İş makinaları tahrik millerinin burulma titreşimleri
Torsional vibrations of motor-machine connecting shafts modelled as a continuous
Yazar: KENAN KOSER
Danışman: PROF.DR. FUAT PASİN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Doktora, 1993
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Tahrik mili = Drive shaft ; Titreşim = Vibration ; İş makineleri = Work machinery

Özet:
ÖZET Kuvvet ve iş makinalarında hareket ve enerjinin iletilmesi çoğu kez peryodik çevrimli mekanizmalar üzerinden sağlanır, liakinaların belirli amaçlar doğrultusunda çalışması esnasında makinaya etkiyen aktif kuvvetlerle periyodik çevrimli mekanizmaya ait hareketli uzuvlardan kaynaklanan atalet kuvvetleri, kuvvet ve iş makinesini bağlayan tahrik millerinde, birden fazla mekanizmanın üzerinde bulunduğu krank millerinde burulma titreşimlerine neden olurlar. Bu çalışmada bir kuvvet makinası ve bir iş makinasın bağlayan tahrik milinin burulma titreşimleri incelenmiştir. Tahrik mili sürekli sistem olarak modellenmiştir. Kuvvet makinası olarak asenkron elektrik motoru öngörülmüştür. İş makinasına ait peryodik çevrimli mekanizmanın değişken kütlesel özellikleri yanında yük momenti karakteristiğinin konuma bağlı olduğu haller gözönüne alınmıştır. Tahrik milinin sürekli sistem modellemesi sonucunda, iş makinesinin bulunduğu sınır şartı itibarıyla klasik sınır şartlarından oldukça farklı yapıda bir sınır değer problemi formüle edilmiştir. Elde edilen sınır değer probleminin, tahrik milinin sürekli Fejim durumundaki davranışına karşı gelen partüküler çözümünü elde etmek amacıyla, matematik analizin bilinen üç farklı çözüm metodu probleme uyarlanmıştır. Örnek problemlerde peryodik çevrimli mekanizmaların farklılığı esas alınmış, harmonik hareket mekanizması, yürek mekanizması ve krank biyel mekanizması olmak üzere üç farklı örnek üzerinde uyarlanan çözüm metotlarının uygulamaları yapılmış, tahrik milinin sürekli rejim durumundaki davranışı incelenmiş ve burulma titreşimlerinin rezonans şartları araştırılmıştır. ıv

Abstract:
TORSIONAL VIBRATIONS OF MOTOR-MACHINE CONNECT I N6 SHAFTS MODELLED AS A CONTINUOUS SYSTEM SUMMARY In machines, motion and energy are widely transmitted by single degree of freedom mechanisms. Mechanisms are generally driven by shafts coupling themselves to a motor. In reciprocating engines 8nd compressors, two or more slider crank mechanisms are connected to a sh8ft called the crankshaft. While the machines are operating, active forces acting on mechanisms and inertia forces due to machanisms' moving members, cause torsional vibrations on driving shafts and crankshafts. Torsional vibrations of the shafts cause severe torsional deformations and angular velocity fluctuations. From this point of view, investigation of resonance and stability conditions of the torsional vibrations of the shafts have been given great Importance. Torsional vibrations of the shafts have been studied by many researchers. The problem of the torsional.vibrations of the shafts of crank and rocker and drag link mechanisms have been investigated by Meyer zur Capellen 1 1 ]. Houben [2] has examined the torsional vibrations of shafts of machines driven by an asynchronous electrical motor. The dynamic effect caused by a sudden temporary change in the natural frequency of the system in which kinematic chains connected to the main drive with one-way cluch or externally controlled coupling has been examined by Vul'fson [3]. Eshleman [4] has investigated torsional response of the crankshafts in diesel engines and in gasoline engine- reciprocating compressor systems. Torsional vibrations of the shafts In a system having variable inertia have been formulated and stability of various systems have been investigated by Krumm 15]. The formulation of the equation of torsional vibrations of the crankshaft for multi-cylinder engine has been given by Pasricha and Carnegie [6]. Dlttrich and Krumm [7] have studied torsional vibration of shafts of machines driven by an internal combustion engine or an electrical motor. The dynamic stability of the torsional vibrations of crankshaft for one cylinder engine has been investigated by Pasricha and Carnegie [8]. Zajaczkowski [9] has examined torsional vibrations of a system composed of two parallel shafts coupled by a number of mechanisms. Dresig and Vul'fson have dealt with the torsional vibrations of the shafts with a number of mechanisms by considering as a continuous system In their book [10]. In the present study, the torsional vibrations of driving shafts have been formulated by considering the shaft as a continuous system. Variable inertia of the driven mechanisms has also been taken into account. Asynchronous electrical motor 1s considered 8S the driving motor. As a result of continuous system modelling of the shaft, a different type of non- homogeneous boundary value problem has been formulated and three different methods are presented for the periodic solutions. The system consists of a motor, connecting shaft and machine which Is shown schematically in Figure 1.M m' G.Ip,p,l,«p j7 (0,t)+M* (4) Jl?(U)]ç(U)^^^9a.O+Gllf(U)=-Hrl(Kl.t)] (5) 2 0(p(l,t) where ( )"and ( /denotes partial derivatives with respect to t and x. v1Around the operating point (CoA.), the asynchronous motor torque Mm can be approximated in the following form, = q> + k(fio-n) (6) where n is the angular velocity, k is the slope of the motor characteristic. It is seen that the wave equation (3) and the boundary condition (4) are linear. On the contrary, the boundary condition (5) which contains Eksergian equation is non-linear. For this reason, boundary condition ( 5) will be linearized with reasonable approachs. Introducing the new variable a, in the form of ş(x,t) = not+a(x,t) (7) and using Taylor expansion of the J(ç) and Mw around D0t, neglecting second and higher order terms of a and derivatives, the wave equation and boundary condition can be formulated in the form shown below: d*a ifa tf dx* (8) Joa(0,t)+ka(0,t)-GIja'(0,t) = Qo (9) J(not)oe(U)+^ 1 d2 J noa(i,t)4[--5 ş = Qot dip q> = Ûot ]a(U) (10) ldJ +GI,a(l,t)=--- = not ] 01(1.0 (14) 1 (U +01,0,(1,.) =.-- (p = fl0t nJ-Mw(not) Here Mw is given by Mw(Dot)= 2, (CmCosfflfÎQt+ dasinmfloO Introducing the non-dimensional quantity T, the wave equation and the boundary conditions are put into the following form (15) (16) (17) J0n§aCT( 0,x ) + kQo at( 0,t ) - GI,^ O.t ) = 0 (18), dJ(T) 2 ld2j(t) 2 dM»(T) J(t)^aCT(l,t)+^n8at(l,t)+[--^^ + -^]a(l)T) 1(0(1), ~ +GIIax(l,T) = ---A-'n8-Mw(T) (19) where a is introduced instead of a v + ( CjSin x + Djcos x) oosnt c c which satisfies the wave equation exactly is considered. If the solution ( 2 1 ) Is inserted into the boundary condition given by equation ( 1 8), one obtains 2p equations with 4p unknowns. From this linear and uncoupled equations, 2p unknowns can be expressed in terms of the remaining 2p unknowns. Thus the solution is converted into the form given by a(x,i)= 2, ( --- Dasin- ^x - ^--^BjSiii- ^x + B»cos- ^x ) anrn i«,l GIj c GIj c c (22) + ( - £ - D-siii x + - - B.M1I x +D-COS x ) cosnt GIj ^ c GI, c ^ c This approximate solution now satisfies the differential equation ( 1 7) and boundary condition (18; exactly. If theexpresion (22) is Inserted into the boundary condition (19), an error function is obtained. Having obtained an error function e, the following Integral Is to be formed. IX'={> (23) which will have to be minimized. Here e shows substruction of two side of the boundary condition (19). The integral given by (23) can be minimized with the following derivatives dJ dJ -.0. -.0 -1.2..., (24) Since the integral given by equation (23) is a definite integral, instead of equations ( 24), the following equations can be written From the equations ( 25), one obtains 2p non- homogeneous linear equations for 2p unknown coefficients. The unknown coefficients are calculated from the equations given by ( 25) using the known numerical techniques. ii) Series Solution: For the solution of the boundary value problems, this method seeks a series solutin in the form «= 2 %?* (26) where fm are chosen as known functions of the independent variables such that they satisfy either the differential equation or boundary conditions or at least, some of the boundary conditions with constants am to be determined. If the series solution is inserted in the differential equation or boundary conditions, this leads to Infinite sytem of equations with infinite many unknowns 8m 1 17l,[ 18]. Such a system is solved by employing the first N equations end unknowns. In our case the series solution which satisfies differential equation (17) and boundary condition ( 1 8) exactly is sellected in the following form. ot(x,t) = 2,(AjSio x + B4cos x)sionT *=1 c c (27) + ( C^sin x + Djcos x) com! c c If the equation (27) is Inserted into the boundary conditions ( 18) an infinite system of linear homogeneous equation and into the boundary condition (19), Infinite system of linear non- homogenous equations for unknowns An, Bn, A*, Dn are obtained. Solving the two systems simultaneously, the unknown constants can be determined.iii) Perturbation Solution: Many physical systems are described by differential equations that can be separated into two parts with one part containing the linear terms and a second part containing the nonlinear or non- autonomous terms relatively small compared to the terms appearing In the first part. The small terms are called perturbations and the solution is sought in the form of a power series of the small parameter E. This approach is known as Perturbation Method [ 1 9],[20]. It is necessary to follow a different approach for the application of perturbation method. The nominal anguter velocity Q have to be choosen as a parameter to be determined. Let us define the small parameter e in terms of the coefficients of reduced moment of inertia as e=ai/ao and expand the angular velocity as a perturbation series around the flo in the following form, n = n0+enl + e2Ck +... @8) Introducing the variable a t)-GI|ax(0.T)=-k(cn14e2n24...) (31) J(T)Q2aCT(l,t)+^n2al(l,t)+[--^-;n2+-^-;]a(U) (32) 1