Presentation on theme: "§ 10 1 Applications and Types of Gear Mechanisms § 10 2 Fundamentals of Engagement of Tooth Profiles § 10 3 The Involute and Its Properties § 10 4 Terminology."— Presentation transcript:
§ 10 1 Applications and Types of Gear Mechanisms § 10 2 Fundamentals of Engagement of Tooth Profiles § 10 3 The Involute and Its Properties § 10 4 Terminology and Definition of Gears Chapter 10 Gear Mechanisms § 10 5 Gearing of Involute Spur Gears §10 6 Introduction to Corrected Gear §10 7 Helical Gears for Parallel Shafts §10 8 Worm Gearing §10 9 Bevel Gears
1) Gear mechanisms are widely used in all kinds of machines to transmit motion and power between rotating shafts. 2) Circular gears have constant transmission ratio whereas, for non-circular gears, the ratio varies as the gears rotate. 3) Depending upon the relative shafts positions, circular gear mechanisms can be divided into planar gear mechanisms and spatial gear mechanisms. 4) In this chapter, only circular gears are considered. Introduction Types of Gear Mechanisms § 10 1 § 10 1 Applications and Types of Gear Mechanisms
Conjugate Profiles Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. (i 12 =ω 1 /ω 2 ) Fundamental Law of Gearing ω2ω2 ω1ω1 O2O2 O1O1 2 1 C1C1 C2C2 P n n K v k1 v k2 v k1k2 n The driving pinion rotates clockwise with angular velocity ω 1 while the driven gear rotates counterclockwise with angular velocity ω 2. The common normal n-n intersects the center line O 1 O 2 at point P. the point P is the instant center of velocity of the gears. i 12 ω 1 / ω 2 O 2 P /O 1 P v 12 O 1 P ω 1 = O 2 P ω 2 The transmission ratio: fundamental law of gearing: T he transmission ratio of two meshing gears is inversely proportional to the ratio of two line segments cut from the center line by the common normal of the tooth profiles through the contact point. § 10 2 § 10 2 Fundamentals of Engagement of Tooth Profiles
pitch circle ω2ω2 ω1ω1 r1r1 r2r2 O2O2 O1O1 2 1 C1C1 C2C2 P n n K point P the pitch point. As the center distance O 1 O 2 is constant, the position of the point P must be fixed if a constant transmission ratio i 12 is required. This implies that, wherever the teeth contact, the common normal n-n of the tooth profiles through the contact point must intersect the center line at a fixed point P, if a constant transmission ratio i 12 is required. pitch circleThe loci of P on the motion planes of both gears are called the circles.
Conjugate Profiles Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. For circular gears, the conjugate profiles are those that provide the desired constant transmission ratio. Generally speaking, for any specific tooth profile, we can find its conjugate profile. Theoretically. there is an infinity of pairs of conjugate profiles to produce any specific transmission ratio. Nevertheless, only a few curves have been used as conjugate profiles in practice. Among them, involutes are used most widely since gears using involutes as teeth profiles, or involute gears as they are called, can be manufactured and assembled easily.
θkθk B K Generation of Involute The involuteis the curve generated by any point on a string which is unwrapped from a fixed cylinder. t t Generating line Base circle O A rkrk rbrb Properties of the Involute 2 The normal of an involute at any point is tangent to its base circle. 3 The tangent point B of the generating line with the base circle is the curvature center of the involute at the point K. The length of the segment BK is the radius of curvature of the involute at the point K. 1 AB = BK; 4 The shape of an involute depends only on the radius of its base circle. § 10 3 § 10 3 The Involute and Its Properties
A1A1 B1B1 O1O1 A2A2 K O2O2 B2B2 O3O3 B3B3 5 No involute exists inside its base circle. Equation of the Involute t t B K O A rkrk θkθk rbrb αkαk rbrb αkαk vkvk cosα k = r b / r k tanα k = BK / r b = AB/r b =r b (θ k +α k ) /r b =θ k +α k r k = r b /cosα k θ k = invα k = tanα k - α k
The common normal N 1 N 2 to the meshing involute profiles through their contact point K must be the common tangent to their base circles. The position of this common tangent remains unchanged as both gears rotate, as does the common normal to the involute profiles. This results in a fixed pitch point P. Therefore, according to the fundamental law of gearing mentioned, the transmission ratio will remain constant. 1. The transmission ratio will remain constant. Gearing of Involute Profiles P O2O2 O1O1 ω2ω2 ω1ω1 r b2 N2N2 N1N1 K/K/ C1C1 C2C2 K i 12 =ω 1 /ω 2 = O 2 P/ O 1 P = constant
2. The direction and magnitude of the reaction force does not change The reaction force is exerted along the line of action if there is no friction. As the position of the line of action stays unchanged during motion for an involute gear pair, the direction and magnitude of the reaction force does not change. N 1 N 2 trajectory of contact line of action α pressure angle P O2O2 O1O1 ω2ω2 ω1ω1 r b2 N2N2 N1N1 K/K/ C1C1 C2C2 K α 3. the separability of the center distance in involute gearing O 1 N 1 P O 2 N 2 P As shown in Fig., the transmission ratio: i 12 = ω 1 /ω 2 = O 2 P / O 1 P = r b2 /r b1 A change in centre distance does not therefore affect the constant transmission ratio of an involute gear pair. This property is called the separability of the center distance in involute gearing.
r rbrb O pnpn Tooth depth h= h a +h f haha hfhf h p rara s e sisi eiei pbpb rfrf pipi Terminology and Definition Addendum circle d a r a Dedendum circle d f r f Tooth thickness s i Spacewidth e i Circular pitch p i = s i + e i Reference circle: Between the addendum circle and the dedendum circle, there is an important circle which is called the reference circle. Parameters on the reference circle are standardized and denoted without subscripts, such as d, s, e and p. Addendum h a Dedendum h f Normal pitch p n = p b Base circle d b r b § 10 4 § 10 4 Terminology and Definition of Gears
Basic Parameters Number of teeth z Module m The module m of a gear is introduced on the reference circle as a basic parameter, which is defined as: m=p/π (as πd = zp then d = zp / ) d=mz (3.25) 3.5 (3.75) Second (6.5) 7 9 (11) Series 28 (30) Modules of involute cylindrical gears GB First Series
Sizes of the teeth and gear are proportional to the module m. m=4 Z=16 m=2 Z=16 m=1 Z=16
Pressure angle α The pressure angle α is taken as a basic parameter to determine the base circle. The pressure angle α is also standardized. It is most commonly 20°. Coefficient of addendum: h a * be standardized: h a * 1 Coefficient of bottom clearance : c * be standardized:c* 0.25 z m α h a * c* are the fundamental parameters which determine the size and shape of a standard involute gear. Parameters of Gear Standard gear 1 m,α, h a *, c * are standardized 2 e = s
4 d a d + 2h a z 2 h a * m 5 d f d 2h f z 2 h a * 2c * m 6 s e p / 2 = m / 2 8 h f =(h a * +c * )m 7 h a =h a * m 3 d = mz Parameters of Gear
B. The Rack Characteristics 1) The involute tooth profile becomes a straight line too and the pressure angle remains the same at all points on the tooth profile. 2) The pitch remains unchanged on the reference line, tip line or any other line, i. e. p i = p =πm e s p pbpb A rack can be regarded as a special form of gear with an infinite number of teeth and its center at infinity. The radii of all circles become infinite and all circles become straight lines, such as the reference line, tip line and root line. α α haha hfhf The Rack and Internal Gears
pbpb N α s e h haha hfhf p B 1)The teeth are distributed on the internal surface of a hollow cylinder. The tooth of an internal gear takes the shape of the tooth space of the corresponding external gear, while the tooth space of an internal gear takes the shape of tooth of the corresponding external gear. 2) d f > d > d a d a d - 2h a d f d + 2h f 2. Internal Gears 3) To ensure that the profile of the tooth on the top is an involute curve, d a >d b. Characteristics: O rfrf r rara rbrb
Proper Meshing Conditions for Involute Gears r b2 r2r2 O2O2 ω2 ω2 r b1 r1r1 r1r1 O1O1 ω1ω1 p b2 pb1pb1 r b2 r2r2 O2O2 ω2ω2 p b1 > p b2 p b1 = p b2 P N1N1 N2N2 B2B2 B1B1 O1O1 ω1ω1 pb1pb1 p b2 § 10 5 § 10 5 Gearing of Involute Spur Gears
To maintain the proper meshing of two pairs of profiles at the same time, the normal distances of the teeth on both gears must be the same. p b1 = p b2 m 1 cosα 1 =m 2 cosα 2 m 1 = m 2 = m α 1 =α 2 =α The proper meshing condition for involute gears: the modules and pressure angles of two meshing gears should be the same. r b2 r2r2 O2O2 ω2 ω2 r b1 r1r1 O1O1 ω1ω1 p b2 pb1pb1 P N1N1 N2N2 B2B2 B1B1
To obtain zero backlash of a gear pair: r2r2 O2O2 r1r1 O1O1 ω1ω1 ω2ω2 P N1N1 N2N2 r b1 r a1 r f2 a Standard mounting Zero backlash C=C*m C r f1 Center Distance and Working Pressure Angle of a Gear Pair 1. There are two requirements in designing a gear pair. 1) The backlash should be zero to prevent shock between the gears. s 1 = e 2 s 2 = e 1 2) The bottom clearance should take the standard value c=c*m 2. Standard(reference) center distance working center distance a =r 1 + r 2 reference center distance a = r 1 + r 2 If two gears are mounted with the reference center distance, then
O2O2 r b2 ω2ω2 r a2 O1O1 ω1ω1 r b1 r a1 r1r1 r2r2 P N1N1 N2N2 a α 3. Center distance a and working pressure angle α 1) Standard mounting(a = a) The reference circles coincide with their pitch circles. r 1 =r 1 r 2 =r 2 α =α c=c * m 2)Non standard mounting ( a >a) The reference circles do not coincide with their pitch circles. r 1 > r 1 r 2 > r 2 α >α c >c * m r 1 =r 1 α = α r 2 =r 2 r b2 O2O2 ω2ω2 O1O1 ω1ω1 r b1 a α P N1N1 N2N2 r2r2 r 2 r1r1 r 1 α > α r 2 >r 2 r 1 >r 1 r b1 r b2 = (r 1 +r 2 )cosα r b1 r b2 = r 1 r 2 cosα a cosα = a cosα
N1N1 r a1 N2N2 α =α v2v2 2 O1O1 1 r f1 ω1ω1 r1r1 P Meshing of a rack and pinion 1 Standard mounting 2 Non standard mounting The pitch line of the rack coincides with its reference line : r 1 = r 1 α α The pitch line of the rack does not coincides with its reference line : r 1 = r 1 α α As mentioned above, α α, and r ' r are characteristics of rack and pinion gearing and differ from those of two spur gears.
Mating Process of a Pair of Gears and Continuous Transmission Condition N1 N1 O1O1 r b1 P r b2 ω2ω2 ω1ω1 O2O2 r a2 N2N2 r a1 B2B2 B1B1 B 1 meshing ends at point B 1 B 2 meshing begins at point B 2 B 1 B 2 the actual line of action N 1 N 2 the theoretical line of action N 1 N 2 meshing limit points 1. Mating Process of a Pair of Gears
pbpb B1B2B1B2 2. Continuous Transmission Condition In order to get a continuous motion transmission, the second pair of teeth must have meshed before the first pair moves out of contact. O1O1 N2N2 N1N1 K O2O2 ω2ω2 ω1ω1 B1B1 B2B2 The condition of continuous motion transmission is B 1 B 2 p b Contact ratio : = B 1 B 2 / p b Theoretically, if 1, a pair of gears can transmit continuously. Considering the manufacture tolerance, the contact ratio should be larger than 1. Actually, the contact ratio should be equal to or larger than the permissible contact ratio[ ]. [ ]
N1N1 N2N2 O1O1 r b1 r b2 O2O2 P Equations of Contact Ratio r a1 B1B1 α a2 α a1 B2B2 r a2 ε α [z 1 (tanα a1 -tanα ) +z 2 (tanα a2 -tanα )]/2π ε α B 1 B 2 / p b (P B 1 +P B 2 ) /π mcos α P B 1 B 1 N 1 -P N 1 r b1 tanα a1 - r b1 tanα z 1 mcos α(tanα a1 -tanα )/2 P B 2 z 2 mcos α(tanα a2 -tanα )/2 The value of the contact ratio indicates the average number of tooth pairs in contact during a cycle to share the load. The higher the contact ratio, the greater the average number of tooth pairs to share the load and the higher the capacity of the gear set to transmit the power.
= p b B1B1 B2B2 Two pairs One pair 0.46 p b 0.54 p b 0.46 p b pb pb pb pb CD
2 The curvature radius of the tooth profile and the tooth thickness of the pinion on the dedendum circle are less than those of the gear. The strength of the pinion is much lower than that of the gear, and contact time of the pinion is more than that of the gear. Standard gears enjoy interchangeability and are widely used in many kinds of machines. However, they also have some disadvantages. 1 It is not fit that aa. When a a, the backlash will increase and the contact ratio will decrease. 3 When z< z min undercutting will occur. Base circle Reference circle Cutter interference In a generating process, it is sometimes found that the top of the cutter enters the profile of the gear and some part of the involute profile near the root portion is removed. Standard gears have some disadvantages § 10 6 § 10 6 Introduction to Corrected Gear
To improve the performance of gears, addendum modification is employed. Manufacturing Methods of Involute Profiles 1. Cutting of Tooth Profiles pinion-shaped shaper cutter rack-shaped shaper cutter The cutting motion is the reciprocation of the cutter while the feed is the movement of the cutter toward the blank. The blank should retreat a little as the cutter goes back to prevent scraping on the finished flank by the cutter.
c*m Reference line r b N 1 P α rbrb r rara N1N1 O1O1 O 1 Reference circle Gear blank ha*mha*m B1B1 B2B2 v N 1 r b Involute 2. Cutting a Standard Gear with Standard Rack-shaped Cutter e = s = p / 2 h a = h a * m ; h f =(h a * + c * )m; The reference line of the cutter should be tangent to the reference circle of the gear 1 The addendum line of the cutter does not exceed the limit point N 1 of the line of action, cutter interference will not occur. 2 Cutter interference will occur if the addendum line of the cutter passes the limit point N 1 of the line of action.
3. Minimum Teeth Number of Standard Gear Without Undercutting To prevent cutter interference, the point B 2 should not pass point N l, PN 1P B 2 PN 1 =rsin α =mzsin α /2 PB 2 =h a * m / sin α =mzsin α /2 4. Methods to Avoid Undercutting 1 Decrease the coefficient of addendum depth h a * h a * z min h a * the transmission characteristics will be influenced and the cutter will not be standard. There are several methods to avoid undercutting
The cutter will be standard. The method commonly used to eliminate undercutting is to cut the gears with profile- shifted, i.e., with unequal addendum and dedendum teeth. 3 Corrected gear Therefore, parameters m,, h a *, c *, of the corrected gear remain the same as those of standard gears, but se the gear is called corrected gear (profile-shifted gear). 2 Increase the pressure angle of cutter r b This procedure will reduce the active length and the contact ratio will reduce too, which will also lead to rougher, noisier gear operation and the cutter will not be standard. z min α N1N1 α O1O1 P Q ha*mha*m xm x min m xm
5. Corrected gear Modification distance xm In cutting the corrected gear, the rackshaped cutter is located a distance xm from the position used for cutting the standard gear. x modification coefficient α N1N1 α O1O1 P Q ha*mha*m xm x min m xm Positive modification( x>0) The cutter is placed further away from the position for cutting a standard gear. positive modification gear Negative modification( x<0) The cutter is placed towards the axis of the blank. negative modification gear
Geometric Dimensions of Corrected Gears 1. Geometric dimensions are identical with that of the standard gear d = mz d b = mzcos p = m 2. Geometric dimensions are not identical with that of the standard gear K J I xm Pitch line of cutter α B2B2 Reference line of cutter K I J πm/2 Reference circle P N1N1 O1O1 α rbrb 1 Tooth thickness and spacewidth Base circle
2 Addendum and dedendum Positive modification gear x>0 Reference circle Standard gear x 0 Negative modification gear x<0
Gearing of a Corrected Gear Pair 1.Proper meshing conditions and condition of continuous transmission Proper meshing conditions m 1 = m 2 α 1 =α 2 Condition of continuous transmission [ ] 2. Centers distance of a pair corrected gear 1) Gearing equation without backlash To keep zero backlash for a corrected gear pair, the following relations should hold, as in the case of standard gears, i.e., s l '= e 2 ', s 2 '= e l ', therefore, p ' s ' 1 + e ' 1 s ' 2 + e ' 2 s ' 1 + s ' 2
(x 1 +x 2 ) The two pitch circles will not overlay on the two reference circles a cos acos a a 2 Shifting coefficient of centers distance y Difference of the centers distance a with standard centers distance a ym = a - a Analysis yShifting coefficient of centers distance
3 Shifting coefficient of addendum depth y Clearance be standard With no backlash If two gears mating with no backlash and remaining standard clearance, therefore Problem (x 1 +x 2 ) > y if x 1 + x 2 0 a ' > a '' a ' =a '' y=x 1 +x 2 Solution No backlash can be assured, the depth of addendum circle is decreased.
3. Types of Corrected Gear Pairs 1 Standard transmission x 1 + x 2 0 and x 1 x 2 0 Types of corrected gear pairs can be divided into three types by the sum of the shifting coefficients x 1 + x 2. z 1 > z min, z 2 > z min As x 1 +x 2 = 0 and the above three equations a = a, = y = 0 y = 0 The pinion should be positive corrected gear( x 1 >0) the gear should be negative corrected gear x 2 <0. Two gears should not be undercutting z 1 + z 2 2z min 2 Zero transmission (height shifting gears transmission x 1 + x 2 0 and x 1 - x 2 0
Since gears are positive corrected gear, the strengths of two gears increase. But the contact ratio decreases since the working pressure angles decrease. 3 Angle shifting gear transmission ( x 1 + x 2 0 ) 1 Positive transmission x 1 +x 2 > 0 As x 1 +x 2 > 0 and the above three equations a > a, > y > 0 y > 0 As x 1 +x 2 > 0 z 1 +z 2 < 2 z min
2 Negative transmission x 1 +x 2 < 0 As x 1 +x 2 < 0 and the above three equations a 0 AS x 1 +x 2 2 z min This transmission is contrary to positive transmission. Since gears are negative corrected the strengths of the two gears decrease.But the contact ratio increases since the working pressure angle decrease.
Spur gear Helical gear Properties : T ooth profiles go into and out of contact along the whole facewidth at the same time Sudden loading and sudden unloading on teeth Vibration and noise are produced. Properties : The tooth surfaces of two engaging helical gears contact on a straight line inclined to the axes of the gears The length of the contact line changes gradually from zero to maximum and then from maximum to zero The loading and unloading of the teeth become gradual and smooth. § 10 7 § 10 7 Helical Gears for Parallel Shafts
Basic Parameters of Helical Gears There are two sets of parameters for a helical gear.One set is on the transverse plane and the other set on the normal plane. The parameters on the normal plane are the standard values. To make use of the formulae for spur gear, the parameter in the equations for spur gears should be replaced by those on the transverse plane of helical gears. Therefore, it is necessary to set up relationships between both sets of parameters. 1. Helix angle β righthanded lefthanded β β helix angle βis the helix angle on the reference cylinder. Basic Parameters of Helical
2. Normal module m n and transverse module m t B β ptpt β πdπd pn pn 3. Normal pressure angle n and transverse pressure angle t 4. Coefficient of addendum h * an h * at and coefficient of bottom clearance( c * n c * t ) h f =(h * an +c n * )m n = (h * at +c t * )m t h a =h * an m n = h * at m t
( Sizes of helical gear Reference diameter Center distance: Modification coefficient Gearing of a pair of helical gears 1. Proper Meshing Conditions for Helical Gears or
2. Contact Ratio for a Helical Gear Pair B B2B2 B2B2 L βbβb βbβb B1B1 B1B1 B1B1 B1B1 B B2B2 B2B2 L Spur gear Helical gear The contact ratio of a helical gear pair is much higher than that of a spur gear pair. transverse contact ratio is the face contact ratio or overlap ratio.
Virtual Number of Teeth for Helical Gears Virtual gearthe tooth profile of the spur gear is equivalent to that of a helical gear on the normal plane. The spur gear is called the virtual gear of the helical gear. The number z v of teeth of the virtual gear is called the virtual number of teeth z v. The minimum number of teeth of the standard helical gear without cutter interference z min = z vmin cos 3 β
The main advantages and disadvantages of helical gears 1. Main advantages Better meshing properties. A much higher total contact ratio. Being more compact means of mechanical power transmission. 2. Main disadvantages The helix angle results in a thrust load in addition to the usual tangential and separating loads. F a =F t tg F a β FnFn FtFt β erringbone gear β 8 ° 20 °
Worm Gearing and its Characteristics Worm gear drives are used to transmit motion and power between non intersecting and non-parallel shafts, usually crossing at a right angle. 90 1) Smooth silent operation as screw drives. 2) Greater speed reduction in a single step. This means compact designs. 3) If the lead angle of a worm is less than the friction angle, the back-driving is self-locking. 4) Lower efficiency due to the greater relative sliding speed. The friction loss may result in overheating and serious wear. Therefore, brass is usually used as the material for the worm wheel to reduce friction and wear. § 10 8 § 10 8 Worm Gearing
Proper Meshing Conditions for Worm Drives mid-plane The transverse plane of a worm wheel passing through the axis of the worm The engagement between a worm and a worm wheel on the mid-plane corresponds to that of a rack and pinion Proper Meshing Conditions The modules and pressure angles of the worm and worm wheel on the mid- plane should be equal to each other. The directions of both helices should be the same.
Main Parameters and Dimensions for Worm Drives 2.The module The series of modules for worms is somehow different from those for gears. 3.The profile angle of worm (pressure angle) Archimedes worm 20º In power transmission 25º In indexing devices 5º or 2º 1. The number of teeth The number of threads on the worm z 1 : usually, z 1 1 ~ 10 the recommended value of z 1 : z The number of teeth on the worm gear z 2 is determined according to the speed ratio and the selected value of z 1. For power transmission, z 2 29 ~ 70.
4. The lead angleγ 1 of the worm 5. reference diameter The mid-diameter d 1 of worm the mid-diameter d 1 of the worm is standardized. The reference diameter d 2 of worm wheel d 2 = mz 2 6. The center distance a of the worm gear pair
Introduction to Bevel Gears Bevel gears are used to transmit motion and power between intersecting shafts. The teeth of a bevel gear are distributed on the frustum of a cone. The corresponding cylinders in cylindrical gears become cones, such as the reference cone, addendum cone and dedendum cone. The dimensions of teeth on different transverse planes are different. For convenience, parameters and dimensions at the large end are taken to be standard values. The shaft angle of a bevel gear pair can be any required value. In most cases, the two shafts intersect at a right angle. 1. Characteristics of Bevel Gears § 10 9 § 10 9 Bevel Gears
2. Types and Applications or Bevel Gears Bevel Gears Straight bevel gears Helical bevel gears Spiral bevel gears are most widely used as they are easy to design and manufacture. operate smoothly and easy to design. operate smoothly and have higher load capacity.
r2r2 O2O2 O1O1 r v1 r1r1 δ1δ1 P 2 1 =90° δ2δ2 Crown gear P P1P1 Back Cone and Virtual Gear of a Bevel Gear Crown gear ----d 2 = 90 the surface of the reference cone becomes a plane. Back conethe cone, the element of which crosses the large end of a bevel gear and is perpendicular to the element of the reference cone. Virtual gear of the bevel gear m v = m α v = α r v = r The tooth profile of the virtual gear is almost the same as that of the bevel gear at the large end. Virtual number of teeth z v The tooth number of the virtual gear
r2r2 O2O2 O1O1 r v1 r1r1 δ1δ1 P 2 1 =90° δ2δ2 Crown gear P P1P1 Virtual number of teeth z v The engagement of bevel gearsThe engagement of spur gears
Proper Meshing Conditions m 1 =m 2, α 1 =α 2 The contact ratio of the bevel gear set. The virtual number of teeth z v should not be less than the minimum number of teeth of the virtual gear. z min =z vmin cosδ Parameters and Dimensions of Bevel Gears The most dimensions of bevel gears are measured at the large end being standardized. 1. The reference diameter is 2. The transmission ratio of a gear pair is 90°
ROuter cone distance δReference cone angle δ aAddendum cone angle bFace width d aAddendum diameter d fdedendum diameter b R d1d1 δa1δa1 δa2δa2 d a2 d2d2 d f 2 δ2δ2 δ1δ1 d 1, d 2Reference diameter Transmission ratio i 12 ω 1 / ω 2 When 90° z 2 /z 1 r 2 / r 1 sinδ 2 /sinδ 1 cotδ 1 i 12 tanδ 2 =90° δ 2 +δ 1 90° 2 haha hfhf O θfθf 1 δ1δ1 r1r1 r2r2 δ2δ2 R