上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch12 journal bearing

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil 607 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumnal Bearings 607 u=U Rotating journal (口+肛d)血k Flow of lubricant (p+ pddk d r dx dz Stationary partial bushing Partial bushing (a) (b) I Figure 12-9 element,and shear forces,due to the viscosity and to the velocity,act upon the top and bottom sides.Summing the forces in the x direction gives ∑E=pdydz-（p+e ar dy dpdx dy di-rdxdz++a dx dz=0 (a) This reduces to dp a y (6 dx From Eq.(12-1),we have ou t=Hay d where the partial derivative is used because the velocity u depends upon both x and y. Substituting Eq.(c)in Eq.(b),we obtain dp a2u = (d) Holding x constant,we now integrate this expression twice with respect to y.This gives au1dp 亦+G 1d迎y2+Cy+C2 2u dx (e] Note that the act of holding x constant means that Ci and C2 can be functions of x.We now assume that there is no slip between the lubricant and the boundary surfaces.This gives two sets of boundary conditions for evaluating the constants Ci and C2: At y=0,u=0 9 At y=h,u=U

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 607 Companies, 2008 Lubrication and Journal Bearings 607 dx h U dy Journal (a) (b) Partial bushing dy h z x y dx  dx dz p dy dz ∂y ( + dy) dx dz ∂ dx (p + dx) dy dz dp u = U Flow of lubricant Stationary partial bushing Rotating journal element, and shear forces, due to the viscosity and to the velocity, act upon the top and bottom sides. Summing the forces in the x direction gives Fx = pdydz − p + dp dx dx dydz − τ dx dz + τ + ∂τ ∂y dy dx dz = 0 (a) This reduces to dp dx = ∂τ ∂y (b) From Eq. (12–1), we have τ = μ ∂u ∂y (c) where the partial derivative is used because the velocity u depends upon both x and y. Substituting Eq. (c) in Eq. (b), we obtain dp dx = μ ∂2u ∂y2 (d) Holding x constant, we now integrate this expression twice with respect to y. This gives ∂u ∂y = 1 μ dp dx y + C1 u = 1 2μ dp dx y2 + C1 y + C2 (e) Note that the act of holding x constant means that C1 and C2 can be functions of x. We now assume that there is no slip between the lubricant and the boundary surfaces. This gives two sets of boundary conditions for evaluating the constants C1 and C2: At y = 0, u = 0 At y = h, u = U (f ) Figure 12–9

608 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 608 Mechanical Engineering Design Figure 12-10 Rotating journal y Velocity of the lubricant. Flow of lubricant 0 0 黑<0 Stationary bushing Notice,in the second condition,that h is a function ofx.Substituting these conditions in Eq.(e)and solving for the constants gives U h dp C1=i-2μdx C2=0 or 1 dp U 4= 2u dx 202-h+) (12-91 This equation gives the velocity distribution of the lubricant in the film as a function of the coordinate y and the pressure gradient dp/dx.The equation shows that the velocity distribution across the film(from y =0 to y =h)is obtained by superposing a para- bolic distribution onto a linear distribution.Figure 12-10 shows the superposition of these distributions to obtain the velocity for particular values of x and dp/dx.In general, the parabolic term may be additive or subtractive to the linear term,depending upon the sign of the pressure gradient.When the pressure is maximum,dp/dx=0 and the velocity is U =y (g) which is a linear relation. We next define O as the volume of lubricant flowing in the x direction per unit time.By using a width of unity in the z direction,the volume may be obtained by the expression Q= u dy Substituting the value of u from Eq.(12-9)and integrating gives Uh h3 dp Q= 2-12μdx The next step uses the assumption of an incompressible lubricant and states that the flow is the same for any cross section.Thus do =0 dx

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 608 © The McGraw−Hill Companies, 2008 608 Mechanical Engineering Design Notice, in the second condition, that h is a function of x. Substituting these conditions in Eq. (e) and solving for the constants gives C1 = U h − h 2μ dp dx C2 = 0 or u = 1 2μ dp dx (y2 − hy) + U h y (12–9) This equation gives the velocity distribution of the lubricant in the film as a function of the coordinate y and the pressure gradient dp/dx. The equation shows that the velocity distribution across the film (from y = 0 to y = h) is obtained by superposing a para￾bolic distribution onto a linear distribution. Figure 12–10 shows the superposition of these distributions to obtain the velocity for particular values of x and dp/dx. In general, the parabolic term may be additive or subtractive to the linear term, depending upon the sign of the pressure gradient. When the pressure is maximum, dp/dx = 0 and the velocity is u = U h y (g) which is a linear relation. We next define Q as the volume of lubricant flowing in the x direction per unit time. By using a width of unity in the z direction, the volume may be obtained by the expression Q =  h 0 u dy (h) Substituting the value of u from Eq. (12–9) and integrating gives Q = Uh 2 − h3 12μ dp dx (i) The next step uses the assumption of an incompressible lubricant and states that the flow is the same for any cross section. Thus d Q dx = 0 h y U u x y Flow of lubricant Stationary bushing Rotating journal dp dx > 0 dp dx = 0 dp dx < 0 Figure 12–10 Velocity of the lubricant

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw--Hill 609 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumnal Bearings 609 From Eq.(i), -号-（你）=0 dx or d (hs dp =6 (12-10) dxu dx which is the classical Reynolds equation for one-dimensional flow.It neglects side leak- age,that is,flow in the z direction.A similar development is used when side leakage is not neglected.The resulting equation is a(h3 ap a(h3 ap 6U ih (12-11) ax u ax azu az ax There is no general analytical solution to Eq.(12-11):approximate solutions have been obtained by using electrical analogies,mathematical summations,relaxation methods, and numerical and graphical methods.One of the important solutions is due to Sommerfelds and may be expressed in the form (12-12 where o indicates a functional relationship.Sommerfeld found the functions for half- bearings and full bearings by using the assumption of no side leakage. 12-7 Design Considerations We may distinguish between two groups of variables in the design of sliding bearings. In the first group are those whose values either are given or are under the control of the designer.These are: 1 The viscosity u 2 The load per unit of projected bearing area,P 3 The speed N 4 The bearing dimensions r,c,B,and I Of these four variables,the designer usually has no control over the speed,because it is specified by the overall design of the machine.Sometimes the viscosity is specified in advance,as,for example,when the oil is stored in a sump and is used for lubricating and cooling a variety of bearings.The remaining variables,and sometimes the viscosity, may be controlled by the designer and are therefore the decisions the designer makes. In other words,when these four decisions are made,the design is complete. In the second group are the dependent variables.The designer cannot control these except indirectly by changing one or more of the first group.These are: 1 The coefficient of friction f 2 The temperature rise AT 3 The volume flow rate of oil O 4 The minimum film thickness ho 5A.Sommerfeld,"Zur Hydrodynamischen Theorie der Schmiermittel-Reibung"("On the Hydrodynamic Theory of Lubrication").Z Math.Physik,vol.50.1904.pp.97-155

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 609 Companies, 2008 Lubrication and Journal Bearings 609 From Eq. (i), d Q dx = U 2 dh dx − d dx h3 12μ dp dx = 0 or d dx h3 μ dp dx = 6U dh dx (12–10) which is the classical Reynolds equation for one-dimensional flow. It neglects side leak￾age, that is, flow in the z direction. A similar development is used when side leakage is not neglected. The resulting equation is ∂ ∂x h3 μ ∂p ∂x + ∂ ∂z h3 μ ∂p ∂z = 6U ∂h ∂x (12–11) There is no general analytical solution to Eq. (12–11); approximate solutions have been obtained by using electrical analogies, mathematical summations, relaxation methods, and numerical and graphical methods. One of the important solutions is due to Sommerfeld5 and may be expressed in the form r c f = φ r c 2 μN P  (12–12) where φ indicates a functional relationship. Sommerfeld found the functions for half￾bearings and full bearings by using the assumption of no side leakage. 12–7 Design Considerations We may distinguish between two groups of variables in the design of sliding bearings. In the first group are those whose values either are given or are under the control of the designer. These are: 1 The viscosity μ 2 The load per unit of projected bearing area, P 3 The speed N 4 The bearing dimensions r, c, β, and l Of these four variables, the designer usually has no control over the speed, because it is specified by the overall design of the machine. Sometimes the viscosity is specified in advance, as, for example, when the oil is stored in a sump and is used for lubricating and cooling a variety of bearings. The remaining variables, and sometimes the viscosity, may be controlled by the designer and are therefore the decisions the designer makes. In other words, when these four decisions are made, the design is complete. In the second group are the dependent variables. The designer cannot control these except indirectly by changing one or more of the first group. These are: 1 The coefficient of friction f 2 The temperature rise T 3 The volume flow rate of oil Q 4 The minimum film thickness h0 5 A. Sommerfeld, “Zur Hydrodynamischen Theorie der Schmiermittel-Reibung” (“On the Hydrodynamic Theory of Lubrication”), Z. Math. Physik, vol. 50, 1904, pp. 97–155.

610 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hil Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition 610 Mechanical Engineering Design N。=0,N=0 N=0,=y N=0=兰 No=N.N =0 N=y,+0-20=N N=y+0-2W=N N=W+0-2N/1=0 N=y+N-20=2y (a) (b) (e) (d) Figure 12-11 How the significant speed varies.(a)Common bearing case.(b]Load vector moves at the same speed as the journal.(c)Lood vector moves at half journal speed,no load can be carried.(d)Joumal and bushing move at same speed,load vector stationary. capacity halved. This group of variables tells us how well the bearing is performing,and hence we may regard them as performance factors.Certain limitations on their values must be imposed by the designer to ensure satisfactory performance.These limitations are specified by the char- acteristics of the bearing materials and of the lubricant.The fundamental problem in bear- ing design,therefore,is to define satisfactory limits for the second group of variables and then to decide upon values for the first group such that these limitations are not exceeded. Significant Angular Speed In the next section we will examine several important charts relating key variables to the Sommerfeld number.To this point we have assumed that only the journal rotates and it is the journal rotational speed that is used in the Sommerfeld number.It has been discovered that the angular speed N that is significant to hydrodynamic film bearing performance is N=INi +N)-2Nl (12-13) where N=journal angular speed,rev/s N=bearing angular speed,rev/s N,=load vector angular speed,rev/s When determining the Sommerfeld number for a general bearing,use Eq.(12-13) when entering N.Figure 12-11 shows several situations for determining N. Trumpler's Design Criteria for Journal Bearings Because the bearing assembly creates the lubricant pressure to carry a load,it reacts to loading by changing its eccentricity,which reduces the minimum film thickness ho until the load is carried.What is the limit of smallness of ho?Close examination reveals that the moving adjacent surfaces of the journal and bushing are not smooth but consist of a series of asperities that pass one another,separated by a lubricant film.In starting a "Paul Robert Trumpler,Design of Film Bearings,Macmillan,New York,1966.pp.103-119

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings 610 © The McGraw−Hill Companies, 2008 610 Mechanical Engineering Design Figure 12–11 How the significant speed varies. (a) Common bearing case. (b) Load vector moves at the same speed as the journal. (c) Load vector moves at half journal speed, no load can be carried. (d) Journal and bushing move at same speed, load vector stationary, capacity halved. W Nb = 0, Nf = 0 (a) N = Nj + 0 – 2(0) = Nj N = Nj + 0 – 2Nj = Nj N = Nj + 0 – 2Nj 2 = 0 N = Nj + Nj – 2(0) = 2Nj Nj W Nb = 0, Nf = Nj (b) Nj Nf W Nb = Nj , Nf = 0 (d) Nj Nb W Nb = 0, Nf = (c) Nj Nj 1 2 Nj 2 This group of variables tells us how well the bearing is performing, and hence we may regard them as performance factors. Certain limitations on their values must be imposed by the designer to ensure satisfactory performance. These limitations are specified by the char￾acteristics of the bearing materials and of the lubricant. The fundamental problem in bear￾ing design, therefore, is to define satisfactory limits for the second group of variables and then to decide upon values for the first group such that these limitations are not exceeded. Significant Angular Speed In the next section we will examine several important charts relating key variables to the Sommerfeld number. To this point we have assumed that only the journal rotates and it is the journal rotational speed that is used in the Sommerfeld number. It has been discovered that the angular speed N that is significant to hydrodynamic film bearing performance is6 N = |Nj + Nb − 2Nf | (12–13) where Nj = journal angular speed, rev/s Nb = bearing angular speed, rev/s Nf = load vector angular speed, rev/s When determining the Sommerfeld number for a general bearing, use Eq. (12–13) when entering N. Figure 12–11 shows several situations for determining N. Trumpler’s Design Criteria for Journal Bearings Because the bearing assembly creates the lubricant pressure to carry a load, it reacts to loading by changing its eccentricity, which reduces the minimum film thickness h0 until the load is carried. What is the limit of smallness of h0? Close examination reveals that the moving adjacent surfaces of the journal and bushing are not smooth but consist of a series of asperities that pass one another, separated by a lubricant film. In starting a 6 Paul Robert Trumpler, Design of Film Bearings, Macmillan, New York, 1966, pp. 103–119.

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 12.Lubrication and Journal T©The McGraw-Hill 611 Mechanical Engineering Elements Bearings Companies,2008 Design,Eighth Edition Lubrication and Joumnal Bearings 611 bearing under load from rest there is metal-to-metal contact and surface asperities are broken off,free to move and circulate with the oil.Unless a filter is provided,this debris accumulates.Such particles have to be free to tumble at the section containing the min- imum film thickness without snagging in a togglelike configuration,creating additional damage and debris.Trumpler,an accomplished bearing designer,provides a throat of at least 200 uin to pass particles from ground surfaces.He also provides for the influence of size (tolerances tend to increase with size)by stipulating ho≥0.0002+0.00004din (a) where d is the journal diameter in inches. A lubricant is a mixture of hydrocarbons that reacts to increasing temperature by vaporizing the lighter components,leaving behind the heavier.This process(bearings have lots of time)slowly increases the viscosity of the remaining lubricant,which increases heat generation rate and elevates lubricant temperatures.This sets the stage for future failure.For light oils,Trumpler limits the maximum film temperature Tmax to Tma≤250°F (b) Some oils can operate at slightly higher temperatures.Always check with the lubricant manufacturer. A journal bearing often consists of a ground steel journal working against a softer, usually nonferrous,bushing.In starting under load there is metal-to-metal contact, abrasion,and the generation of wear particles,which,over time,can change the geo- metry of the bushing.The starting load divided by the projected area is limited to W D ≤300psi (c) If the load on a journal bearing is suddenly increased,the increase in film temper- ature in the annulus is immediate.Since ground vibration due to passing trucks,trains, and earth tremors is often present,Trumpler used a design factor of 2 or more on the running load,but not on the starting load of Eq.(c): nd≥2 (d) Many of Trumpler's designs are operating today,long after his consulting career is over:clearly they constitute good advice to the beginning designer. 12-8 The Relations of the Variables Before proceeding to the problem of design,it is necessary to establish the relationships between the variables.Albert A.Raimondi and John Boyd,of Westinghouse Research Laboratories,used an iteration technique to solve Reynolds'equation on the digital computer.This is the first time such extensive data have been available for use by designers,and consequently we shall employ them in this book. 70p.cit,pp.192-194. A.A.Raimondi and John Boyd,"A Solution for the Finite Journal Bearing and Its Application to Analysis and Design.Parts I,II,and III,"Trans.ASLE,vol.1.no.1,in Lubrication Science and Technology. Pergamon,New York,1958.pp.159-209. See also the earlier companion paper,John Boyd and Albert A.Raimondi,"Applying Bearing Theory to the Analysis and Design of Journal Bearings,Part I and II,"J.Appl.Mechanics,vol.73.1951.pp.298-316

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 12. Lubrication and Journal Bearings © The McGraw−Hill 611 Companies, 2008 Lubrication and Journal Bearings 611 bearing under load from rest there is metal-to-metal contact and surface asperities are broken off, free to move and circulate with the oil. Unless a filter is provided, this debris accumulates. Such particles have to be free to tumble at the section containing the min￾imum film thickness without snagging in a togglelike configuration, creating additional damage and debris. Trumpler, an accomplished bearing designer, provides a throat of at least 200 μin to pass particles from ground surfaces.7 He also provides for the influence of size (tolerances tend to increase with size) by stipulating h0 ≥ 0.0002 + 0.000 04d in (a) where d is the journal diameter in inches. A lubricant is a mixture of hydrocarbons that reacts to increasing temperature by vaporizing the lighter components, leaving behind the heavier. This process (bearings have lots of time) slowly increases the viscosity of the remaining lubricant, which increases heat generation rate and elevates lubricant temperatures. This sets the stage for future failure. For light oils, Trumpler limits the maximum film temperature Tmax to Tmax ≤ 250◦ F (b) Some oils can operate at slightly higher temperatures. Always check with the lubricant manufacturer. A journal bearing often consists of a ground steel journal working against a softer, usually nonferrous, bushing. In starting under load there is metal-to-metal contact, abrasion, and the generation of wear particles, which, over time, can change the geo￾metry of the bushing. The starting load divided by the projected area is limited to Wst l D ≤ 300 psi (c) If the load on a journal bearing is suddenly increased, the increase in film temper￾ature in the annulus is immediate. Since ground vibration due to passing trucks, trains, and earth tremors is often present, Trumpler used a design factor of 2 or more on the running load, but not on the starting load of Eq. (c): nd ≥ 2 (d) Many of Trumpler’s designs are operating today, long after his consulting career is over; clearly they constitute good advice to the beginning designer. 12–8 The Relations of the Variables Before proceeding to the problem of design, it is necessary to establish the relationships between the variables. Albert A. Raimondi and John Boyd, of Westinghouse Research Laboratories, used an iteration technique to solve Reynolds’ equation on the digital computer.8 This is the first time such extensive data have been available for use by designers, and consequently we shall employ them in this book.9 7 Op. cit., pp. 192–194. 8 A. A. Raimondi and John Boyd, “A Solution for the Finite Journal Bearing and Its Application to Analysis and Design, Parts I, II, and III,” Trans. ASLE, vol. 1, no. 1, in Lubrication Science and Technology, Pergamon, New York, 1958, pp. 159–209. 9 See also the earlier companion paper, John Boyd and Albert A. Raimondi, “Applying Bearing Theory to the Analysis and Design of Journal Bearings, Part I and II,” J. Appl. Mechanics, vol. 73, 1951, pp. 298–316