Active Control of Input Power Flow to the Cylindrical Shells

(Department of Naval Architecture&Ocean Engineering,Huazhong University of Science&Technology,Wuhan 430074,China)

1 Introduction

Cylindrical shell is widely used in numerous fields such as domestic,industrial and martial environments.As a structure used commonly,the cylinder is always stimulated by lopsided forces and moments.The more important thing is that,vibration of the cylinders will also stimulate other equipments attached to the shell.This not only makes a lot of noise,but also reduces the performance of the apparatuses,threatens the safety and life-span of the structure.Furthermore,with the vibration of the structure,the whole system may be damaged,and this will induce an unpredictable results.Therefore,it is of great importance to analyze and control the vibration of the cylinders.

Many scholars have done large amounts of researches on the vibration control of cylindrical shells.Most of the works have been done are concentrated on the passive control methods such as the usage of the damping materials and the discontinuity of the structure.The former induces a heavier weight and a larger volume of structures and the latter does not perform receivable at a wide frequency bands.Compared with this,it is more attractive to implement an active control for the structures with the weight constraints which prevent the increase of mass or damping treatments,or static stiffness requirements which prevent the fitting of flexible elements.

For the method of active control,much work has been done and a lot of available theories have been obtained.It is easier to implement the active control in frequency domain than in time domain.However,modal models are only appropriate for structures that have low modal density,because the estimated models are not exact and are extremely sensitive to modeling errors,particularly at high modal density.While,power flow contains the state of the structure,considers the magnitude and phase offset of the force and response at the same time.We can describe the dynamics in wave model and control the vibration of the structure in the view of energy,then,a global optimal control effect can be attained.

Relevant theories about the method of structure power flow are introduced firstly by Goyder and White[1-3].Because of its brief form and prominent merit,the method of structure power flow has become a useful tool to analyze the structure vibration and has caused a wide attention of a lot of scholars.For the research of cylinder power flow,Langley[4]considered that the vibration power flow was associated with helical wave motion.The relationship between the energy flow direction and the helical wave angle is denoted visually.Zhang[5]investigated the input and transmission power flow of the infinite cylindrical shell,periodic ring-stiffened cylindrical shell and periodic viscoelastic cylindrical shell stimulated by the external force.Xu[6]studied the input and transmission power flow of the cylindrical shell coupled with fluid.

For the active control methods based on energy,Yan[7]studied the multiple dynamic vibration absorbers.The input power flow to the plate is described in the form of the mobility function.By changing the tuning ratio and damping ratio,minimum value of the objective function can be obtained.Pan et al[8]investigated the control method for radial displacements and transmitting power flow of the locations where the sensors are mounted.Brennan[9]developed a modal of beam,which is controlled by a secondary force array with the wave approach.The flexural vibration of the beam is controlled with different strategies.The algorithm is relatively easy,which reforms the objective function into the Hermitian quadratic form to get the optimum control forces.The frequency dispersion of the vibrational waves in cylindrical shells filled with fluid is investigated by Brévart et al[10].The square of the radial modal response is treated as the objective function and the algorithm mentioned in Ref.[9]is used to implement the global optimal control.The difference of power flow through the system with and without control is then used to evaluate the performance of control approach.Xu[11]performed the theoretical research of active control for the vibration of infinite elastic cylindrical shell.The transmission loss of the structure power flow is studied and used to evaluate the performance of the active control method.Zhu[12]proposed three strategies based on power flow to control the structure,and the algorithm is then used to control the vibration of a reduced building structure model which has single degree of freedom.

In this paper,theory analysis will be performed for the active control of the infinite elastic cylindrical shell,which is stimulated by a prime radial circumference harmonic line force.Control forces(secondary forces)are also line forces applied apart from the primary force,and they have the same frequency with the prime one.The total power flow put into the structure by the primary and secondary forces is described in the form of Hermitian quadratic function by using mobility functions.The optimal control forces are obtained by using feed-forward quadratic optimal theory.Moreover,the influence of locations where the secondary forces are applied,and the magnitudes of the control forces needed are discussed.

2 Forced vibration of cylindrical shell

An infinite thin-walled cylindrical shell as shown in Fig.1 is considered.Supposing the wall thickness of the shell is h.R is the mean radius of the shell;ρ,μ and E are the mass density,Poisson ratio and Young’s modulus of the shell material,respectively;u,v and w are the shell displacements in the x,θ and r directions respectively.

The wave motions in the shell wall are described by the Flügge shell Eq.[13]:

where Lijdenote differential operators,Fjare the terms about the external forces exerted on the shell in the directions of coordinate x-,θand r-axes.

Supposing that the shell is stimulated by the radial harmonic line force:

where Fpis the amplitude of the force,n is the order of the circumferential modes,ω is the circular frequency,δ is the Dirac delta function.

The normal mode shapes assumed for the displacement of the shell wall,which are associated with the axial wave number kns,are given by:

Substituting Eqs.(2),(3)into Eq.(1)and taking the Fourier transform gives following equations:Let matrix T be the inverse of matrix L,the spectral displacements can be obtained as:Substituting Eq.(5)into Eq.(3),omitting exp （iωt） for brevity,gives the space displacement as:

3 Active control method for input power flow

Structure power flow contains not only the magnitudes of the forces and displacements but also the phase relationship between them.To implement the control with total input power flow as the objective function will reduce the vibration of the structure efficiently.For the cylindrical shells,the structure power flow stimulated by the primary force can be controlled by offering proper secondary forces(including the magnitudes,the phases and their application locations).

Supposing that the secondary forces are the radial harmonic line forces applied at the cross-sections X=[x1,x2,x3…]Twith their amplitudes expressed as:

where Fcjare the amplitudes of the secondary forces applied at the cross-section xj(j=1,2,3…).

The radial displacement amplitude vector Wpof the section where the primary force is applied and the vector Wcof the sections where the secondary forces are applied can be described as:

where αpp,αccare the point mobility matrices,and αpc,αcpare transfer mobility matrices.The elements of the matrices can all be obtained by using Eq.(6).

Supposing the total power flow input by the primary and secondary forces is Pt.According to the definition of power flow and Eq.(8),Ptcan be described as:

where the superscript*denotes the complex conjugate,the superscript H denotes the transpose of the complex conjugate(Hermitian transpose).Acc=Im α（cc）,

As shown in Eq.(9),the total input power flow Ptis expressed into the Hermitian quadratic form,it is a positive definite quadratic function of the control forces(secondary forces).This positive definite quadratic relationship guarantees that there will be only one extremum and this extremum is a minimum.The optimal value of the control forces amplitudes,which educes the minimum value of the total input power flow,can be obtained by using feed-forward quadratic optimal theory[14]:

where Rcc=Re（ αcc）,Rcp=Re（ αcp）;Vc=[Vc1,Vc2,Vc3…]is the velocity amplitude vector of the cross-sections where the secondary forces are applied.

By using Eq.(10),the optimum control forces,which educe the minimum total input power flow,can be obtained.

The objective function mentioned above can be expressed as follows:T

Like Eq.(9),Eq.(11)is a positive definite quadratic function of the control forces,too.By using the algorithm which educes Eq.(10),the following expression which develops the minimum value of Eq.(11)can be obtained:

4 Numerical computation and results discussion

The material and geometry parameters of the shell considered in this paper are shown as follows:Young’s modulus E=2.0×1011N/m2,Poisson ratio μ=0.3,density ρ=7 800kg/m3,mean radius of the shell R=1m,ratio of wall thickness to radius h/R=0.05.The discussions will focus on the circumferential mode order n=1,which represents the typical behavior of the system.The amplitude of the primary force is supposed to be Fp=1N.

The non-dimensional frequency Ω and non-dimensional input power flow Pt′are defined as:

4.1 Application of a single secondary force

First,Bccis set to be zero.Then,the relationship between the control distance and the control effect can be found out clearly.Where,the control distance indicates the axial intervals between the sections where the primary and secondary forces are applied.

4.1.1 The effect of the control distance on total input power flow

From Fig.2,it can be found that,how the normalized total input power flow varies as a function of the ratio xc=x1/R at the non-dimensional frequencies Ω=0.9 and Ω=1.0.The figure indicates that,with the increasing of the control distance,the total power flow put into the shell varies periodically.Furthermore,it can be observed that the fluctuant period of the total input power flow has close correlation with the structure waves exist in the shell.

By using the coefficient matrix [L3×3]mentioned in Eq.(4),the wave numbers for a given non-dimensional frequency can be obtained.Thereinto the results of these wave numbers,the pure real value represents propagating wave[16].For the example discussed in this paper,there exists two propagating waves in the cylindrical shell at the normalized frequency Ω=0.9.The corresponding non-dimensional wave numbers for these waves are ks=3.71 and kl=0.81,respectively.The ratio λ=2π/k is defined as the non-dimensional wavelength of the propagating wave in this paper.Then,the following expressions are obtained:λs=2π/ks=1.7 and λl=2π/kl=7.8.By analyzing,the connections among the normalized propagating wavelength,the control distance and the control effect are detected and represented as follows:when the control force is applied at the distance xc=0.5nλs（n=1,2,3…）,the better control effect can always be obtained(accordant with the trough of the curve in Fig.2);when the control distance is xc=2.5nλs(n=1,2,3…),the total input power flow with control tends to be zero.Where,λsis the non-dimensional wavelength corresponded to the larger real wave number.In addition,the relationship between λsand λlcan be described as:λl≈5λs.At the normalized frequency Ω=1.0,the same connections mentioned above are found to be established all the same.

From Fig.2,it can still be found that,at the origin of coordinate xc=0,for both of the two non-dimensional frequencies Ω=0.9 and Ω=1.0,there exists Pt′=0.As the increasing of xc,all curves vary periodically.And besides,the force with higher frequency is applied,the shorter period the curves in Fig.2 have,and the control effect becomes more sensitive to the control distance.

By analyzing the results，following conclusions can also be obtained:For the reason that,at all frequencies,the curves,which indicate the input power flow with control,are all getting start with zero at the origin of coordinate xc=0,thus,a perfect control effect can be obtained at a wide frequency band when the control force is applied very near to the primary force(e.g.xc＜0.1).With the increasing of the control distance,the control effect gets more sensitive to the position where the control force is applied at some frequencies.It means that,the control effect will not be acceptable at a wide frequency band with a large control distance when only one control force is applied.To illuminate these,the curves of total input power flow versus driving frequency are drawn in Fig.3.The non-dimensional input power flow are plotted in the figure for the control distance xc=0.05 and xc=3.5.To show the control effect,the input power flow without active control is also plotted in the figure.The graphics shows clearly that,the total input power flow for the control distance xc=0.05 is much smaller and more stable than that for xc=3.5,especially at relative low frequencies(e.g.Ω＜2.3).For the characteristics of the total input power flow with control discussed above,we can control the input power flow to the shell with a small control distance or implement the control at a certain long distance aiming at the given frequency.

4.1.2 The connections between the control distance and the magnitude of the control force needed

Fig.4 shows how the non-dimensional control force amplitude Fcn=Fc1/Fpchanges with the increasing control distance xc,when the shell is stimulated at the normalized frequency Ω=0.9.Fc1denotes the optimal value of the control force amplitude calculated by Eq.(10).From Fig.4,the relationship between Fcnand xccan be found out clearly.

In Fig.4,it can be found out that,Fcnvaries periodically with the increasing of the normalized control distance xc.By comparing the figure with Fig.2,it can be found that the curve for frequency Ω=0.9 in Fig.2 has the same period with the curve in Fig.4.The trough of the curve for frequency Ω=0.9 in Fig.2 is just corresponding to the wave crest of the curve in Fig.4,which means that,when the cylindrical shell is stimulated at some frequency,more control energy is needed to carry out the control at the locations where a preferable control effect can be achieved.In addition,when the control is implemented with one secondary force,the inequality Fcn≤1 is valid at any frequency.

4.2 Application of secondary forces array

When the control is implemented with one secondary force,the control effect will not be good enough at a wide frequency band except that the control force is applied near to the primary force.To overcome this problem,more than one secondary force can be used to control the total power flow put into the shells.

By using the conclusion obtained in section 4.1,the appropriate locations can be chosen to apply the control forces.Then,just as what will be shown in the following text,a better control effect can be obtained by using multi-force control.For the sake of comparability,we still set Bccto be zero in this section.

The condition of three secondary forces applying is considered here,and two groups of control locations are chosen to give a comparison.From Fig.5,it can be found,how the non-dimensional total input power flow Pt′varies as a function of the normalized frequency Ω at two groups of locations for the multi-force conditions(with and without control are both given in it).In the figure,the ratios xcj=xj/R （ j=1,2,3）indicate the locations where the control forces are applied.

There are two groups of xcjin Fig.5,which are used to denote the short and long distance control respectively.By comparing them with Fig.3,it is clear to find that,not only at the short but also at the long control distance,the control effect with three secondary forces applying is much better than that with one secondary force.Multi-secondary force control guarantees the control effect at a wide frequency band,especially for the condition of a short distance control,just as we see in Fig.5,using the method can get the total input power flow reduced more than 95%,and make Pt′tend to be zero at the short control distance.

4.3 Multi-secondary forces control with different strategies

Up to now,all the conclusions obtained are based on the condition Bcc=0.In this part,the characteristics of the input power flow controlled by multi-secondary force under the circumstance Bcc≠0,are discussed.

For the active control of input power flow to the cylindrical shell stimulated by a primary force at certain frequency,the reduction ratio of the total input power flow CE is used to assess the control effect in this part,which is described as:

where P0is the input power flow stimulated by the primary force without any control force applied.

The curves A and C in Fig.6 show the secondary forces amplitudes needed with strategy 1,from which it can be found that,the maximum value is more than 16 times of the amplitude of the primary force.Compared with one-force-control,multi-force-control can get a better control effect at the cost of extra high control energy when Bcc=0.Different from this,by using strategy 2,the amplitudes of the control forces needed are much smaller(curves B and D).Moreover,when Ω＜0.8,the sum of the three control forces amplitudes needed to implement optimal control meets the formula:Fcnt≈Fp.

To control the input power flow with strategy 2,the control force amplitude needed is much smaller than that with strategy 1.Meanwhile,by the results listed in Fig.7,it can be found that,a receivable control effect can also be attained with this strategy.The curves A,B in Fig.7 indicate the control effect with two strategies when the control forces are applied near to the primary force.It shows that,with strategy 2,the control effect is comparable with that by using strategy 1,and reduction of the input power flow is more than 90%.What needs to be mentioned is that,the control forces amplitudes needed to achieve this effect are much smaller than those by using strategy 1,which can be found by the curves A and B in Fig.6.

The coefficient β of the weighting matrix Bccis of great importance.The objective function with a smaller value of β,will induce a better control effect but more control energy at the same time.The influence of the coefficient β can be realized intuitively from Fig.8.

Fig.8 is constructed with Pt/P0and Fcnt/Fpas ordinate and β as abscissa(the lines with symbol*indicate Fcnt/Fp,the others indicate Pt/P0)to show the influence of different β.The control forces are located at xc1=2.5,xc2=3.0,xc3=3.5.From this figure,it can be found clearly that,with the increasing of β,the control forces needed decrease and the control effect is getting worse at the same time.When β→∞,Pt/P0is almost up to 1 but will not equal to 1 because of the existence of Ptin the objective function.When β is set to be zero,the objective of control is just the total input power flow,the results will be the same with those that be derived with strategy 1.

5 Conclusions

The active control method of the total input power flow to the hollow cylindrical shells is investigated in this paper.The objective function(the total input power flow)is expressed in the Hermitian quadratic form by using mobility function.The optimum set of secondary forces is obtained and the results by using two strategies are fully examined and compared.By calculating and analyzing,following conclusions can be obtained:

When the control is implemented with one secondary force,a nice control effect can be attained on the condition that the control force is applied near to the primary force(e.g.xc＜0.1).

Both the total input power flow after control and the secondary force amplitude needed vary periodically with the increasing of the control distance.The two sets of varying data have the same period,and the trough of the curve,which indicates the controlled power flow,is just corresponding to the wave crest of the curve which denotes the control force amplitude needed.

When the cylindrical shell is controlled with one secondary force,the control force amplitude needed will not be larger than that of primary force at all the frequencies.

When the cylindrical shell is controlled with one secondary force at certain frequency,a relatively fine control effect can be attained on the condition that the control force is applied at the control distances xc=0.5nλs（n=1,2,3…）;if the control distances satisfy xc=2.5nλs（n=1,2,3…）,an excellent control effect can be obtained.In addition,there exists a relationship between λsand λl:λl≈5λs.

Compared with one-force-control,multi-forces-control will lead to much more reduction of the total input power flow at a wide frequency band.Especially at the short control distance,reduction of the total input power flow is more than 90%.

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