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Torsional vibration suppression by wave absorption controller

JOURNAL OF

SOUND AND

VIBRATION

?Corresponding author.Tel.:+81298617085;fax:+81298617275.

E-mail address:m.saigo@aist.go.jp(M.Saigo).

Online computer simulation of a large dof structural system having properties similar to the actual controlled system has been used to avoid treating an irrationonal transfer function inherently existed in wave propagation[12–14].This‘‘imaginary’’system is connected virtually to the real system by an actuator satisfying the continuity condition between real and imaginary systems.This strategy realizes an in?nite structural system free of wave re?ections in the controlled real system if a suitable process is conducted to clear vibrating energy in the imaginary system at appropriate timing.When the imaginary system does not have enough degrees of freedom to absorb all vibration energy of the real system,the imaginary system will give the solution which re?ects at the end of the imaginary system.This solution returns to the real system vibration energy absorbed from the real system before.For this,the process initializes the imaginary system where de?ection and velocity of all elements are set to zero except for the end element of the connecting side. Initialization should be done before the re?ecting wave from the end of the imaginary system reaches the real system.Control effectiveness in a lumped torsional system has been shown both theoretically and experimentally [14],but control performance in experiments was not satisfactory.This was due primarily to the initialization reaction,i.e.,the successive connection of the initialized imaginary system to the real system makes a torque jump leading to computation error in the imaginary system due to suddenly increased amplitude.

We studied the impedance-matching transfer function of wave-absorption control to a lumped torsional system needing no initialization.The equation of motion of the end element of a rotating torsional system corresponds to that of a spring-and-mass system with a?xed boundary condition to which the usual impedance-matching condition cannot be applied.For this,we investigate two strategies—a new imaginary system having an impedance-matching condition at the end element and a modi?ed impedance-matching condition directly applicable to the real end element of the?xed boundary condition.The characteristic wave solution,an irrational function,is approximated with fractional polynomials by curve?tting and used to make up a so-called IIR?lter with impedance-matching condition.We also conducted experiments for2and3dof systems.

2.Control law

2.1.Equation of motion

The1D torsional vibration system considered consists of torsional bars and rigid discs(Fig.1).Rigid lines represent the real system and dotted lines the imaginary system.Our control compensates for torque K mt1f mt1,i.e.,generated at the(imaginary)connecting torsional bar between real and imaginary discs,by an actuator,and absorbs vibration energy in the real system propagated to the imaginary system.

Consider a(m+n)-dof torsional vibration system in which the controlled(real)system is m-dof and the imaginary system n-dof.The equation of motion is expressed as

€f 1t

I0;1

f1à

f2?à

€f m à

K mà1

I mà1

f mà1t

K m

I mà1;m

f mà

K mt1

I m

f mt1

|???????{z???????}

?à

T mà1

I mà1

T m

I m

Nomenclature

K i,K spring constant of i th torsional bar for nonuniform and uniform systems

I i,I moment of inertia of i th disc for nonuni-

form and uniform systems t time

T i disturbance torque on i th disc

f i angle of i th torsional bar

o0speci?c frequency for homogeneous sys-teme?

?????????

K=I

M.Saigo,N.Tanaka/Journal of Sound and Vibration295(2006)317–330 318

€f m t1àK m I m f m |{z}b

tK m t1I m ;m t1f m t1àK m t2

I m t1f m t2

?0......

€f m tn àK m tn à1I m tn à1f m tn à1tK m tn I m tn à1;m tn f m tn

?01I i ;j 1I i t1I j

,e1T

where K i ;I i ;f i are the spring constant of i th torsional bar,the moment of inertia of i th rigid disc,and the torsional angle of i th torsional bar.External disturbance on the i th disc is expressed as T i .The moment of inertia of the left-end disc and the external disturbance on it are represented by I 0and T 0.

Vibration energy in the real system propagates to the imaginary system based on propagation properties when Eq.(1)is satis?ed.Elements whose suf?xes exceed (m +1)in Eq.(1)are virtual,so we compensate for the term relating to f m +1(shown ?|{z}a

in Eq.(1))as control acceleration.Previously [14],equations of motion

including variables whose suf?xes exceed (m +1)were solved by online calculation,where variable f m (shown ?|{z}

in Eq.(1))is measured.In this paper,the imaginary system with impedance-matching characteristic at the end element is introduced,which gives a transfer function with no initialization.Because the transfer function uses a wave solution,the imaginary system is con?ned as a uniform system whose wave solution is known.The real system is not required to be uniform.2.2.Wave-absorption condition

The i th equation of motion of the uniform system with no external disturbance and in?uence of boundary condition (called as an inner element)is

€f i ào 20f i à1t2o 20f i ào 20f i t1

?0ei a 1;0T;

o 20?K =I .

(2)

Laplace transformation of Eq.(2)is

àF i à1es Tte2ts 2=o 20TF i es TàF i t1es T?0,

(3)

I 0

I m ?1

I m

I m+1

I m+2

I m+n

Fig.1.Real and imaginary torsional vibration systems connected.

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319

where F i (s )is Laplace transformation of f i .Substituting general solution F i (s )?b (s )i (F i+1(s )?b (s )F i (s ))into Eq.(3),we obtain the speci?c roots

b es T?1ts 2=e2o 20T?s =o 0??????????????????????????

1ts 2=e4o 20Tq 1ts 2=e2o 20T?s =o 0??????????b 0es Tq b es Tt;b es Tà(4)and the general solution

F i es T?c 1es Teb tTi tc 2es Teb àTi F ti es TtF à

i es T,

(5)

where c 1(s )and c 2(s )are arbitrary constants determined by boundary conditions.

Introducing s ?j o (j is the imaginary unit),when b 0(j o )in Eq.(4)is positive,b +(j o )represents a positive propagating solution to higer numbered elements and b à(j o )a negative propagating solution to lower numbered elements.The condition of existence of propagating solution 0p b 0(j o )gives the limit frequency as

o p 2?????????

K =I p 2o 0.(6)Eq.(6)shows that the torsional bar-and-rigid disc wave-absorption controller must have a speci?c

frequency o 0??????????K =I p ,i.e.,greater than half of the disturbance frequency of the controlled system.From Eq.(4),we obtain

b teàT ?1(7)which means that the steady-state wave amplitude is constant regardless of frequency.

Wave-absorption control at the boundary is simpler than elsewhere in the system because waves from outside of the boundary need not be considered.Impedance-matching control is a well-known wave-absorption strategy at the boundary [15,16],and so we study the impedance-matching condition for a lumped torsional system.Because the equation of motion for a uniform torsional bar-and-rigid disc is the same as that of a uniform spring-and-mass system (Fig.2),we use the latter to make it easier to take impedance into consideration.We regard K ,I ,torsional spring constant and moment of inertia of torsional system,as k ,m ,spring constant and mass of spring-and-mass system mathematically.

The mechanical impedance for the positive propagating solution is de?ned as the ratio of the spring force between i th and (i +1)th masses to the velocity of i th mass as

z t

es T?K F i es TàF i t1es Ts F i es T?K

e1àb tTs

?K ˉz tes T.(8)

Substituting solution b +given by Eq.(4)into Eq.(8),we obtain z tes T???????

IK p àj o =e2o 0Tt????????????????????????????1ào 2=e4o 20Tq &'.(9)

Using Eq.(8),Eq.(3)is transformed as

e1ts 2=o 20TF i àF i à1?F i t1àF i ?eb tà1TF i ?àˉz ts F i .

(10)When the right-hand term àˉz

ts F m ,putting i ?m in Eq.(10),is used as an impedance-matching wave control force at the end element,the equation of motion of the end element should satisfy Eq.(11)to be in a positive

(a)

(b)

1 i ?1 i i+1 m

Fig.2.Uniform mass-and-spring system equivalent to torsional system:(a)?xed–free boundary and (b)?xed–?xed boundary.

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320

wave propagating state

e1ts 2=o 20TF m àF m à1?0.

(11)

The equation of motion of the end element of the free boundary condition (Fig.2(a))satis?es the above

relation,but that of the ?xed boundary condition (Fig.2(b))does not,shown as

e2ts 2=o 20TF m àF m à1?0.

(12)

From Eq.(1),the end element of a uniform torsional system corresponds to Eq.(12),so we must consider the

generalized impedance-matching condition of wave absorption applied to the end element of the ?xed boundary condition.The reason why an uniform torsional vibration system,with both ends free,corresponds to a ?xed–?xed spring-and-mass system is that Eq.(1)is represented by a torsional rather than a rotation angle,that is,rigid rotation disappears in Eq.(1).As seen from the above consideration,the wave-absorption condition at the end element is obtained by making wave solution b +obtained from the equation of motion of an inner element satisfy that of the end element.Taking this into consideration,Eq.(10),also putting i ?m ,is transformed as

2ts 2=o 20àáF m es TàF m à1es T?eàˉz tt1=s Tás F m es T?b tF m es T.(13)Since the left-hand side of Eq.(13)is the same as Eq.(12),Eq.(13)represents the equation of motion of the end element when eàˉz

tt1=s Ts F m es Tis the control term.Substituting relation b tF m es T?F m t1es Tinto Eq.(13)apparently gives the same relation of three adjacent elements as the equation of motion of an inner element.

The wave-absorption condition on an arbitrary element is therefore equivalent to satisfying the wave progressive solution in its equation of motion.The so-called usual impedance-matching condition is a speci?c case of the end element of the free boundary condition.

As Eqs.(10)and (13)show,it is more convenient for the lumped torsional system considered here to deal with the relation of control force to displacement rather than to velocity,i.e.,stiffness or compliance rather than impedance.

Eqs.(10)and (13)give two wave-absorption strategies for a lumped torsional system—one to apply the generalized impedance-matching condition (GIP)given by Eq.(13)and the other to introduce an imaginary system with an impedance-matching condition at the end (IIP)where the free boundary condition is conveniently used.GIP is better than IIP method for a uniform controlled system,because a smaller order controller is constructed.When the controlled system is nonuniform,however,we must introduce an imaginary system to construct a virtual uniform system.In this case,a 1dof imaginary system with

: Control Force

(a)

m 21

k ( m ? m )

Fig.3.GIP and IIP method:(a)GIP and (b)IIP.

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321

123G a i n G1NC

G3NC G1C G3C

-540

-360-1800

0.5

1.52

Normalized frequency

00.5

1.52

Normalized frequency

P h a s e (d e g )

Fig.4.Response of 3dof mass-and-spring system:G1(3)NC,P1(3)NC,gain and phase of element 1(3)with no control and G1(3)C,P1(3)C,gain and phase of element 1(3)with wave-absorption control.

012340

1.50.5

Normalized frequency

1.50.5

Normalized frequency

G a i n

-540

-360-1800

P h a s e (d e g )

Fig.5.Response of 3dof mass-and-spring system with normal damping at end element (damping coef?cient ???????

IK p ):G1(3)D,P1(3)D,gain and phase of element 1(3)with normal damping and G1(3)C,P1(3)C,gain and phase of element 1(3)with wave-absorption control.

M.Saigo,N.Tanaka /Journal of Sound and Vibration 295(2006)317–330

322

impedance-matching characteristic constructs a wave-absorption system because a 2dof uniform system constructed by 1real and 1dof imaginary systems satisfy Eq.(11)regarding m as m +1.If a nonuniform controlled system has two same end-side elements,we use GIP because Eq.(13)is satis?ed.Again,wave-absorption control for a lumped system can be constructed for any system at the end element.Fig.3shows these strategies schematically.

Fig.4shows the response of a 3dof uniform spring-and-mass system for a sinusoidal displacement input at the left end element 1together with wave absorption control at the right end element 3and without control.

0.999

0.999511.00051.0010

Frequency (Hz)

E r r o r

-180

-120-600

024

68101214

Frequency (Hz)P h a s e (d e g

)

Fig.6.Transfer function of GIP (b +):phase:phase of Eq.(4);GER,PER,approximation error of gain and phase (GER ?GAP/G,PER ?PAP/P;GAP,approximated gain,PAP,approximated phase).

0.50.60.70.80.911.164

81012

Frequency (Hz)64

2081012

Frequency (Hz)

G a i n

-180

-120-600

P h a s e (d e g )

Fig.7.Transfer function of GIP controller for sampling time 1and 4ms:G0,G1,G4,gain of continuous system,digital ?lter with sampling time 1and 4ms;P0,P1,P4,phase of continuous system,digital ?lter with sampling time 1and 4ms.

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323

The controlled amplitude remains constant corresponding to the input amplitude up to wave propagation limit frequency (o =o 0?2).The controlled phase of element 1changes from 01to à1801and element 3from 0to à5401up to the wave propagation limit frequency.The uncontrolled phases of elements 1and 3also change from 01to à1801and 01to à5401stepwise.Here,the uncontrolled phase is depicted as continuous functions,unusual in vibration textbooks,which helps in understanding wave-controlled characteristics—phase shift characteristics inherently exist in the spring-and-mass vibration system.The uncontrolled phase and amplitude approach the wave controlled one if a velocity-dependent normal damping is introduced in place of wave-absorption control (Fig.5),with damping coef?cient equivalent to z ???????IK p used.2.3.Wave-absoption control ?lter

The wave-absorption control ?lter is designed for a uniform torsional system whose parameters are the same as those of the experiment detailed in the next section.Characteristic root b +(s )is approximated by

00.511.522.5G a i n

-180

-120-600P h a s e (d e g )

2081012

Frequency (Hz)

81012

Frequency (Hz)Fig.8.Transfer function of IIP controller for sampling time 1and 4ms:G0,G1,G4,gain of continuous system,digital ?lter with sampling time 1and 4ms;P0,P1,P4,phase of continuous system,digital ?lter with sampling time 1and 4ms.

Fig.9.Experimental apparatus.

M.Saigo,N.Tanaka /Journal of Sound and Vibration 295(2006)317–330

324

2dof 1st 4.28Hz

Time (s)

A n g l e (d e g )

2dof 2nd 7.18Hz

Time (s)

0.4

-0.4

0.2-0.20

A n g l e (d e g )

0.4

-0.4

0.2-0.20

(a)

(b)

Fig.10.Response of controlled and uncontrolled amplitudes of 2dof uniform system by GIP:(a)at the ?rst resonance frequency and (b)at the second resonance

frequency.

Time (s)

A n g l e (d e g )

5678

Time (s)

A n g l e (d e g )

0.4

-0.4

0.2-0.20

3.5

2.5

3.0

0.5

1.5

(a)

(b)

Fig.11.Response of controlled and uncontrolled amplitudes of 2dof uniform system by IIP:(a)at the ?rst resonance frequency and (b)at

the second resonance frequency.

M.Saigo,N.Tanaka /Journal of Sound and Vibration 295(2006)317–330

325

six-order fractional polynomials.Fig.6shows b +(s )and the approximated representation for frequency o using the spring constant of torsional bar K ?21:4Nm and moment of inertia I ?0:0123kgm 2(o 0?6:64Hz).Gain is constant at unity and the phase changes from 0to à1801.The approximated function is obtained by the invfreqs function of MATLAB,which agrees well with original function b +(j o )in all frequency ranges except for the vicinity of wave propagation limit frequency 2o 0.By zero-order hold z -transformation of this approximated transfer function with sampling periods 1and 4ms,an IIR digital control ?lter for GIP is obtained (Fig.7).This transfer function was calculated by using the freqz function of MATLAB.

Similarly,an imaginary system with impedance-matching condition is designed as an IIR digital ?lter.Because the 1dof imaginary system having a free end condition is suf?cient for impedance-matched wave absorption,the transfer function is expressed as

K F m t1es TF m es T?K 1ts 2=o 2

0tes =o 0Tz tes T=??????IK p h i à1.(14)By zero-order hold z -transformation of Eq.(14)with sampling period 1and 4ms,we obtain an IIR digital ?lter (Fig.8)with approximation errors at o 0p o for the 4ms sampling period.Since no characteristics of

Time (s)

A n g l e (d e g )

Time (s)

0.5

1.5

2.5

3.5

4.5

0.30.20.10-0.1-0.2-0.30.2-0.4-0.2-0.6-0.8

00.40.60.80.20.1-0.2-0.3-0.1-0.4-0.5

00.30.40.5234

567

135

(a)

(b)

(c)

Fig.12.Response of controlled and uncontrolled amplitudes of 3dof uniform system by GIP:(a)at the ?rst resonance frequency,(b)at the second resonance frequency and (c)at the third resonance frequency.

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326

such errors are seen in Fig.6,this may be due to resonance phenomena of the 1dof imaginary system resembling characteristics in Fig.5.By losing the wave propagation compensation characteristic,the roughly approximated controller approaches normal damping.3.Experiments

We have conducted experiments by using the IIR ?lters of GIP and IIP (Fig.9).The uniform controlled vibration system consists of discs 200mm in diameter and 20mm thick and torsional bars 4mm in diameter and 100mm long for 2and 3dof uniform systems.AC servomotors are used for drive and control.The drive motor has a rated torque of 0.9Nm and a rated speed of 3000rev/min and the control motor has a rated torque of 0.16Nm and a rated speed of 3000rev/min.The measurement system consists of rotary encoders,a torsional angel converter,low-and high-pass ?lters,and a personal computer (CPU clock:333MHz).Torque disturbance is applied by torque ?uctuation of the AC drive motor in constant speed mode,which is 4times,2

times,and 1time per rotation,so,resonance may occur at a rotation speed of 14,12,and 1

1of the natural

Time (s)

A n g l e (d e g )

Time (s)

A n g l e (d e g )

Time (s)

A n g l e (d e g )

2.55

432

0.40.2-0.2-0.4

-0.3-0.2-0.10.10.20.30

0(a)

(b)

(c)

Fig.13.Response of controlled and uncontrolled amplitudes of 3dof uniform system by IIP:(a)at the ?rst resonance frequency,(b)at the second resonance frequency and (c)at the third resonance frequency.

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327

frequency.Torque magnitude cannot be adjusted.Experiments are conducted for 2and 3dof with a sampling period of 4ms.

Figs.10and 11show controlled and uncontrolled amplitudes of 2dof uniform system by GIP and IIP,with (a)at the ?rst resonance frequency and (b)at the second resonance frequency.Amplitudes are suppressed considerably by control and phase shifts between f 1and f 2of the controlled amplitudes are seen.Figs.12and 13show controlled and uncontrolled amplitudes of 3dof uniform system by GIP and IIP,with (a)at the ?rst resonance frequency,(b)at the second resonance frequency,and (c)at the third resonance frequency.Amplitudes are suppressed considerably by control and the phase between f 1and f 3of controlled amplitudes is shifted.These results are greatly superior to those produced by online simulation of the imaginary system in Ref.[14](Appendix A),which con?rm the high control performance of the developed wave-absorption control ?lter.Fig.14shows wave propagation characteristic b +in the controlled system.The experimental phase shift is obtained by using a band-pass ?lter at the corresponding frequency.The phase shift for the 3dof system is calculated as half of the shift between f 1and f 3.The observed phase shift angle between adjoining elements agrees well with Eq.(4)—further evidence of the wave control performance of the developed control ?lter.Although control performance between GIP and IIP differs in Figs.7and 8,we can see no marked difference in experiments (Figs.10–14).4.Conclusions

We have developed wave-absorption control ?lters for the end element of a lumped torsional system and have demonstrated its effectiveness in experiments,with the following major results:

(1)The wave-absorption ?lter for the end element of a lumped torsional system has been developed by using

the characteristic root of the equation of motion of an inner element with no in?uence of boundary condition.It is quite useful for a uniform controlled system experimentally,and it is also applicable to a nonuniform system with two same end elements.

(2)The wave-absorption ?lter for the end element of a lumped torsional system constructed by using the 1dof

imaginary system and impedance-matching condition is useful for both a uniform system and a nonuniform system.

(3)The impedance-matching condition of wave-absorption control at the end element is a speci?c condition

for a free end boundary condition,and the general wave-absorption condition is to satisfy a characteristic root on the equation of motion of the end element.

(4)The wave-absorption control ?lter constructed using only the characteristic root of the equation of motion

of an inner element is more suitable than that using the characteristic root and the imaginary system because of the lesser order of the transfer function with practically no loss of control performance.

(5)The characteristic root of wave propagation is usefully approximated by six-order fractional polynomials

to construct digital control ?lters.

-180

-150-120-90-60-300

Normalized frequency

P h a s e d i f f e r e n c e (d e g )

Fig.14.Wave propagation characteristics in controlled system;&,experimental result of GIP ?lter;W ,experimental result of IIP ?lter;~,resonance point (m –n :n th resonance of m dof system).

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328

Appendix A

Experimental results of wave absorption control with online simulation of imaginary system [14]are shown below for comparison with those in this paper (Fig.A1).

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Time (s)

A m p l i t u d e (r a d )

-3.E-033.E-03-2.E-032.E-03-1.E-031.E-030.E-000

246810

Time (s)

A m p l i t u d e (r a d )

46810

Time (s)

A m p l i t u d e (r a d )

Time (s)

A m p l i t u d e (r a d )

(a)

(b)

(d)

(c)

Fig.A1.Timing chart of 3dof real system experiment at a speed of (a)a quarter of ?rst resonance,(b)?rst resonance,(c)second resonance,and (d)third resonance.

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