equations (8) and (9) take the form

φ (τ, ɛ) = φ 0 (T 0, T 1) + ɛ φ 1 (T 0, T 1)

(14)

ψ (τ, ɛ) = ψ 0 (T 0, T 1) + ɛ ψ 1 (T 0, T 1)

(15)

According to equation (13), the time derivatives about τ are transformed in terms of T₀ and T₁, i.e.

\frac{d}{d τ} = D 0 + ɛ D 1

(16)

\frac{d 2}{d τ 2} = D_{0}^{2} + 2 ɛ D 0 D 1

(17)

where

D i = \partial / \partial T i

and (i = 0, 1).

Substituting equations (14)–(17) into equations (11) and (12), then equating coefficients of the same order of ɛ on both sides, the following differential equations are obtained.

Order ( $ɛ 0$ )

D_{0}^{2} φ 0 + ω_{ϕ β}^{2} φ 0 = - d 2 D_{0}^{2} Ω 1 - d 3 D_{0}^{2} Ω 2

(18)

D_{0}^{2} ψ 0 + ψ 0 = - d 7 D_{0}^{2} Ω 1 - d 8 D_{0}^{2} Ω 2

(19)

Order ( $ɛ 1$ )

D_{0}^{2} φ 1 + ω_{ϕ β}^{2} φ 1 = - 2 D 0 D 1 φ 0 - ζ 11 D 0 φ 0 + d 1 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) ψ 0 + d 1 (D 0 ψ 0) 2 + d 1 D_{0}^{2} ψ 0 ψ 0 + d 1 (D 0 Ω 1 + D 0 Ω 2) 2 - d 4 (D 0 Ω 1) 2

(20)

D_{0}^{2} ψ 1 + ψ 1 = - 2 D 0 D 1 ψ 0 - ζ 22 D 0 ψ 0 + (d 5 D_{0}^{2} Ω 1 + d 6 D_{0}^{2} Ω 2) ψ 0 - d 10 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) φ 0 + d 10 D_{0}^{2} φ 0 ψ 0 - d 9 (D 0 Ω 1) 2 - d 6 (D 0 Ω 1 + D 0 Ω 2) 2

(21)

Equations (18)–(21) can vary with the joint motion, which is an important difference from the flexible structure.

The solutions to equations (18) and (19) take the complex form

φ 0 = A 1 (T 1) \exp (j ω ϕ β T 0) - g 1 + cc

(22)

ψ 0 = A 2 (T 1) \exp (jT 0) - g 2 + cc

(23)

where the coefficients

A 1 (T 1)

and

A 2 (T 1)

are functions of the slow time T₁,

cc

denotes the complex conjugate of preceding term,

g 1 = (d 2 D_{0}^{2} Ω 1 + d 3 D_{0}^{2} Ω 2) / (2 ω_{ϕ β}^{2})

, and

g 2 = (d 7 D_{0}^{2} Ω 1 + d 8 D_{0}^{2} Ω 2) / 2

Substituting equations (22) and (23) into equations (20) and (21) gives

D_{0}^{2} φ 1 + ω_{ϕ β}^{2} φ 1 = - 2 j ω ϕ β A' 1 \exp (j ω ϕ β T 0) - ζ 11 j ω ϕ β A 1 \exp (j ω ϕ β T 0) + d 1 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) [A 2 \exp (jT 0) - g 2] + d 1 [- A_{2}^{2} \exp (2 jT 0) + A 2 \bar{A} 2] + d 1 [g 2 A 2 \exp (jT 0) - A_{2}^{2} \exp (2 jT 0) - A 2 \bar{A} 2] + d 1 (D 0 Ω 1 + D 0 Ω 2) 2 - d 4 (D 0 Ω 1) 2

(24)

D_{0}^{2} ψ 1 + ψ 1 = - 2 jA' 2 \exp (jT 0) - ζ 22 jA 2 \exp (jT 0) + (d 5 D_{0}^{2} Ω 1 + d 6 D_{0}^{2} Ω 2) [A 2 \exp (jT 0) - g 2] - - d 10 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) [A 1 \exp (j ω ϕ β T 0) - g 1] - d 10 ω_{ϕ β}^{2} {A 1 A 2 \exp [j (1 + ω ϕ β) T 0] + A 1 \bar{A} 2 \exp [j (ω ϕ β - 1) T 0] - 2 g 2 A 1 \exp (j ω ϕ β T 0)} - d 9 (D 0 Ω 1) 2 - d 6 (D 0 Ω 1 + D 0 Ω 2) 2

(25)

where

()' \equiv \partial () / \partial T 1

To obtain solutions of equations (24) and (25), the solvability conditions should be determined.

4. Analysis of internal resonance

Whether internal resonance can be successfully established for the flexible manipulator should be investigated. Since only the second-order nonlinear terms exist in equations (5) and (6), the 2:1 internal resonance condition is analyzed. Therefore, a detuning parameter σ is introduced as

ω ϕ β = 2 - ɛ σ

(26)

and

2 T 0

and

(ω ϕ β - 1) T 0

are expressed as

2 T 0 = ω ϕ β T 0 + σ T 1

(27)

(ω ϕ β - 1) T 0 = T 0 - σ T 1

(28)

Substituting equations (27) and (28) into equations (24) and (25) gives

D_{0}^{2} φ 1 + ω_{ϕ β}^{2} φ 1 = - 2 j ω ϕ β A' 1 \exp (j ω ϕ β T 0) - ζ 11 j ω ϕ β A 1 \exp (j ω ϕ β T 0) + d 1 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) [A 2 \exp (jT 0) - g 2] + d 1 {- A_{2}^{2} \exp [j (ω ϕ β T 0 + σ T 1)] + A 2 \bar{A} 2} + d 1 {g 2 A 2 \exp (jT 0) - A_{2}^{2} \exp [j (ω ϕ β T 0 + σ T 1)] - A 2 \bar{A} 2} + d 1 (D 0 Ω 1 + D 0 Ω 2) 2 - d 4 (D 0 Ω 1) 2

(29)

D_{0}^{2} ψ 1 + ψ 1 = - 2 jA' 2 \exp (jT 0) - ζ 22 jA 2 \exp (jT 0) + (d 5 D_{0}^{2} Ω 1 + d 6 D_{0}^{2} Ω 2) [A 2 \exp (jT 0) - g 2] - - d 10 (D_{0}^{2} Ω 1 + D_{0}^{2} Ω 2) [A 1 \exp (j ω ϕ β T 0) - g 1] - d 10 ω_{ϕ β}^{2} {A 1 A 2 \exp [j (1 + ω ϕ β) T 0] + A 1 \bar{A} 2 \exp [j (ω ϕ β - 1) T 0] - 2 g 2 A 1 \exp (j ω ϕ β T 0)} - d 9 (D 0 Ω 1) 2 - d 6 (D 0 Ω 1 + D 0 Ω 2) 2

(30)

From equations (29) and (30), the solvability conditions are

- 2 j ω ϕ β A' 1 - ζ 11 j ω ϕ β A 1 - 2 d 1 A_{2}^{2} \exp (j σ T 1) = 0

(31)

- 2 jA' 2 - ζ 22 jA 2 + g 3 A 2 - d 10 ω_{ϕ β}^{2} A 1 \bar{A} 2 \exp (- j σ T 1) = 0

(32)

where

g 3 = d 5 D_{0}^{2} Ω 1 + d 6 D_{0}^{2} Ω 2

For convenience, $A 1 (T 1)$ and $A 2 (T 1)$ are expressed in the polar form as

A 1 = \frac{1}{2} a 1 \exp (j θ 1)

(33)

A 2 = \frac{1}{2} a 2 \exp (j θ 2)

(34)

where a₁, a₂,

θ 1

, and

θ 2

are real functions of T₁; and a₁ and a₂ are defined as the modal amplitudes.

Substituting equations (33) and (34) into equations (31) and (32), then separating the result into real and imaginary parts, gives

a' 1 = - \frac{1}{2} ζ 11 a 1 + \frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \sin γ

(35)

a' 2 = - \frac{1}{2} ζ 22 a 2 - \frac{1}{4} d 10 ω_{ϕ β}^{2} a 1 a 2 \sin γ

(36)

a 1 θ' 1 = \frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \cos γ

(37)

a 2 θ' 2 = - \frac{1}{2} g 3 a 2 + \frac{1}{4} d 10 ω_{ϕ β}^{2} a 1 a 2 \cos γ

(38)

where

γ = θ 1 - 2 θ 2 - σ T 1

(39)

Eliminating $θ 1$ and $θ 2$ from equations (37)–(39), we obtain

a 1 γ' = \frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \cos γ + a 1 g 3 - \frac{1}{2} d 10 ω_{ϕ β}^{2} a_{1}^{2} \cos γ - σ a 1

(40)

Therefore, the solvability conditions are reduced to the solution of equations (35), (36), and (40).

To investigate the energy transfer between the flexible manipulator and vibration absorber, the undamped case (i.e. $ζ 1 = ζ 2 = 0$ ) is studied first. Then, equations (35) and (36) can be written as

a' 1 = \frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \sin γ

(41)

a' 2 = - \frac{1}{4} d 10 ω_{ϕ β}^{2} a 1 a 2 \sin γ

(42)

letting

υ = \frac{2 d 1}{d 10 ω_{ϕ β}^{3}} = \frac{2 m 22}{m 11 l_{2}^{2} ω_{ϕ β}^{3}}

(43)

Multiplying equation (41) by a₁ and equation (42) by $va 2$ , then adding and integrating, leads to

a_{1}^{2} + υ a_{2}^{2} = ς

(44)

where ς is a constant of integration, which is proportional to the initial energy in the system.

In equation (43), $υ > 0$ . It is shown in equation (44) that a₁ and a₂ are always bounded. As a result, if the modal amplitude a₁ is periodic, then the modal amplitude a₂ will be periodic and anti-phase with a₁. Equation (44) indicates that, in the absence of damping, the energy in the system is continuously exchanged between the fundamental mode of the flexible manipulator and the vibration mode of the vibration absorber. It is proven that internal resonance can be successfully established for the flexible manipulator undergoing rigid motion. In this study, the frequency of the vibration absorber is adjustable and thus can be used to establish and maintain the internal resonance relationship.

5. Vibration control based on internal resonance

Although vibration energy can be transferred via internal resonance from the flexible manipulator to the vibration absorber, it cannot be dissipated in the absence of damping. To reduce vibration, damping of the vibration absorber should be taken into account.

In the presence of damping (i.e. $ζ 11 > 0$ and $ζ 22 > 0$ ), the steady-state response is

a' 1 = a' 2 = γ' = 0

(45)

As a result, from equations (35), (36) and (40)

- \frac{1}{2} ζ 11 a 1 + \frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \sin γ = 0

(46)

- \frac{1}{2} ζ 22 a 2 - \frac{1}{4} d 10 ω_{ϕ β}^{2} a 1 a 2 \sin γ = 0

(47)

\frac{1}{2 ω ϕ β} d 1 a_{2}^{2} \cos γ + a 1 g 3 - \frac{1}{2} d 10 ω_{ϕ β}^{2} a_{1}^{2} \cos γ - σ a 1 = 0

(48)

By inspection, it is easy to find that the system possesses an infinite number of equilibrium points defined by

a 1 = 0, a 2 = 0, γ \in R

(49)

Therefore, by evaluating the Jacobian, one can ascertain the stability of the system.

The Jacobian matrix for this case is

[μ 1 000 μ 2 0000]

(50)

where

μ 1 = - \frac{1}{2} ζ 11

and

μ 2 = - \frac{1}{2} ζ 22

The corresponding eigenvalues are $(μ 1, μ 2, 0)$ . Since $μ 1 < 0$ and $μ 2 < 0$ , the modal amplitudes a₁ and a₂ are stable, as indicated by the negative eigenvalues.

By way of numerical integrations of equations (35), (36), and (40), it is found that, in the presence of damping, the vibration energy is continually exchanged between the flexible manipulator and vibration absorber, and dissipated simultaneously by the damping of the latter.

6. Theoretical simulations

6.1. Numerical simulations

To verify the theoretical analysis above, a two-link manipulator with final flexible link was studied, as shown in Figure 1. Its parameters are listed in Table 1.

Table 1.

Parameters of the two-link flexible manipulator.

Component	Parameter	Value
Link 1	l ₁	0.4 m
	δ	4 mm
	s ₁	70 mm
	s ₂	50 mm
	Mass density	7800 kg/m³
	Elastic modulus	210 GPa
Link 2	l ₂	0.8 m
	h	30 mm
	b	5 mm
	Mass density	7800 kg/m³
	Elastic modulus	210 GPa
Branch link	m ₃	0.04 kg
	r_c ₃	0.3 m
Motor at o₂	Mass	3.55 kg
Branch link servomotor	Mass	0.65 kg

Suppose the rigid motions of the flexible manipulator are

q 1 = \frac{π}{6} \sin (\frac{\sqrt{2}}{10} t), q 2 = \frac{π}{6} \cos (\frac{π}{10} t + \frac{π}{2})

(51)

If the flexible manipulator is not equipped with the vibration absorber, given the initial disturbance $w (l 2, 0) = 80 mm$ , its end-effector deformation is obtained, as shown in Figure 2. Due to the small amount of structural damping in the flexible link, the vibration response of the end-effector decreases very slowly.

Figure 2.

Response of the end-effector.

To control the vibration of flexible manipulator, a vibration absorber based on a servomotor is attached to the flexible link. In this example, the fundamental frequency of the flexible manipulator is 4.34 Hz. According to the 2:1 internal resonance condition, the frequency of the vibration absorber is tuned to 2.17 Hz. In the absence of damping, by numerically integrating equations (35), (36), and (40), the modal amplitudes a₁ and a₂ are obtained, and are shown in Figure 3. As can be seen, the peaks and troughs of these two modal amplitudes are exactly anti-phase with each other. Since the vibration energy is proportional to the square of the amplitude, the vibration energy of a₁ is the largest in the peak and the smallest in the trough, as is the case for a₂. Therefore, if a₁ descends and a₂ ascends, it means that the vibration energy is being transferred from a₁ to a₂, and vice versa. This phenomenon indicates that internal resonance has been established and the vibration energy is continuously exchanging between the two vibration modes.

Figure 3.

Relationship of a₁ and a₂.

The end-effector deformation of the flexible manipulator is shown in Figure 4. As can be seen, it descends almost to zero abruptly, resulting in the beat phenomenon. Also, a similar phenomenon arises in the vibration responses of the vibration absorber, as shown in Figure 5. For comparison, these two figures are shown in Figure 6. It is found that when the amplitude of the flexible manipulator decreases abruptly, the amplitude of the vibration absorber increases abruptly at the same time, and vice versa. It is again verified that internal resonance has been successfully established, and the vibration energy is being exchanged between the flexible manipulator and vibration absorber.

Figure 4.

Undamped response of the flexible manipulator.

Figure 5.

Undamped response of the vibration absorber.

Figure 6.

Comparison of vibration responses.

When damping of the vibration absorber is taken into account, e.g. $k d = 0.0049$ , by numerically integrating equations (35), (36), and (40), the end-effector deformation of the flexible manipulator is shown in Figure 7. As can be seen, the deformation almost decreases to zero at 4 s. At the same time, the rotation angle of the vibration absorber increases to the maximum, as shown in Figure 8. It is proven that the vibration energy of the flexible manipulator is being transferred into the vibration absorber. However, it can be seen in Figure 7 that there is still a small amount of vibration energy that cannot be effectively dissipated due to insufficient damping of the vibration absorber.

Figure 7.

Damped response of the flexible manipulator.

Figure 8.

Damped response of the vibration absorber.

When the damping of the vibration absorber is increased, e.g. $k d = 0.0083$ , the end-effector deformation of the flexible manipulator is as shown in Figure 9. Compared with Figure 7, the residual vibration response of the flexible manipulator becomes very small. Also, compared with the uncontrolled case, the vibration response of the flexible manipulator has been effectively controlled, as shown in Figure 10. After 6 s, the deformation has been reduced by 80%. It is verified that this control method is effective in reducing the vibration of the flexible manipulator.

Figure 9.

Damped response of the flexible manipulator.

Figure 10.

Comparison of vibration responses.

6.2. Virtual prototyping simulations

To examine the correctness of the theoretical analysis and provide good guidance for experimental investigation, virtual prototyping technology is used in this section to validate the proposed method via ANSYS (Pennsylvania, USA), ADAMS (MSC Software Corporation, Newport Beach, CA), and MATLAB software (MathWorks Corporation, Massachusetts, USA). Parameters of the flexible manipulator and vibration absorber are listed in Table 1.

Suppose the rigid motions of the flexible manipulator are the same as those given by numerical simulations. If the flexible manipulator is not equipped with the vibration absorber, given the initial disturbance $w (l 2, 0) = 80 mm$ , its end-effector deformation is shown in Figure 11. As can be seen, it attenuates very slowly due to the small amount of structural damping, resulting in bad tracking accuracy.

Figure 11.

End-effector deformation of the flexible manipulator without the vibration absorber.

To control the vibration of the flexible manipulator, the vibration absorber is attached to the flexible manipulator. The fundamental frequency of the flexible manipulator is 4.34 Hz and the frequency of the vibration absorber is tuned to 2.17 Hz. In the absence of damping, i.e. $k d = 0$ , the end-effector deformation of the flexible manipulator and the rotation angle of the vibration absorber are obtained at the 2:1 internal resonance condition, as shown in Figure 12 and Figure 13, respectively. The beat phenomenon can be observed, and the peaks of one envelope curve coincide with the troughs of the other. In particular, the end-effector deformation of the flexible manipulator descends to the minimum at about t = 2 s, 6 s, 10 s, and 13 s, and the rotation angle of the vibration absorber ascends to the maximum at the same time. It is verified that internal resonance has been successfully established, and the vibration energy is being exchanged between the flexible manipulator and the vibration absorber.

Figure 12.

Undamped response of the flexible manipulator.

Figure 13.

Undamped response of the vibration absorber.

When the damping of the vibration absorber is taken into account, letting $k d =$ 8 (the same unit as that in ADAMS software), the end-effector responses are obtained, and are shown in Figure 14.

Figure 14.

End-effector deformation under damping.

For the convenience of comparison, the end-effector deformation without the vibration absorber (as shown in Figure 11) and that with the desired damping (as shown in Figure 14) are incorporated into Figure 15. It can be seen that the flexible manipulator’s end-effector deformation has been effectively reduced based on internal resonance.

Figure 15.

Comparison of vibration responses with and without an absorber.

Through the virtual prototyping simulations above, it is demonstrated that the proposed method is effective in controlling nonlinear vibration of the flexible manipulator undergoing rigid motion.

7. Experimental investigation

In this section, experimental investigations are conducted to examine the feasibility of the proposed method.

7.1. Experimental setup

An experimental setup was designed and developed as shown in Figures 16 and 17, including the manipulator, the actuator components, the detection devices, the vibration absorber, and the control system.

Figure 16.

Scheme of the experimental setup.

Figure 17.

Photograph of the experimental setup.

The manipulator consists of two links: one is rigid and the other is flexible. These links possess the same parameters as those in the virtual prototyping simulations. Each joint is composed of a motor, a reducer, and a flange, as shown in Figure 18. To enhance the stiffness of the experimental setup, joint 1 is installed on the high stiffness support and the support is fixed on the cast-iron platform. The accelerometer is stuck to the end-effector of the flexible link and used to measure the acceleration signals. The dynamic signal analyzer is used to collect and process the acceleration signals, then transfer them to the computer.

Figure 18.

Photograph of the joints and supports.

7.2. Experimental investigation

7.2.1. Measuring vibration responses

In the absence of the vibration absorber, the vibration responses of the flexible manipulator undergoing rigid motions were measured. Two disturbances are exerted on the end-effector of the flexible manipulator at about t = 0 s and 14 s during motion. The acceleration signals of the end-effector are detected by the accelerometer and processed by the dynamic signal analyzer, as shown in Figure 19. Due to the small amount of structural damping in the flexible link, the vibration responses of the end-effector decrease very slowly. After each disturbance, it takes about 10 s to decrease to 20% of the initial amplitude.

Figure 19.

Responses of the end-effector without the vibration absorber.

7.2.2. Verifying 2:1 internal resonance

To establish internal resonance, a vibration absorber based on a servomotor was attached to the flexible link. The fundamental frequency of the flexible manipulator was measured to be 4.55 Hz via fast Fourier transformation. Using the tuning module of the PMAC controller (Delta Tau Data System Corporation, Chatsworth, CA, USA), when $k p = 3000$ , the frequency of vibration absorber was measured to be 2.237 Hz. In this case, the ratio of the fundamental frequency of the flexible manipulator to the frequency of the vibration absorber is close to 2:1. The end-effector responses of the flexible manipulator and the rotation angles of the vibration absorber were measured, when subjected to the abovementioned two disturbances, in the absence of damping, i.e. $k d = 0$ , as shown in Figures 20 and 21. As can be seen, the envelope curve of the end-effector responses decreases abruptly at some time when the envelope curve of the rotation angles increases abruptly. The beat phenomenon can be observed, and the peaks of one envelope curve coincide with the troughs of the other. It is verified that internal resonance has been successfully established, and vibration energy is being exchanged between the flexible manipulator and the vibration absorber.

Figure 20.

Undamped responses of the flexible manipulator ( $k p = 3000, k d = 0$ ).

Figure 21.

Undamped responses of the vibration absorber ( $k p = 3000, k d = 0$ ).

When $k p = 2000$ , the frequency of the vibration absorber was measured to be 1.736 Hz. In this case, the ratio of the fundamental frequency of the flexible manipulator to the frequency of the vibration absorber is far from 2:1. The end-effector responses of the flexible manipulator and the rotation angles of the vibration absorber were measured when subjected to the abovementioned two disturbances, in the absence of damping, as shown in Figure 22 and Figure 23. As can be seen, no beat phenomena can be observed. Internal resonance cannot be established. Therefore, there is no energy exchange between the flexible manipulator and the vibration absorber.

Figure 22.

Undamped response of the flexible manipulator ( $k p = 2000, k d = 0$ ).

Figure 23.

Undamped response of the vibration absorber ( $k p = 2000, k d = 0$ ).

7.2.3. Vibration control effect

When damping of the vibration absorber is taken into account for a 2:1 internal resonance condition, i.e. $k d \neq 0$ , the vibration absorber is expected to dissipate the vibration energy of the flexible manipulator. To this end, letting $k d = 5000$ , the end-effector responses of the flexible manipulator undergoing two disturbances are shown in Figure 24. It is seen that the vibration responses of the end-effector can be eliminated effectively and rapidly by the damping of the vibration absorber.

Figure 24.

Damped response of the flexible manipulator ( $k p = 3000, k d = 5000$ ).

To compare the control effect, the vibration responses of the end-effector with and without the vibration absorber are shown in Figure 25. Take the first disturbance at t = 0 s as an example. In the absence of the vibration absorber, it takes about 2.4 s to decrease the end-effector vibration responses by 60% and about 10 s to decrease the end-effector vibration responses by 90%. In the presence of the vibration absorber with $k d = 5000$ , it takes about 0.64 s to decrease the end-effector vibration responses by 60% and about 2.3 s to decrease the end-effector vibration responses by 90%.

Figure 25.

Comparison of vibration responses.

From the above experiments, it is verified that the proposed control method is feasible and effective in reducing nonlinear vibration of the flexible manipulator.

8. Conclusions

To attenuate nonlinear vibration of the flexible manipulator, a vibration control method based on internal resonance is proposed. A vibration absorber is designed based on a servomotor and used to establish the internal resonance relationship with the flexible manipulator. To satisfy the 2:1 internal resonance conditions, the frequency of the vibration absorber should be tuned to half the fundamental frequency of the flexible manipulator. By means of perturbation analysis, it is proven that internal resonance can be successfully established for a flexible manipulator undergoing rigid motion. At the internal resonance state, the vibration energy can be exchanged between the flexible manipulator and vibration absorber. In the presence of damping, the vibration energy of the flexible manipulator can be transferred to and dissipated by the vibration absorber via internal resonance. Through numerical simulations and experimental investigation, it is verified that the proposed method is effective for the control of nonlinear vibration of the flexible manipulator. Future work will focus on adaptive adjustment of the frequency and damping of the vibration absorber.

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 is supported by National Natural Science Foundation of China (grant number 51675017) and Pre-Research Foundation of GEH PLA (grant number 9140A34030315KG18081).

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