Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 2863–2878

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ymssp

Active HN control of the vibration of an axially moving cantileverbeam by magnetic force

Liang Wang, Huai-hai Chen n, Xu-dong He

MOE Key Lab of Structure Mechanics and Control for Aircraft, Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics,

Nanjing 210016, China

a r t i c l e i n f o

Article history:

Received 21 July 2009

Received in revised form

5 November 2009

Accepted 15 May 2011Available online 12 June 2011

Keywords:

Active control

Vibration

Noncontact magnetic force

Axially moving beam

HN

70/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ymssp.2011.05.009

esponding author.

ail address: chhnuaa@nuaa.edu.cn (H.-h. Chen

a b s t r a c t

An HN method for the vibration control of an iron cantilever beam with axial velocity

using the noncontact force by permanent magnets is proposed in the paper. The

transverse vibration equation of the axially moving cantilever beam with a tip mass is

derived by D’Alembert’s principle and then updated by experiments. An experimental

platform and a magnetic control system are introduced. The properties of the force

between the magnet and the beam have been determined by theoretic analysis and

tests. The HN control strategy for the suppression of the beam transverse vibration by

initial deformation excitations is put forward. The control method can be used for the

beam with constant length or varying length. Numerical simulation and actual

experiments are implemented. The results show that the control method is effective

and the simulations fit well with the experiments.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Structures such as tapes, band saws, elevator hoist cables, and robot arms, steel strip from rolling machine and flexibletubes from aerial tankers can all be modeled as axially moving beams. These kinds of dynamic problems have been widelystudied in the past 30 years. Zajaczkowski and Lipinski [1] and Zajaczkowski and Yamada [2] analyzed the dynamicstability of an Euler-Bernoulli beam under periodically sliding motion. Vu-Quoc L et al. [3] investigated dynamics of slidinggeometrically-exact beams. Behdinan et al. [4] used non-linear theory to study the dynamics of flexible sliding beams andobtained some new formulae. Zhu and Ni [5] derived the energy and stability characteristics of translating beams andstrings.

Lightweight and flexible structures often suffer from the drawback of large vibration due to low stiffness. Manyresearchers have set forth methods to solve the problem. Zhu et al. [6] used a pointwise control strategy to control thevibration of a beam with varying length under initial deformation excitations. Zhu et al. [7,8] researched the stabilitymargin of the controller for the translating string and beam. Many researchers use piezoelectric systems for the vibrationsuppression or shape control [9–17]. But here we mainly concern about the control method by noncontact magneticforces. Kojima and Nagaya [18] derived the time response of a cantilever beam with an iron tip mass subjected to analternating electromagnetic force. Lu et al [19], Shih et al. [20], and Liu and Chang [21] studied the dynamics of a simplysupported beam with a time varying axial magnetic load. Their results showed that the vibration of the beam can becontrolled with a magnetic field of appropriate amplitude and frequency.

ll rights reserved.

).

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782864

The multiple scale method or other numerical methods were used to solve the temporal equation of motion [19–21].Zhou et al. [22] investigated the nonlinear dynamic and control characteristics of giant magnetostrictive smart laminatedbeams with numerical simulations. Majewska et al. [23] discussed the application of actuators utilizing the magneticshape memory effect for vibration control of beam-like structures. Niu and Zhang [24] studied the vibration control of abeam through electro-magnetic constrained layer damping which consists of electromagnet layer, permanent magnetlayer and viscoelastic damping layer. When the coil of the electromagnet is electrified with proper control strategy, theelectromagnet can exert magnetic force opposite to the direction of structural deformation so that the structural vibrationis attenuated. Cheah et al. [25] proposed a method to suppress the vibration of a structure with a novel eddy currentmechanism by forming a damper with a magnet and conductor. Hirunyapruk et al. [26] investigated an adaptive tunedvibration absorber (ATVA) designed as a three-layer beam with Magneto-Rheological fluid in the middle layer. The shearstiffness of the ATVA can be changed by altering the magnetic field applied to the fluid. Routaray et al [27] designed aneddy current damper that uses a conducting disc vibrating within the magnetic field generated by an electromagnet. Theyattempted to control the vibration of a cantilever beam by adjusting the current supply to the eddy current damper. Nandiet al. [28] studied the vibration control of a structure using one-sided magnetic actuator and a digital Proportional-Derivative controller. Barun and Santosha [29] exploited the effect of the application of an alternating magnetic field toreduce the transverse vibration of a cantilever beam with tip mass. They derived the governing equation of motion byD’Alembert’s principle and reduced it to a nondimensional temporal form by the generalized Galerkin method. Theydetermined the instability region and frequency response curves of the system by the multiple scale method. They alsostudied the influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity ofthe beam material on the frequency response curves.

All the researchers [18–29] mentioned above used electromagnet for the vibration control, in which complex controlcircuits were needed, and lumped permanent magnets are also used to generate control force with variable direction. Thepointwise force control [6] is theoretically good but is very difficult to be carried out in practice. In this paper, we try to usepermanent magnet for the transverse vibration control of an axially moving iron beam and attempt to implement thepointwise control. A noncontact loading system is introduced and an on-line control method is proposed. The simulationand test results are given.

2. Beam model

Fig. 1 shows an Euler–Bernoulli beam of mass per unit length r, and flexural rigidity EI, translating horizontally througha rigid, prismatic joint. The length external to the joint channel is L(t), where t is the time. The beam is assumed to beextensible with an arbitrarily prescribed translational velocity v(t). One has _LðtÞ ¼ vðtÞ, where the overdot denotes timedifferentiation. A lumped mass me is attached to the tip of the beam. We consider here small transverse vibration of thebeam about its trivial equilibrium. Relative to the fixed coordinate system shown in Fig. 1, the transverse displacement ofthe beam particle instantaneously located at spatial position x, where 0oxoL(t), is w(x,t). A noncontact control system islocated at a, where a is a constant value, and the control force is f(t).

The transverse vibration equation of the axially moving cantilever beam with tip mass is derived by the D’Alembert’sprinciple. Fig. 2 shows the transverse forces acting on a beam element of infinitesimal length. Fig. 2(a) denotes inertia force

v(t)

x

me

f(t)a

L(t)

w(x,t)

Fig. 1. Schematic of a translating beam subject to a control force.

Inertia force caused by

transverse acceleration

Inertial force

Coriolis acceleration

Inertia force

centripetal acceleration

wxxInertial force

of axial acceleration the transverse component

dx

vwxρ ⋅

dx

v2 dx⋅ρ

dx

2 vwxρ ⋅

dx

dx∂2t∂2wρ

Transverse shear forceThe transverse

component of axial force

Transverse control force

dx

dxEIwxxxx⋅

dx

dxf (t) ⋅

dx

dxD (N wx) ⋅ (x-a)δ⋅ dx⋅ ⋅dx

caused by caused bycaused by

Fig. 2. Transverse forces on a micro beam element.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2865

contributed by transverse acceleration; Fig. 2(b) denotes the inertial force caused by Coriolis acceleration which isproduced by beam rotation and axial movement; Fig. 2(c) denotes the inertia force contributed by centripetal acceleration;Fig. 2(d) denotes the inertia force caused by transverse component of axial acceleration; Fig. 2(e) denotes the transversecomponent of axial inertial force; Fig. 2(f) denotes transverse shear force associated with flexure rigidity of the beam;Fig. 2(g) denotes the transverse control force.

Based on Fig. 2 and by establishing the transverse force balance, the governing equation is derived as

rð €wþ2v _wxþv2wxxþ _vwxÞ�DðNwxÞþEIwxxxx ¼ f ðtÞdðx�aÞ ð1Þ

where N¼�½meþrðL�xÞ� _v, D(Nwx)¼NxwxþNwxx, and the subscript sign x used to indicate the differentiation to theposition coordinate x.

The boundary conditions are

wð0Þ ¼ 0 ð2Þ

wxð0Þ ¼ 0 ð3Þ

wxxðLÞ ¼ 0 ð4Þ

me €wðLÞþ2mev _wxðLÞþmev2wxxðLÞ�EIwxxxðLÞ ¼ 0 ð5Þ

Galerkin’s method is employed to truncate the governing equation to a set of time-dependent ordinary differentialequations. The solution of the displacement can be expressed as a superposition of every mode, and it has a form as

wðx,tÞ ¼Xn

j ¼ 1

qjðtÞfjðx,tÞ ð6Þ

where n is the number of modes, qj(t) are generalized coordinates, and fj(x,t) are instantaneous orthonormaleigenfunctions and have forms as

fjðx,tÞ ¼ Cj ðcosbjx�coshbjxÞ�þAjðsinbjx�sinhbjxÞh i

ð7Þ

where Cj is a function of the length of the beam, and obtained fromZ L

0fifjrdxþmefiðLÞfjðLÞ ¼ dij,

where dij is the Kronecker delta defined by dij¼1 if i¼ j and dij¼0 if iaj. bj is a function of the length of the beam L, andsatisfies

1þcosbjLcoshbjL¼meUbjL

rLðsinbjLcoshbjL�cosbjLsinhbjLÞ ð8Þ

where L is a function of t.Substituting Eq. (6) into Eq. (1) and Eqs. (2)–(5), multiplying the governing equation by fi(x,t), integrating it over the

domain [0,L(t)], and using the boundary conditions, yield the discretized equations of motion of the controlled translatingbeam with time-dependent coefficient matrices

MðtÞ €qðtÞþCðtÞ _qðtÞþKðtÞqðtÞ ¼ FðtÞ ð9Þ

where q¼[q1,q2........,qn]T is the vector of generalized coordinates, M is the mass matrix, C is the damping matrix, K is thestiffness matrix, and F is the control force vector. Entries of these matrices are given by

mij ¼ dij ð10Þ

cij ¼

Z L

02rð _f ifjþvf0ifjÞdxþ2með

_f iþvf0iÞ9x ¼ LfjðL,tÞ ð11Þ

kij ¼

Z L

0r½ €f ifjþ2vf0ifjþv2fifjþ _vf

0

ifj�dx� _v

Z L

0½meþrðL�xÞ�f0if

0

jdx

þmeð€f iþ2v _f

0

iþ _vf0

iÞ9x ¼ LfjðL,tÞþ

Z L

0EIfifjdx ð12Þ

Fi ¼ f ðtÞfiða,tÞ ð13Þ

where the prime denotes the differentiation to the position coordinate x.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782866

3. Experiment platform

The platform, as shown in Fig. 3, contains table, motor, lead screw, driving block, prismatic joint, beam, and a set of controlsystem. The motor drives the beam moving through the joint by the driving block on the lead screw. The velocity of the beam isadjustable due to the tunable rotation speed of the motor. An accelerometer is installed on the right tip of the beam. The controlsystem shown in Fig. 4 contains a noncontact magnet vibration exciter, an eddy current displacement sensor, a laserdisplacement sensor, a laser velocity sensor and a signal processor (PXI). The noncontact magnet vibration exciter consists of anordinary vibration exciter , a C-type clamp connected to the vibration exciter, two armature irons and two sets of permanentmagnets connected to the two arms of the C-type clamp, respectively. The beam is sliding through the gap of two armatureirons. The laser velocity sensor and the eddy current displacement sensor are used to measure the velocity and displacement ofthe beam at the control point, respectively. The laser displacement sensor is used to obtain the displacement of the C-typeclamp. The movement of the clamp can be controlled by the system. Fig. 5 is a photo of the system.

Initially, the beam is placed in the median of the gap of two armature irons and the resultant magnetic force is adjustedto zero.

4. Model updating and magnetic force measurement

The width and the height of the cross section of the beam are d¼0.03 m and h¼0.012 m, respectively. The length of thebeam varies between 0.771 and 1.305 m. The mass density and Young’s modulus are r¼7800 kg/m3 and E¼210�109 N/m2,respectively. The mass of the accelerometer is the tip mass and me¼0.005 kg.

4.1. Correction of boundary condition

The beam in Fig. 3 is simplified as a cantilever beam model. As the boundary support is complex, there is a gap betweenbeam and prismatic joint, so the nominal length of the beam L in Fig.1 is not exact the one out of the prismatic joint, it need

Prismatic joint

Beam

Motor Driving block

Lead screw Table

Laser velocity sensor

Noncontact magneticvibration exciter

Eddy current displacement sensor

Laser displacement

sensor

Acceleration transducer

NI-PXI

Fig. 3. Schematic diagram of experiment platform.

Laser velocity

Noncontact magnetic

Laserdisplacementsensor

NI-PXI

Eddy current displacement sensor

Permanent magnets

C-type clamp

Armature iron

Ordinary vibration exciter

vibration exciter

Beam

sensor

Fig. 4. Schematic diagram of the control system.

Fig. 5. Photo of control system.

0.7 0.8 0.9 1 1.1 1.2 1.30.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Length [m]

Cor

rect

ion

of le

ngth

[m]

Fig. 6. Fitting curve of length correction (*: experimental data, —: fitting result).

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2867

to be corrected. Here a small length is plus to nominal length of the beam to make the theoretical natural frequency equalto the experimental one. To correct the length of the beam, chose a set of lengths between 0.771 and 1.305 m, thefrequency response functions of every length are measured by experiments. Then, the theoretical length l for every firstorder natural frequency is calculated by Eq. (14) so the difference between L and l is obtained

l¼ 1:8751EI

rAð2pf Þ2

!1=4

ð14Þ

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782868

where f is the first order nature frequency of the beam and the unit is Hz. To obtain the relation between nominal L and thecorrection length, Polynomial fitting method is used while the fitting order is determined by the cross validation method[30]. Finally, the correction of length, 0.0239 m, is obtained and is shown in Fig.6.

4.2. Correction of damping ratio

The damping ratio is different as the length of the beam is changing. The following method is used to correct the firstorder damping ratio. Chose a set of lengths of the beam between 0.771 and 1.305 m hang a weight to the free end of thebeam to make the beam having a deformation similar to the first order modal shape. The free acceleration response of thebeam tip is measured after the static load is suddenly released. The decay coefficient, a¼ox, is obtained by the modulationenvelope of the response curve, where o is the first order natural frequency and x the damping ratio. To obtain the relationbetween the beam length and the damping ratio polynomial with its order determined by the cross validation method [30]is used. Finally, the damping ratio is expressed as a quadratic function of the beam length as

xðlÞ ¼ 0:0949l2�0:2404lþ0:1590 ð15Þ

The damping ratio fitting curve is shown in Fig. 7.

0.7 0.8 0.9 1 1.1 1.2 1.30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Length [m]

Dam

ping

ratio

Fig. 7. Fitting curve of damping ratio (*: experimental data, —: fitting result).

C-type clamp

Permanent magnet

Force sensor

Iron disc

Fixed table

Armature

iron

d i

Fig. 8. Schematic diagram of magnetic force measurement.

Fig. 9. Photo of instrument for magnetic force measurement.

Table 1Magnetic forces of different distances.

Distance (mm) Magnetic force (N) Distance (mm) Magnetic force (N)

1 13.7020 0.6551 11 9.6444 1.5553

2 13.5600 0.6399 12 9.2029 1.6385

3 13.3010 0.6747 13 8.7610 1.7613

4 12.9640 0.7521 14 8.3590 1.7947

5 12.5200 0.8647 15 7.9730 1.8757

6 12.1620 0.9635 16 7.5960 1.8855

7 11.5460 1.0881 17 7.0250 1.9309

8 11.1170 1.1957 18 6.8580 1.9571

9 10.6927 1.2936 19 6.5160 1.9576

10 10.0805 1.4672 20 6.3790 1.9634

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2869

4.3. Magnetic force measurement

The measurement of the magnetic force of one side arm of the controller is explained in this section. The schematicdiagram for the measurement is shown in Figs. 8 and 9. A 0.25 kg iron disc is connected to the force sensor. The distancebetween armature iron and the iron disc is denoted as di and the magnetic force is denoted as Fi.

The forces are measured while the clamp is moved vertically and sinusoidally. The resulting magnetic forces accordingto di are shown in Table 1.

The function between the distance and the magnetic force is also set up by the polynomial fitting method used in theabove section. The standard deviation of each fitting order is shown in Fig. 10 and the fitting result is shown in Fig. 11 andthe obtained function is

FiðdiÞ ¼ 0:0006d4i �0:022d3

i þ0:2703d2i �1:4864diþ4:9496 ð16Þ

1 2 3 4 5 60

0.05

0.1

0.15

0.2

Fitting order

Sta

ndar

d de

viat

ion

Fig. 10. The relationship of fitting order with standard deviation.

6 7 8 9 10 11 121

1.2

1.4

1.6

1.8

2

Distance [mm]

Attr

actio

n [N

]

Fig. 11. Fitting curve of magnetic force and distance (*: experimental data, —:fitting result).

K

Gu y

w z

Fig. 12. The HN closed-loop system.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782870

5. HN method

The control system shown in Fig. 12 [31] can be described by the following state-space equations:

_x ¼ AxþB1wþB2u

z¼ C1xþD12u

y¼ C2xþD21w ð17Þ

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2871

where

A¼�1 0

0 m11

" #�10 �1

k11 c11

" #, B2 ¼

1 0

0 m11

" #�10

l�1=2f1ðaÞ

" #, C2 ¼

l�1=2f1ðaÞ 0

0 l�1=2f1ðaÞ

" #,

D12 ¼ 0½ �, D21 ¼1 0

0 1

� �, x¼ q _q

h iT, u¼ f

� �and m11, c11 and k11 are the elements in Eqs. (10)–(12), respectively.

In Fig.12, K is the controller, w is the exogenous input (such as commands, disturbances), u is the actuator input, z is theregulated output (at which performance is evaluated), and y is the sensed (or controlled) output.

The HN control problem includes determining the controller K to make the HN norm of the closed-loop transferfunction Gwz from w to z is minimized over all realizable controllers K, that is, it needs to find a realizable K so that:GwzðKÞ:1 is minimized for the system.

5.1. Smallest r

The solution says that there exists an admissible controller such that :GwzðKÞ:1or, where r is the smallest numbersuch that the following four conditions hold [31]

1.

SNcZ0 solves the following central HN controller algebraic Riccati equation (HCARE)

S1cAþAT S1cþC1T C1�S1cðB2B2

T�r�2B1B1

TÞS1c ¼ 0: ð18Þ

2.

SNeZ0 solves the following central HN filter (or estimator) algebraic Riccati equation (HFARE)

S1eATþAS1eþB1BT

1�S1eðCT2C2�r�2C1

T C1ÞS1e ¼ 0: ð19Þ

3.

lmaxðS1cS1eÞor2, where lmaxðXÞ is the largest eigenvalue of X.

4. The Hamiltonian matricesA q-2B1BT

1-B2BT2

-CT1C1 -AT

" #,

AT q-2CT1C1-CT

2C2

�B1BT1 -A

" #do not have eigenvalues on the jo-axis.

It is assumed that the length of the beam is 1.09 m and not changed. It can be obtained that the smallest r¼10166.

5.2. Closed-loop system

The state estimator equation is as follows [31]:

_̂x¼ ðAþq-2B1B1TSfc-B2Kc-KeC2Þx̂þKey

u¼ -Kcx̂ ð20Þ

where x̂ is the estimation of x.Defining a new state variable

x0 ¼x

e

� �

where e¼ x-x̂The closed-loop state-space equations are showed in the following form:

_x0 ¼ A0x0þB0wz¼ C0x0 ð21Þ

where

A0 ¼A�B2Kc B2Kc

�r�2B1B1T S1c A�KeC2þr�2B1B1

T S1c

" #

B0 ¼B1

B1�KeD21

" #

C0 ¼ C1þD12Kc �D12Kc� �

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782872

5.3. Optimal controller and estimator

In this section, it is also assumed that the length of the beam is 1.09 m and not changed. The control force is theresultant of the up and down magnetic forces. According to Section 4.3, in the linear region of the magnetic force, themaximum resultant is about 1.5N. According to the maximum range, it is chosen that

B1 ¼1=l�ð1=2Þf1ðaÞ 0

0 1=l�ð1=2Þf1ðaÞ

" #, C1 ¼ 0 11l�ð1=2Þf1ð1Þ

h i

Then it is obtained that the controller and the estimator are

Kc ¼0 0

0:1545 13:755

� �, Ke ¼ 106 2:5206 �0:0223

�0:0223 0:0002

� �

6. Vibration control

6.1. Precise method of control force

The transverse deflection of the beam at control point is defined as dc. The relative displacement between the beam andthe clamp is defined as d. The transverse displacement of the C-type clamp is defined as D and D¼dcþd.

When the desire control force is known, the relative displacement d can be derived by Eq. (16). The principle togenerate control force applied to the beam is shown in Fig. 13. When the beam is vibrating, the control force can bechanged by adjusting the relative displacement d.

6.2. Vibration control of beam with constant length

Now we consider the beam with a fixed length of 1.09 m. The distance between control position and the fixed end ofthe beam is 0.365 m. A heavy mass hung at the free end is cut down to give an initial static transverse deformationexcitation to the beam. Figs. 14 and 15 show the responses with and without control of experimental and theoreticalresults, respectively. The comparison of the experimental and theoretical responses under control is shown in Fig. 16. It isproved that the control scheme is effective.

6.3. Vibration control of an axially moving beam

When the beam is moving along the axis with a constant velocity and the initial length is 1.09 m, a heavy mass hung atthe free end is cut down to give an initial transverse deformation excitation to the beam. The distance between controlposition and the fixed end of the beam is 0.365 m.

The optimal controller is Time-Varying as the system is Time-Varying which is due to the varying length of the beam.But the length of the beam does not change much; the controller for fixed length in Section 5 is used here.

Figs. 17–19 show the results when the axial velocity of the beam is v¼0.94 cm/s. Figs. 20–22 show the results when theaxial velocity of the beam is v¼2.06 cm/s. Figs. 23–25 show the results when the axial velocity of the beam is v¼2.9 cm/s.It is verified that the control scheme is effective both in experiment and theory. It can be found that the experimentalresults have more noise and the response curve become rougher as the velocity increasing because of violent vibration of

No control force

The control force is downward

The control force is upward

dt

dd

db

dc

vc

dc

vc

D

D

Fig. 13. Principle to generate control force.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 14. Experimental results.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 15. Theoretical results.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

TheoreticalExperimental

Fig. 16. Comparison of the experimental and theoretical results under control.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2873

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 17. Experimental result when v¼0.94 cm/s.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 18. Theoretical result when v¼0.94 cm/s.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

TheoreticalExperimental

Fig. 19. Comparison of the experimental and theoretical results under control when v¼0.94 cm/s.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782874

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 20. Experimental result when v¼2.06 cm/s.

0 1 2 3 4 5 6-20

-10

0

10

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 21. Theoretical result when v¼2.06 cm/s.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

TheoreticalExperimental

Fig. 22. Comparison of the experimental and theoretical results under control when v¼2.06 cm/s.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2875

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 23. Experimental result when v¼2.9 cm/s.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

ControlledUncontrol

Fig. 24. Theoretical result when v¼2.9 cm/s.

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

Time [s]

Acc

lera

tion

[m/s

2 ]

TheoreticalExperimental

Fig. 25. Comparison of the experimental and theoretical results under control when v¼2.9 cm/s.

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–28782876

L. Wang et al. / Mechanical Systems and Signal Processing 25 (2011) 2863–2878 2877

the motor. When the velocity of the beam increases, the motor vibrate more violent and excite the table, so the responsecurves have extra noise.

There is some error in correction of boundary condition, so there is delay between numerical and experimental resultsin Figs.16, 19, 22, and 25.

7. Conclusions

Vibration control of the transverse oscillation of an axially moving beam is a challenging problem in practice. In thispaper we try to solve it by a noncontact method and obtain the following conclusions.

1.

The proposed noncontact magnet vibration exciter in the paper is proved to be effective for the vibration control of thecantilever beam with or without an axial velocity. The HN controller is used to suppress the vibration and the optimalcontroller and estimator are designed.

2.

The theoretical model updated by experiment can predict the controlled response which fits well with theexperimental result.

3.

The control system can be used to any iron structures with or without motion. 4. The control system only can be applied on structures with iron parts. The experiment platform vibration is not taken

into account in the paper.

5. Variable velocity for the beam movement and the updating to the noncontact magnetic exciter to increase the control

force will be studied in the future.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant 10672078.

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