輪機中英文翻譯--船舶螺旋槳軸系扭振的流體阻尼

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1、　　中文4550字　　HYDRODYNAMIC DAMPING OF THE TORSIONAL VIBRATIONS OF THE SYSTEM　　SHAFT-SHIP'S PROPELLER　　For ships' power plant

2、s with an internal combustion engine, whose important component is the propeller and shaft, a very topical question is the refinement of existing methods and the devising of new methods of calculating torsional resonance

3、 vibrations. This will make it possible to accurately determine the state of dynamic stress of the installation, and consequently also to determine the possibility of fatigue failure of its most heavily loaded elements.&

4、lt;/p>　　At present, calculations of the torsional vibrations of ships' propeller shafts, amounting to obtaining the amplitude response of the system, are carried out by taking into account the damping w

5、hich, as a rule, is due to friction in the internal combustion engine, in elastic couplings, and the friction of the propeller against the water [1]. Investigations in recent years showed that the damping of torsional vi

6、brations in consequence of energy dissipation in the material of the propeller shafts 　　Damping of the propeller in the ship's power plant is the most important form of damping outside the engine

7、 in all forms of vibrations, except the so-called motor vibrations at which the amplitudes of the free vibrations in the shaft section are small. However, it is very difficult to obtain a formula for calculating the damp

8、ing effect of the propeller, because it depends on an entire complex of factors such as the geometry of the propeller, the number of blades, the vibration frequency, etc. Th　　The methods used at pres

9、ent in the investigation and calculation of torsional vibrations of ships' propeller shafts are based, as a rule, on the approximate method worked out by Terskikh [2]; this method entails the replacement of the real

10、friction in the system by two nominal components, one of which has the properties of linear friction, and consequently, has a quadratic dependence of work on amplitude, and the other has the properties of dry friction wi

11、th linear dependence of work on amplitud　　In practice, friction in the elements of a ship's propeller shaft is not linear; however, the nonlinear problem of the damping properties of different co

12、mponents of the ship's power plant, especially of the propeller, led to difficulties in the solution of nonlinear differential equations, and in view of this, various approximate methods are used in practice, but the

13、y are not very accurate.　　Therefore, one of the ways of improving the accuracy of the calculations of torsional resonance vibrations is to continue the theoretical and experimental investigations, wh

14、ose object would be to determine more accurately the elements of the propeller shaft and to solve the nonlinear problem of torsional resonance vibrations.　　The question of taking into account the ene

15、rgy dissipation due to hysteresis losses in the material in the calculation of mechanical vibrations has received sufficient attention; this included the elaboration of physically substantiated methods of calculating the

16、 vibrations of systems, which was done very successfully by the use of asymptotic methods of nonlinear mechanics [3].　　Up to the present, however, there do not exist any reliable methods of calculati

17、ng vibrations for other kinds of energy losses (structural and aerodynamic damping). 　　Pisarenko [4] suggested a new approach which makes it possible to solve the problem of taking into account not

18、 only the hysteretic energy dissipation In the material, but also stnlctural as well as aerodynamic kinds of damping, on the basis of a single method whose essence is that all kinds of energy losses in the vibrating syst

19、em, regardless of their origin, are represented in the form of some hysteresis loops, separately for each kind of loss, and the areas of the hysteresis loops then charact　　Here we have to proceed fro

20、m the following nonlinear correlations between stress and relative deformation in any single cyclically deformed element (spring) with peak value of deformation for the ascending and descending motion leading in a cycle

21、to the formation of the hysteresis loop [5]:　　(1)　　where is the modulus of elasticity of the material; is the logarithmic decrement of the vibrations. Arrows

22、pointing to the right refer to the ascending branch of the hysteresis loop; arrows pointing to the left refer to the descending branch.　　By introducing into (1) the decrement as a function of the fac

23、tor on which it depends, we can generalize the approach used in taking hysteresis losses in the material into account to the case of taking other energy losses into account, losses that are due to any arbitrary causes, b

24、ecause in all cases energy is dissipated which was integrally accumulated in the vibrating system and which consists of the sum of the energies of unit volumes of the cyclically deformed material of an elastic element &l

25、t;/p>　　By integrating with respect to the volume of the cyclically deformed material, we can take into account any energy losses, summing them as hysteresis loops having the same shape whose magnitude depen

26、ds on the level of the vibration decrements contained in the equation of the loop and obtained experimentally as a function of some factor. The above-mentioned hysteresis loops characterizing the energy losses in a unit

27、volume of cyclically deformed mate; rial with deformation amplitude may, general　　(2)　　where is the vibration decrement characterizing the energy dissipation

28、 in the cyclically deformed material itself, which, as was shown above, depends on the deformation amplitude; is the vibrational decrement characterizing energy losses in fixed joints (structural damping), which, as a r

29、ule, depends on the magnitude of the reactive moment acting in the node; is tile aerodvnamic vibration decrement, which may depend on various factors in dependence on the nature of the environment and its int

30、　　Thus, proceeding from relation (2) and taking into account the decrements it contains as functions of the respective factors, we may, by the methods of nonlinear mechanics [3], envisaging energy losses to be

31、taken into account by integration with respect to the volume of tile cyclically deformed material of the vibrating system, construct the amplitude response of the latter in the resonance and near-resonance zones that are

32、 of interest to us; this will make it possible to evaluate the state of 　　Extending the above approach to the case of hydrodynamic damping of a propeller with critical vibrations of the propeller sha

33、ft, we devise, in analogy with expression (1), equations describing the outline of the hysteresis loop corresponding to the given type of damping, and we adopt them as initial ones:　　(3)<

34、/b>　　where G is tile shear modulus; , hydrodynamic vibration decrement; , amplitude of the cyclic torsional deformation; and , running value of the relative shear deformation.

35、　　In order to simplify further calculations, we represent formula (3) in the form　　(4)　　where is the stress,　　=

36、 (5)　　is the frictional stress,　　(6)　　We assume that the cyclic deformation of the material ychanges with time cosinusoidall

37、　　(7)　　Where　　(8)　　Taking (7) into account, we write expression (6) in the form

38、　　(9)　　Shear deformation of an element of the material of a circular rod, situated at the distancefrom the center of its cross section, is determined by the formula</

39、p>　　(10)　　where is the angle of torsion of the rod; x is the axis of coordinates in the direction of the axis of the rod.　　The peak value of she

40、ar at the given point is equal to　　(11) If we substitute (10) into (5), and (11) into (9), we obtain:　　(12) (13)　　Taking (12) and (13) into

41、 account, we write expression (3) in the form　　(14)　　The magnitude of the torque acting in the cross section of the rod is determined by the formula

42、　　(15)　　For a rod with completely circular cross section and outer radius =4, Eq. (15), taking (14) into account, assumes the form　　(16)<

43、/b>　　or　　(17)　　where is the polar moment of inertia;　　(18) Since is the elastic torque in an a

44、rbitrary cross section of the rod, Eq. (17) may be written in abbreviated form as follows:　　(19)　　where is the "braking" torque due to hydrodynamic d

45、amping of the propeller in consequence of its friction with the water, which, in accordance with expression (17), may be represented in the following manner:　　(20)<p

46、>　　Where　　(21)　　is the full angle of torsion of a rod with length at any instant, determined by the expression　　（22）

47、 not taking into account the energy dissipation in consequence of the hydrodynamic damping of the propeller subjected to torsional vibrations; is its peak value.　　In fact, the angle of rotation of th

48、e end section of the rod with length , taking hydrodynamic damping into account, can be determined on the basis of expressions (19), (20), and (22) by the formula　　(23)

49、　　We denote the second term on the right-hand side of expression (23):　　(24) Then Eq. (23), characterizing the true angle of rotation of the end section of the rod at any instant , may be wr

50、itten in abbreviated form as follows: 　　(25) In investigations and calculations, the propeller shaft is reduced to a system consisting of concentrated masses that have only inertial properties, and o

51、f joints that have only elastic properties.　　Figure I shows the initial structural simulator of the investigated system propeller shaft-propeller, which is an elastic rod with a disk at the end. The

52、mass of the shaft, which has only elastic properties, may be neglected in comparison with the mass of the disk.　　The forced torsional vibrations are effeeted under the influence of small periodic ang

53、ular displacements of the constraint, proportional to the small parameter [6]in the plane parallel to the plane of the disk: 　　(26) where is the peak value of the angle of rotation of the constrain

54、ed section of the rod, ; w is the angular frequency of the vibrations of the constrained rod.　　Then the expression of the total angle of rotation of the end section of the rod, i .e .of the disk, at

55、any instant may be written in the form　　(27) where is the full angle of torsion determined by Eq. (25), and taking this into account,　　we write expression (27) as follows: </

56、p>　　(28) If we apply to the system the torque of the inertial forces of the disk, using thereby　　Lagrange's second-order equation, we obtain the differential equation of the motion

57、of the disk at the end of the rod:　　Fig.1. Structural simulator of a torsional　　vibration system with one degree of freedom.　　(29)</p&g

58、t;　　where c is the torsional rigidity of the rod, 　　(30) L, length of the rod; and I, moment of inertia of the mass of the disk relative to the x axis of the rod, which is perpendicular to

59、 the plane of the disk.　　For the component o~ the angle of rotation of the end section of the rod , the values in the ascending motion and in descending motion are different, in consequence of whic

60、h Eq. (29) is nonlinear. The given nonlinearity of "hysteretic" origin, which is due to hydrodynamic damping, is very slight, and it is expedient to express this by introducing the small parameter into the corr

61、esponding term of Eq. (29): ：　　（31）　　Where　?。?2）We introduce the following notation: 　　（3

62、3）　　（34）　　where p is the natural angular frequency of the torsional vibrations of the system.　　In view of (33) and (34), Eq. (31

63、) may he written in the form　?。?5）　　where ；　　（36）　　Following the methods of Kry

64、lov and Bogolyubov [7], we will seek the general solution of Eq. (35) with weak distortion in the form of an expansion into a series with powers of the small parameter :　　（37）</

65、p>　　Where ; is the angular frequency of the distorting force; , phase of the vibrations; and , phase shift.　　The magnitudes u and are functions of time, and they are determined from th

66、e differential equations　　（38）　　Fig. 2. Dependence of the hydrodynamic decrement on the frequency of the forced torsional vibrations. (The dots show the experi

67、mental 　　values of the hydrodynamic decrements.)　　where　　With vibrations in the resonance zones, when the phase shift is a constant magni

68、tude, it may be assumed that the amplitude u and the phase do not depend on the phase shift ,and they may be determined by using the differential equation (38).　　The terms and u2 of the series (37)

69、 are periodic functions of with the period 2.Thus, the solution of Eq. (35) reduces to finding the functions u1(u, T); u2(u, T) .... ,At(u); A2(u) ..... B1(u) ; B2(u) .....　　We will omit the procedu

70、re of solving Eq. (35), since its method was explained in detail in [3], and we present the final expressions for determining the phase shift in the first approximation and for plotting the amplitude response of the inve

71、stigated system, taking hydrodynamic damping into account: 　?。?9）　　（40）　　In Esq. (39) and (40) the unknown is , which i

72、s determined on the basis of (18) from the expression　　The hydrodynamic decrement contained in this formula may be represented in the form　　（41）<

73、;p>　　where is the component of the hydrodynamic decrement that depends on the frequency of the forced torsional vibrations; , component of the hydrodynamic decrement that depends on the rotational frequency of the pr

74、opeller shaft with the propeller carrying out forced torsional vibrations; and , component of the hydrodynamic decrement that depends on the speed of the flow of water past the propeller in forced torsional vibrations.&l

75、t;/p>　　The above components of the hydrodynamic decrement may in turn be represented by the following expressions : 　?。?2）　　Where</b&

76、gt;　　（43）　　is the proportionality factor, sec; , slope of the straight line to the axis of the abscissas, obtained experimentally (Fig. 2); , pack value of the

77、 shear deformation rate　　（44）　　where is the proportionality factor, ,　?。?5）　　is the slope of the

78、straight line to the axis of abscissas obtained by experiment (Fig. 3a); n, rotational frequency of the propeller, ;　?。?6）　　where is the proportionality facto

79、r, 　　（47）　　is the slope of the straight line to the axis of abscissas obtained by experiment (Fig. 3b); V is the flow rate of water past the propeller, </p&

80、gt;　　Let us examine in more detail the expression for . It is known from Esq. (10) and (21) that　　；　　Then the shear deformation may be expressed in

81、the following way: 　?。?8）　　where , in solving Eq. (35) in the first approximation, is equal to　　（49）<p&g

82、t;　　With this taken into account, the expression for shear deformation assumes the form　?。?0）　　If we substitute (50) into (42) and differentiate, we obtain: &l

83、t;/p>　?。?1）　　For the case of the peak value of the shear deformation rate, which is being investigate here, . Then we have　　（52）</b&g

84、t;　　Substituting (52), (44), and (46) into Eq. (18), we write: 　　（53）　　If we integrate expression (53) and substitute r = d/2 into the r e

85、 s u l t , where d is the shaft diameter, we obtain finally : 　　（54）　　where w = p in the first approximation, in accordance with (38).　　If

86、 we substitute (54) into (39) and (40), we have:　?。?5）　　Fig. 3. Dependence of the hydrodynamic decrement on the rotational frequency of　　t

87、he propeller (a) and on the flow rate around the propeller (b). (Notation is　　the same as in Fig. 2.)　　Fig. 4. Theoretical amplitude responses of the investigated system for diff

88、erent rotational frequencies of the propeller: 1) = 620.27，n = 0,V = 0； 2) = 618.83，n = 1.67 rev.，V = O; 3) = 617.39 ，n = 3.33 rev., V = 0; 4) = 615.95 ， n = 5.00 rev., V = O;5) = 614.58 ， n = 6.67 rev'，V = 0; 6)

89、 = 613.08 ,，n =8.33 rev.， V = 0; 7) = 611.70, n = i0 rev. ， V = O. (For comparison, tile dots show the experimentally obtained resonance frequencies.) 　?。?6）

90、　where the coefficeints kl, k2, k3 are calculated by Esq. (43), (45), and (47) on the basis of experimental data.　　Equations (55), (56)enable us to determine the phase shift and to plot the amplitude

91、 response with critical vibrations of the propeller shaft, taking hydrodynamic damping of the propeller into account.　　For experimental investigations with the object of confirming the above theoreti

92、cal conclusions, the device K-80 was built; it is described in detail in [8].　　To calculate the vibration decrements, we used the energy method, according to which relative energy dissipation is dete

93、rmined by the formula [9] 　?。?7）　　where W is the total power expended on the excitation of torsional vibrations, W; , power expended on overcoming friction in

94、 the device itself, W; f, frequency of the steady-state torsional vibrations of the propeller, Hz; and , potential energy of the twisted shaft, corresponding to the amplitude of the steady-state vibrations.&l

95、t;p>　　We represent the correlation between the relative energy dissipation and the logarithmi decrement of the vibrations in the form [I0] 　　（58）　　Fig. 5. T

96、heoretical amplitude responses of the investigated .system for different flow rates: 1） = 620.27 , V = 0, n = 0; 2) = 619.18, V = 0.5m., n = 0; 3) = 618.08, V = 1.0 m, n = 0; 4) = 616.99, V = 1.5 m., n = 0; 5) = 615.94

97、, V = 2.0 m. , n = 0; 6) = 614.86, V = 2.5 m., n =0. (The meaning of the dots is the same as in Fig. 4.)　　Fig. 6. Theoretical amplitude responses of the investigated system: 1） = 620.27, n = 0, V = 0

98、; 2) = 618.83, n = 1.67 rev. , V = 0; 3) =618.08, n = O, V = 1 m.; 4) = 611.33, n = 1.67 rev., V =1 m. (file meaning of the dots is the same as in Fig. 4.)　　where is the vibration decrement. Takin

99、g Eqs. (57) and (58) into account, we write the expression for determining the hydrodynamic decrement of the propeller in the following form　?。?9）　　Figure 2 sh

100、ows the experimentally obtained dependence of the hydrodynamic decrement on the frequency of the forced torsional vibrations of the propeller; it is a straight line at the angle a1 to the axis of abscissas.&l

101、t;p>　　Figures 3a, b show the experimentally determined dependences of the hydrodynamic decrement on the rotational frequency of the propeller and on the flow rate of the water past the propeller; they are straight lin

102、es at the angles a2 and a3, respectively, to the axis of abscissas.　　Using the experimentally found angles a1, a2, and a3, we calculate by Esq. (43), (45), and (47) the coefficients kl, k2, and k3, a

103、nd with their aid we can determine the phase shift by Eqs.(55)and(56)and plot the amplitude responses of the investigated system for different rotational frequencies of the propeller and different flow rates of the water

104、 past the propeller.　　Figures 4-6 present the resonance curves obtained by calculation.　　A comparison of the resonance frequencies obtained by calculation using the formulas of t

105、he theoretical section with those experimentally determined shows the error of calculation lies within the limits 0.3-0.4%.This ensures an accuracy sufficient for engineering calculations.　　CONCLUSIO

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