Awrejcewicz J. Numerical Simulations of Physical and Engineering Processes

Подождите немного. Документ загружается.

Numerical Simulations of Seawater Electro-Fishing Systems

363

can often be manipulated on manufactured equipment, In general, operators should

reduce pulse frequency to the range of 15-30 Hz, while trying to maintain acceptable catch

rate, if injury rate has to be significantly reduced. Pulse duration is related to duty cycle.

At a given peak voltage or amplitude, changing pulse duration will change the average

voltage (area under the waveform curve), meaning that the fish is subjected to more

electrical energy. It is possible that longer pulse duration (e.g., 6-8 ms) contributes more to

added stress than injury, compared to shorter pulse duration (e.g., 2-4 ms). Experimental

results of sea bass after exposure to electro-fishing in laboratory tanks are presented in

Figure 15 and 16. These figures illustrate differences in sea bass fish (two sizes: 10 and 30

cm) in terms of electro-taxis and tetanus threshold values after electrical exposure.

Tetanus threshold values decreased significantly (P<0.05) for higher frequencies in both

sizes while electro-taxis was not influenced by the electrical exposure. It is worth noting

that, these values decreased with the fish size. All fish were immobilized during the

electrical exposure. However, after 5 minutes, they recovered the opercular movements

and swimming ability.

Results of electro-fishing exposure (frequency: 25-75-125 Hz; duty cycle: 5-20-40%) on

carcass quality characteristics are reported in Table 6, Fig.15 -16. No effects on carcass

quality characteristics were identified for any of the fish exposed to the experimental

treatments. Fish were inspected for hemorrhages in the skin, external damage, internal

haemorrhaging, blood spotting and damage of the spines. No differences were found after

electro-fishing on other carcass quality characteristics (QIM, colour, shear force, rigor

mortis).

Treatments

25-5 25-20 25-40 75-5 75-20 75-40 125-5 125-20 125-40

Rse

df 18

pH 6.4 6.1 6.4 6.1 6.4 6.2 6.1 6.2 6.3 0.19

Colour:

L* 34.8 36.4 35.5 36.5 36.1 36.1 35.5 35.8 36.0 2.65

a* -1.9 -1.6 -1.7 -2.7 -1.5 -1.5 -1.5 -2.5 -1.8 0.16

b* 6.0 7.6 6.3 5.4 5.1 5.1 6.4 6.1 6.5 1.74

Croma 6.3 7.8 7.3 6.7 5.3 5.3 6.6 6 6.7 1.63

Hue angle 107.3 102.2 109.6 107.4 106.9 106.9 105.1 109.8 106.2 15.98

Cooking

yield (%)

98.76 98.00 97.96 98.62 99.02 98.93 97.66 97.78 98.10 0.96

Maximum

force (N)

9.0 8.5 8.7 9.2 8.3 7.5 8.5 8.9 9.0 4.34

Total

amount of

work (J)

0.125 0.095 0.122 0.104 0.090 0.088 0.103 0.100 0.101 0.0001

Table 6. Results of electro-fishing exposure on carcass quality characteristics of sea bass

Numerical Simulations of Physical and Engineering Processes

364

Fish lenght (cm): 10

0,00

5,00

10,00

15,00

20,00

25,00

125-20 75-20 125-20

Treatments

Galvanotaxis

Galvanonarcosis

Fish lenght (cm): 30

0,00

5,00

10,00

15,00

20,00

25,00

125-20 75-20 125-20

Treatments

V/m

Galvanotaxis

Galvanonarcosis

a, b < P<0.05

Fig. 15. 16. Electric-induced electro-taxis and tetanus of sea bass after electro-fishing

exposure (frequency:25-75-125 Hz; duty cycle: 20)

Numerical Simulations of Seawater Electro-Fishing Systems

365

5. Conclusions

The main problem in sea water electro-fishing is the high electric current demand in the

equipment, brought about by the very high ionic concentration of salt water. The solution of

this problem is to reduce the current demand as much as possible by using pulsed direct

current, the pulses being as small as possible. For example, if pulse duration is reduced to 1

or 2 milliseconds, and pulse frequency is kept below 30 hertz (pulses per second), this will

allow the operator to increase the amplitude, or height, of the pulses with the voltage

control. Fish generally respond best when the peak voltage is higher and the average

voltage (area under each pulse curve) is lower. If the fish don't respond, then average

voltage is increased (i.e., pulse frequency and/or pulse duration) is increased until they do

respond. It is usually better to increase frequency first, followed by duration. Ultimately, if

none of this may work, the power source (generator) is may be inadequate. In this case, one

can experiment with smaller electrodes (reduced surface area) to further reduce the demand

for current. The numerical simulations of a non homogeneous electric field (fish and water)

permit to estimate the current gradient in the open sea and to evaluate the attraction

capacity of fish using an electro-fishing device. An area of about 30 m

suitable for electro-

taxis is estimated for a voltage of 90 V on a circular anode and two linear cathodes which are

5 m far from the centre of the anode. Tank simulations are, instead, carried out in a uniform

electric field, generated by two parallel linear electrodes. The convenience of using an

uniform field is given by the need of finding threshold values of current field which are

independent from the position of the fish in the tank. Numerical simulations allow to

compare the electric field in the water and inside fish. The current field inside fish is

resulted smaller in a tank compared to the open sea. This means that, in practice, in the open

sea situation, the efficacy of an electro-fishing system is stronger, in terms of attraction area.

Numerical simulations carried out using a group of 30 fish, both in open sea and in the tank,

showed the presence of a “group effect”, increasing the electric field intensity in the water

around each fish. In this situation, each single fish has a greater current field compared to a

fish group.

6. Acknowledgement

This study was funded by the Region Friuli Venezia Giulia, Innovation Projects 2010.

7. References

Ainslie, B.J. & Post, A.J. (1998). Effects of pulsed and continuous DC electrofishing on juvenile

rainbow trout. North American Journal of Fisheries Management, 18, 905-918

Beaumont, W.R.C.; Taylor, A.A.L.; Lee, M.J. & Welton, J.S. (2002). Guidelines for electric fishing

best practice, R&D Technical Report W2-054/TR. Environmental Agency, Bristol,

UK.

Blancheteau, M. (1971). La peche electrique en eau del mer. II - choix du stimulus approprié a la

peche a l’electricité en mer. Rev. Trav. Inst. Pêches Marit., 35(1), 13-20

Hamrin, S.; Heggberget, T.G.; Rassmussen, G. & Salveit, S.J. (1989). Electrofishing – Theory

and practice with special emphasis on salmonids. Hydrobiologia, 173, 9-43

Codecasa, R.; Specogna, R. & Trevisan, R. (2007). Symmetric Positive-Definite Constitutive

Matrices for Discrete Eddy-Current Problems. IEEE T. Magn., 42 (2), 510-515

Numerical Simulations of Physical and Engineering Processes

366

Dalbey, S.R.; McMahon, T.E. & Fredenberg, W (1996). Effect of electrofishing pulse shape and

electrofishing-induced spinal injury to long-term growth and survival of wild rainbow

trout. North American Journal of Fisheries Management, 16, 560-569

Diner, N. & Le Man, R. (1971). La peche electrique en eau del mer: III – Etude du champ electrique

necessaire a la taxie anodique du poisson. Rev. Trav. l'Inst. Pêches Marit., 35(1), 21-34

Kolz, A.L. (1989). A power transfer theory for electrofishing. In: A.L. Kolz, J.B. Reynolds (Eds),

Electrofishing, a power related phenomenon. U.S. Fish and Wildlife Service, Technical

Report, 22, 1-11

Kolz, A.L. (1993). In water electrical measurements for evaluating electrofishing systems. U.S Fish

and Wildlife Service, Biological Report 11

Kurk, G. (1971). La peche electrique en eau de mer: I – Peche a l’électricité avec lumiére artificielle et

pompe. Rev. Trav. Inst. Pêches Marit., 35(1), 5-12

Kurk, G.( 1972). Device for electric sea-fishing. United States Patent Office, N. 3, 693,276

Le Men, R. (1980). Comportement de poissons marins dans un champ electrique – perspectives

d’application a la peche. Rev. Trav. Inst. Pêches Marit., 44(1), 5-83

McMichael, G.A. (1993). Examination of electrofishing injury and short-term mortality in hatchery

rainbow trout. North American Journal of Fisheries Management, 13, 229-233

Miranda, L.E. & Dolan, C.R. ( 2003). Test of a power transfer model for standardized electrofishing.

T. Am. Fish. Soc., 132, 1179-1185

B.; Slinde, E. & Arildsen, J. (2006). Pre or post mortem musle activity in Atlantic salmon (Salmo

salar). The effect on rigor mortis and the physical properties of flesh, Aquaculture, 257,

504-510

Sharber, N.G. & Carothers, S.W. ( 1988). Influence of electrofishing pulse shape on spinal injuries

in adult rainbow trout. North American Journal of Fisheries Management, 8, 117-122.

Specogna, R. & Trevisan, F.( 2005). Discrete constitutive equations in A-χ geometric eddy-currents

formulation. IEEE T. Magn., 41(4), 1259-1263

Specogna, R. & Trevisan, F.(2006). Voltage Source in A-χ discrete geometric approach to eddy

currents. Eur. J. Appl. Phys., Vol. 6, 97-101

Stravisi, F. (1983). The vertical structure annual cycle of the mass field parameters in the Gulf of

Trieste. Boll. Oceanol. Teor. Appl., 1 (3), 239-250

Van Harreveld, A. (1938). On galvanotropism and oscillotaxis in fish. J. Exp. Biol., 15, 197-208

Numerical Analysis of a Rotor Dynamics

in the Magneto-Hydrodynamic Field

Jan Awrejcewicz

and Larisa P. Dzyubak

Technical University of Łódź

National Technical University “Kharkov Polytechnic Institute”

Poland

Ukraine

1. Introduction

In general, rotating machinery elements are frequently met in mechanical/mechatronical

engineering, and in many cases their non-linear dynamics causes many harmful effects, i.e.

noise and vibrations. In particular, nonlinear rotordynamics plays a crucial role in

understanding various nonlinear phenomena and in spite of its long research history (see

for instance (Tondl, 1965; Someya, 1998; Rao, 1991; Gasch et al., 2002; Muszyńska, 2005) and

the references therein) it still attracts attention of many researchers and engineers. Since the

topics related to nonlinear rotordynamics are broadband and cover many interesting aspects

related to both theory and practice, in this chapter we are aimed only on analysis of some

problems related to rotor suspended in a magneto-hydrodynamics field in the case of soft

and rigid magnetic materials.

The magnetic, magneto-hydrodynamic and also piezoelectric bearings are used in many

mechanical engineering applications in order to support a high-speed rotor, provide

vibration control, to keep lower rotating friction losses and to potentially avoid flutter

instability. There are a lot of publications devoted to the dynamics analysis and control of a

rotor supported on various bearings systems. The conditions for active close/open-loop

control of a rigid rotor supported on hydrodynamic bearings and subjected to harmonic

kinematical excitation are presented in (Kurnik, 1995; Dziedzic & Kurnik, 2002). The

methodology for modeling lubricated revolute joints in constrained rigid multibody systems

is described in (Flores et al., 2009). The hydrodynamic forces, used in the dynamic analysis

of journal-bearings, which include both squeeze and wedge effects, are evaluated from the

system state variables and included into the equations of motion of the multibody system.

To analyze the dynamic behavior of rub-impact rotor supported by turbulent journal

bearings and lubricated with couple stress fluid under quadratic damping the authors of

(Chang–Jian & Chen, 2009) have used the system state trajectory, Poincaré maps, power

spectrum, bifurcation diagrams and Lyapunov exponents. It was detected the dynamic

motion as periodic, quasi-periodic and chaotic types.

In (Zhang & Zhan, 2005; Li et al., 2006) a rotor–active magnetic bearings (rotor–AMB) systems

with time-varying stiffness are considered. Using the method of multiple scales a governing

nonlinear equation of motion for the rotor-AMB system with 1-dof is transformed to the

averaged equation and then the bifurcation theory and the method of detection function are

used to analyze the bifurcations of multiple limit cycles of the averaged equation.

Numerical Simulations of Physical and Engineering Processes

368

Fig. 1. The cross–section diagram of a rotor symmetrically supported on the magneto–

hydrodynamic bearing

In the present chapter 2-dof nonlinear dynamics of the rotor supported on the magneto-

hydrodynamic bearing (MHDB) system is analyzed in the cases of soft and rigid magnetic

materials. In the case of soft magnetic materials the analytical solutions have been obtained

by means of the method of multiple scales (Nayfeh & Mook, 2004). Rigid magnetic materials

possess hysteretic properties which are realized in the frames of the present work by means

of Bouc-Wen hysteretic model. This model allows simulating hysteretic loops of various

forms for systems from very different fields (Awrejcewicz & Dzyubak, 2007). Chaotic

regions and the amplitude level contours of the rotor vibrations have been obtained in

various control parameter planes.

2. Mathematical model of the rotor suspended in the magneto-hydrodynamic

field

Consider a uniform symmetric rigid rotor (Fig. 1) which is supported by a magneto–

hydrodynamic bearing system. The four–pole legs are symmetrically placed in the stator. F

is the electromagnetic force produced by the kth opposed pair of electromagnet coils. This

force is controlled by electric currents

0kk

ii i





can be expressed in the form



AN i















Numerical Analysis of a Rotor Dynamics in the Magneto-Hydrodynamic Field

369

where i

denotes bias current in the actuators electric circuits,



is the magnetic

permeability of vacuum, A is the core cross–section area, N is the number of windings of the

electromagnet,



is the air gap in the central position of the rotor with reference to the

bearing sleeve, l is the total length of the magnetic path, the constant value





0ss





denotes the magnetic permeability of the core material; the values of the magnetic induction

and magnetizing force H

define the magnetic saturation level.



is the angle between

axis x and the kth magnetic actuator. Q

is the vertical rotor load identified with its weight,



) are the radial and tangential components of the dynamic oil–film action, respectively.

Equations of motion of the rotor are represented in the following form (Kurnik, 1995;

Dziedzic & Kurnik, 2002; Osinski, 1998)

 



 



** * * * * * * *

** * * * * * * * *

,, cos , sin cos ,

,, sin , cos sin ,

rkkx

rkky

mx P P F Q t

my P P F Q Q t



     









   







 



2* *

*** *

,, 2

PC arctg

pq p

 



 

 



 













  





















 

** *

,PC







 









Here m

denotes the rigid rotor mass, (x

, y

) are the Cartesian coordinates of the rotor center;





Qt,



Qt are the external excitation characterizing bearing housing movements. We

are considering vibrations of the rotor excited by harmonic movements of the bearing

foundation in the vertical direction









****

sin

Qt Q t,

where Q

and



are the amplitude and frequency of the external excitation, respectively.

Constant C

is defined as

scc







Parameters





, R

, L

denote oil viscosity, relative bearing clearance, journal radius and

total bearing length, respectively. (





) are the polar coordinates,

















are the functions conditional



To represent the equations of motion in a dimensionless form the following changes of

variables and parameters are introduced:



 ,

















 ,







*2 *







 ,







Numerical Simulations of Physical and Engineering Processes

370

*2 *







***



 ,





 ,

**2*



 ,

**2*



 ,

**2*



 ,

**2*





**2*





 ,

where



is the rotation speed of the rotor; c

is the bearing clearance.

Thus the dimensionless equations of motion take the form









 

,, cos , sin ,

,, sin , cos sin ,

xP P F

PPFQQt



   







   

(1)







,, 2

PC arctg

pq p





 





 













  





















 

,.PC





 









Here

cosx







, siny





 ,













xx yy















cos







sin







;

the magnetic control forces are expressed as follows



Fxxx



  







Fyyy



  



where



are the coordinates of the rotor static equilibrium,



and



are the control

parameters.

3. Soft magnetic materials

In this section, we consider 2–dof dynamics of the rotor in the MHDB system without taking

hysteresis into account.

3.1 The non-resonant case

The right–hand sides of Eqs (1) were expanded in Taylor’s series and the origin was shifted

to the location of the static equilibrium





for the convenience of the investigation. The

linear and quadratic terms were kept. So, the reformed equations of motion are as follows:



11 2 3 4 5 6 7

21234567

2cos.

x x y x x y xx xy xy xy yy

y y x y x y xx xy xy xy yy F t

        



 

       

         

      

(2)

We seek the first–order solution for small but finite amplitudes in the form

Numerical Analysis of a Rotor Dynamics in the Magneto-Hydrodynamic Field

371









 

101 201

,,,

xxTT xTT

yyTT yTT



 



(3)

where



is the small, dimensionless parameter related to the amplitudes and





(

n=0, 1) are the independent variables. It follows that the derivatives with respect to t

become expansions in terms of the partial derivatives with respect to T

according to

01 2

012

dTT

DD D

dt T t T t T t





 



  



 

22 22

01 2 0 01 1 02

DD D D DD DDD

  

     , where







To analyze the non–resonant case the forcing term is ordered so that it appears at order



Thus, we recall in (2)









. Substituting (3) into (2) and equating coefficients of

similar powers of



we obtain

Order





01 1 01

01 1 01 0

cos .

DxxDy

DyyDxf T







 

  

(4)

Order









2 22

0 2 2 02 0 11 11 11 11 21

3101 411 5101 6101 7101

2 22

02 2 02 0 11 21 11 11 21

3101 411 5101 6101 7101

Dx x Dy D Dx x Dy x y

xDx xy xDy yDx yDy

Dy y Dx D Dy y Dx x y

xDxxyxDyyDxyDy

    



    





 

  



 

  

(5)

The solution of (4) is expressed in the form























  

111 10 21 20 1 0

1111 10 221 20 2 0

exp exp exp ,

xATiTATiT iT CC

yATiTATiT iT CC

 









     



(6)

where CC denotes the complex conjugate of the preceding terms,

and A

are the arbitrary

functions of

at this level of approximation,















222



















222













, (n=1, 2).



are assumed to be distinct and



are the roots of the characteristic equation

100

det

00 1





































4222



 

 =





4222





, (7)

Numerical Simulations of Physical and Engineering Processes

372

1,2 1







 ,

3,4 2







 ,

1,2

42 2 4



  

    

3,4

42 2 4



  

    

Substitution of (6) into (5) gives



02 2 02 1 1 1 1 1 1 10

22 12 22 20

02 2 02 1 1 1 2 1 1 10

22 2 22 2 20

2exp

2exp,

2exp

2exp.

Dx x Dy i A A A i T

iA A A iT CC

Dy y Dx i A A A i T

iAAA iT CC

    

 

    







  





      







 





      





(8)

The terms, which do not influence solvability conditions, are not presented in the last

equations and replaced by dots.

To determine the solvability conditions of (8), following to the method of undetermined

coefficients we seek a particular solution in the form









211 10 12 20

exp expxP iT P iT















221 10 22 20

exp exp

PiTPiT







(9)

with unknowns P

, P

and P

. Substitution of expressions (9) into (8) and collection of

coefficients at





exp iT



and





exp iT



yields





121

nn nn n

PiP R

 

 





122

nn n n n

iP PR

  

  (n=1,2), (10)

where









 

11 1 1 11 11 12 2 2 12 22

21 1 1 1 2 1 1 22 2 2 2 2 2 2

2,2 ,

2,2 .

RiAA ARiAA A

RiAAARiAAA



  

 







 





       

Taking into account the characteristic equation (7), the determinant  of the set of linear

algebraic equations relative to P

, P

(10) is equal to zero



222

 

  









According to Kronecker–Kapelly’s theorem, the set of linear algebraic equations is

compatible if and only if the matrix rank of the linear set is equal to the extended matrix

rank. Therefore, the solvability conditions are













(n=1,2),