VIBRATION CONTROL OF A STRUCTURE BY USING A TUNABLE ABSORBER AND AN OPTIMAL VIBRATION ABSORBER UNDER

Journal of Sound and

Article No.jsvi.1999.2443,available online at https://www.sodocs.net/doc/2718310121.html, on

VIBRATION CONTROL OF A STRUCTURE BY USING A TUNABLE ABSORBER AND AN OPTIMAL VIBRATION

ABSORBER UNDER AUTO-TUNING CONTROL

K.N AGAYA,A.K URUSU,S.I KAI AND Y.S HITANI

Department of Mechanical Engineering,Gunma Uni v ersity,Kiryu,Gunma376-8515,Japan (Received16April1998,and in,nal form1June1999)

Vibrations of machines and structures vanish perfectly at a certain frequency when they have a vibration absorber without damping.But if forced frequencies vary from the anti-resonance frequency,their vibration amplitudes increase signi"cantly.Then the absorber without damping cannot be applied to the structure subjected to variable frequency loads or the loads having high-frequency components.The present article discusses a method of vibration control of

a structure by using the vibration absorber without damping.In this method,

a variable sti!ness vibration absorber is used for controlling a principal mode.The

sti!ness is controlled by the microcomputer under the auto-tuning algorithm for creating an anti-resonance state.The optimal vibration absorber with damping is also utilized for controlling higher modes.The analyses and the algorithm for the auto-tuning control are developed.A method to obtain optimal parameters has been presented for the vibration absorber which controls higher modes.In order to validate the control method and the analysis,experimental tests have been carried out.

1999Academic Press

1.INTRODUCTION

Various vibration absorbers and dampers have been presented in a number of papers[1}4].Recently,complicated phenomena of vibration absorbers such as non-linear vibrations have been discussed in references[5,6].Vibration absorbers have been used for suppressing vibrations of various structures[7,8],and machines such as o!shore platform[9],valves[10],multi-stage pumps[11],and shock absorbers of machines[12].The absorbers as just mentioned are the vibration control elements for passive control,so in the design,maximum amplitudes of the vibrating body are minimized in a wide frequency zone by using optimal absorber parameters.Recently,active and active/passive vibration absorbers have been discussed[13}15]in which rigidity and damping are varied by using actuators. Najet et al.[16]gave an interesting tunable absorber for a two-degree-of-freedom system in which the optimal poles were given by the use of time-delay feedback. Recently,Sun et al.[17]gave a survey of passive,adaptive and active tuned

774K.NAGAYA E1A?.

Vibrations of a mass having a vibration absorber without damping vanish perfectly when the frequency of a sinusoidal force acting on the mass is equal to the natural frequency of the absorber.Since the phenomenon is just opposite to the resonance,the phenomenon is called&&anti-resonance'',and the frequency in which the vibration amplitude becomes minimum is called&&anti-resonance frequency''in the present paper.When the forced frequency is di!erent from the anti-resonance frequency,the vibration amplitude of the mass increases signifcantly.For the reason as just mentioned,the absorber using the principle has not been used in practical problems.In the absorber,if the sti!ness of the absorber is tunable, vibrations of the mass can be suppressed perfectly from a theoretical stand point. The principle has been used for controlling vibrations for a rigid body[17,18]by use of the tunable vibration absorber.The methods in references[16}18]are applicable in case of a rigid body and sinusoidal exciting force.

In practical structures,exciting forces have complicated waves with respect to time,which are di!erent from the sinusoidal waves in general.This implies that the forces involve higher frequency components,and hence the higher vibration modes are generated for#exible structures.The tunable vibration absorbers for the rigid body under sinusoidal forces cannot be applied directly to such a problem.Then all resonance peaks in the frequency domain have to be suppressed in the design of absorbers for#exible structures.In order to reduce resonance peaks,and to have #at response curve in a frequency domain,vibration absorbers with damping have been widely used[7}15].However,the amplitudes at a low-frequency region within the"rst resonance frequency are greater than the displacement due to the static load whose intensity is equal to the amplitude of vibration load.Hence,the maximum amplitude ratio in the frequency domain is greater than unit[the amplitude of displacement due to a vibration load with amplitude P/(the displacement due to the static load with intensity P)'1]when the vibration absorbers with damping are used.In addition,the number of vibration absorbers has to be equal to that of control modes.Hence,many absorbers are required. In the present article,a method of vibration control for structures has been presented with consideration of higher modes of vibrations.In our method,the principal vibration mode is controlled by use of auto-tuning anti-resonance control of the tunable vibration absorber without damping,and the higher modes are suppressed by the optimal vibration absorber with a magnetic damper.Theoretical results under the control have been presented.In order to validate the present control method and analyses,experimental tests have been carried out.

2.RESPONSE OF A STRUCTURE WITH VIBRATION ABSORBERS

The purpose of this paper is to suppress vibrations of structures with consideration of higher modes.When a structure is excited by a cyclic exciting force whose principal frequency is smaller than the"rst resonance frequency,the vibration amplitude with the forced frequency is signi"cant,but the components of higher frequencies are also involved in the amplitude.When an ordinary vibration

AUTO-TUNING VIBRATION CONTROL OF STRUCTURES775 cannot be suppressed(the amplitude ratio is greater than unit).The tunable vibration absorber without damping suppresses vibrations for the principal mode, but it cannot suppress higher modes.In order to suppress vibrations in a wide frequency region,the present article proposes a system consisting of a tunable vibration absorber and a vibration absorber with a magnetic damper.The tunable absorber suppresses the principal mode,and the vibration absorber with the damper suppresses higher vibration modes.When the vibration absorbers with damping are used,the number of vibration absorbers has to be equal to that of the control modes in the usual design of absorbers.The vibrations of higher modes can be suppressed by small damping,because a damping force is in proportion to the vibrating speed.The present authors presented a method for suppressing higher modal vibrations by using the fewer number of vibration absorbers with consideration of the damping phenomena just mentioned[19].In the method,the absorbers control higher modes whose number is greater than the number of vibration absorbers.The method is used to design the absorber in the present article.

Figure1shows the geometry of the structure with the absorbers in which one of the absorber has no damping(the tunable absorber for anti-resonance is called absorber1),and the other has the magnetic damper(the absorber for higher modes is called absorber2).The locations of the absorbers are important,because the system parameters vary with the sti!ness of the tunable absorber.The optimal parameters of the vibration absorber with damping vary with the system parameters.This implies that both the tunable absorber and the vibration absorber with damping cannot be used simultaneously.In order to use the absorber for higher modes,the tunable absorber(absorber1)lies on the nodal point of the second mode of the structure,and so the motion of absorber1does not a!ect the second mode,because,the second mode vibration is not generated due to the excitation at the second nodal point.Absorber2lies at the anti-nodal point of the second mode.The motion of absorber2due to the second modal vibration also

does not a!ect the"rst mode(the phenomena will be clari"ed in the analysis).Then absorber2can be designed without consideration of absorber1,and the constant optimal parameters of the absorber are applicable.For the"rst mode,since absorber2is not laid on the nodal point of the"rst mode,it a!ects the"rst modal vibration of course,but its e!ect can be involved in the auto-tuning control of the "rst mode.

In order to indicate the validity of the present system,consider a simple beam with both built-in edges.The theoretical model is shown in Figure1.The vibration response can be obtained by using the transfer matrix method whose nature has been discussed in references[20,21].The matrix equation between the j th and (j#1)th element of the beam is

<*H"F H<0H\ ,(1) where

F H"1!?H

?

H

2EI H

?

H

6EI H

01

!?

H

EI H

!?

H

2EI H

001?H0

00010

00001

(2)

and where R denotes the right side,?the left side,w the displacement of the beam, the bending slope,1the bending moment,Q the shearing force,?H the element length and EI H the#exural rigidity.The equation between the right side R and the left side?at an arbitrary point j is

<0H"P H<*H,(3) where

P H" 100000100000100M H 001f H00001 H(4) and where M H is the element mass at the point j,f H the load amplitude acting on the 776K.NAGAYA E1A?.

The equation of motion of the vibration absorber is

m d u d t #c d u d t !d w d t #k (u !w )"0,(5)

where m is the mass,k the spring constant,c the damping coe $cient,and u the displacement of the mass of the vibration absorber,respectively.Substituting w "=

e S R ,u "; e S R into equation (5),one obtains Q "B ( )m

,(6)where B ( )"p #2 i p

! #2 i ,(7)

2 "c /m ,p "k /m

,i "(!1.Hence,the point transfer matrix at absorber 2is P H

" 100000100000100(M H #B m ) 001000001 .(8)

The size and weight of absorber 1are not small,and so a rotary inertia must be considered.The point transfer matrix with such consideration is P H " 1000001

0000!J 100(M H #B m ) 0

01000001 H ,(9)

where J is the moment of inertia of absorber 1,and where

B ( )"p p

! .(10)

Since both ends of the beam are built-in,the boundary conditions are

AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 777

T ABLE 1

Dimensions of the beam

Material

Aluminum Shape of the cross-section Channel Size of the cross-section

21)5;60)4;3)3(mm)Length of the beam

980(mm)Young 's modulus

7)06;10 (N/m )Density

2)7;10 (kg/m )Total mass of the beam

0)845(kg)Moment of the cross-section

1)2;10\ (m )Mass of the attached plate at the forcing point 0)5(kg)

where 0denotes the left end,and N the number of the point at the right end of the beam,from which one obtains the state variables at point 0:

Q "a a !a a det

,1 "a a !a a det ,det "a a !a a

,where a GH is the element of i th row and j th column of the transfer matrix 1,"P ,F ,2P F .The state vector

Numerical calculations are carried out for the beam with vibration absorbers used in the experiment as mentioned below.An aluminum beam with channel shape is used.Table 1shows the dimensions of the beam.The dimensions for the variable sti !ness absorber (see Figure 2)used in the calculation are also depicted in Table 2.The nodal point of the second mode is calculated as x "46cm from the left end of the beam where the auto-tuning absorber is attached (hereafter,the point is called as point A).Numerical calculations are carried out under the assumption that the sinusoidal force with amplitude 10N acts on the point measured from 24)5cm from the left end of the beam.Figure 3depicts the compliance versus the forced angular frequency at point A.The resonance frequency of the beam varies with the sti !ness of the absorber (the natural frequency of the absorber).The anti-resonance frequency also varies with the sti !ness of the absorber.However,since the variable sti !ness absorber lies on the nodal point of the second mode,the second resonance frequency has no dependence on the sti !ness of the absorber.This implies that absorber 2can be designed without consideration of sti !ness variation of absorber 1.The total mass of the absorber is large,and so the resonance peaks of higher modes more than the third mode are signi "cantly small in comparison with the "rst two peaks.If the anti-resonance frequency is tuned to

778K.NAGAYA E 1A ?.

Figure 2.Geometry of the anti-resonance absorber (variable sti !ness absorber)

T ABLE 2

Dimensions of the variable sti +ness absorber for anti -resonance control

Attached mass at the tip of beam

0)6(kg)Eigenfrequency of the absorber

13}29(Hz)(variable)Spring of the absorber

Beam made of a stainless bar Total mass of the absorber 4)23(kg)

be the principal frequency of the forced load,the vibration amplitude of the "rst mode of the beam becomes zero when there is no damping.Since the present system has a damper in absorber 2for suppressing higher modes,the "rst modal vibration does not become zero,and so the small damping is desirable for absorber 2.

From Figure 3,it is ascertained that the combination of the tunable absorber and the absorber with damping is able to control vibrations.

3.METHOD OF AUTO-TUNING OF ABSORBER 1FOR

THE ANTI-RESONANCE CONTROL

In the present paper,the "rst modal vibration is controlled by the tunable (variable sti !ness)absorber under the anti-resonance control.Figure 2shows the geometry of absorber 1for the anti-resonance (variable sti !ness absorber).The variable sti !ness absorber consists of the mass,beam,rotary encoder,motor and ball screw as shown in Figure 2.In the absorber,the spring constant at mass (1)varies by moving the middle support of beam (2).The moving support consists of a Te #on plate with a hole in which a circular stainless bar is inserted.The ball nut is AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 779

Figure 3.Theoretical response of the beam for various values of the rigidity of the anti-resonance vibration absorber.

The angle of rotation of the motor is detected by the rotary encoder (3)attached to the ball screw.

In order to have the anti-resonance state,a method of auto-tuning with two steps is presented.The acceleration signal is measured by the acceleration sensor.It is converted into the displacement signal by using the integration circuits.The displacement signal is input to the digital signal processor (DSP).In the "rst step control,the principal frequency is calculated,and the angle of rotation of the motor is calculated.The motor is rotated until the calculated angle,where the angle of rotation is detected by the rotary encoder.If the theoretical anti-resonance frequency coincides with the actual frequency,one obtains the anti-resonance state by using the theoretical expression.At the anti-resonance frequency,the response curve is sharp as shown in Figure 3,and so a few theoretical error has considerable e !ects on the amplitude of the beam.Hence,it is di $cult to create the anti-resonance state by the control based on the theoretical analysis.The theoretical analysis can be applied for obtaining approximate rotation angles of the motor.Then the second step (precise auto-tuning)control is required.

The second step auto-tuning control is performed as follows.The support of the absorber beam is moved a little from the position settled by the control based on the analysis.The vibration amplitude after the support movement is compared with that before the movement,and the support is moved in the direction where the vibration amplitude decreases.The support movement (the rotation angle of the 780K.NAGAYA E 1A ?.

Figure 4.Flow chart of control of the anti-resonance vibration absorber.

By repeating the control,one obtains the anti-resonance state automatically by using the DSP.Figure 4shows the algorithm of this control.

4.METHOD TO OBTAIN OPTIMAL PARAMETERS

OF VIBRATION ABSORBER 2

When the anti-resonance control is performed,the "rst mode vibration is controlled to be signi "cantly small.However,higher modes cannot be controlled.In order to control higher modes,the damper with large damping is desirable,but it a !ects the "rst mode.This implies that,small damping is desirable for the control of higher modes.Hence,the optimal vibration absorber is used for the higher modes.When the optimal vibration absorbers is used,vibrations of higher modes are suppressed by use of the optimal damping coe $cient.The damping is not so large,because the mass of the absorber damps vibrations.In the usual design of vibration absorbers,the number of absorbers has to be equal to that of the control modes.For the problem,the present author proposed a method of control of higher modes by use of the small number of absorbers.We controlled the "rst "ve modes successfully by use of only three absorbers in the problem of plate vibrations [19].In the present article,the method is also applicable to the design of the absorber.The e !ect of the absorber is large when the absorber lies on the anti-nodal point where the mode shape has a maximum value.Then absorber 2lies near the anti-nodal point of the second mode (The point was calculated as x "65cm measured from the left end of the beam by using the analysis mentioned above.Hereafter,the point is called point B).The vibrations of higher modes can be suppressed by small damping,because the damping force is in proportion to the vibrating speed.Then the absorber can control not only the second mode but also AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 781

T ABLE 3

Dimensions of the absorber for higher modes

Initial values

Optimal values Spring constant

10 (N/m)2)51;10 (N/m)Damping coe $cient

10(N s/m)9)07(N s/m)Attached mass

0)2(kg)0)205(kg)

Total mass of the absorber 0)327(kg)damper e !ects are not considered in the design of vibration absorbers.However,when higher modes are considered,the optimal absorber parameters are di !erent from those of ordinary absorbers [19].In the present article,absorber 2is designed with consideration of higher modes.In order to have the optimal parameters of the absorber,the frequency response curve in higher modes except the region of the "rst mode (which is controlled by the anti-resonance control)is considered.The frequency response curve near higher peaks is divided into n -pieces for n -higher peaks in the considered region.Each divided curve is integrated with respect to the frequency in the considered frequency region from G to G for i th piece.Their summation is taken to be a cost function

=N "L G G G f (m ,k ,c , )d ,(13)

where f (m ,k ,c , )is the vibration amplitude of displacement at the considered point in the beam.Numerical values of f (m ,k ,c , )is calculated by equation (12).Hence,one obtains the values of the cost function =N by use of the numerical integration scheme.When we consider one absorber,the parameters of the absorber are m ,k and c .The optimal parameters can be obtained by the iterations of the following expression:

X O "X O ! O *=N *X O (q "1,2,3)(14)

By repeating the calculation of equation (14),one obtains the optimal parameters m "X ,k "X and c "X which make the cost function =N minimum.The method as just mentioned has advantages,because it controls higher modes by using the fewer number of absorbers.All higher modes more than the second mode will be controlled by using the present method.There is no guarantee that equation

(14)will converge to the truly optimal solution,so that the calculation of equation

(14)is performed for some initial values.The optimal values are decided when the solutions converge to the same value.

Table 3depicts the calculated optimal values of the absorber for the present beam.One example of initial values (before the optimization)are also shown in the

782K.NAGAYA E 1A ?.

Figure 5.Theoretical response of the beam with optimal vibration absorbers for the second mode.-----not optimized;**optimized.

table.By using the values,the compliance of the beam at point B are calculated.The results are depicted in Figure 5.There is a large peak at the second resonance frequency when using the initial values before the optimization (see the dotted line),while the second resonance peak is suppressed when using the optimal values (see the solid line).Then the present method is applicable for designing the vibration absorber with the damper.Since the variable sti !ness absorber lies on the nodal point of the second mode,the e !ect of variation of the tunable absorber on the second mode is almost zero in the "gure.Hence,absorber 2can be designed without considering variations of tunable absorber 1,and "xed parameters are applicable for the absorber 2.The tunable absorber also a !ects the modes greater than the third mode.But,although the damping e !ect varies with the sti !ness of tunable absorber,the higher modes are suppressed by the #uid damper e !ects of course.

5.EXPERIMENTS

In order to validate the present control method and analyses,experimental tests have been carried out for the same model mentioned in the numerical calculation.Figure 6illustrates the geometry of the control system.A magnetic damper is used in absorber 2.The rectangular aluminum plate conductor is inserted between two permanent magnets in which the N-pole of one magnet faces the S-pole of the other as shown in Figure 7.When the plate conductor vibrates,the magnetic force is AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 783

Figure 6.Geometry of the control

system.

Figure 7.Geometry of the magnetic vibration absorber for higher modes.

plate conductor.The damping coe $cient can be varied by adjusting the air gap between the magnets,hence one obtains the optimal damping coe $cient shown in Table 3.The analysis of damping coe $cients was reported in our previous report

[3].The spring constant of the absorber is also varied by adjusting the length of the plate spring shown in Figure 7.Then the optimal values of the mass and the spring constant in Table 3are settled.The second resonance frequency has no dependence on the variation of the sti !ness of the auto-tuning absorber (anti-resonance absorber)as shown in Figure 3,so the absorber with constant parameters is applicable of course as mentioned above.The "rst step control for obtaining the anti-resonance state requires the relationship between the eigenfrequency of the variable sti !ness absorber and the length of the absorber beam measured from the tip mass.The relationship is non-linear as shown in Figure 8.Then the relationship is obtained as an expression by making use of the least-squares method.The solid line in Figure 8illustrates the result obtained by the expression which is used in the 784K.NAGAYA E 1A ?.

Figure 8.Relation between the length of the beam of the absorber and the natural

frequencies.

Figure 9.Shape of the exciting force generated by a magnet.

The acceleration sensors were attached at two points at x "46cm (point A)and x "65cm (point B)from the left end of the beam,and detected signals were input to the FFT analyzer as shown in Figure 6.The forced excitation was applied at the point x "24)5cm from the left end of the beam by an electromagnet.Since the electromagnet generates attractive force only,and the force is in proportion to the square of the current,the exciting force becomes the square of half sine pulse as shown in Figure 9under the sinusoidal input current.This implies that the force has various frequency components.Then the absorber to suppress higher modes is needed.

5.1.EXPERIMENTAL RESULTS WITHOUT VIBRATION ABSORBERS FOR HIGHER MODES

IN CASE OF THE VARIABLE STIFFNESS ABSORBER BEING UNCONTROLLED

Experimental tests have been carried out for the beam with variable sti !ness AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 785

Figure 10.Experimental time-response curves for the beam without control and without absorber for higher modes in case of "rst resonance frequency

excitation.

Figure 11.Experimental time-response curves for the beam without control and without absorber for higher modes in case of second resonance frequency excitation.

the experimental time response curve at both the points (points A and B)under the excitation of the "rst resonance frequency (f "25Hz).The response curves under the excitation of the second resonance frequency (f "58Hz)are also shown in Figure 11.The frequency spectrum are also shown in Figures 12and 13,where the value (dB)illustrates 10log w .In Figures 12and 13,two typical peaks (the "rst and second modes)are observed.In addition,some of sub-harmonic and super-harmonic peaks due to the non-linear exciting force are also observed.

5.2.EXPERIMENTAL RESULTS WITH THE VIBRATION ABSORBER FOR HIGHER MODES IN

CASE OF THE VARIABLE STIFFNESS ABSORBER BEING CONTROLLED

Figure 14depicts the time-response curve of the beam with both the variable-sti !ness absorber under the anti-resonance control and the magnetic absorber when the exciting load has the "rst resonance frequency (f "25Hz).The 786K.NAGAYA E 1A ?.

Figure 12.Experimental frequency spectrum for the beam without control and without absorber for higher modes in case of "rst resonance frequency

excitation.

Figure 13.Experimental frequency spectrum for the beam without control and without absorber for higher modes in case of second resonance frequency

excitation.

Figure 14.Experimental time-response curves for the beam with control and with absorber for AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 787

Figure 15.Experimental frequency spectrum for the beam with control and with absorber for higher modes in case of "rst resonance frequency

excitation.

Figure 16.Experimental time-response curves for the beam with control and with absorber for higher modes in case of second resonance frequency

excitation.

Figure 17.Experimental frequency spectrum for the beam with control and with absorber for 788K.NAGAYA E 1A ?.

Figure 18.Experimental time response at point A of the beam with control and with absorber for the results under the control subjected to second resonance frequency excitation.The amplitude under the present control at the resonance frequency is suppressed within only 8)6per cent of the uncontrolled result (see Figure 10).Especially,sub-harmonic and super-harmonic motions are also supressed signi "cantly as shown in Figure 15.

The present control is also applicable when the system parameters and forced frequency vary.In Figure 18,the result is shown when the forced frequency varies suddenly from the anti-resonance state.It shows that the amplitude of the beam increases with the variation of the forced frequency,but after about 2s the anti-resonance state is created again by the auto-tuning control,and hence the amplitude is suppressed.

The frequency response at the point A of the beam is depicted by the solid line in Figure 19,where the anti-resonance states are created.Figure 20also shows the response at the point B.In the "gures,the dashed line is the result without control.The "gures show that the vibration amplitudes under the present control are suppressed signi "cantly in a wide frequency range involving "rst and second resonance frequencies.Especially,when using the usual vibration absorbers,the amplitudes for the low-frequency region are almost the same as those without control,because the e !ects of inertia force for low-frequency is small.Hence it is di $cult to control amplitude ratios for low-frequencies less than one in the usual vibration absorbers or the PID active control.However,when the present method is used,the vibration amplitudes of the beam for the low-frequency region are signi "cantly small.This is the advantage of using the present method.

In the present problem,since the "rst two peaks in the frequency response curve are signi "cant,the higher modes greater than the third mode are not considered in the design of absorber 2because the principal frequency of the exciting force is considered to be within 100Hz (6000rpm).In usual machines and structures,the principal frequencies are not large,so the e !ect of the "rst mode is signi "cant,the second mode is the next,and components higher than the third mode are small AUTO-TUNING VIBRATION CONTROL OF STRUCTURES 789

Figure 19.Frequency response at point A in the beam.**with control;-----without

control.Figure 20.Frequency response at point B in the beam.**with control;-----without control.even if the exciting force involves high-frequency components.Hence,higher modes may be suppressed by the damper e !ect of the second absorber as mentioned 790K.NAGAYA E 1A ?.

AUTO-TUNING VIBRATION CONTROL OF STRUCTURES791

6.CONCLUSION

A method of vibration control of structures has been presented.In the method, a variable sti!ness absorber was presented for controlling vibrations of the"rst mode by using an auto-tuning anti-resonance control.The algorithm by use of a microcomputer was presented for the auto-tuning control.In order to control vibrations higher than the second mode,a vibration absorber with a damper was used,and its design method to obtain optimal parameters of the absorber was presented.

Experimental tests have been carried out for a#exible beam,and it is clari"ed that the vibrations can be controlled to be signi"cantly small by use of the present method.

In the ordinary vibration absorbers,the amplitude ratio in the low-frequency region is greater than unit,while the present method enables one to reduce the amplitude ratio to0)5.This is the essential advantage of using the present method. In addition,the number of vibration absorbers has to be equal to that of the control modes in the usual design method of absorbers.The absorber designed by the present method enables one to suppress vibrations of the higher modes by use of a fewer number of absorbers.

It is also clari"ed that the present method is applicable when the system parameters and forced frequencies with higher components vary during vibrations. Although the variable sti!ness absorber is controlled by the servo-motor,the vibration control is carried out based on the passive control.This implies that the control energy is signi"cantly small in comparison with active controls. Therefore,the present method has advantages over the previous methods.

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系泊系统的设计和探究

赛区评阅编号（由赛区组委会填写）： 2016年高教社杯全国大学生数学建模竞赛 承诺书 我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》（以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载）。 料 我们的报名参赛队号（12位数字全国统一编号）： 参赛学校（完整的学校全称，不含院系名）： 参赛队员 (打印并签名) ：1. 2.

3. 指导教师或指导教师组负责人 (打印并签名)： （指导教师签名意味着对参赛队的行为和论文的真实性负责） 日期：年月日 送全国评阅统一编号（赛区组委会填写）： 全国评阅随机编号（全国组委会填写）： （请勿改动此页内容和格式。此编号专用页仅供赛区和全国评阅使用，参赛队打印后装订到纸质论文的第二页上。注意电子版论文中不得出现此页。）

系泊系统的设计和探究 摘要 本文利用牛顿力学定律，力矩平衡原理、非线性规划、循环遍历法等方法对系泊系统进行了设计与探究。通过对系泊系统各组件和浮标运用牛顿经典力学体系进行分析，得到了各个情况下的钢桶倾斜角度、锚链状态、浮标吃水深度和游动区域。 ?， 。当风 对于第二问，求解当海面风速为36m/s时，浮标的吃水深度和游动区域、钢桶以及钢管的倾斜角度和锚链形态。利用第一问中的力学方程和程序，求得钢桶的倾角为19.5951?和四节钢管的倾斜角度依次为19.756?、19.755?、19.916?、20.076?。浮标的游动区域为以锚在海面上的投影为圆心，半径为18.8828m的圆。由于部分数据与问题二中钢桶的倾斜角度不超过5?，锚链在锚点与海床的夹角不超过16?的要求不符，所以通过调节重物球的质量使钢桶的倾斜角度和锚链在锚点与海床的夹角处在要求的范围之内。借助MATLAB程序中的循环遍历法，可以求得重物球的质量3770kg。

2016数学建模A题系泊系统设计

系泊系统的设计 摘要 对于问题一，建立模型一，已知题目给出的锚链长度与其单位长度的质量，得到悬链共210环。对各节锚链，钢桶，四节钢管受力分析得出静力平衡方程，使用分段外推法，可以得到静力平衡下的迭代方程。其中锚对锚链的拉力大小方向为输入变量，迭代的输出变量为浮标的位置和对钢管的拉力，在给定的风速下，输入和输出满足关系2)2(25.1cos 水v h T -=α，αθcos cos 11T T =，通过多层搜索算法得出最符合的输入输出值，即可得到给定风速下浮标的吃水深度，浮标拉力、锚链与海床夹角。利用MATLAB 软件编程求解模型得到：风力12m/s 时，钢桶与竖直方向上的角度1.9863度，从下往上四节钢管与竖直方向夹角为1.9652度、1.9592度、1.9532度、1.9472度，浮标吃水0.7173m ，以锚为圆心浮标的游动区域16.5125m ，锚链末端切线与海床的夹角3.8268度。风力24m/s 时，锚链形状，钢桶与竖直方向上的夹角3.9835度，从下往上四节钢管与竖直方向夹角为3.9420度、3.9301度、3.9183度、3.9066度，浮标吃水0.7244m ，以锚为圆心浮标的游动区域18.3175m 。锚链末端切线与海床夹角15.9175度。 对于问题二的第一小问，使用模型一求解，当风速36m/s 时，锚链末端切线与海床夹角26.3339度，浮标吃水0.7482m ，浮标游动区域为以锚为圆心半径为18.9578m 的圆形区域，从下往上四节钢管与竖直方向倾斜角度为8.4463度、8.4225度、8.3989度、8.3753度，钢桶与竖直方向倾斜角度为8.5294度。为满足问题二的要求，在模型一的基础上把重物球质量作为变量，建立模型二，将钢桶倾斜角小于5度和锚链前端夹角小于16度当做两个约束条件，通过MATLAB 编程求解得到满足约束条件要求的重物球质量取值范围为3700kg 到5320kg 。 对于问题三，首先取不同水深、水速、风速三种情况，建立模型三，即在模型一的基础上增加水流对系统产生的影响。在三种情况下，找到合适的锚链型号、锚链长度，重物球质量，对吃水深度、游动区域、钢桶的倾斜角三个目标进行优化达到最小。通过MATLAB 编程实现该模型三得到结果：选用Ⅲ型锚链，锚链长度为27.24m ，重物球质量为2580kg 。 关键词：平面静力系分析 多层搜索算法 遗传算法 逐步外推法 多目标优化

系泊系统的设计和探究

系泊系统的设计和探究 This model paper was revised by the Standardization Office on December 10, 2020

赛区评阅编号（由赛区组委会填写）： 2016年高教社杯全国大学生数学建模竞赛 承诺书 我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》（以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载）。 我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。 我们知道，抄袭别人的成果是违反竞赛章程和参赛规则的，如果引用别人的成果或资料（包括网上资料），必须按照规定的参考文献的表述方式列出，并在正文引用处予以标注。在网上交流和下载他人的论文是严重违规违纪行为。 我们以中国大学生名誉和诚信郑重承诺，严格遵守竞赛章程和参赛规则，以保证竞赛的公正、公平性。如有违反竞赛章程和参赛规则的行为，我们将受到严肃处理。 我们授权全国大学生数学建模竞赛组委会，可将我们的论文以任何形式进行公开展示（包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等）。 我们参赛选择的题号（从A/B/C/D中选择一项填写）： 我们的报名参赛队号（12位数字全国统一编号）： 参赛学校（完整的学校全称，不含院系名）： 参赛队员 (打印并签名) ：1.

2. 3. 指导教师或指导教师组负责人 (打印并签名)： （指导教师签名意味着对参赛队的行为和论文的真实性负责） 日期：年月日 （请勿改动此页内容和格式。此承诺书打印签名后作为纸质论文的封面，注意电子版论文中不得出现此页。以上内容请仔细核对，如填写错误，论文可能被取消评奖资格。） 赛区评阅编号（由赛区组委会填写）： 2016年高教社杯全国大学生数学建模竞赛 编号专用页 赛区评阅记录（可供赛区评阅时使用）： 送全国评阅统一编号（赛区组委会填写）： 全国评阅随机编号（全国组委会填写）：

数学建模a题系泊系统设计

摘要 本题要求观测近海观测网的组成，建立模型对其中系泊系统进行设计，在不同风速和水流的情况下确定锚链，重物球，钢管及浮标等的状态，从而使通讯设备的工作效果最佳。求解的具体流程如下： 针对问题一，分别对系统中的受力物体在水平方向和竖直方向上的力进行分析，找出锚链对锚无拉力时的临界风速，运用力矩平衡求出钢管与钢桶的倾斜角度。对于锚链，将其等效为悬链线模型，根据风速不同判断锚链的状态，从而求出结果。 ?时能够正常工针对问题二，需要调节重物球的质量，使通讯设备在36m m 作。为了确定重物球的质量，首先将实际风速与临界风速进行比较，判断此时系统中各物体的状态，与题目中已知数据进行比较。在钢桶倾斜角度达到临界角度时，计算锚链与海床的夹角并于题中数据进行比较，计算重物球的质量。在浮标完全没入海面时，计算相应条件下重物球的质量，从而确定满足条件的重物球的质量范围。 针对问题三，要求在不同条件下，求出系泊系统中各物体的状态。以型号I 锚链为例，当水流方向与风速方向相同时，系统条件最差，分析在不同水深条件下的系泊系统设计。由题中已知条件确定系统设计的限制条件，对系统各物体进行受力分析，以使整体结果最小，即可得出最优的系泊系统设计。 # 》 关键词：悬链线多目标非线性规划 @

一、问题重述 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。水声通讯系统安装在一个长1m、外径30cm 的密封圆柱形钢桶内，设备和钢桶总质量为100kg。钢桶上接第4节钢管，下接电焊锚链。钢桶竖直时，水声通讯设备的工作效果最佳。若钢桶倾斜，则影响设备的工作效果。钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。为了控制钢桶的倾斜角度，钢桶与电焊锚链链接处可悬挂重物球。 系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。 问题1某型传输节点选用II型电焊锚链，选用的重物球的质量为1200kg。现将该型传输节点布放在水深18m、海床平坦、海水密度为×103kg/m3的海域。若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。 | 问题2在问题1的假设下，计算海面风速为36m/s时钢桶和各节钢管的倾斜角度、锚链形状和浮标的游动区域。请调节重物球的质量，使得钢桶的倾斜角度不超过5度，锚链在锚点与海床的夹角不超过16度。 问题3 由于潮汐等因素的影响，布放海域的实测水深介于16m~20m之间。布放点的海水速度最大可达到s、风速最大可达到36m/s。请给出考虑风力、水流力和水深情况下的系泊系统设计，分析不同情况下钢桶、钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。 二、模型假设 1.不考虑流体对锚链的作用，忽略锚链本身的伸长，锚链沿长度均匀分布； 2.假设风是二维的，只存在平行于水平面的风速，不存在垂直方向上的分量；

数学建模A题系泊系统设计完整版

数学建模A题系泊系统 设计 HEN system office room 【HEN16H-HENS2AHENS8Q8-HENH1688】

系泊系统的设计 摘要 本题要求观测近海观测网的组成，建立模型对其中系泊系统进行设计，在不同风速和水流的情况下确定锚链，重物球，钢管及浮标等的状态，从而使通讯设备的工作效果最佳。求解的具体流程如下： 针对问题一，分别对系统中的受力物体在水平方向和竖直方向上的力进行分析，找出锚链对锚无拉力时的临界风速，运用力矩平衡求出钢管与钢桶的倾斜角度。对于锚链，将其等效为悬链线模型，根据风速不同判断锚链的状态，从而求出结果。 ?时能够正常工作。为针对问题二，需要调节重物球的质量，使通讯设备在36m m 了确定重物球的质量，首先将实际风速与临界风速进行比较，判断此时系统中各物体的状态，与题目中已知数据进行比较。在钢桶倾斜角度达到临界角度时，计算锚链与海床的夹角并于题中数据进行比较，计算重物球的质量。在浮标完全没入海面时，计算相应条件下重物球的质量，从而确定满足条件的重物球的质量范围。 针对问题三，要求在不同条件下，求出系泊系统中各物体的状态。以型号I锚链为例，当水流方向与风速方向相同时，系统条件最差，分析在不同水深条件下的系泊系统设计。由题中已知条件确定系统设计的限制条件，对系统各物体进行受力分析，以使整体结果最小，即可得出最优的系泊系统设计。 关键词：悬链线多目标非线性规划 一、问题重述 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。水声通讯系统安装在一个长1m、外径30cm的密封圆柱形钢桶内，设备和钢桶总质量为100kg。钢桶上接第4节钢管，下接电焊锚链。钢桶竖直时，水声通讯设备的工作效果最佳。若钢桶倾斜，则影响设备的工作效果。钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。为了控制钢桶的倾斜角度，钢桶与电焊锚链链接处可悬挂重物球。 系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。

系泊系统的研究设计

Internal Combustion Engine &Parts 0引言 近海系泊系统作为气象监控，海洋探测的主要载体工具，对工程的实际应用有一定的积极作用。为了开发海洋资源，促进经济的发展，满足日益增长的能源需求量，深海油气的开发已成为必然趋势，而系泊系统的设计是深海平台开发的关键问题之一，其设计的目的是保证选择的型号满足工作功能需求，建立相关的代数模型研究在不同情况下的设计方案。 1系泊系统模型 1.1系泊系统平衡模型 忽略作用在锚链上的流动力， 可以将整个锚链简化成悬链线的形式，浮标系统和系泊系统在经过持续海风时浮 标停止运动，系泊系统达到平衡，运用牛顿运动定律和平 行四边形状规则对系统内的各个物体进行受力分析，得到关系表达式为 第一钢管对浮标的拉力为T ，T 与竖直方向多的夹角为θ，浮标所受的浮力为F f ，所受海风荷载为F N ，浮标本身的重力为G ，浮标沉在水中的百分比为α，浮标的直径为d ，沉在水中的高度为h f ，海水的密度为ρ，风速为V ，根据海风荷载近似公式为 求。同时满足轮胎加防滑链极限包络大于 15mm 间隙要求。图3图4Z 向空间 >80mm 图5铆接设计图6铆接设计 ③翼子板下部设计。翼子板下部设计成台阶形。当下部长度>120mm ，需设计成双台阶形（图4），便于两个安装点安装。当翼子板下部刚度不足时，翼子板下部增加下部加强板。翼子板下部设计注意几点：1）翼子板下部加强板材料DC05，料厚为0.7-1.0。2）翼子板下部加强板与翼子板下部安装面打3-5焊点相连。④翼子板后部设计。翼子板后部由于前门总成运动包络影响，翼子板翻边向内小于45度，利于成型。通常安装面设计成台阶形或平 面。翼子板后部安装孔设计圆孔或开口U 形孔，U 形更利于装配。 ⑤翼子板上部设计。 翼子板上部通常安装点采有台阶设计。根据冲压工艺反馈，翻边高度设计为<40mm ，防止出现面品问题。 为满足行人保护要求，通常会从以下几种方式考虑。通常翼子板支架设计成几字形，并在支架两开孔落化（图4）。翼子板上部与边梁高度控制在80mm 以上（图4）。翼子板翻边在满足刚度前提下，尽量开豁口。随着新工艺的发展，翼子板结构，可以采铆接技术，具体设计如图5、6所示。 3制造过程控制 ①翼子板周边件产品要符合设计精度要求。②翼子板过涂装通常有两种方式，一种是在焊装装配 随车身一同电泳，另一种是单件单独电泳。建议采用第一 种，防止总装配出现难调整问题。③总装装配严格按工艺装配相关件。按装配顺序装配完成，再作微调。4结语总之保证翼子板品质，要从设计、制造过程着手。随着 铝合金翼子板、塑料翼子板应用，翼子板设计、制造过程有所变化，但总体设计理念不变。参考文献: [1]袁亮，秦信武，苗布和.汽车前翼子板的布置和结构设计[J].设计研究，2012，04，28. 系泊系统的研究设计 王莎莎;晏明莉 （河南师范大学，新乡453007） 摘要:本文所建立的近海系泊系统模型主要用于研究系泊系统在不同环境下的内在关系，进而给出适应不同情况的设计方案。影 响系泊系统的主要因素分别为锚链的型号，长度和重物球的质量，以及锚链的倾斜角度，浮标的吃水深度和游动区域，通过对系统中各个部分进行受力分析，建立相关的代数模型。该模型通过建立平衡方程组，采用最优控制策略、二分法收敛性和迭代判断，反复迭代计算求解。分别把锚链的型号，长度和重物球的质量作为自变量，把锚链的倾斜角度，浮标的吃水深度和游动区域作为因变量，建立模型进而得出不同环境下系泊系统的设计方案。 关键词:系泊系统设计；最优控制策略；二分法；平衡方程组；迭代计算

数学建模系泊系统的设计

系泊系统的设计 摘 要 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成，其中系泊系统由钢管、钢桶、重物球及锚链共同组成。此种系泊系统承受风、浪、流的作用及锚链的作用力，运动特性十分复杂。因此，针对海洋环境中水声通讯系统的要求，分析风浪中浮标的动力问题并设计出既安全又经济的系泊系统，对保证水声通讯系统的工作效果来说意义重大。 本文运用了两种方法对锚链进行了受力分析，首先对单一材质的锚链进行分析，从而得出了经典悬链方程，对不同段不同材质的锚链进行分段受力分析，得出了不同段不同材质的悬链方程，该方程的得出极大的方便了计算浮标锚泊系统的初始状态，为动力分析奠定基础；其次利用牛顿法对锚链受力问题进行了数值求解，得到当海面风速为12/m s 加大到24/m s 时，每节钢管的倾斜角度也随之变大，浮标的吃水深度也不断增大，浮标的游动区域增加的更为明显。当风速加大为36/m s 时，钢桶的倾斜角已超过5度，为使钢桶倾斜角小于5度，须将重物球的质量增加至1783kg 。 再考虑风力、水流力、潮汐（波浪）等动力因素时，可以将问题进行简化，即直接考虑在水深18m 的情况下由于波浪的作用（准确的说是2m 波浪的作用），可使整个浮标漂浮于水面上（20m 情形），也可使整个浮标沉于水面下（16m 情形）。最后通过对浮标的受力分析，可得到浮标的动力控制方程，采用数值方法，可以得到在风速为36/m s ，水流速度为1.5/m s 时，倾斜角、吃水深度的数值解。 关键词： 浮标；系统；设计；动力分析

一．问题重述 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。水声通讯系统安装在一个长1m、外径30cm 的密封圆柱形钢桶内，设备和钢桶总质量为100kg。钢桶上接第4节钢管，下接电焊锚链。钢桶竖直时，水声通讯设备的工作效果最佳。若钢桶倾斜，则影响设备的工作效果。钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。为了控制钢桶的倾斜角度，钢桶与电焊锚链链接处可悬挂重物球。系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。 问题1某型传输节点选用II型电焊锚链22.05m，选用的重物球的质量为1200kg。现将该型传输节点布放在水深18m、海床平坦、海水密度为1.025×103kg/m3的海域。若海水静止，分别计算海面风速为12m/s和24m/s时钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。 问题2在问题1的假设下，计算海面风速为36m/s时钢桶和各节钢管的倾斜角度、锚链形状和浮标的游动区域。请调节重物球的质量，使得钢桶的倾斜角度不超过5度，锚链在锚点与海床的夹角不超过16度。 问题3 由于潮汐等因素的影响，布放海域的实测水深介于16m20m之间。布放点的海水速度最大可达到1.5m/s、风速最大可达到36m/s。请给出考虑风力、水流力和水深情况下的系泊系统设计，分析不同情况下钢桶、钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域。 二．模型假设与符号说明 1.模型的假设 由于浮标在海洋受海风、气流、海浪等的作用，气象及水文条件多变，在此，我们对整个问题作适当假设，在下文中，当涉及到具体问题时，我们将有针对性地给出必要的补充。 （1）锚链重力远远大于锚链所受到的流体作用力； （2）海床平坦，无凹凸不平情况； （3）浮标始终是垂直于海平面的，无倾斜或跌倒现象； （4）海面风速均匀、恒定，且风向始终平行于海平面； （5）海水流速均匀（流速与水深无关），且方向恒定（海流无垂直分量）； （6）锚链在整个过程中是不可弹性形变。

数学建模a题系泊系统设计完整版

数学建模a题系泊系统 设计 集团标准化办公室：[VV986T-J682P28-JP266L8-68PNN]

系泊系统的设计 摘要 本题要求观测近海观测网的组成，建立模型对其中系泊系统进行设计，在不同风速和水流的情况下确定锚链，重物球，钢管及浮标等的状态，从而使通讯设备的工作效果最佳。求解的具体流程如下： 针对问题一，分别对系统中的受力物体在水平方向和竖直方向上的力进行分析，找出锚链对锚无拉力时的临界风速，运用力矩平衡求出钢管与钢桶的倾斜角度。对于锚链，将其等效为悬链线模型，根据风速不同判断锚链的状态，从而求出结果。 ?时能够正常工针对问题二，需要调节重物球的质量，使通讯设备在36m m 作。为了确定重物球的质量，首先将实际风速与临界风速进行比较，判断此时系统中各物体的状态，与题目中已知数据进行比较。在钢桶倾斜角度达到临界角度时，计算锚链与海床的夹角并于题中数据进行比较，计算重物球的质量。在浮标完全没入海面时，计算相应条件下重物球的质量，从而确定满足条件的重物球的质量范围。 针对问题三，要求在不同条件下，求出系泊系统中各物体的状态。以型号I 锚链为例，当水流方向与风速方向相同时，系统条件最差，分析在不同水深条件下的系泊系统设计。由题中已知条件确定系统设计的限制条件，对系统各物体进行受力分析，以使整体结果最小，即可得出最优的系泊系统设计。 关键词：悬链线多目标非线性规划 一、问题重述 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。水声通讯系统安装在一个长1m、外径30cm的密封圆柱形钢桶内，设备和钢桶总质量为100kg。钢桶上接第4节钢管，下接电焊锚链。钢桶竖直时，水声通讯设备的工作效果最佳。若钢桶倾斜，则影响设备的工作效果。钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。为了控制钢桶的倾斜角度，钢桶与电焊锚链链接处可悬挂重物球。

系泊系统的设计

系泊系统的设计 发表时间：2020-01-18T09:59:04.837Z 来源：《基层建设》2019年第28期作者：武佳雷宇张曼[导读] 摘要：系泊系统不论是在船舶航行，还是在海洋资源的综合利用与开发中，均得到了广泛应用，因而，系泊系统的设计问题十分具有现代意义。 华北理工大学数学建模实验室 063000摘要：系泊系统不论是在船舶航行，还是在海洋资源的综合利用与开发中，均得到了广泛应用，因而，系泊系统的设计问题十分具有现代意义。本文隔离系统各组成部分，逐一进行受力分析和力矩分析，构造相应的刚体力学方程组，并根据海水深度联系各参数最终建立系泊系统状态模型。 关键词：系泊系统状态模型；受力平衡；力矩平衡 0引言 当海面风速一定且海水静止时，系泊系统的状态，即钢桶和各节钢管的倾斜角度、锚链形状、浮标的吃水深度和游动区域与系泊系统各部分之间的受力平衡和力矩平衡以及海水深度的约束密切相关。因此，可以隔离该系统的各组成部分，逐一进行力学分析，并最终根据海水深度，联系各参数建立系泊系统状态模型。具体数值参考2016年高教社杯全国大学生数学建模竞赛A题。系泊系统可分为浮标、钢管、钢桶和重物球、锚链四个部分，由题中锚链长度和型号计算得锚链共有210个链环，为了方便表述，对系统内部由上到下的构件进行标记：表1 各构件编号 1力学方程与模型的建立 1.1对浮标的力学分析[1] 漂浮在水面上的浮标，受到来自水平方向的风力、海水对它的浮力、其余组件对它作用力以及自身的重力，与夹角为 。已知浮标的高为，质量为，直径为，海水的密度为，设浮标的吃水深度为，根据重力、浮力公式，以及近海风荷载的近似计 算公式，可得。此时，浮标受到速度为的海风作用在海面上达到平衡，受力分解后，其在水平方向和垂直方向的受力均平衡。于是整理可得关于浮标的完整的刚性力学方程组[2]。其中，其在竖直面的投影高度即为浮标的吃水深度。 1.2对钢管的力学分析没入水中的钢管，由于海水静止，因此忽略水流力水平方向的作用。以与浮标相接的第一节钢管为例，其受到浮标对它的反作用力 、其余组件对它的作用力、海水对它的浮力以及自身的重力。为保持力矩平衡，钢管不发生旋转的现象，浮标对它的反作用力应相对于它的中心轴更偏向竖直方向。 此时有相对于竖直方向的夹角与和的夹角相等，即：，根据牛顿第三定律：。此时，钢管在这些力的共同作用下保持平衡。已知每节钢管的长度为 ，直径为，质量为，根据受力平衡、力矩平衡和重力、浮力公式，可得到相应的刚性力学方程组。其中，其在竖直面的投影高度为。 对于余下的三节钢管，依次进行同样的力学分析。得到对应的完整的刚性力学方程组，以及各钢管在竖直面的投影高度为、 、。 1.3对钢桶和重物球的力学分析没入水中的钢桶和重物球，由于海水静止，则忽略水流力水平方向的作用。对于重物球，其受到钢桶对其的作用力、海水对它的浮 力以及自身的重力。对于钢桶，其受到重物球对其的作用力（方向竖直向下）、其余部件对它的作用力（与竖直方向夹角为 ）和（与夹角为）、海水对它的浮力以及自身的重力.根据牛顿第三定律有。此时，钢桶在这些力的共同作用下保持平衡。已知钢桶的长度为，重物球的质量，设备和钢桶的总质量为，且外径为，根据受力平衡、力矩平衡（注意判断力矩方向）、牛顿第三定律则和阿基米德定律可得到对应的完整的刚性力学方程组。 此时，其在竖直面的投影高度为。 1.4对锚链的力学分析选取编号为的链环，联系该链环以及其上的其余部件构成一个新的整体，对该整体进行受力平衡的分析，其受到海风给它的风力、余下部分给它的拉力、海水给它的浮力以及自身所受的总重力，与水平方向夹角为。通过对整体的分析，简化了作用力与反作用力不断迭代的过程，更加直观的得到受力平衡时的状态。锚链密度为，每节链环的长度和单位长度的质量，再根据前式得到整体所受的总重力，受到海水的总浮力，整理后，可得到关于这个整体的完整的刚性力学方程组。 此时，各链环与竖直方向的夹角即可描述出锚链的形状。各节链环在竖直面的投影高度分别为： 1.5建立系泊系统状态模型 已知水深为，则有