Lecture 6 Friction. Friction Phenomena Types of Friction

Transcription

1 Leture 6 Frition Tangential fores generated between ontating surfaes are alled frition fores and our to soe degree in the interation between all real surfaes. whenever a tenden eists for one ontating surfae to slide along another surfae, the frition fores developed are alwas in a diretion to oppose this tenden In soe tpes of ahines and proesses we want to iniize the retarding effet of frition fores. Eaples are bearings of all tpes, power srews, gears, the flow of fluids in pipes, and the propulsion of airraft and issiles through the atosphere. In other situations we wish to aiize the effets of frition, as in brakes, luthes, belt drives, and wedges. Wheeled vehiles depend on frition for both starting and stopping, and ordinar walking depends on frition between the shoe and the ground. Frition fores are present throughout nature and eist in all ahine so atter how auratel onstruted or arefull lubriated. A ahine or proess in whih frition is sall enough to be negleted is said to be ideal. When frition ust be taken into aount, the ahine or proess is tered real. In all real ases where there is sliding otion between parts, the frition fores result in a loss of energ whih is dissipated in the for of heat. Wear is another effet of frition. Frition Phenoena Tpes of Frition (a) Dr Frition. Dr frition ours when the unlubriated surfaes of two solids are in ontat under a ondition of sliding or a tenden to slide. A frition fore tangent to the surfaes of ontat ours both during the interval leading up to ipending slippage and while slippage takes plae. The diretion of this frition fore alwas opposes the otion or ipending otion. This tpe of frition is also alled Coulob frition. The priniples of dr or Coulob frition were developed largel fro the eperients of Coulob in 1781 and fro the work of Morin fro 1831 to Although we do not et have a oprehensive theor of dr frition, in Art. 6/3 we desribe an analtial odel suffiient to handle the vast ajorit of probles involving dr frition. (b) Fluid Frition. Fluid frition ours when adjaent laers fluid (liquid or gas) are oving at different veloities. This otion auses fritional fores between fluid eleents, and these fores depend on the relative veloit between laers. When there is no relative veloit, there is no fluid frition. Fluid frition depends not onl on the veloit gradients within the fluid but also on the visosit of the fluid, whih is a easure of its resistane to shearing ation between fluid laers. Fluid frition is treated in the stud of fluid ehanis and will not be disussed further in this book. () Internal Frition. Internal frition ours in all solid aterials whih are subjeted to lial loading. For highl elasti aterials the reover fro deforation ours with ver little loss of energ due to internal frition. For aterials whih have low liits of elastiit and whih undergo appreiable plasti deforation during loading, a onsiderable aount of internal frition a aopan this 47

2 deforation. The ehanis of internal frition is assoiated with the ation of shear deforation, whih is disussed in referenes on aterials siene. Dr Frition Mehanis of Dr Frition Consider a solid blok of ass resting on a horizontal surfae, as shown in Fig. 1a.We assue that the ontating surfaes have soe roughness. The eperient involves the appliation of a horizontal fore P whih ontinuousl inreases fro zero to a value suffiient to ove the blok and give it an appreiable veloit. The free-bod diagra of the blok for an value of P is shown in Fig.1b, where the tangential frition fore eerted b the plane on the blok is labeled "F'. This frition fore ating on the bod will alwas be in a diretion to oppose otion or the tenden toward otion of the bod. There is also a noral fore N whih in this ase equals g, and the total fore R eerted b the supporting surfae on the blok is the resultant of N and F. A agnified view of the irregularities of the ating surfaes, Fig.1, helps us to visualize the ehanial ation of frition. Support is neessaril interittent and eists at the ating hups, The diretion of eah of the reations on the blok, R1, R2, R3, et. depends not onl Figure 1 48

3 on the geoetri profile of the irregularities but also on the etent of loal deforation at eah ontat point. The total noral fore N is the su of the n-oponents of the R's, and the total fritional fore F is the su of the t-oponents of the R's. when the surfaes are in relative otion, the ontats are ore nearl along the tops of the hups, and the t-oponents of the R's are saller than when the surfaes are at rest relative to one another. This observation helps to eplain the well known fat that the fore P neessar to aintain otion is generall less than that required to start the blok when the irregularities are ore nearl in esh. If we perfor the eperient and reord the frition fore F as a funtion of P, we obtain the relation shown in Fig. 1d. when P is zero, equilibriu requires that there. be no frition fore. As p is inreased the frition fore ust be equal and opposite to p as long as the blok does not slip. During this period the blok is in equilibriu, and all fores ating on the blok ust satisf the equilibriu equations. Finall, we reah a value of P whih auses the blok to slip and to ove in the diretion of the applied fore. At this sae tie the frition fore dereases slightl and abruptl. It then reains essentiall onstant for a tie but then dereases still ore as the veloit inreases. Stati Frition The region in Fig. 1d up to the point of slippage or ipending otion is alled the range of stati frition, and in this range the value of the frition fore is deterined b the equations of equilibriu. This frition fore a have an value fro zero up to and inluding the aiu value. For a given pair of ating surfaes the eperient shows that this aiu value of stati frition F a is proportional the noral fore N. Thus. we a write.. equ. 1 where μ s is the proportionalit onstant, alled the oeffiient of stati frition. Be aware that Eq. 1 desribes onl the liiting or aiu value Of the stati frition fore and not an lesser value. Thus, the equation applies onl to ases where otion is ipending with the frition fore at its peak value. For a ondition of stati equilibriu when otion is not ipending, the stati frition fore is Kineti Frition After slippage ours, a ondition of kineti frition aopanies the ensuing otion. Kineti frition fore is usuall soewhat less than the aiu stati frition fore. The kineti frition fore F k, is also proportional to the noral fore. Thus... equ. 2 49

4 where μ k is the oeffiient of kineti frition. It follows that μ k is generall less than μ s. As the veloit of the blok inreases, the kineti frition dereases soewhat, and at high veloities, this derease a be signifiant. Coeffiients of frition depend greatl on the eat ondition of the surfaes, as well as on the relative veloit, and are subjet to onsiderable unertaint. Beause of the variabilit of the onditions governing the ation frition, in engineering pratie it is frequentl diffiult to distinguish between a stati and a kineti oeffiient, espeiall in the region of transition between ipending otion and otion. Well-greased srew threads under ild loads, for eaple, often ehibit oparable fritional resistane whether the are on the verge of turning or whether the are in otion. In the engineering literature we frequentl find epressions for aiu stati frition and for kineti frition written sipl as,f=μn. It is understood fro the proble at hand whether aiu stati frition or kineti frition is desribed. Although we will frequentl distinguish between the stati and kineti oeffiients, in other ases no distintion will be ade, and the frition oeffiient will be written sipl as p,. In those ases ou ust deide whih of the frition onditions, aiu stati frition for ipending otion or kineti frition, is involved. We ephasize again that an probles involve a stati frition fore whih is less than the aiu value at ipending otion, and therefore under these onditions the frition relation Eq. 1 annot be used. Figure 1 shows that rough surfaes are ore likel to have larger angles between the reations and the n-diretion than do soother surfaes. Thus, for a pair of ating surfaes, a frition oeffiient reflets the roughness, whih is a geoetri propert of the surfaes. With this geoetri odel of frition, we desribe ating surfaes as "sooth" when the frition fores the an support are negligibl sall. It is eaningless to speak of a oeffiient of frition for a single surfae. Fators Affeting Frition Further eperient shows that the frition fore is essentiall independent of the apparent or projeted area of ontat. The true ontat area is uh saller than the projeted va1ue, sine onl the peaks of the ontating surfae irregularities support the load. Even relativel sall noral loads result in high stresses at these ontat points. As the noral fore inreases, the true ontat area also inreases as the aterial undergoes ielding, rushing, or tearing at the points of ontat. A oprehensive theor of dr frition ust go beond the ehanial eplanation presented here. For eaple, there is evidene that oleular attration a be an iportant ause of frition under onditions where the ating surfaes are in ver lose ontat. Other fators whih influene dr frition are the generation of high loal teperatures and adhesion at ontat points, relative hardness of ating surfaes, and the presene of thin surfae fils of oide, oil, dirt, or other substanes, 50

5 Tpes of Frition Probles We an now reognize tine following three tpes of probles enountered in appliations involving dr frition. The first step in solving a frition proble is to identif its tpe. (1) In the first tpe of proble, the ondition of ipending otion is known to eist. Here a bod whih is in equilibriu is on. the verge of slipping. and the frition fore equals the liiting stati frition F a = μ s N. the equations of equilibriu will, of ourse, also hold. (2) In the seond, tpe of proble, neither the ondition of ipending otion nor the ondition of otion is known to eist. To deterine the atual frition onditions, we first assue stati equilibriu and then solve for the frition fore F neessar for equilibriu. Three outoes are possible: (a) F < (F a = μ s N): Here the frition fore neessar for equilibriu an be supported, and therefore the bod is in stati equilibriu as assued. We ephasize that the atual frition fore F is less than the liiting value F a given b Eq. 1 and that F is deterined solel b the equations of equilibriu. (b) F = (F a = μ s N): Sine the frition fore F is at its aiu value F a otion ipends, as disussed in proble tpe (1). The assuption of stati equilibriu is valid. () F > (F a = μ s N): Clearl this ondition is ipossible, beause the surfaes annot support ore fore than the aiu μ s N. The assuption of equilibriu is therefore invalid, and otion ours. The frition fore F is equal to μ s N fro Eq. 2. (3) In the third tpe of proble, relative otion is known to eist between the ontating surfaes, and thus the kineti oeffiient of frition learl applies. For this proble tpe, Eq.2 alwas gives the kineti frition fore diretl. The foregoing disussion applies to all dr ontating surfaes and to a liited etent, to oving surfaes whih are partiall lubriated. 51

6 Eaples Eaple 1 Deterine the aiu angle θ whih the adjustable inline a have with the horizontal before the blok of ass begins to slip. The oeffiient of stati frition between the blok and the inlined surfae is μ s. Solution The free-bod diagra of the blok shows its weight W = g, the noral fore N, and the frition fore F eerted b the inline on the blok. The frition fore ats in the diretion to oppose the slipping whih would our if no frition were present. Equilibriu in the - and -diretions requires Eaple 2 Deterine the range of values whih the ass o a have so that the 100-kg blok shown in the figure will neither start oving up the plane nor slip down the plane. The oeffiient of stati frition for the ontat surfaes is

7 Eaple 3 The three flat bloks are positioned on the 30 inline as shown, and a fore P parallel to the inline is applied to the iddle blok. The upper blok is prevented fro oving b a wire whih attahes it to the fied support. The oeffiient of stati frition for eah of the three pairs of ating surfae. is shown. Deterine the aiu value whih P a have before an slipping takes plae. 53

8 Probles 54

9 55

10 Sanned b CaSanner

11 Sanned b CaSanner

12 Sanned b CaSanner

13 Sanned b CaSanner

14 Sanned b CaSanner

15 Sanned b CaSanner

16 Sanned b CaSanner

17 Center of Mass and Centroids Center of Mass A bod of ass in equilibriu under the ation of tension in the ord, and resultant W of the gravitational fores ating on all partiles of the bod. -The resultant is ollinear with the ord Suspend the bod at different points -Dotted lines show lines of ation of the resultant fore in eah ase. -These lines of ation will be onurrent at a single point G As long as diensions of the bod are saller opared with those of the earth. - we assue unifor and parallel fore field due to the gravitational attration of the earth. The unique Point G is alled the Center of Gravit of the bod (CG) 1

18 Center of Mass and Centroids Deterination of CG - Appl Priniple of Moents Moent of resultant gravitational fore W about an ais equals su of the oents about the sae ais of the gravitational fores dw ating on all partiles treated as infinitesial eleents. Weight of the bod W = dw Moent of weight of an eleent -ais = dw Su of oents for all eleents of bod = dw Fro Priniple of Moents: dw = ӯ W dw W dw W z zdw W Moent of z ais??? = 0 or, 0 Nuerator of these epressions represents the su of the oents; Produt of W and orresponding oordinate of G represents the oent of the su Moent Priniple. 2

19 Center of Mass and Centroids Deterination of CG Substituting W = g and dw = gd d d In vetor notations: Position vetor for eleental ass: Position vetor for ass enter G: z Densit ρ of a bod = ass per unit volue Mass of a differential eleent of volue dv d = ρdv ρ a not be onstant throughout the bod r r d dv dv dv dv z zd dw W zdv dv r r dw W z i j zk i j zk zdw W 3

20 Center of Mass and Centroids Center of Mass: Following equations independent of g r d r d d z zd (Vetor representation) dv dv dv dv z zdv dv Unique point [= f(ρ)] :: Centre of Mass (CM) CM oinides with CG as long as gravit field is treated as unifor and parallel CG or CM a lie outside the bod 4

21 Center of Mass and Centroids Setr CM alwas lie on a line or a plane of setr in a hoogeneous bod Right Cirular Cone CM on entral ais Half Right Cirular Cone CM on vertial plane of setr Half Ring CM on intersetion of two planes of setr (line AB) 5

22 Center of Mass and Centroids Centroid - Geoetrial propert of a bod - Bod of unifor densit :: Centroid and CM oinide d d z zd dl L dl L z zdl L Lines: Slender rod, Wire Cross-setional area = A ρ and A are onstant over L d = ρadl Centroid and CM are the sae points 6

23 Center of Mass and Centroids Centroid Areas: Bod with sall but onstant thikness t Cross-setional area = A ρ and A are onstant over A d = ρtda Centroid and CM are the sae points d d z zd da A da z A zda A Nuerator = First oents of Area 7

24 Center of Mass and Centroids Centroid dv V Volues: Bod with volue V ρ onstant over V d = ρdv Centroid and CM are the sae point d d z zd dv V dv V z zdv V Nuerator = First oents of Volue 8

25 Center of Mass and Centroid :: Guidelines (a) Eleent Seletion for Integration - Order of Eleent - First order differential eleent preferred over higher order eleent - onl one integration should over the entire figure A = da = ld A = d d V = dv = πr 2 d V = dddz 9

26 Center of Mass and Centroids :: Guidelines (b) Eleent Seletion for Integration - Continuit - Integration of a single eleent over the entire area - Continuous funtion over the entire area Continuit in the epression for the width of the strip Disontinuit in the epression for the height of the strip at = 1 10

27 Center of Mass and Centroids :: Guidelines () Eleent Seletion for Integration - Disarding higher order ters - No error in liits :: Vertial strip of area under the urve da = d :: Ignore 2 nd order triangular area 0.5dd 11

28 Center of Mass and Centroids :: Guidelines (d) Eleent Seletion for Integration - Coordinate sste - Convenient to ath it with the eleent boundar Curvilinear boundar (Retangular Coordinates) Cirular boundar (Polar oordinates) 12

29 Center of Mass and Centroids :: Guidelines (e) Eleent Seletion for Integration - Centroidal oordinate (,, z ) of eleent -,, z to be onsidered for lever ar :: not the oordinates of the area boundar Modified Equations A da A da z z A da V dv V dv z z V dv 13

30 Center of Mass and Centroids :: Guidelines Centroids of Lines, Areas, and Volues 1.Order of Eleent Seleted for Integration 2.Continuit 3.Disarding Higher Order Ters 4.Choie of Coordinates 5.Centroidal Coordinate of Differential Eleents dl L A V da dv dl z L A V da dv z z z z zdl L A V da dv 14

31 Eaple on Centroid :: Cirular Ar Loate the entroid of the irular ar Solution: Polar oordinate sste is better Sine the figure is setri: entroid lies on the ais Differential eleent of ar has length dl = rdө Total length of ar: L = 2αr -oordinate of the entroid of differential eleent: =rosө dl L dl L z zdl L For a sei-irular ar: 2α = π entroid lies at 2r/π 15

32 Eaple on Centroid :: Triangle Loate the entroid of the triangle along h fro the base Solution: da = d ; /(h-) = b/h Total Area, A = ½(bh) A da A da z z A da = 16

33 Center of Mass and Centroids Coposite Bodies and Figures Divide bodies or figures into several parts suh that their ass enters an be onvenientl deterined Use Priniple of Moent for all finite eleents of the bod X Mass Center Coordinates an be written as: X Y Z z -oordinate of the enter of ass of the whole s an be replaed b L s, A s, and V s for lines, areas, and volues 17

34 Centroid of Coposite Bod/Figure Irregular area :: Integration vs Approiate Suation - Area/volue boundar annot be epressed analtiall - Approiate suation instead of integration Divide the area into several strips Area of eah strip = hδ Moent of this area about - and -ais = (hδ) and (hδ) Su of oents for all strips divided b the total area will give orresponding oordinate of the entroid A A Aura a be iproved b reduing the thikness of the strip A A 18

35 Centroid of Coposite Bod/Figure Irregular volue :: Integration vs Approiate Suation - Redue the proble to one of loating the entroid of area - Approiate suation instead of integration Divide the area into several strips Volue of eah strip = AΔ Plot all suh A against. Area under the plotted urve represents volue of whole bod and the -oordinate of the entroid of the area under the urve is given b: A A Aura a be iproved b reduing the width of the strip V V 19

36 Eaple on Centroid of Coposite Figure Loate the entroid of the shaded area Solution: Divide the area into four eleentar shapes: Total Area = A 1 + A 2 - A 3 - A

37 Center of Mass and Centroids Theore of Pappus: Area of Revolution - ethod for alulating surfae area generated b revolving a plane urve about a non-interseting ais in the plane of the urve Surfae Area Area of the ring eleent: iruferene ties dl da = 2π dl Total area, L dl is the -oordinate of the entroid C for the line of length L Generated area is the sae as the lateral area of a right irular linder of length L and radius Theore of Pappus an also be used to deterine entroid of plane urves if area reated b revolving these a non-interseting ais is known A 2 dl A 2 L If area is revolved through an angle θ<2π θ in radians or A L 21