230 Rules of Thumb for Mechanical Engineers 113 Po=(=) 6PE*2 These equations describe Hertz contact for spheres. No- tice that these results are nonlinear and that the maximum pressure increases as the load is raised to the % power. The corresponding surface stresses can be calculated as: Figure 5. Hertz contact of spheres. - uzl + iiz2 = 6 - -1_r2 2R [I - (1- r2/ a2 )312] I-2v a2 where 1/R = l/R1 + 1/R2. That is, a pressure distribution 5, = - po which gives a constant plus 1.2 term is needed to cancel the potential interpenetration of the spheres. Comparing Equa- tions 6 and 7 reveals that the contacting spheres induce an ellipsoidal pressure distribution and n: PO (2a2 - r2) = 6 - 1 r2 4 aE* 2R + 2v (1 - r2/ a2)112} (12) (13) 2 2 112 - 6, =-Po (1-r /a ) inside the contact patch (r < a) and - - (1 - 2v)a2 (J z-0 - This equation must be valid for any r < a requiring: e -Po 3r2 a= IT POR 2 E* and Global equilibrium requires: Finally: outside the contact patch (r > a). These stresses are shown in Figure 6. Notice that the radial stress is tensile outside the circle and that it reaches its maximum value at r = a. This is the maximum tensile stress in the whole body. 0.25 0.501 l -2.0 -1.0 0.0 1 .o 2.0 Position @/a) Figure 6. Surface stresses induced by circular point (*) contact (v = 0.3). Tribology 231 The stresses along the z-axis (r = 0) can be calculated by first evaluating the stress due to a ring of point force along r = r and integrating from r = 0 to r = a. For example: 6, (O,z)= Making the substitution a2 - 1.2 = u2 and using integration leads to: 0,(o,z)=0~(o,z)=po + } 1 a2 2 z2+a2 Elliptical Contact When solids having unequal curvatures in two directions are brought into contact, the contours of constant separa- tion are ellipses. When the principal curvatures of the two bodies are aligned, the axes of the ellipse correspond to these directions. Placing the x and y axes in the directions of prin- cipal curvature, the equation comparable to Equation 7 is: - 1 1 u,1 +Ez2 =6 x2 2R’ 2R” y2 where Symmetry dictates that the maximum shear stress in the body occurs along r = 0. Manipulation of the above equa- tions leads to: 3 a’ =Po (1+v) 1 tan-’- [ ( E :I 2zz+a2] For v = 0.3, the maximum shear stress is about 0.31~~ and occurs at a depth of approximately z = 0.48a. The stresses are plotted for v = 0.3 in Figure 7. and Ri are the curvatures in the x direction and Rr are the curvatures in the y direction. The contact area is an ellipse, and the resulting pressure distribution is semi-ellipsoidal given by: x2 y2 P(X¶ Y)“ PoJ - 2- 2 The actual calculation of a and b is cumbersome. Howev- er, for mildly elliptical contacts (Greenwood [5]), the con- tact can be approximated as circular with: stress (dpJ -1 .o -0.5 0.0 0.5 Re = JR‘ R“ (18) with 6 and po given by Equations 9 and 10. R2, respectively, the effective radii are given by: Note that for contact of crossed cylinders of radii R1 and 111 +- R’ R, = -=- 111 -=-+- R” = R, Figure 7. Subsurface stresses induced by circular point contact (v = 0.3). so that if Rl = R2, the contact patch is circular and the equa- tions of the previous section (“Contact of Spheres”) hold. 232 Rules of Thumb for Mechanical Engineers Effect of Friction on Contact Stress If contacting cylinders are loaded tangentially as well as normally and caused to slide over each other, then a shear stress will exist at the surface. This shear stress is equal in magnitude to the coefficient of friction p multiplied by the normal contact pressure. The shear stress acts to oppose the tangential motion of each cylinder. Thus, if the top cylin- der moves from the left to the right relative to the bottom cylinder, then the tangential traction on the bottom cylin- der is given by: The Westergaard stress function that yields these surface tractions is: as can be verified using the equations of Westergaard [ 171. Contours of the in-plane maximum shear stress for the combined shear and normal tractions are shown in Figures 8 and 9 for p = 0.1 and p = 0.4, respectively. The indenter is sliding over the surface from left to right. Notice that in- creasing the coefficient of friction increases the maximum shear stress while changing its location to be nearer the sur- face and off the z-axis towards the leading edge of contact. For the higher value of p, T,, occurs on the surface. Tangentially loading spheres so that they slide with re- spect to each other has similar effects. The subsurface maximum shear stress is increased and moved closer to the surface toward the leading edge of contact. In addition, the surface tensile stress is decreased at the leading edge of con- tact and increased at the trailing edge of contact. Position (x/a) -2 -1 0 1 2 0.0 0.5 31.0 h 3.5 \ - Level tauma A 0.3 9 0.27 8 0.24 7 0.21 6 0.18 5 0.15 4 0.12 3 0.09 2 0.06 1 0.03 Figure 8. Stress contours of z,, /po for frictional Hertz contact with p = 0.1. The load is sliding from left to right. Position (x/a) -2 -1 0 1 2 0.0 0.5 $1.0 v El .5 2.0 2.5 3.0 0) - Level tauma 9 0.36 8 0.32 7 0.28 6 0.24 5 0.2 4 0.16 3 0.12 2 0.08 1 0.04 3.5 t \ Figure 9. Stress contours of z,, /po for frictional Hertz contact with p = 0.4. The load is sliding from left to right. Yield and Shakedown Criteria for Contacts The maximum shear stress values illustrated above can be used as initial yield criteria for contacts. However, rolling contacts that are loaded above the elastic limit can sometimes develop residual stresses in such a way that the body reaches a state of elastic shakedown. Elastic shakedown implies that there is no repeated plastic defor- mation and the resulting deleterious fatigue effects. Shake- down occurs when the sum of the residual stresses and the live stresses do not violate yield anywhere. Whenever the loads are such that it is possible for such a residual stress state to be developed, then the body does shakedown. This idea makes it possible for shakedown limits to be calculated. As an example, for two-dirnen- sional contacts without friction, the maximum shear stress Tribology 233 is about T,, = 0.3 po, implying that initial yield occurs when 0.3% = k or po = 3.3k where k is the yield stress in shear. However, it can be shown that for po < 4k, elastic shake down is reached so that subsurface plastic deformation does not continue throughout life. Recalling that the max- hum contact pressure inmases with the square root of load makes this shakedown effect very important. More details on the concept of residual stress-induced shakedown and additional effects that can lead to shakedown such as strajn hardening can be found in Johnson [ 101. The discussion of surface roughness and the techniques used to quantify it are discussed here. Some general work in this area includes Thomas [ 151 and Greenwood [6]. The length scale of interest is smaller than that consid- ered in the Hertz contact calculations in which contact stresses are calculated to discover what is happening inside the body such as the location of first yield or cracking. Now we are going to focus on the surface of the bodies. One con- venient manner of characterizing this surface is to measure its surface roughness. However, it is important to note that mechanical properties are also different near the surface than they are in the bulk of the material. Table 1 Surface Roughness for Various Finishing Processes ProcessS RMS Roughness (Microns) Grinding Fine grinding Polishing Sum finishing 0.8 - 0.4 0.25 0.1 0.025 - 0.01 Definition of Surface Roughness consider a surface profile whose heiit is given as a func- The root-mean-square roughness or standard deviation tion of position z(x). The datum is chosen so that: is defined by: I,Lz(x)dx = 0 The average roughness Ra is defmed as: The RMS roughness is always greater than the average roughness so that: where the absolute value implies that peaks and valleys have the same contribution. and 234 Rules of Thumb for Mechanical Engineers CJ= 1.2Ra for most surfaces. Qpical RMS values for finishing process- es are given in Table 1. Of course, widely different surfaces could give the same R, and RMS values. The type of statistical quantity needed will depend on the application. One quantity that is used in practice is the bearing area curve which is a plot of the sur- face area of the surface as a function of height. If the surface does not deform during contact, then the bearing area curve is the relationship between actual area of contact and approach of the two surfaces. This concept leads to discussion of con- tact of actual rough surface contacts. (See also Figure 10.) Distance along surface (mm) Figure 10. A typical rough surface. Contact of Rough Surfaces Much can be gleaned from consideration of the contact of rough surfaces in which it is found that the real area of contact is much less than the apparent area of contact as il- lustrated by the contact pressure distributions shown in Figure 11. The smooth solid line is the contact pressure for contact with an equivalent smooth surface, while the line showing pressure peaks is the contact pressure for contact with a model periodic rough surface. The dashed line is the moving average of the rough surface contact pressure, and it is very similar to that of the Hertz contact with the smooth surface. Thus, the subsurface stresses are similar for the smooth and rough surface contacts, and yield and plas- tic flow beneath the Surface iS not Strongly dependent on the surface roughness. This conclusion is the reason that many hertzian contact designs are based on calculated smooth surface pressure distributions with an accompanying call-out on surface roughness. Position along surface, x/a Figure 1 1, Line contact pressure distribution for periodic rough surface. Life Factors The fact that some of the information contained in the rough surface stress field can be inferred from the come sponding Hertz stresses and the surface profile has lead to the development of life factors for rolling element bearings. In these life factor equations, there are terms that account for near-surface metallurgy, surface roughness, lubrica- tion, as well as additional effects. These life factors were summarized recently by longtime practitioners in the bear- ing design field. This sum can be found in Zaretsky [ 181 and is written in a format that can be applied easily by the practicing engineer. Tribology 236 Consider a block of weight W resting on an inclined plane. As the plane is tilted to an angle with the horizontal 8, the weight can be resolved into force components per- pendicular to the plane N and parallel to the plane F. If 8 is less than a certain value, say, e,, the block does not move. It is inferred that the plane resists the motion of the block that is driven by the component of the weight parallel to the plane. The force resisting the motion is due tofriction be- tween the plane and block. As 8 is increased past e, the block begins to move because the tangential force due to the weight F overcomes the frictional force. It can be shown ex- perimentally that es is approximately independent of the size, shape, and weight of the block. The ratio of F to N at slid- ing is tan 8, = p, which is called the coeflcieat offriction. The frictional resistance to motion is equal to the coefficient of fiction times the compressive normal force between two bodies. The coefficient of friction depends on the two materials and, in general, 0.05 e p c 00. The idea that the resistance to motion caused by friction is F = pN is called the Amontons-Coulomb Law of sliding friction. This law is not like Newton’s Laws such as F = ma. The more we study the friction law, the more complex it be- comes. In fact, we cannot fully explain the friction law as evidenced by the fact that it is difficult or even impossible to estimate p for two materials without performing an ex- periment. As a point of reference, p is tabulated for sever- al everyday circumstances in Table 2. As noted in the rough surface contact section, the real area of contact is invariably smaller than the apparent area of con- tact. The most common model of friction is based on as- suming that the patches of real contact area form junctions in which the two bodies adhere to each other. The resistance to sliding, or friction, is due to these junctions. Assuming that the real area of contact is equal to the ap- plied load divided by the hardness of the softer material, and that the shear stress required to break the junctions is the yield stress in shear of the weaker material, leads to: where p is the coefficient of friction, zy is the yield stress in shear, and H is the hardness. For most materials, the hard- ness is about three times the yield stress in tension and the Tresca yield conditions assume that the yield stress in shear is about half of the yield stress in tension. Substitut- ing leads to p =5 1/6. The simple adhesive law of friction is attractive in that it is independent of the shape of the bodies and leads to the force acting opposite the direction of motion, resulting in energy dissipation. Its weakness is that it incorrectly pre- dicts that p is always equal to 1/6. This can be explained in part by the effect of large localized contact pressure on material ppedes and the contribution of plowing and ther- mal effects. Table 2 Typical Coefficients of Friction Physical Situation CI Rubber on cardboard (try it) 0.5-0.8 Brake material on brake drum 1.2 Dry tire on dry mad 1 Wet tire on wet mad 0.2 Copper on steel, dry 0.7 Ice on wood 0.05 Source: Bowden p] The rubbing together of two bodies can cause material re- moval or weight loss from one or both of the bodies. This phenomenon is called wea,: Wear is a very complex process. It is much more complex even than friction. The complex- ity of wear is exemplified in Table 3 where wear rate is shown with p for several material combinations. Radical dif- ferences in wear rate occur over relatively small ranges of the coefficient of friction. Note that wear is calculated from wear rate by multiplying by the distance traveled. There are several standard wear configurations that can be used to obtain wear coefficients and compare material choices for a particular design. A primary source for this information is the Wear Contr-ol Handbook by Peterson and Winer [ 121. 236 Rules of Thumb for Mechanical Engineers Table 3 Friction and Wear from Pin on Ring Tests Materials P Wear rate - x lwl* 1 Mild steel on mild steel 0.62 157,000 2 6W40 leaded brass 0.24 24,000 3 PTFE (Teflon) 0.1 8 2,000 4 Stellite 0.60 320 5 Ferritic stainless steel 0.53 270 6 Polyethylene 0.65 30 7 Tungsten carbide on itself 0.35 2 Rings are hardened tool steel except in tests 1 and 7 (Halling m). The load is 400 g and the speed is 180 cmlsec. cm8 cm As in friction, the most prominent wear mechanism is due to adhesion. Assuming that some of the real contact area junctions fail just below the surface leading to a wear par- ticle results in: ws V=k- H where V is the volume of material removed, W is the nor- mal load, s is the horizontal distance traveled, H is the hardness of the softer material, and k is the dimensionless wear coefficient repsenting the probability that a junction will form a wear particle. In this form, wear coefficients vary from - I@ to - There is a wealth of information on wear coefficients published biannually in the proceedings of the International Conference on Wear of Materials, which is sponsored by the American Society of Mechani- cal Engineers. Lubrication is the effect of a third body on the contact- ing bodies. The third body may be a lubricating oil ur a chem- ically formed layer fiom one or both of the contacting bod- ies (oxides). In general, the coefficient of friction in the presence of lubrication is reduced so that 0.001 e p e 0.1. Lubrication is understood to fall into three regimes de- pendent on the component configuration, load, and speed. Under relatively modest loads in conformal contacts such as a journal bearings, moderate pressures exist and the de formation of the solid components does not have a large ef- fect on the lubricant pressure distribution. The bodies are far apart and wear is insignificant. This regime is known as hydrodynamic lubrication. As loads are increased and the geometry is noncon- forming, such as in roller bearings, the lubricant pressure greatly increases and the elastic deformation of the solid components plays a role in lubricant pressure. This regime is known as elastohydrodynamic lubrication, provided the lubrication film thickness is greater than about three times the surface roughness. Once the film thickness gets small- er than this, the solid bodies touch at isolated patches in a mechanism known as boundary lubrication. Here, the in- tense pressures and temperatures make the chemistry of the lubricant surface interaction important. The lubricant film thickness is strongly dependent on lu- bricant viscosity at both high and low temperatures. Nondi- mensional formulas are available for designers to use in dis- tinguishing the regimes of lubrication. Once the regime of lubrication is determined, additional formulas can be used to estimate the maximum contact pressue as well as the min- imum film thickness. The maximum contact pressure can then be usedin the life factors of Zaretsky [18], and the rnjn- hum film thickness can be used in the consideration of lu- bricant film breakdown. While these formulas are too nu- merous to summarize here, a primary some is Hamrock [8] in which all of the requisite formulas are defined. Tribology 237 1. Bhushan, B. and Gupta, B. K., Handbook of Tribolo- gy: Materials, Coatings, and SurjGace Treatments. New York McGraw-Hill, 199 1. 2. Blau, P. J. (Ed.), Friction, Lubrication, and Wear Tech- nology. ASM Handbook, Vol. 18, ASM, 1992. 3. Bowden, E P. and Tabor, D., ne Friction and Lubri- cation of Solids: Part Z. Oxford Clarendon Press, 1958. 4. Bowden, E P. and Tabor, D., Friction: An Zntmduction to Tribology. Melbourne, FL: Krieger, 1982. 5. Oreenwood, J. A., “A Unified Theory of Surface Rough- ness,” Proaxdm ’ gs of the Royal Society, A393,1984, pp. 6. Greenwood, J. A., “Formulas for Moderately Elliptical Hertzian Contact,” Journal of Tribology, 107(4), 1985, 7. Halling, J. (Ed.), Principles of Tribology. MacMillan Press, Ltd., 1983. 8. Hamrock, B. J., Fundamentals of Fluid Film Lubrica- tion. New York McGraw-Hill, 1994. 9. Hutchings, I. M., T~bology: Friction and Wear of En- gineering Materials. Boca Raton: CRC Press, 1992. 133- 157. pp. 501-504. 10. Johnson, K. L., Contact Mechanics. Cambridge: Cam- bridge, 1985. 11. Jost, P., “Lubrication (Tribology) Education and Re- search,” Technical report, H.M.S.O., 1966. 12. Peterson, M. B. and Wmer, W. 0. (Eds.), Wear Control Handbook. ASME, 1980. 13. Rabinowicz, E., Friction and Wear of Materials. New York: Wiley, 1965. 14. Suh, N. P., Tribophysics. Englewood Cliffs: Prentice- Hall, 1986. 15. Thomas, T. R. (Ed.), Rough Surfaces. London: Long- man, 1982. 16. Timoshenko, S. P. and Goodier, J. N., Theoiy of Elas- ticity, 3rd Ed. New York: McGraw-Hill, 1970. 17. Westergaard, H. M., “Bearing Pressures and Cracks,” Journal of Applied Mechunics, 6(2), A49-A53, 1939. 18. Zaretsky, E. V. (Ed.), STLE Life Factors for Roller Bearings. Society of Tribologists and Lubrication En- gineers, 1992. Lawrence D. Norris, Senior Technical Marketing Engineer-Large Commercial Engines, Allison Engine Company, Rolls-Royce Aerospace Group Vibration Definitions, Terminology, and Symbols 239 Solving the One Degree of Freedom System 243 Solving Multiple Degree of Freedom Systems 245 Vibration Measurements and Instrumentation 246 Table A: Spring Stiffness 250 Table B: Natural Frequencies of Simple Systems 251 Table C: Longitudinal and Torsional Vibration of Uniform Beams 252 Table D: Bending (Transverse) Vibration of Uniform Beams , 253 Table E: Natural Frequencies of Multiple DOF Systems 254 Table F: Planetary Gear Mesh Frequencies 255 Table G: Rolling Element Bearing Frequencies and Bearing Defect Frequencies 256 Table H: General Vibration Diagnostic Frequencies 257 References 258 238 Vibration 239 This chapter presents a brief discussion of mechanical Vibrations and its associated terminology. Its main emphasis is to provide practical “rules of thumb” to help calculate, measure, and analyze vibration frequencies of mechanical systems. Tables are provided with useful formulas for computing the vibration frequencies of common me- chanical systems. Additional tables are provided for use with vibration measurements and instrumentation. A num- ber of well-known references are also listed at the end of the chapter, and can be referred to when additional infor- mation is required. Vibration Definitions, Terminoloay, and Symbols Beating: A vibration (and acoustic) phenomenon that occurs when two harmonic motions (XI and x2) of the same amplitude (X), but of slightly different frequencies are ap- plied to a mechanical system: XI = x cos at The resultant motion of the mechanical system will be the superposition of the two input vibrations x1 and x2, which simplifies to: x=2xcos - tcos a+- t (3 ( :) This vibration is called the beating phenomenon, and is illustrated in Figure 1. The frequency and period of the beats will be, respectively: Am 27c 2x A63 fb = - cycles/ sec Tb = - SeC A common example of beating vibration occurs in a twin engine aircraft. Whenever the speed of one engine varies slightly from the other, a person can easily feel the beating in the aircraft’s structure, and hear the vibration acoustically. Critical speeds: A term used to describe resonance points (speeds) for rotating shafts, rotors, or disks. For ex- ample, the critical speed of a turbine rotor occurs when the rotational speed coincides with one of the rotor’s nat- ural frequencies. VV vv V Figure 1. Beating phenomenon. Damped natural frequency (q or fd): The inherent fre- quency of a mechanical system with viscous damping (friction) under free, unforced vibration. Damping de- creases the system’s natural frequency and causes vibratory motion to decay over time. A system’s damped and un- damped natural frequencies are related by: Damping (c): Damping dissipates energy and thereby “damps” the response of a mechanical system, causing vi- bratory motion to decay over time. Damping can result from fluid or air resistance to a system’s motion, or from friction between sliding surfaces. Damping force is usually pro- portional to the velocity of the system: F = ai, where c is the damping coeficient, and typically has units of lb-sec/in or N-sec/m. Damping ratlo (6): The damped natural frequency is related to a system’s undamped natural frequency by the follow- ing formula: a, =and- The damping ratio (<) determines the rate of decay in the vibration of the mechanical system. No vibratory oscilla- tion will exist in a system that is overdamped (< > 1 .O) or [...]... S., vibrationfor Engineers Upper Saddle River, NJ: Prentice-Hall, 199 2 4 Thomson, W,T., neory of vibration with Applications Upper Saddle River, NJ: Prentice-Hall, 197 2 5 Thomson, W T., “Vibration” in StandardHapldbookfor Mechanical Engineers, T Baumeister and L S Marks (Editors) New York McGraw-Hill, 196 7 6 Meirovich, L., Analytical Methods in vibration New York: The Macmillan Co., 196 7 Solution Methods... 260 260 262 Polymers Ceramics Mechanical Testing Tensile Testing Fatigue Testing Hardness Testing Creep and Stress Rupture Testing 262 264 265 266 268 2 69 270 273 276 2 79 Forming Casting Case Studies Failure Analysis Corrosion References 2 59 281 284 284 284 285 286 287 288 2 89 290 290 291 292 ... Interrelationship between the phase angle of displacement, velocity, and acceleration [9] (Reprinted by permission of the Institution of Diagnostic Engineers. ) 242 Rules of Thumb for Mechanical Engineers Resonance: When the frequency of the excitation force (forcing function) is equal to or very close to the natural frequency of a mechanical system, the system is excited into resonance During resonance, vibration... 2N,3N ttN NSFor 2SF, 3N, 6 N SN 2s F either slv or FNf 2F, 2K N, 2N, 3N nN N,2 N , 3 N 17N N 2N 0 4 5N NN N,2N,3 N ttN N N . material choices for a particular design. A primary source for this information is the Wear Contr-ol Handbook by Peterson and Winer [ 121. 236 Rules of Thumb for Mechanical Engineers Table. acceleration [9] . (Reprinted by permission of the Institution of Diagnostic Engineers. ) 242 Rules of Thumb for Mechanical Engineers Resonance: When the frequency of the excitation force. Applied Mechunics, 6(2), A 49- A53, 193 9. 18. Zaretsky, E. V. (Ed.), STLE Life Factors for Roller Bearings. Society of Tribologists and Lubrication En- gineers, 199 2. Lawrence D. Norris,