Bearing Design in Machinery Episode 2 Part 4 doc

Similar to the hydrodynamic journal bearing, the load of the hydrostatic journal bearing is also divided into two components, W x and W y , and the displacement of the bearing center e is divided into two components, e x and e y . In Chapter 7, the two components of the journal bearing stiffness are defined [Eq. (7-31)], and the cross-stiffness components are defined in Eq. (7-32). Cross- stiffness components can result in bearing instability, which was discussed in Chapter 9. 10.12 APPLICATIONS An interesting application is the hydrostatic pad in machine tool screw drives (Rumberger and Wertwijn, 1968). For high-precision applications, it is important to prevent direct metal contact, which results in stick-slip friction and limits the machining precision. Figure 10-7 shows a noncontact design that includes hydrostatic pads for complete separation of the sliding surfaces of screw drive. Another important application is in a friction testing machine, which will be described in Chapter 14. 10.13 HYDRAULIC PUMPS An example of a positive-displacement pump that is widely used for lubrication is the gear pump. The use of gear pumps is well known in the lubrication system of automotive engines. Gear pumps, as well as piston pumps, are positive-displace- ment pumps; i.e., the pumps deliver, under ideal conditions, a fixed quantity of liquid per cycle, irrespective of the flow resistance (head losses in the system). However, it is possible to convert the discharge at a constant flow rate to discharge at a constant pressure by installing a pressure relief valve that maintains a constant pressure and returns the surplus flow. A cross section of a simple gear pump is shown in Fig. 10-8a. A gear pump consists of two spur gears (or helix gears) meshed inside a pump casing, with one of the gears driven by a constant-speed electric motor. The liquid at the suction side is trapped between the gear teeth, forcing the liquid around the casing and finally expelling it through the discharge. The quantity of liquid discharged per revolution of the gear is known as displacement, theoretically equal to the sum of the volumes of all the spaces between the gear teeth and the casing. However, there are always tolerances and small clearance for a free fit between the gears and casing. The presence of clearance in pump construction makes it practically impossible to attain the theoretical displacement. The advantages of the gear pump, in comparison to other pumps, are as follows. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. 3. It can handle very high-viscosity fluid. 4. It can generate very high heads (or outlet pressure) in comparison to centrifugal pumps. 5. It is self-priming (unlike the centrifugal pump). It acts like a compres- sor and pumps out trapped air or vapors. 6. It has good efficiency at very high heads. 7. It has good efficiency over a wide speed range. 8. It requires relatively low suction heads. The flow rate of a gear pump is approximately constant, irrespective of its head losses. If we accidentally close the discharge valve, the discharge pressure would rise until the weakest part of the system fails. To avoid this, a relief valve should be installed in parallel to the discharge valve. When a small amount of liquid escapes backward from the discharge side to the suction side through the gear pump clearances, this is referred to as slip. The capacity (flow rate) lost due to slip in the clearances increases dramatically with the clearance, h 0 , between the housing and the gears (proportional to h 3 0 ) and is inversely proportional to the fluid viscosity. An idea about the amount of liquid lost in slippage can be obtained via the equation for laminar flow between two parallel plates having a thin clearance, h 0 , between them: Q ¼ lh 3 0 12mb Dp ð10-49Þ where Q ¼ flow rate of flow in the clearance ðslip flow rateÞ Dp ¼ differential pressure ðbetween discharge and suctionÞ b ¼ width of fluid path ðnormal to fluid pathÞ h 0 ¼ clearance between the two plates m ¼ fluid viscosity l ¼ length along the fluid path This equation is helpful in understanding the parameters affecting the magnitude of slip. It shows that slip is mostly dependent on clearance, since it is proportional to the cube of clearance. Also, slip is proportional to the pressure differential Dp and inversely proportional to the viscosity m of the liquid. Gear pumps are suitable for fluids of higher viscosity, for minimizing slip, and are widely used for lubrication, since lubricants have relatively high viscosity (in comparison to water). Fluids with low viscosity, such as water and air, are not suitable for gear pumps. Piston pumps are also widely used as high-pressure positive-displacement (constant-flow-rate) hydraulic pumps. An example of the multipiston pump is Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. shown in Fig. 10-8b. The advantage of the piston pump is that it is better sealed and the slip is minimized. In turn, the efficiency of the piston pump is higher, compared to that of a gear pump, but the piston pump requires valves, and it is more expensive. 10.14 GEAR PUMP CHARACTERISTICS The actual capacity (flow rate) and theoretical displacement versus pump head are shown in Fig. 10-9. The constant theoretical displacement is a straight horizontal line. The actual capacity (flow rate) reduces with the head because the ‘‘slip’’ is proportional to the head of the pump (discharge head minus suction head). When the head approaches zero, the capacity is equivalent to the theoretical displace- ment. 10.14.1 Hydraulic Power and Pump E⁄ciency The SI unit of power _ EE is the watt. Another widely used unit is the Imperial unit, horse power [HP]. Brake power, _ EE b (BHP in horsepower units), is the mechanical shaft power required to drive the pump by means of electric motor. In the pump, this power is converted into two components: the useful hydraulic power, _ EE h , and the frictional losses, _ EE f . The useful hydraulic power can be converted back to work done by the fluid. A piston or hydraulic motor can do this energy conversion. In the pump, the friction losses are dissipated as heat. Friction FIG. 10-9 Gear pump Q–H characteristics. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. losses result from friction in the bearings, the stuffing box (or mechanical seal), and the viscous shear of the fluid in the clearances. In Fig. 10-10, the curves of the various power components _ EE versus pump head H p are presented in horsepower [HP] units. The frictional horsepower [FHP] does not vary appreciably with increased head; it is the horizontal line in Fig. 10- 10. The other useful component is the hydraulic horsepower [HHP]. This power component is directly proportional to the pump head and is shown as a straight line with a positive slope. This component is added to constant FHP, resulting in the total brake horsepower [BHP]. The BHP curve in Fig. 10-10 is a straight line, and at zero pump head there is still a definite brake horsepower required, due to friction in the pump. In a gear pump, the friction horsepower, FHP, is a function of the speed and the viscosity of the fluid, but not of the head of the pump. Because FHP is nearly constant versus the head, it is a straight horizontal line in Fig. 10-10. On the other hand, HHP is an increasing linear function of H p (see equation for hydraulic power). (This is approximation, since Q is not constant because it is reduced by the slip.) The sum of the friction power and the hydraulic power is the brake horsepower. The brake horsepower increases nearly linearly versus H p , as shown in Fig. 10-10. Since FHP is constant, the efficiency Z is an increasing function versus H p . The result is that gear pumps have a higher efficiency at high heads. FIG. 10-10 Power and efficiency characteristics of the gear pump. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. The head of the pump, H p , generated by the pump is equal to the head losses in a closed-loop piping system, such as in the hydrostatic bearing system. If the fluid is transferred from one tank to another at higher elevation, the head of the pump is equal to the head losses in the piping system plus the height difference DZ. The head of the pump, H p , is the difference of the heads between the two points of discharge and suction: H p ¼ H d À H s ð10-50Þ Pump head units are of length (m, ft). Head is calculated from the Bernoulli equation. The expression for discharge and suction heads are: H d ¼ p d g þ V 2 d 2g þ Z d ð10-51Þ H s ¼ p s g þ V 2 s 2g þ Z s ð10-52Þ where H d ¼ head at discharge side of pump ðoutletÞ H s ¼ head at suction side of pump ðinletÞ p d ¼ pressure measured at discharge side of pump ðoutletÞ p s ¼ pressure measured at suction side of pump ðinletÞ g ¼ specific weight of fluid ðfor water; g ¼ 9:8 Â10 3 ½N=m 3 Þ V ¼ fluid velocity g ¼ gravitational acceleration Z ¼ height The pump head, H p , is the difference between the discharge head and suction head. In a closed loop, H p is equal to the head loss in the loop. The expression for the pump head is H p ¼ p d À p s g þ V 2 d À V 2 s 2g þðZ d À Z s Þð10-53Þ The velocity of the fluid in the discharge and suction can be determined from the rate of flow and the inside diameter of the pipes. In most gear pumps, the pipe inside the diameters on the discharge and suction sides are equal. In turn, the discharge velocity is equal to that of the suction. Also, there is no significant difference in height between the discharge and suction. In such cases, the last two Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. terms can be omitted, and the pump is determined by a simplified equation that considers only the pressure difference: H p ¼ p d À p s g ð10-54Þ 10.14.2 Hydraulic Power The hydraulic power of a pump, _ EE h , is proportional to the pump head, H p , according to the following equation: _ EE h ¼ QgH p ð10-55Þ The SI units for Eq. (10-53), (10-54), and (10-55) are _ EE h ½w Q ½m 3 =s g ½N=m 3  H ½m p ½N=m 2 or pascals] The pump efficiency is the ratio of hydraulic power to break power: Z ¼ _ EE h _ EE b ð10-56Þ The conversion to horsepower units is 1 HP ¼ 745.7 W. In most gear pumps, the inlet and outlet pipes have the same diameter and the inlet and outlet velocities are equal. In Imperial units, the hydraulic horsepower (HHP) is given by HHP ¼ QDp 1714 ð10-57Þ Here, the units are as follows: Dp ½psi¼gðH p À H s Þ and Q ½GPM In imperial units, the efficiency of the pump is: Z ¼ HHP BHP ð10-58Þ The BHP can be measured by means of a motor dynamometer. If we are interested in the efficiency of the complete system of motor and pump, the input power is measured by the electrical power, consumed by the electric motor Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. that drives the pump (using a wattmeter). The horsepower lost on friction in the pump, FHP, cannot be measured but can be determined from FHP ¼ BHP ÀHHP ð10-59Þ 10.15 FLOW DIVIDERS Using many hydraulic pumps for feeding the large number of recesses of hydrostatic pads in machines is not practical. A simple solution of this problem is to use constant pressure and flow restrictors. However, flow restrictors increase the power losses in the system. Therefore, this method can be applied only with small machines or machines that are operating for short periods, and the saving in the initial cost of the machine is more important than the long-term power losses. Another solution to this problem is to use flow dividers. Flow dividers are also used for distributing small, constant flow rates of lubricant to rolling-element bearings. It is designed to divide the constant flow rate of one hydraulic pump into several constant flow rates. In hydrostatic pads, each recess is fed at a constant flow rate from the flow divider. The advantage of using flow dividers is that only one hydraulic pump is needed for many pads and a large number of recesses. The design concept of a flow divider is to use the hydraulic power of the main pump to activate many small pistons that act as positive-displacement pumps (constant-flow-rate pumps), and thus the flow of one hydraulic pump is divided into many constant flow rates. A photo of a flow divider is shown in Fig. 10-11a. Figure 10-11b presents a cross section of a flow divider made up of many rectangular blocks connected together for dividing the flow for feeding a large number of bearings. The contact between the blocks is sealed by O-rings. The intricate path of the inlet and outlet of one piston is shown in this drawing. For a large number of bearings, the flow divider outlets are divided again. An example of such a combination is shown in Fig. 10-12. 10.16 CASE STUDY: HYDROSTATIC SHOE PADS IN LARGE ROTARY MILLS Size reduction is an important part of the process of the enrichment of ores. Ball- and-rod rotary mills are widely used for grinding ores before the enrichment process in the mines. Additional applications include the reduction of raw- material particle size in cement plants and pulverizing coal in power stations. In rotary mills, friction and centrifugal forces lift the material and heavy balls against the rotating cylindrical internal shell and liners of the mill, until they fall down by gravity. The heavy balls fall on the material, and reduce the particle size by impact. For this operation, the rotation speed of the mills must be slow, Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. For successful operation, rolling bearings as well as hydrodynamic bearings require precision machining. For rolling-element bearings, any out-of-roundness of the trunnion or the bearing housing would deform the inner or outer rings of the rolling bearing. This undesired deformation would adversely affect the performance of the bearing and significantly reduce its life. Moreover, large- diameter rolling-element bearings are expensive in comparison to other alter- natives. For a hydrodynamic bearing, the bearing and journal must be accurately round and fitted together for sustained performance of a full hydrodynamic fluid film. Any out-of-roundness in the bearing or journal results in a direct contact and excessive wear. In addition, rotary mills rotate at relatively low speed, which is insufficient for building up a fluid film of sufficient thickness to support the large trunnion. An alternative that is often selected is the hydrostatic bearing system. As mentioned earlier, hydrostatic bearings can operate with a thicker fluid film and therefore are less sensitive to manufacturing errors and elastic deformation. 10.16.1 Self- Aligning and Self-Adjusted Hydrostatic Shoe Pads A working solution to the aforementioned problems of large bearings in rotary mills has been in practice for many years, patented by Arsenius, from SKF (see Arsenius and Goran, 1973) and Trygg and McIntyre (1982). It is in the form of self-aligning hydrostatic shoe pads that support the trunnion as shown in Fig. 10- 13. These shoe pads can pivot to compensate for aligning errors, in all directions. Hydrostatic pads that pivot on a sphere for universal self-aligning are also used in small bearings. When two hydrostatic shoe pads support a circular trunnion (Fig. 10-13a), the load is distributed evenly between these two pads. In fact, the location of the two pads determines the location of the trunnion center. However, whenever three or more pads are supporting the trunnion, the load is no longer distributed evenly, and the design must include radial adjustment of the pads, as shown in Fig. 10- 13b. The load capacity is inversely proportional to h 3 0 , where h 0 is the radial clearance between the face of the hydrostatic pad and the trunnion running surface. Due to limitations in precision in the mounting of the pads, the clearance h 0 is never equal in all the pads. Therefore, the design must include adjustment of the pad height to ensure that the load is distributed evenly among all the pads. Adjustment is required only for the extra pads above the first two pads, which do not need adjustment. Therefore, each of the extra hydrostatic pads must be designed to move automatically in the radial direction of the trunnion until the load is divided evenly among all pads. This way, the pads always keep a constant clearance from the trunnion surface. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. Since the pads are self-aligning and self-adjusting, the foundation’s construction does not have to be precise, and a relatively low-cost welded structure can be used as a bed to support the set of hydrostatic pads. The design concept is as follows: The surface of the pads is designed with the same radius of curvature as the trunnion outside surface. The clearance is adjusted, by pad radial displacement, which requires additional lower piston and hydraulic oil pressure for radial displacement. Explanation of the control of the pad radial motion will follow shortly. If sufficient constant-flow-rate of oil is fed into each pad from external pumps, it is then possible to build up appropriate pressure in each of the pad FIG. 10-13 Hydrostatic shoe pads: (c) Self-aligning ball support with pressure relief. (d) Master and slave shoe pads. (From Trygg and McIntyre (1982), reprinted with permission from CIM Bulletin.) Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. recesses for separating completely the mating surfaces by means of a thin oil film. A major advantage of hydrostatic pads is that the fluid film thickness is independent of the trunnion speed. The fluid film is formed when the trunnion is stationary or rotating, and the mating surfaces are completely separated by oil film during start-up as well as during steady operation. All pads have universal angular self-aligning (see Fig. 10-13b). This is achieved by supporting each pad on a sphere (hard metal ball), as shown in Fig. 10-13c, where the pad has a spheroid recess with its center coinciding with the sphere center. In this way, it can tilt in all directions, and errors in alignment with the trunnion outside surface are compensated. However, the spheroid pivot arrangement under high load has a relatively high friction torque. This friction torque, combined with the inertia of the pad, would result in slow movement and slow reaction to misalignment. In fact, in large hydrostatic pads the reaction is too slow to adequately compensate the variable misalignment during the rotation of the trunnion. To improve the self-alignment performance, part of the load on the metal ball is relieved by hydrostatic pressure. The bottom part of the pad has been designed as a piston and is pressurized by oil pressure. The oil pressure relieves a portion of the load on the metal ball, and in turn the undesired friction torque is significantly reduced, as shown schematically in Fig. 10-13c. In Fig. 10-13b, the radial positions of two inner pads determine the location of the axis of rotation of the trunnion; therefore, these two pads do not require radial adjustments, and they are referred to as master shoe pads. Any additional shoe pads require radial adjustment and are referred to as slave shoes. The design of the master and slave shoe pads with the hydraulic connections is shown in Fig. 10-13d. In the slave shoe, there is radial adjustment of the pad clearance with the trunnion surface. The radial adjustment requires an additional lower piston, as shown in Fig. 10-13d. The radial motion of the lower piston is by means of hydrostatic oil pressure. The oil is connected by an additional duct to the space beneath the lower pad. There is a hydraulic duct connection, and the pressures are equalized in the two spaces below the two pads and in the pad recess (in contact with the trunnion surface) of the master and slave shoes. Since there is a constant flow rate, this equal pressure is a load-dependent pressure. If the area of the lower piston is larger than the effective pad area, the lower piston will push the piston and shoe pad in the radial direction (in the slave shoe) and adjust the radial clearance with the trunnion until equal load capacity is reached in all pads. The recess pressure is a function of the load and the pad effective area. As the load increases, the film thickness diminishes and the pressure rises. It is desirable to limit the pressure and the size of the pad. This can be achieved by increasing the number of pads. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. [...]... elliptical shape In addition to shorter trunnion length, this design eliminates expensive castings followed by expensive precise machining, which are involved in the manufacturing process of the conventional hydrodynamic design In this case, the casting can be replaced by a relatively low-cost welded construction The hydrostatic shoes are relatively small, and their machining cost is much lower in comparison... bearing failure or excessive wear The concept of this design is to apply self-aligning hydrostatic shoes, preferably four shoes for each bearing One important advantage of the design is that the length of the trunnion is much shorter in comparison to that in hydrodynamic bearing design The shortening of the trunnion results in several advantages 1 2 3 4 5 It reduces the weight of the trunnion and thus... these self-adjusting hydrostatic pads claim that there are major advantages in this design: It made it possible to significantly reduce the cost and to reduce the weight of the bearing and trunnion as well as the length of the complete mill in comparison to hydrodynamic bearings Most important, it improved the bearing performance, namely, it reduced significantly the probability of bearing failure or excessive... increase power losses The preferred design is to use one central pump with flow dividers A standby pump in parallel is usually provided, to prevent loss of production in Copyright 20 03 by Marcel Dekker, Inc All Rights Reserved drops, the mill rotation is stopped Temperature monitoring is included to protect against overheating of the oil 10.16 .2 Advantages of Self-Aligning Hydrostatic Shoe Pads Several... comparison to that of large bearings Moreover, the hydrostatic design operates with a thicker oil film and provides self-aligning bearings in all directions These improvements prevent unexpected failures due to excessive wear or seizure This aspect is important because of the high cost involved in rotary mill repair as well as loss of production Copyright 20 03 by Marcel Dekker, Inc All Rights Reserved ... simplifies the feed into and from the mill It reduces the total length of the mill and its weight, resulting in reduced bending moment, and thus the mill can be designed to be lighter It will, in fact, reduce the cost of the materials and labor for construction of the mill It reduces the elastic deformation, in the radial direction, of the trunnion Stiffer trunnion has significant advantages in bearing operation,... recesses (both are supplied by one pump) In this way, the loaddependent pressure in the piston cylinder of the slave shoe will be the same as that in the master shoe, resulting in equal load capacity of each shoe pad at all times This design can operate with certain deviation from roundness of the trunnion For example, if there is a depression (reduced radius) in the trunnion surface, when this depression... four or six shoes are used The combination of a master shoe and slave shoe operates as follows: The effective areas of the two pads are equal If the clearance is the same in both shoes, the hydrostatic recess pressures must also be equal in the two pads In this case, the load on both shoe pads is equalized There is hydraulic connection between the lower piston cylinders of the master and slave recesses... and lift the piston until there are equal recess pressures and load capacity in the two pads Similarly, if there is a bulge (increased radius) on the surface of the trunnion, the process is reversed In this way, the radial loads on the master and slave shoes are automatically controlled to be equal (with a minimal delay time) In conclusion, the clearances between the pads and trunnion surface are automatically... perfectly round Cross-sectional views of the slave and master shoes and an isometric view of the slave shoe are shown in Fig 10- 14 In the master pad, there is one oil inlet and there is a hydraulic connection to the slave shoe The pad recess of the slave and master shoes is of a unique design For stable operation, it is important that the pad angular misalignment be immediately corrected Each pad has . slow, Copyright 20 03 by Marcel Dekker, Inc. All Rights Reserved. For successful operation, rolling bearings as well as hydrodynamic bearings require precision machining. For rolling-element bearings, any. Cross- stiffness components can result in bearing instability, which was discussed in Chapter 9. 10. 12 APPLICATIONS An interesting application is the hydrostatic pad in machine tool screw drives (Rumberger. the trunnion or the bearing housing would deform the inner or outer rings of the rolling bearing. This undesired deformation would adversely affect the performance of the bearing and significantly