Tablalı yürek mekanizmalarının boyutlandırılmasında yeni bir kriter

Abstract

ÖZET Genel olarak, yürek mekanizmalarının dizaynında, kal- kışPbeklenıe ve iniş açıları ile strokun belirlenmesinden şGnraskonstrüktör tarafından çeşitli kriterlerden bir veya birkaçına göre optimum mekanizma- boyutlarının tespiti yapı lır. Bu çalışmada, doğrusal hareket temin eden kuvvet ka palı düz tablalı yürek mekanizmalarında, `sürtünmeyi yenmek için harcanan enerjinin (kayıp enerjinin) minimum yapılması kriteri` esas alınarak optimum mekanizma boyutlarının tayi nine çalışılmıştır a Birinci bölümde, konunun tanıtılmasından sonra yürek mekanizmalarında optimum boyutların tespiti veküllanilan kri terler hakkında daha önce yapılan araştırmaların bir özeti verilmiştir*, îkinci bölümde J önce tablalı yürek mekanizmalarında kuvvet analizi yapılmış ^yürek-tabla arasındaki normal kuvvet ile tij-yatak arasındaki tepki kuvvetleri hesap edilmiştir» Sonra yürek-tabla ve tij-yatak arasındaki sürtünmeyi yenmek için harcanan enerji miktarı geçiş eğrisinin bir fonksiyone li olarak tayin edilmiştir. Çalışmanın üçüncü bölümü, sürtünmeyi yenmek için har canan enerjiyi minimum yapacak eğrinin bulunmasına ayrılmış- tır0Birinci alt' bölümde, bazı kabuller yapılarak.kayıp enerji ifadesi basitleştirilmiştir «îkinci alt bölümdesmekanizmada aktif dış kuvvetin etkin olması halinde kayıp enerjiyi mini- IllIV ım'yapaeak geçiş eğrisinin tayini ele alınmxşskarşılaşılan varyasyon probleminin çözümü neticesinde parabol'fradi sinoid eklindeki bileşik geçiş eğrisinin -optimum sonuç verdiği tespit edilmiş t ir.Fakat muhtelif geçiş eğrileri için kayıp enerji miktarlarını orijinal ifadesinden hesaplıyarak yapı lan mukayeseler neticesinde, bu halde kayıp enerjinin eğri nin cinsinden ziyade taban dairesi yarıçapına bağlı olduğukalkış eğrisinin seçiminin^varyasyon problemine göre değilB taban dairesi yarıçapının mümkün olduğu kadar küçük seçile bilmesi göz önünde tutularak yapılması gerektiği sonucuna varılmıştır »Üçüncü alt bölüm ise mekanizmada atalet kuvvet lerinin etkin olması haline ayrılmıştır.Karşılaşılan varyas yon probleminin. Euler-Poissön denklemi olarak ikinci merte beden lineer olmayan bir diferansiyel denklem elde edilmiş ve bu denklemin Pertürbasyon Metodu ile yaklaşık çözümü ne ticesinde bir geçiş eğrisi elde edilmiş tir,. Bu alt bölümde yapılan mukayeselerin neticesinde ise 9 ikinci alt bölümdeki- nin aksine^geçiş eğrisinin kayıp enerji üzerinde taban dai resi yarıçapına nazaran daha etkin ve. kayıp ener j inin mini- nimum olması kriterinin anlamlı ve kullanılabilir olduğu tespit edilmiştir. Dördüncü bolümde ise, elde edilen sonuçlar ana Jıatla- rıylâ verilmiştir.

SUMMâEY In the design of cam mechanisms, once the rise, return and dwell angles and the stroke are determined the designer is faced with the task of selecting suitable rise and return motion curves. The choice of the motion curve is effected by ^criteria such as 9 smooth and slient operations, small magnitude of maximum acceleration, suitable vibration characteristics etc or.combination of these criteria.For this purpose, either the standart motion curves are compared with respect to the.selected criteria or starting with a given criterion. the mo tion curve which would yield the optimum solution is investi- ? gated o In some cases, for a mechanism to satisfy the required design characteristics, dimensions such as^eceentiri city, base circle radius etca have` to be determined in the best possible way e During the relative motion of two members in contact, due to friction wear, temperature rise and energy losses appear « Problems created by friction and energy losses can be better understood. if it is realised that one third to one half of the total energy production in the world is being used in over coming. friction. Also statistically it has been shown that failure of % 70 of the machine members, is solely due to fric tion wear. ; In a similar way, the major cause of failure in' cam mechanisms is the geometrical. deterioration of the cam profile due to wear which is directly related to the amount of work done in- overcoming friction,, vVI In this study, plate' cams with flat-face followers, dch are subject to greater amount of friction in compari son to plate cams with roller followers 9 are investigated» Optimum dimensions of the cam mechanism is determined with respect to the criterion of minimizing the energy losses in overcoming frictional forces »Since minimizing the total en ergy loss will minimize the wear of the cam profile 9 the cam mechanism will become more ef icient and useful working life will be prolonged» In the first chapter., after an introduction to the subject, various criteria used in determining the optimum cam dimensions and motion curves are discussed and a liter ature survey of previous work is presented* In the second chap ter, firsts the force analysis is made in plate cams with flat-face followers «.Expression for the normal force between the cam and flat-face follower., the. reaction forces between the follower stem and follower bear ing are derived «It has been shown that the sense of direction of the reaction forces and the lines of action of the fric tion forces change as function of the cam rotate angle (p. For this reasons depending on the value of displacement de rivative with respect to cam rotation angle, three different groups öf expressions are formulated for the normal force N and the reaction forces Rf9 B..Later s an expression for the total energy required to overcome the friction forces between the cam and the flat-face follower, the follower stem and bearing is obtained in the following formsVII Where £cc and/j^are the coefficients of friction between the cam and flat-face and between the follower stem and follower bearing9 re is the base circle radius, s is displacement of follower, tf is the rise angle and 's'sds/djp. In the third chapter of this study,a method for ob taining the motion curve that would minimi ze the total energy loss is presented o Since the reaction and the friction forces change their directions and lines of action respectively de pending. on - the cam rotation angle an analytical solution of the problem is practically impossible. However, if the fr-ic- tional effects are neglected as an initial assumption and the normal and reaction forces are calculated, using suitable co- - efficent of £riction,'frictional forces can be determined and 'hence a relatively simplified expression for the total energy loss is obtained o In the first analysis B the case where the active exter nal force is predominating is considered and a motion curve that would minimise the total energy loss expression is searched for0This formulation leads- to a' variational problem, the solution of which is found to be a parabola minimizing the total energy loss espression0Buts,in the plate cam with flat-face followers,, it is realized that this curve would not be satisfying the necessary boundary condition namely! s (») s 0. Therefore in -the last portion of the rise curve a suitable curve is added that would satisfy the requiredboundary condition,, A `comparison is made amongst the various motion curves given in Appendices 1 and II to determine the most suitable form of the curve to be added to last portion of the parabola that would minimize the total energy loss 0 A parametric study is conducted with base circle radius and follower stem- bearing length taken as pa-VIII rameters9and an optimum curve form is determined «These results are summarized in Fig: 3. 3, where the stem bearing length is nlotted against the base circle radiuSoWhen the selected pa rameters define a point in the left part of diagram9it is seen' that a motion curve composed of a cubic curve plus a simple harmonic minimize the total energy loss,, whereas if the point lies in right part of the diagram then a parabola plus a simple harmonic would minimize the total energy loss0 When the active external force predominates, it has been ob-' served that the base circle radius rather than the motion curve effects the total energy loss «Therefore it is recom mended to select the base circle radius as small as possible and then to apply the minimum total energy loss criterion to choose a motion curve,. As a second case, inertia forces are considered to predominate and the variational- problem encountered is for mulated as Euler-Poisson equation which is a second order nonlinear differantial equation. Approximate, solution of this nonlinear differantial equation is obtained by the perturba tion method» Since this differantial equation is of second order the solution is made to satisfy the following boundary conditions s(0)=0,s(o»)slj,where Sss/h and h is the stroke. Furthermore to satisfy an additional boundary condition name ly? &w)sO,it is found that a relation given by has to be realized amongst the design parameters provided by the constructor o In this expression ,E and %a^e nondimensional active external forces nondimensional weight of stem and flat- face j, nondimensional initial tension in the spring respectivelyIX and F@ is defined by ' Also the noadimens ional frequency parameters is defined as L /r to where. k is the spring stifness which provides force closed property of the mechanismsm is total mass of stem plus flat- face and to is angular velocity of the canuThe motion curve is finally obtained by inserting the relation between the design parameters into the 'solution, in the following form;... Sm7IZrjjk { eos'^ (^] ~Cos % % ] ' ' ' In this equation8to avoid undercutting and to have a force closed mechanism Fs, n and re have to be selected suitably and therefore nomograms are provided in.'Figs:307-_ 3015 to choose suitable Fe, n tr0 values for various rise an gles oUnl ike to the first, caseshere the iner t ial forces pre dominate and the motion curve which is found by application of the minimum total energy loss criterion minimize the to tal energy loss irrespective of the base circle radius o Also a comparison is made from the view point of contact stresses amongst the various motion curves» It has been observed that the motion curve found by the proposed criterion would also produce the minimum contact stresses.. In the fourth chapter, the results are outlined and the conclusions are presented»