An active control strategy for vibration control of an axially translating beam

Abstract

Axially translating beams are widely seen in engineering applications. An active vibration control strategy based on a modified fuzzy sliding mode control is developed for controlling and stabilizing the motion of the beam. Geometric nonlinearity of the beam is considered. In the development of the control strategy, the governing equation of the beam is transformed into a multi-dimensional dynamic system with the Galerkin method of 6th order. An active control strategy is developed corresponding to the dynamic system, such that the control strategy can be used for multi-dimensional systems. Numerical simulations are performed with application of the control strategy developed. The effectiveness of the active control strategy is demonstrated in controlling and stabilizing the chaotic motion of the translating beam.

Keywords

Active vibration control chaotic motion control strategy fuzzy sliding mode control nonlinear dynamics translating beam

1. Introduction

Axially translating beams and engineering structures with translating materials have been investigated extensively for their stability, vibration control, and control strategy applications in mechanical and industrial engineering. Significant progress has been accomplished and numerous findings of the investigations have been applied in industrial practices. To increase the efficiency of ultra-high-rise elevators, for example, the linear dynamics and stability of a class of horizontally and vertically translating beams with an attached spring-inertia-damper were investigated by Zhu and Ni (2000). It was concluded by the authors that the stabilization of a translating medium with varying length required the suppression of the amplitudes of the responses.

The effects of the translating speed of the power transmission chains, aerial cableways, textile fibers and paper sheets on their vibration features were also investigated by researchers and engineers. Wickert (1992) quantitatively studied the nonlinear behavior and translating speed of an axially translating beam and string. The author even claimed that higher-order equilibria of the translating beam can help complete the dynamic investigation of the beam.

In the dynamics investigations of translating media, such as high-speed magnetic tapes, band saw and transport cables, the linear equation of motion of a translating beam with constant length between boundaries was derived by some researchers based on the Euler-Bernoulli beam and Lagrangian frame of reference (Zhu, 2002). The rate of change of energies from the constant length of the beam as well as the dynamic stability was also investigated by the researcher. With regard to the existing vibration, which is harmful to cranes, strips in a thin metal-sheet production line and deployable robot, the nonlinear model of a tensioned beam with constant length between boundaries was established by Yang et al. (2005). A robust adaptive vibration control strategy was developed to simply suppress and reduce the undesired vibration of the beam. In the investigation of possibly large vibration of a slender beam, which is held by two different types of supports, the nonlinear governing equations of a Euler-Bernoulli beam were established (Humer, 2013). It was recognized by the author that the frequency of the vibration became lower as the length of the beam became shorter, which was explained by the larger inertia of the portion of the beam.

Generally, the governing equations of the axially translating media are in the forms of partial differential equations. In solving the governing equations, much attention of the previous studies has been paid to dynamic systems of the translating beam with single or two dimensions on translating beams, though some studies suggested the considerations of multi-dimensional dynamic system (Wickert, 1992). The solutions are then employed in developing the control strategies or techniques for controlling the linear and nonlinear vibrations of the axially translating media.

In engineering applications, it is desirable to control the nonlinear or irregular responses of translating beams. Numerous control techniques and theories are available in the field. In 1992, a control theory, namely the theory of sliding modes control (SMC), was proposed. This control theory provides high efficiency in the control of nonlinear systems operated under conditions of uncertainties (Utkin, 1992). A decade later, an improved control strategy, which was developed on the basis of SMC, was developed with implementation of fuzzy logic theories and named as fuzzy sliding mode control (FSMC). Many researchers used the strategy for controlling the nonlinear vibrations of engineering systems, and demonstrated the effectiveness of this control strategy in suppressing the nonlinear responses of the systems (Yau and Kuo, 2006; Kuo et al., 2007; Yau et al., 2010). The FSMC strategy was also used in controlling the chaotic response of a micro mechanical resonator under electrostatic forces applied at both sides of the resonator, modeled as a beam (Haghighi and Markazi, 2009). In their studies, the FSMC strategy demonstrated high efficiency in stabilizing the vibrations of the targeted system. It should be noticed that, the existing FSMC strategy is merely suitable for applying in the systems of first-order Galerkin discretization. Solutions of higher reliability and accuracy require higher order discretization in developing for the solutions. Therefore, the existing FSMC strategy is not applicable to the systems with higher-order discretization. Corresponding to a multiple-dimensional dynamic system, an active control strategy based on FSMC strategy is desired.

In the present research, an active vibration control strategy based on modified FSMC strategy is to be developed to control and stabilize the nonlinear responses of a translating beam which has fixed boundaries. The equations of motion of the beam are established based on von Karman-type equations. In developing the solutions of the beam, the equations in the form of partial differential equations are non-dimensionalized and transformed into six ordinary differential equations via a six-order Galerkin method. Corresponding to the derived multi-dimensional dynamic system, a control strategy is developed based on the FSMC strategy. Applicability and effectiveness of the control strategy developed is evident in the numerical simulations conducted. Some cases of chaotic motion of the beam are considered and the nonlinear motions are suppressed and stabilized with application of the control strategy.

2. Equations of motion

The axially translating beam considered in this research is sketched in Figure l. The equations of motion of the beam are to be derived based on Hamilton’s principle and von Karman-type equations. As can be seen from the figure, the translating beam with pinned-pinned boundaries is allowed to move axially at a constant rate b and thickness h. The x axis is along the mid-plane of the beam. The displacement of any point of the beam along the x- and z- axes are designated as u and w.

Starting from the origin at the left support of the beam, a position vector, r, of any point $(x (t), z)$ of the translating beam without deformation is given as

r = x (t) i + z k,

(1)

where

i

and

k

are the unit vectors of the fixed Cartesian coordinate shown in the figure.

Thus, the displacement field of the translating beam can be derived as

R = (x (t) + u_{0} (x (t), t) - z \frac{\partial w_{0} (x (t), t)}{\partial x (t)}) \times i + (z + w_{0} (x (t), t)) k,

(2)

where

u_{0} (x (t), t)

and

w_{0} (x (t), t)

are the displacement components along the x- and z- directions respectively, of a point in the mid-plane (

z = 0

Taking full differentiation of R with respect to time t, one may obtain

\frac{dR}{d t} = (\frac{d x (t)}{d t} + \frac{d u_{0} (x (t), t)}{d t} - z \frac{\partial}{\partial x (t)} (\frac{d w_{0} (x (t), t)}{d t})) \times i + \frac{d w_{0} (x (t), t)}{d t} k,

(3)

where the derivative of

x (t)

with respect to time is equal to the translating rate of the beam, and the full derivative of w₀ is

\frac{d w_{0} (x (t), t)}{d t} = \frac{\partial w_{0} (x (t), t)}{\partial t} + v_{0} \frac{\partial w_{0} (x (t), t)}{\partial x (t)}

(4)

Hence, the kinetic energy of the translating beam over a volume V of the beam is expressible as

T = \int_{V} \frac{1}{2} ρ (\frac{dR}{d t}) 2 d V = b \int_{- h / 2}^{+ h / 2} \int_{0}^{l_{0}} \frac{1}{2} ρ (\frac{dR}{d t}) 2 d x d z,

(5)

where ρ denotes the density of the material of the beam.

The von Karman-type equations of strains of large deflection associated with the displacement field, normal to the cross section of the beam along the x direction, in equation (2) can thus be given by

ɛ_{11} = \frac{\partial u_{0} (x (t), t)}{\partial x (t)} + \frac{1}{2} (\frac{\partial w_{0} (x (t), t)}{\partial x (t)}) 2 - z \frac{\partial 2 w_{0} (x (t), t)}{\partial x 2 (t)},

(6)

Therefore, the total strain energy of the beam can be given by

E = \int_{V} \frac{1}{2} Q_{11} ɛ_{11}^{2} d V = b \int_{- h / 2}^{+ h / 2} \int_{0}^{l_{0}} \frac{1}{2} Q_{11} ɛ_{11}^{2} d x (t) d z,

(7)

where

Q_{11}

represents the elastic coefficient in the same direction with

ɛ_{11}

The virtual work done by the force due to damping effect is defined as

W = - b \int_{0}^{l} c (\frac{d w_{0} (x (t), t)}{d t}) d x (t),

(8)

where c denotes the damping coefficient.

In the following analysis, Hamilton’s principle will be employed to obtain the nonlinear equations of motion for the translating beam. The mathematical statement of Hamilton’s principle is given by

\int_{t_{1}}^{t_{2}} (δ L + δ W) d t = 0,

(9)

where the total Lagrangian function L is given by

L = T - E .

(10)

For the sake of clarity, hereafter, use x, u₀, w₀ to replace $x (t)$ , $u_{0} (x (t), t)$ , and $w_{0} (x (t), t)$ respectively. Substituting equations (5), (7), and (8) into equation (10), and the nonlinear equations of motion of the translating beam are derived as follows:

A_{11} \frac{\partial 2 u_{0}}{\partial x 2} + A_{11} \frac{\partial w_{0}}{\partial x} \frac{\partial 2 w_{0}}{\partial x 2} - I_{0} \frac{d 2 l}{d t 2} - I_{0} \frac{d 2 u_{0}}{d t 2} = 0,

(11a)

A_{11} \frac{\partial w_{0}}{\partial x} \frac{\partial 2 u_{0}}{\partial x 2} + A_{11} \frac{\partial u_{0}}{\partial x} \frac{\partial 2 w_{0}}{\partial x 2} + \frac{3}{2} A_{11} (\frac{\partial w_{0}}{\partial x}) 2 \frac{\partial 2 w_{0}}{\partial x 2} - D_{11} \frac{\partial 4 w_{0}}{\partial x 4} - c \frac{d w_{0}}{d t} - I_{0} \frac{d 2 w_{0}}{d t 2} = 0,

(11b)

where

A_{11} = Q_{11} h, I_{0} = ρ h, D_{11} = \frac{h 3}{3} Q_{11} .

(12)

Associated with the nonlinear dynamic equations and the boundary conditions (Nayfeh and Mook, 1979; Abou-Rayan and Nayfeh, 1993), the strain can be obtained as

\frac{\partial u_{0}}{\partial x} = - \frac{1}{2} (\frac{\partial w_{0}}{\partial x}) 2 + \frac{1}{2 l_{0}} \int_{0}^{l_{0}} (\frac{\partial w_{0}}{\partial x}) 2 d x + \frac{I_{0}}{A_{11}} \frac{d 2 x}{d t 2} (x - \frac{l_{0}}{2}) .

(13)

Then, substituting equation (13) into equation (11b), the nonlinear differential governing equation of the translating beam in z direction is derived as

- I_{0} \frac{d 2 w_{0}}{d t 2} - D_{11} \frac{\partial 4 w_{0}}{\partial 4 x} - c \frac{d w_{0}}{d t} + \frac{A_{11}}{2 l} \frac{\partial 2 w_{0}}{\partial x 2} [\int_{0}^{l} (\frac{\partial w_{0}}{\partial x}) 2 d x] = 0 .

(14)

3. Non-dimensionalization

To validate the governing equation equation (14) and facilitate the numerical simulations in the consequent sections, the following non-dimensional variables are introduced:

\bar{t} = \sqrt{\frac{Q_{11} I}{I_{0} {bl}_{0}^{4}}} t = τ t, \bar{x} = \frac{x}{l_{0}},

(15)

and

{\bar{w}}_{0} = \frac{w_{0}}{h}, \frac{d {\bar{w}}_{0}}{d \bar{t}} = \frac{1}{τ h} \frac{d w_{0}}{d t}, \frac{d 2 {\bar{w}}_{0}}{d \bar{t} 2} = \frac{1}{τ 2 h} \frac{d 2 w_{0}}{d t 2}, {\bar{l}}_{0} = \frac{1}{h} l_{0}, \bar{c} = c / (\frac{Q_{11}}{h τ}) .

(16)

With the non-dimensional variables shown in equations (15) and (16) introduced into equation (14), the non-dimensional governing equation of the translating beam can be expressed as

\frac{d 2 {\bar{w}}_{0}}{d \bar{t} 2} = \frac{1}{A} (- D \frac{d {\bar{w}}_{0}}{d \bar{t}} - G \frac{\partial 4 {\bar{w}}_{0}}{\partial \bar{x} 4} + F \frac{\partial 2 {\bar{w}}_{0}}{\partial \bar{x} 2} [\int_{0}^{1} (\frac{\partial {\bar{w}}_{0}}{\partial \bar{x}}) 2 d x]),

(17)

where

A = \frac{I_{0}}{ρ h}, D = \frac{c}{ρ h τ}, G = \frac{D_{11}}{ρ h τ 2 l_{0}^{4}}, F = \frac{1}{2} \frac{{hA}_{11}}{ρ τ 2 l_{0}^{4}} .

(18)

4. Series solutions

Based on the Galerkin method of discretization, the transverse displacement ${\bar{w}}_{0}$ is expanded in a series form, in terms of a set of comparison functions as

{\bar{w}}_{0} = \sum_{n = 1}^{\infty} ϕ_{n} (\bar{x}) {\bar{w}}_{n} (\bar{t}) .

(19)

Corresponding to the pinned-pinned boundaries of the translating beam, $ϕ_{n} (\bar{x})$ can be given as follows:

ϕ_{n} (\bar{x}) = \sin (n π \bar{x}) .

(20)

Substitute the series solution of equation (19) into equation (17), and to assist presentation, replace $φ_{n}$ , w_n, ${\overset{\cdot}{w}}_{n}$ , ${\overset{··}{w}}_{n}$ , v, and t for $ϕ_{n} (\bar{x})$ , ${\bar{w}}_{n} (\bar{t})$ , $\frac{d w_{n}}{d t}$ , $\frac{d 2 w_{n}}{d t 2}$ , $\bar{v}$ and $\bar{t}$ respectively. With the application of the Galerkin method at $n = 6$ , the discretized governing equations of the translating beam with pinned-pinned boundaries can be obtained as the following:

{\overset{··}{w}}_{1} = \frac{2}{A} (\frac{8}{3} {Avw}_{2} + \frac{16}{15} {Avw}_{4} + \frac{24}{35} {Avw}_{6} + \frac{1}{2} A π 2 v 2 w_{1} - \frac{1}{2} π 4 {Gw}_{1} + \frac{12}{35} {Dvw}_{6} + \frac{8}{15} {Dvw}_{4} + \frac{4}{3} {Dvw}_{2} - \frac{1}{2} D {\overset{\cdot}{w}}_{1} - \frac{9}{4} π 4 {Fw}_{1} w_{3}^{2} - \frac{25}{4} π 4 {Fw}_{1} w_{5}^{2} - \frac{1}{4} π 4 {Fw}_{1}^{3} - 4 π 4 {Fw}_{1} w_{4}^{2} - 9 π 4 {Fw}_{1} w_{6}^{2} - π 4 {Fw}_{1} w_{2}^{2}),

(21a)

{\overset{··}{w}}_{2} = \frac{2}{A} (\frac{40}{21} {Avw}_{5} + \frac{24}{5} {Avw}_{3} - \frac{8}{3} {Avw}_{1} + 2 A π 2 v 2 w_{2} - 8 π 4 {Gw}_{2} + \frac{12}{5} {Dvw}_{3} - \frac{4}{3} {Dvw}_{1} + \frac{20}{21} {Dvw}_{5} - \frac{1}{2} D {\overset{\cdot}{w}}_{2} - 9 π 4 {Fw}_{2} w_{3}^{2} - 25 π 4 {Fw}_{2} w_{5}^{2} - π 4 {Fw}_{2} w_{1}^{2} - 16 π 4 {Fw}_{2} w_{4}^{2} - 36 π 4 {Fw}_{2} w_{6}^{2} - 4 π 4 {Fw}_{2}^{3}),

(21b)

{\overset{··}{w}}_{3} = \frac{2}{A} (\frac{48}{7} {Avw}_{4} + \frac{8}{3} {Avw}_{6} - \frac{24}{5} {Avw}_{2} + \frac{9}{2} A π 2 v 2 w_{3} - \frac{81}{2} π 4 {Gw}_{3} + \frac{4}{3} {Dvw}_{6} - \frac{12}{5} {Dvw}_{2} + \frac{24}{7} {Dvw}_{4} - \frac{1}{2} D {\overset{\cdot}{w}}_{3} - \frac{81}{4} π 4 {Fw}_{3}^{3} - \frac{225}{4} π 4 {Fw}_{3} w_{5}^{2} - \frac{9}{4} π 4 {Fw}_{3} w_{1}^{2} - 36 π 4 {Fw}_{3} w_{4}^{2} - 81 π 4 {Fw}_{3} w_{6}^{2} - 9 π 4 {Fw}_{3} w_{2}^{2}),

(21c)

{\overset{··}{w}}_{4} = \frac{2}{A} (- \frac{48}{7} {Avw}_{3} - \frac{16}{15} {Avw}_{1} + \frac{80}{9} {Avw}_{5} + 8 A π 2 v 2 w_{4} - 128 π 4 {Gw}_{4} + \frac{40}{9} {Dvw}_{5} - \frac{24}{7} {Dvw}_{3} - \frac{8}{15} {Dvw}_{1} - \frac{1}{2} D {\overset{\cdot}{w}}_{4} - 36 π 4 {Fw}_{4} w_{3}^{2} - 100 π 4 {Fw}_{4} w_{5}^{2} - 4 π 4 {Fw}_{4} w_{1}^{2} - 64 π 4 {Fw}_{4}^{3} - 144 π 4 {Fw}_{4} w_{6}^{2} - 16 π 4 {Fw}_{4} w_{2}^{2}),

(21d)

{\overset{··}{w}}_{5} = \frac{2}{A} (- \frac{40}{21} {Avw}_{2} - \frac{80}{9} {Avw}_{4} + \frac{120}{11} {Avw}_{6} + \frac{25}{2} A π 2 v 2 w_{5} - \frac{625}{2} π 4 {Gw}_{5} - \frac{40}{9} {Dvw}_{4} - \frac{20}{21} {Dvw}_{2} + \frac{60}{11} {Dvw}_{6} - \frac{1}{2} D {\overset{\cdot}{w}}_{5} - \frac{225}{4} π 4 {Fw}_{5} w_{3}^{2} - \frac{625}{4} π 4 {Fw}_{5}^{3} - \frac{25}{4} π 4 {Fw}_{5} w_{1}^{2} - 100 π 4 {Fw}_{5} w_{4}^{3} - 225 π 4 {Fw}_{5} w_{6}^{2} - 25 π 4 {Fw}_{5} w_{2}^{2}),

(21e)

{\overset{··}{w}}_{6} = \frac{2}{A} (- \frac{24}{35} {Avw}_{1} - \frac{8}{3} {Avw}_{3} - \frac{120}{11} {Avw}_{5} + 18 A π 2 v 2 w_{6} - 648 π 4 {Gw}_{6} - \frac{12}{35} {Dvw}_{1} - \frac{60}{11} {Dvw}_{5} - \frac{4}{3} {Dvw}_{3} - \frac{1}{2} D {\overset{\cdot}{w}}_{6} - 81 π 4 {Fw}_{6} w_{3}^{2} - 225 π 4 {Fw}_{6} w_{5}^{2} - 9 π 4 {Fw}_{6} w_{1}^{2} - 144 π 4 {Fw}_{6} w_{4}^{2} - 324 π 4 {Fw}_{6}^{3} - 36 π 4 {Fw}_{6} w_{2}^{2}) .

(21f)

5. Control strategy development

With the developed governing equations, boundary conditions and the solutions of the governing equations, an active vibration control strategy can be developed. Based on previous works (Utkin, 1992; Haghighi and Markazi, 2009; Kuo et al., 2007; Dai and Sun, 2012), the proposed active control strategy is developed for the vibration control of a multi-dimensional nonlinear dynamic system shown in equation (21).

For a nonlinear governing equation in the following general form:

\overset{··}{w} = Φ (w, \overset{\cdot}{w}, t),

(22)

denote U as the control input and

Δ P (w, \overset{\cdot}{w})

the unknown external disturbance applying on the beam, the governing equation, (22), for the beam with the control input and the external disturbance can be given by

\overset{··}{w} = Φ (w, \overset{\cdot}{w}, t) + U + Δ P (w, \overset{\cdot}{w}) .

(23)

With application of the control, it is expected that the vibration of the beam can be controlled.

If the n^th Galerkin method is applied in the discretization of the governing equation given in equation (23), a series of 2nd order ordinary differential equations considering the control input U and the unknown external disturbance will be derived as follows:

{{\overset{\cdot}{w}}_{1, 1} = w_{1, 2} {\overset{\cdot}{w}}_{1, 2} = ϕ_{1} (W, t) + u_{1} + Δ f_{1} (W, t) {\overset{\cdot}{w}}_{2, 1} = w_{2, 2} {\overset{\cdot}{w}}_{2, 2} = ϕ_{2} (W, t) + u_{2} + Δ f_{2} (W, t) : {\overset{\cdot}{w}}_{i, 1} = w_{i, 2} {\overset{\cdot}{w}}_{i, 2} = ϕ_{i} (W, t) + u_{i} + Δ f_{i} (W, t) : {\overset{\cdot}{w}}_{n, 1} = w_{n, 2} {\overset{\cdot}{w}}_{n, 2} = ϕ_{n} (W, t) + u_{n} + Δ f_{i} (W, t),

(24)

where

ϕ_{i} (W, t)

, u_i, and

Δ f_{i} (W, t)

represent the expressions of

Φ (w, \overset{\cdot}{w}, t)

, U, and

Δ P (w, \overset{\cdot}{w})

after the application of the Galerkin discretization. With the Galerkin discretization, the column vector W in equation (24) is given below:

W = [w_{1, 1} w_{1, 2} w_{2, 1} w_{2, 2} \dots w_{i, 1} w_{i, 2} \dots w_{n, 1} w_{n, 2}] T

Considering the response of a point in the translating beam along the x direction, based on equation (24) and the expression in equation (19), the non-dimensional response of the selected point w_p can be given as

w_{p} = \sum_{n = 1}^{\infty} ϕ_{n} (x_{p}) w_{n} (t),

(25)

where x_p denotes the location of the selected point.

For a desired vibration or reference vibration expressed as

w_{r} = Ψ (t),

(26)

the control input U can be given as

U = U_{eq} - U_{r},

(27)

where

U_{eq}

and U_r are expressible as

U_{eq} = - (({\overset{\cdot}{w}}_{p} - \overset{\cdot}{Ψ}) + κ (w_{p} - Ψ)) and U_{r} = k_{fs} U_{fs} .

(28)

In equation (28), κ designates the control parameters governing the sliding surface,

k_{fs}

is given as

| Δ F (w, \overset{\cdot}{w}) | < k_{fs} \in R +

, and the value of

U_{fs}

depends on the fuzzy rule shown in Table 1 (Utkin, 1992).

Table 1.

The fuzzy rule of $U_{fs}$ .

$U_{fs}$		$U_{eq}$
$U_{fs}$		1	2/3	1/3	0	−1/3	−2/3	−1
$\frac{d U_{eq}}{dt}$	1	−1	−1	−1	−1	−2/3	−1	0
	2/3	−1	−1	−1	−2/3	−1	0	1/3
	1/3	−1	−1	−2/3	−1	0	1/3	2/3
	0	−1	−2/3	−1	0	1/3	2/3	1
	−1/3	−2/3	−1	0	1/3	2/3	1	1
	−2/3	−1	0	1/3	2/3	1	1	1
	−1	0	1/3	2/3	1	1	1	1

With the control strategy demonstrated in equation (23) to equation (28), the control of an axially translating beam governed by the general governing equation, equation (22), can be realized.

Take the axially translating beam governed by equation (17) as an example. Use the control strategy developed and apply the control input as shown in equation (27), the governing equation with the control input for the beam can be given by the following expression:

{\overset{··}{w}}_{0} = \frac{1}{A} (- D {\overset{\cdot}{w}}_{0} - G \frac{\partial 4 w_{0}}{\partial x 4} + F \frac{\partial 2 w_{0}}{\partial x 2} [\int_{0}^{1} (\frac{\partial w_{0}}{\partial x}) 2 d x]) + U + Δ F (w, \overset{\cdot}{w}) .

(29)

With the application of the Galerkin discretization of 6th order, equation (29) may have the following form:

{{\overset{\cdot}{w}}_{1, 1} = w_{1, 2} {\overset{\cdot}{w}}_{1, 2} = \frac{2}{A} (\frac{8}{3} {Avw}_{2, 1} + \frac{16}{15} {Avw}_{4, 1} + \frac{24}{35} {Avw}_{6, 1} + \frac{1}{2} A π 2 v 2 w_{1, 1} - \frac{1}{2} π 4 {Gw}_{1, 1} + \frac{12}{35} {Dvw}_{6, 1} + \frac{8}{15} {Dvw}_{4, 1} + \frac{4}{3} {Dvw}_{2, 1} - \frac{1}{2} {Dw}_{1, 2} - \frac{9}{4} π 4 {Fw}_{1, 1} w_{3, 1}^{2} - \frac{25}{4} π 4 {Fw}_{1, 1} w_{5, 1}^{2} - \frac{1}{4} π 4 {Fw}_{1, 1}^{3} - 4 π 4 {Fw}_{1, 1} w_{4, 1}^{2} - 9 π 4 {Fw}_{1, 1} w_{6, 1}^{2} - π 4 {Fw}_{1, 1} w_{3, 1}^{2}) + u_{1} + Δ f_{1} (W, t),

(30a)

{{\overset{\cdot}{w}}_{2, 1} = w_{2, 2} {\overset{\cdot}{w}}_{2, 2} = \frac{2}{A} (\frac{40}{21} {Avw}_{5, 1} + \frac{24}{5} {Avw}_{3, 1} - \frac{8}{3} {Avw}_{1, 1} + 2 A π 2 v 2 w_{2, 1} - 8 π 4 {Gw}_{2, 1} + \frac{12}{5} {Dvw}_{3, 1} - \frac{4}{3} {Dvw}_{1, 1} + \frac{20}{21} {Dvw}_{5, 1} - \frac{1}{2} {Dw}_{2, 2} - 9 π 4 {Fw}_{2, 1} w_{3, 1}^{2} - 25 π 4 {Fw}_{2, 1} w_{5, 1}^{2} - π 4 {Fw}_{2, 1} w_{1, 1}^{2} - 16 π 4 {Fw}_{2, 1} w_{4, 1}^{2} - 36 π 4 {Fw}_{2, 1} w_{6, 1}^{2} - 4 π 4 {Fw}_{2, 1}^{3}) + u_{2} + Δ f_{2} (W, t),

(30b)

{{\overset{\cdot}{w}}_{3, 1} = w_{3, 2} {\overset{\cdot}{w}}_{3, 2} = \frac{2}{A} (\frac{48}{7} {Avw}_{4, 1} + \frac{8}{3} {Avw}_{6, 1} - \frac{24}{5} {Avw}_{2, 1} + \frac{9}{2} A π 2 v 2 w_{3, 1} - \frac{81}{2} π 4 {Gw}_{3, 1} + \frac{4}{3} {Dvw}_{6, 1} - \frac{12}{5} {Dvw}_{2, 1} + \frac{24}{7} {Dvw}_{4, 1} - \frac{1}{2} {Dw}_{3, 2} - \frac{81}{4} π 4 {Fw}_{3, 1}^{3} - \frac{225}{4} π 4 {Fw}_{3, 1} w_{5, 1}^{2} - \frac{9}{4} π 4 {Fw}_{3, 1} w_{1, 1}^{2} - 36 π 4 {Fw}_{3, 1} w_{4, 1}^{2} - 81 π 4 {Fw}_{3, 1} w_{6, 1}^{2} - 9 π 4 {Fw}_{3, 1} w_{2, 1}^{2}) + u_{3} + Δ f_{3} (W, t),

(30c)

{{\overset{\cdot}{w}}_{4, 1} = w_{4, 2} {\overset{\cdot}{w}}_{4, 2} = \frac{2}{A} (- \frac{48}{7} {Avw}_{3, 1} - \frac{16}{15} {Avw}_{1, 1} + \frac{80}{9} {Avw}_{5, 1} + 8 A π 2 v 2 w_{4, 1} - 128 π 4 {Gw}_{4, 1} + \frac{40}{9} {Dvw}_{5, 1} - \frac{24}{7} {Dvw}_{3, 1} - \frac{8}{15} {Dvw}_{1, 1} - \frac{1}{2} {Dw}_{4, 2} - 36 π 4 {Fw}_{4, 1} w_{3, 1}^{2} - 100 π 4 {Fw}_{4, 1} w_{5, 1}^{2} - 4 π 4 {Fw}_{4, 1} w_{1, 1}^{2} - 64 π 4 {Fw}_{4, 1}^{3} - 144 π 4 {Fw}_{4, 1} w_{6, 1}^{2} - 16 π 4 {Fw}_{4, 1} w_{2, 1}^{2}) + u_{4} + Δ f_{4} (W, t),

(30d)

{{\overset{\cdot}{w}}_{5, 1} = w_{5, 2} {\overset{\cdot}{w}}_{5, 2} = \frac{2}{A} (- \frac{40}{21} {Avw}_{2, 1} - \frac{80}{9} {Avw}_{4, 1} + \frac{120}{11} {Avw}_{6, 1} + \frac{25}{2} A π 2 v 2 w_{5, 1} - \frac{625}{2} π 4 {Gw}_{5, 1} - \frac{40}{9} {Dvw}_{4, 1} - \frac{20}{21} {Dvw}_{2, 1} + \frac{60}{11} {Dvw}_{6, 1} - \frac{1}{2} {Dw}_{5, 2} - \frac{225}{4} π 4 {Fw}_{5, 1} w_{3, 1}^{2} - \frac{625}{4} π 4 {Fw}_{5, 1}^{3} - \frac{25}{4} π 4 {Fw}_{5, 1} w_{1, 1}^{2} - 100 π 4 {Fw}_{5, 1} w_{4, 1}^{3} - 225 π 4 {Fw}_{5, 1} w_{6, 1}^{2} - 25 π 4 {Fw}_{5, 1} w_{2, 1}^{2}) + u_{5} + Δ f_{5} (W, t),

(30e)

{{\overset{\cdot}{w}}_{6, 1} = w_{6, 2} {\overset{\cdot}{w}}_{6, 2} = \frac{2}{A} (- \frac{24}{35} {Avw}_{1, 1} - \frac{8}{3} {Avw}_{3, 1} - \frac{120}{11} {Avw}_{5, 1} + 18 A π 2 v 2 w_{6, 1} - 648 π 4 {Gw}_{6, 1} - \frac{12}{35} {Dvw}_{1, 1} - \frac{60}{11} {Dvw}_{5, 1} - \frac{4}{3} {Dvw}_{3, 1} - \frac{1}{2} {Dw}_{6, 2} - 81 π 4 {Fw}_{6, 1} w_{3, 1}^{2} - 225 π 4 {Fw}_{6, 1} w_{5, 1}^{2} - 9 π 4 {Fw}_{6, 1} w_{1, 1}^{2} - 144 π 4 {Fw}_{6, 1} w_{4, 1}^{2} - 324 π 4 {Fw}_{6, 1}^{3} - 36 π 4 {Fw}_{6, 1} w_{2, 1}^{2}) + u_{6} + Δ f_{6} (W, t),

(30f)

where u₁, u₂, u₃, u₄, u₅ and u₆ are derived as follows through the 6 ^th -order Galerkin method:

u_{1} = \frac{2}{π} U, u_{2} = 0 \cdot U, u_{3} = \frac{2}{3 π} U, u_{4} = 0 \cdot U, u_{5} = \frac{2}{5 π} U, u_{6} = 0 \cdot U .

It should be noted that due to the pinned-pinned boundary of the translating beam, the application of the Galerkin method based on equation (20) makes the coefficients of u₂, u₄, and u₆ as zero. In the next section, it will be demonstrated in the numerical simulation that the actual response of the axially translating beam at a selected point can be well synchronized with a desired reference signal in the case that the coefficients of u₂, u₄, and u₆ are zero.

6. Numerical simulations

To demonstrate the applicability and effectiveness of the control strategy developed in the previous section, numerical simulations are conducted for controlling an axially translating beam governed by equation (17). The nonlinear responses of the beam are focused in this research. With the numerical simulations performed, a chaotic motion is found when the beam is translating at certain rates. The proposed active control strategy not only effectively reduces the amplitude of the chaotic motion, but also stabilizes the motion so that the response of the translating beam is controlled to a desired periodic motion. To facilitate the numerical simulation, the 4th-order P-T method (Dai, 2008), is implemented.

The parameters used for the simulations for the responses of the axially translating beam are given as follows:

v_{0} = 0.4 m / s, c = 0 N / ((m / s) m 2), Q_{11} = 1.1809 \times 1010 Pa,

l_{0} = 0.5 m, b = 0.02 m, h = 0.002 m, ρ = 1800 kg / m 3 .

The non-dimensionalized initial conditions, corresponding to the displacements described by equation (30) after the implementation of the 6th-order Galerkin method, are taken as

w_{1, 1} (0) = 0.01, w_{1, 2} (0) = 0, w_{2, 1} (0) = 0.002, w_{2, 2} (0) = 0, w_{3, 1} (0) = 0, w_{3, 2} (0) = 0, w_{4, 1} (0) = 0, w_{4, 2} (0) = 0, w_{5, 1} (0) = 0, w_{5, 2} (0) = 0, w_{6, 1} (0) = 0.002, w_{6, 2} (0) = 0 .

If the response of a point at $0.35 m$ along the x-axis of the beam is selected, based on equations (19) and (30) the non-dimensional response of the selected point w_p can be derived as

w_{p} = \sum_{n = 1}^{6} ϕ_{n} w_{n, 1} = 0.8090170164 w_{1, 1} - 0.9510564931 w_{2, 1} + 0.3090168873 w_{3, 1} + 0.5877853737 w_{4, 1} - 1.000000000 w_{5, 1} + 0.5877853737 w_{6, 1}

(31)

Chaos occurs when the control parameters and the unknown external disturbance take the following values:

w_{r} = 1.4 \sin (0.4966 t), κ = 2750, k_{fs} = 375, Δ F (w, \overset{\cdot}{w}) = 0.001 \sin (w_{p}) .

(32)

The response of the beam translating at a speed

v_{0} = 0.4 m / s

is shown in Figure 2, corresponding to the non-dimensional time from

t = 0

t = 550

. During this period of time, the control strategy is applied at

t = 250

. It can be seen in Figure 2 that the maximum amplitude of the vibration of the beam can exceed 3. Considering that the displacement shown in the figure is non-dimensional, the amplitude is very large. Thus, reduction and stabilization of the chaotic motion may improve the operation of the beam.

Starting from $t = 250$ until $t = 550$ , the control strategy developed in the previous section is continuously applied. As can be seen from Figure 2, the nonlinear response of the beam is indeed controlled with significantly reduced amplitude and the motion of the beam is stabilized, after the application of the control strategy. In other words, the response of the beam is well synchronized with the reference signal applied as described in equation (32). However, although the control strategy is applied right at $t = 250$ , as is noticed in the numerical simulations, a short period of time or a few more cycles are needed for the beam motion to be actually controlled after the application of the control strategy. Usually, it may take about $100$ non-dimensional time units for the multi-dimensional dynamic system to be satisfactorily stabilized.

Figure 3 shows the phase diagram of the motion of the beam at the selected point before the application of the control strategy, to demonstrate the chaotic response of the axially translating beam. As can be seen from Figure 3, the maximum amplitude of the vibration of the axially translating beam is more than 3.5, or in other words 3.5 times the thickness of the beam. Therefore, an effective control strategy, corresponding to the six-dimensional dynamic system, is needed for the vibration control of the beam.

It may also be significant to see the displacements of the beam corresponding to each of the solutions after the application of the 6th-order Galerkin method. The responses of the beam represented by w₁, w₂, w₃, w₄, w₅ and w₆ are shown in Figure 4(a–f). As can be seen from the figures, the responses of the beam are all chaotic before the application of the control strategy. It is interesting to notice that the amplitudes of the responses are monotonically reduced from in Figure 4 (a–f), as expected. However, it should be noticed that the maximum amplitude of the response of the first vibration mode shown in Figure 4(a) is close to four times the thickness of the beam, while the maximum amplitude shown in Figure 4(f) is close to 0.5 times the thickness of the beam. Based on the numerical results, the response of the sixth vibration mode of the beam is not negligible. Therefore corresponding to the selected axially translating speed in this case, at least n = 6 should be chosen in the Galerkin method of discretization.

After the application of the control strategy, the responses of the beam at the translating speed $v_{0} = 0.4 m / s$ are shown in Figures 5 and 6. In Figure 5 the responses of the selected point, w_p, are shown, for the period of time from $t = 250$ to $t = 550$ . It can be seen from Figure 5 that a short period of time is needed for stabilizing the beam, while in the studies (Haghighi and Markazi, 2009; Yau et al., 2010), such a period is very short and the response of the dynamic system in these studies is almost synchronized with the reference signal right after the control strategy is applied. However, in the previous studies, the control strategy employed, i.e. FSMC strategy, cannot be employed in the vibration control of a multi-dimensional nonlinear dynamic system.

From Figure 6(a–f), the displacements are shown in terms of w₁, w₂, w₃, w₄, w₅ and w₆. Based on these figures, through the application of the control strategy, each of the displacements of the axially translating beam are gradually stabilized from a chaotic motion into a periodic one. However, it can be seen from these figures, the amplitudes of the displacements are different and not monotonically reduced from w₁ to w₆, though they are all stabilized eventually. Also, it can be seen from Figure 6(b) that stabilization of w₂ takes longer than the others. It should be noticed that all these stabilizations including the variations contribute to the control and stabilization of the translating beam as described by w_p of equation (31). It should be noticed the maximum amplitude of the response of the first vibration mode shown in Figure 6(a) is about 1 time the thickness of the beam, while the maximum amplitude shown in Figure 6(f) is about 0.3 times the thickness of the beam. That is, after the vibration of the beam has been stabilized, the response of the sixth vibration mode of the beam has become more significant. Thus corresponding to the selected axially translating speed in this case, a six-dimensional nonlinear dynamic system should be derived in the Galerkin method of discretization.

Figure 7 shows the comparison between the actual response of the beam w_p and the reference signal w_r applied. It can be seen that the reference signal w_r is perfectly periodic with respect to time t. It can also be seen from the figure that the maximum amplitude of the beam’s response slightly varies after the stabilization of the beam with the application of the control strategy. However, as shown in the figure, the maximum amplitude of the beam is very close to that of the reference signal and the response of the beam is fairly close to periodic following the pattern of the reference signal, as desired.

Figure 8 shows the control input U. Initially, the control input displays a non-periodic wave diagram for a short period of time corresponding to the time period of stabilization of the beam motion after the application of the control strategy. The maximum value of the control input in this period is close to $15000$ , and once the system is stabilized, the control input displays a periodic wave diagram as shown in the figure, and the maximum value of the control input is significantly decreased to about $7500$ .

7. Conclusion

An active control strategy is developed in this research to control and stabilize the nonlinear motion of an axially translating beam, on the basis of a modified fuzzy sliding mode control strategy. The governing equation of the geometrically nonlinear beam is converted into a six-dimensional dynamic system and the control strategy is then developed in relating to the dynamic system converted with the application of the Galerkin method. The control strategy developed may therefore be applied to control multi-dimensional dynamic systems. The control strategy may also be used to control the nonlinear systems with external disturbances. Such a control strategy is not seen in the current literature. The numerical simulation with application of control strategy shows the effectiveness of the strategy. The research conducted will be beneficial to the engineering applications of vibration control for multi-dimensional nonlinear dynamic systems.

Footnotes

Acknowledgement

The authors would like to acknowledge with great appreciation the support from the National Science and Engineering Research Council of Canada (NSERC) and National Natural Science Foundation of China (Grant No. 11272270). The support of the Nonlinear Science research team at the Sino-Canada Research Centre for Noise and Vibration Control were significant to the performance of this research.

Funding

The students involved in the research are paid by some of the findings available to the research team, as indicated in the acknowledgement.

References

Abou-Rayan

Nayfeh

Mook

Nayfeh

(1993) Nonlinear response of a parametrically excited bucked beam. Nonlinear Dynamics 4: 499–525.

Dai

(2008) Nonlinear Dynamics of Piece Constant Systems and Implementation of Piecewise Constant Arguments, New Jersey: World Scientific.

Dai

Sun

(2012) Controlling a nonlinear laminated composite cantilever beam with a modified fuzzy sliding mode control strategy. Journal of Applied Nonlinear Dynamics 1: 287–307.

Haghighi

Markazi

AHD

(2009) Chaos prediction and control in MEMS resonators. Community of Nonlinear Science 15: 3091–3099.

Humer

(2013) Dynamic modeling of beams with non-material, deformation-dependent boundary conditions. Journal of Sound and Vibration 332: 622–641.

Kuo

Shieh

Lin

Shih

(2007) Design of an adaptive fuzzy sliding-mode controller for chaos synchronization. International Journal of Nonlinear Science 8: 631–636.

Nayfeh

Mook

(1979) Nonlinear Oscillation, New York: Wiley-Interscience.

Utkin

(1992) Sliding Modes in Control and Optimization, Berlin: Springer-Verlag.

Wickert

(1992) Non-linear vibration of a travelling tensioned beam. International Journal of Mechanics 27: 503–517.

10.

Yang

Hong

Matsuno

(2005) Energy-based control of axially translating beams: varying tension, varying speed, and disturbance adaption. IEEE Transactions on Control Systems Technology 13: 1045–1054.

11.

Yau

Kuo

(2006) Fuzzy sliding mode control for a class of chaos synchronization with uncertainties. International Journal of Nonlinear Sciences and Numerical Simulation 7: 333–338.

12.

Yau

Wang

Hsieh

Cho

(2010) Nonlinear analysis and control of the uncertain micro-electro-mechanical system by using a fuzzy sliding mode control design. Computers and Mathematics with Applications 61: 1912–1916.

13.

Zhu

(2002) Control volume and system formulations for translating media and stationary media with moving boundaries. Journal of Sound and Vibration 254: 189–201.

14.

Zhu

(2000) Energetic and stability of translating media with an arbitrarily varying length. Journal of Vibration and Acoustics 122: 295–304.