Elastokinetikte dinamik çarpan hesabı
Dynamic factor in elastokinetics
- Tez No: 21990
- Danışmanlar: PROF. DR. İBRAHİM BAKIRTAŞ
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Ani yükleme, Dinamik çarpanlar, Elastokinetik, Esnek sistemler, Impulsive loading, Dynamic factor, Elastokinetics, Elastic systems
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Yüksek lisans diploma tezi olarak sunulan bu çalışmada elastik sistemlerin dinamik etkiler altındaki davranışı, başka bir deyimle elastokinetik problemleri incelenmiş ve bu problemlerde dinamik çarpan adı verilen oranın aldığı değerler gösterilmiştir. Çalışma dört ana bölümden oluş maktadır. Birinci ana bölümde elastokinetik problemleri, üç ana başlık altında sınıflandırılmıştır. Bu ana başlıkların her biri, bir ana bölümün konusu olarak incelenmiştir. Bu bölümde dinamik çarpanın da tanımı yapılmıştır. ikinci ana bölümde, eylemsizlik kuvvetlerinden doğan etkiler ele alınarak, iki örnekle bu tip problemlerde dinamik çarpanın aldığı değerler hesaplanarak özellikleri açıklanmıştır. Bu ana bölümde 111,121 ve E33 numaralı kaynaklardan yararlanılmıştır. Üçüncü ana bölümde elastokinetikte diğer bir problem türü olan ani yükleme ve çarpışma problemleri ayrı ayrı birer alt bölümde incelenmiş ve dinamik çarpanın aldığı değerler hesaplanmıştır. Bu ana bölümde E13,[23,E43, ES3 ve E63 numaralı kaynaklardan yararlanılmıştır. Son olarak, dördüncü ana bölümde elastik titreşim problemleri ele alınmıştır. Bu tür problemlerin çeşitli sınıflandırma şekilleri açıklandıktan sonra, önce tek serbestlik dereceli sistemler sönümlü ve sönümsüz hallerde ayrı ayrı ele alınarak dinamik çarpan hesaplanmış ve özellikleri belirtilmiştir. İkinci alt başlıkta sonsuz serbestlik dereceli sistemler ele alınarak, bu halde çubukların titreşim hareketi incelenmiştir. özel bir uygulama olarak baca problemi alınmış, açısal frekans önce diferansiyel denklem kuvvet serisine açılarak, daha sonrada Rayleigh oranı yöntemiyle hesaplanarak sonuçlar karşılaştırılmıştır. Bu bölümde son olarak çubuklarda dinamik çarpanın ifadesi çıkartılarak, baca probleminde bulunan birinci ve ikinci modun açısal frekans değerleri ile, kabul edilen bir deprem ivme spekturumu için dinamik çarpanın hesabı yapılarak bulunan değerler grafik olarak gösterilmiştir. Bu ana bölümde El 3, E23, E43, E63, [73, E83 E93,E1Q3 ve [113 numaralı kaynaklardan yararlanılmıştır.
Özet (Çeviri)
SUMMARY DYNAMIC FACTOR IN ELASTOKI NETİ CS The term dynamic refers, to loads which change in time with variations in magnitude, direction and point of application, to inertia forces which is given rise by- accelerated motions of bodies on which dynamic loads are imposed and, to suddenly applied loads and effects caused by collisions. According to the D'Alembert principle every dynamic problem can be reduced to statical problem by adding some effects. The ratio between dynamic displacements and static displacements gives us a di mensi onl ess ratio which is named as DYNAMIC FACTOR. In this M. S thesis, in order to analyse behaviour of elastic systems, dynamic effects are classified in three groups and each of them are examined in different chapters. This classification is made in that way ; ID Strains in structural elements making accelerated motions: Inertia forces. 2D Impulsive loads and collision problems. 3D Vibrations of elastic systems. In chapter 2, strains caused by inertia forces are examined with two example. In first problem a lift, which is going down with Vo velocity, is considered and strain in cable, when it is stopped in time to, is calculated. Dynamic factor, which shows the dynamic aspects of strain, is also given for this problem. In second problem, a simple supported shaft is examined. In chapter 3, impulsive loads and collision problems are examined. First, as a subchapter impulsive loads are analysed. For a chosen elastic system, equation of motion is written and system is analysed for different loading types. For each loading displacement equation is found and also dynamic factor is shown both formulated and graphically. VİFirst loading is the case of suddenly applied external force of constant magnitude. Dynamic factor for this type of loading is found as V <t> = (1- cos tot) <3.10) As it can be seen in the formula, maximum value of dynamic factor is equal to 2. That is, dynamic loading can make two times greater effect than statical loading. Second loading is the case of external force increas ing with constant velocity. For this case, if initial conditions are zero, displacement equation is K(t)= -S - (cot- sin tot) C3.14D k CO Here, it is seen that motion is composed of two parts as x =- ; - t and x =(-ez/)c co) sin cot. First motion indicates the case of linearly increasing force, and second motion indicates dynamic effects of loading. Third loading is the case of a ramp loading. If we consider a ramp loading with rise time tr, applied to system which is at rest prior to application of the load, dynamic factor is 0< t < tr for first phase ¥ (t) = U- - sin <* 1 (3.19D ^ L cp tr J t > tr for second phase“.. r, 2 Sin co<tr/2D,. tr.1,_ __,. V (t)= Jl- i. cos co(t - _ ) I (3. 223 for t > tr maximum value of y is J 1 ”, V 2(1- cos cotr),.-“- w =1+ (3.263 max üi t r Fourth loading which is examined in this chapter is rectangular impulse. This loading can be considered as summation of two loading. First loading is suddenly applied load with constant magnitude starting from t=0 and second loading is again suddenly applied force with constant magnitude but starting from t=td. For second phase, initial conditions are * Ctd)»» (td)=0 *~ ' 2 2 vtiFor first phase, dynamic factor is same with suddenly applied load with constant magnitude in other words, step loading's dynamic factor for first phase ;. 0 < t < td V < t ) = 1 - cos o>t ( 3. 33 ) For second phase dynamic factor is t > td W <t>= f 2 sin -İ^p sin co(t--|İ-)1 (3.34) For this type of loading maximum value of dynamic factor is, for 0 < t < td if td > T/2 ise yt = 2 max if td < T/2 ise y < 2 max ”. ^., _. cot d _. rct d for t > t d tu - 2 sin - rr- = 2 san - =5- max 2 1 Another impulsive loading is short -duration impulsive loading. This type of loading can be considered as a rectangular impulsive load. If td is very short, dynamic factor is td yt <t)= td co sin cot = Zn - =. sin cot (3.36) and maximum value of it, is ys (t> = td co (3.373 max Last impulsive load, which is examined in this chapter, is sine-wave impulse. This loading has two phase. In first phase system, is subjected to a harmonic loading, starting from rest. When t=td, load acting on system is come to an end. After t=td that is second phase, free vibration occurs. To find solution for second phase, initial conditions can be obtained from first phase as fol 1 ows ; * (td>=* (td) ve « (td)=« (td) 2 1 2 1 Dynamic factor for first phase; 0 < t 5td V(t)= |sin cot -(-)sin cot] (3.46) if co > to maximum value of dynamic factor occurs in the second phase, for this phase, if cotd=n maximum value of dynamic factor is found as follows. Vİİİ?(?#?] w =,., 2 cos (3.49) max - -,. 2 -m.(-*-) Last subject of this subchapter is sudden discharge. If a system at rest is subjected to an external load F, and discharged at t=to initial conditions can be written as fol 1 ows for first phase t=0 * =0 ve « =0 ^ 11 for second phase t=to * =» ve k =k r 12 12 After load is discharged dynamic factor is found as - 1/2 2 (1 - cos uto) I <3.61) If to/T is equal to 1/2» dynamic factor reaches to its maximum value, namely, becomes 2. This value coincides with the value of dynamic factor for the case of suddenly applied load. In second subchapter collision problems are examined. If a body of mass M collides with an elastic system which consists of mass and spring, they move on together and their subsequent motion must be such that the total momentum of the system remains unchanged. If body falls from height of h, its velocity at the time of collision is - M v =Mgh or v = V 2 g h 2 o © After impact, they move on together and their common velocity is, M vo + 0 =( M+m ) v v=Mv/(M+m)= v /(1+p) /Li=m/M (3.63) o o Maximum displacement in elastic system is described by <5d. In that case kinetic energy and potential energy Mg <5d is stored in the spring of elastic system. From the principle; conservation of the energy i(M+m)v2+M g 6d = İ k 6d2 2 a 2 w h... -, i,.,2 (3. 64 ) M g tttt- * + M g 6d = - k 6d a (1+/J) 2 equations are obtained. Statical displacement is ^T- =“IT- IXif equation C3.64D is rearranged, then the following equation is obtained. Oil k j- O X Xj - Xj2 <3.65> 2 os,”v + 2 ös od = öd ( 1 +/U ) And from this equation dynamic factor is found as follows (3.67) W = 1 + /l + Al?, ^ y <5s(i+ij) This equation is the fundemantal equation of collision problem. Application of this formula is shown with three example. In last chapter vibrations of elastic systems are analysed. If we apply an external force to upset the stable equilibrium of an elastic system, inertia forces resulting from the motion, will vibrate the system. The classification of vibration motion can be made with different ways. The number of independent coordinates Cboth linear and angular D which is required to fully locate in space all the constituent masses of the moving system at any instant of time is the number of degrees of freedom of the system. Number of degrees of freedom can be finite or infinite. Another classification can be made according to applied force. If external load is removed after excitation, vibration is named as fr&& vibration. But if external load continues acting on system, this type of variation is named as forced vibration. In tree vibration, most important characteristic is natural frequency of system. In this study single degree of freedom systems and infinite degree of freedom systems are analysed. For the forced vibration of an undamped systems with single degree of freedom, general equation of di spl acement i s t üj <t~r> F<T)drj (4.11) - f I-in ı a I J If external load is a harmonic load as F(t)=Fo sin o>t with initial conditions «COD = «COD = 0, displacement equation becomes XFo 1 « (t) = -12. sin cot (4.12) JC - - - 2 and dynamic factor is 1 _ y, <t)= sin cot (4.13) -,.2 1 -m and maximum value of the dynamic factor is V = (4.14) max - - _ 2 1 -t-*-]' For the forced vibration of a damped system with single degree of freedom, general equation of displacement can be obtained from superposition of homogenous solution and particular solution. k <t)= e“Ç [« cos cot + - ° sin co.t| (4.24> h 1 O d CO d f K +<C0« ^ I « cos co,t + - ° - si n co,t I [_ O d CO d J x (t)= - - f e-Ç”<t.-T> sin w (t_T) F(T)drl(4.25) p m co I J d J d v o ' If external load is a harmonic load as F(t)=Fo sin cot with initial conditions «COD = «COD = O. from particular solution, displacement equation becomes - -*2- Ü1 -(-£-) ]sln:rt-2!:(-£-] c°s a] K(t)= 6a (4.26) - N2i2, -,.2 and dynamic factor becomes sin (cSt-£>) V <t)= - (4.28 -.2,2. - -2 \ [-(?#?)] ?(--§-] maximum value of the dynamic factor is 1 y, = (4.29) max i i \|H4-)T ?(*-£-) XITo analyse systems with infinite degrees of freedom C continuous systems]) a cantilever beam is chosen» m(x) : mass of unit length p(x,t) : external load acting on unit length EI Cat) : bending rijidity y ( k, t ) : di spl acement of the beam By summing all forces acting vertically and arranging it, partial differential equation of motion is obtained as fol 1 ows ? d2 r.T,. d2y(x,t)“1 ^,.,,. d2y(x,t) 1 EI(«) - - I + p(x,t)= m(«> - - ? C4. 36) J dtt L dn J &t In case of free vibration there is no effect of external loading, therefore, differential equation is reduced to form as follows, dZ fr-x/ x d2y<*,t> ], x a~y(.K,t),. ”, EI<«) - - ? = m(n) ? C4.37) J ax L 3x J dt By using transformation y(x,t)= Y<») Z(t) C4.38) d Z?t) +</ 2(t)= O C4.41) dt2 f EICx) d Y(M) 1 = w2 m(«) YCjO 0<*<L C4. 42) differ ental equations are obtained. Equation C4. 42D can be solved by using geometric and dynamic boundary conditions. As an numerical example a smokestack is analysed. For this purpose equation C4. 42D rearranged and a homogeneous, ordinary differential equation of fourth order with polynominal coefficients is obtained. To solve it, power series method is used. From this solution, for first two modes, numerical values of on. and u>z are calculated. An approximate way to calculate fundamental frequency is Rayleigh's method. This method is also used in numerical example to verify the result which is obtained by using power series solution. xiiLast subject of this study is determining the expresion of dynamic factor in continuous systems _£_ [ EI(«) d yC»'t) 1 + m(«) d y(»»t>., o (4.62) dxz I die2 J at2 by using transformation CD yC«,t3= E *iC«3 ZtCO C4.63Z) i 2 2 ü! f EIC*> d Yl“(a!) 1 - co2 m(«) Yi(«)=0 (4-64) equation is obtained. By using ortogonality properties and Lagrange's equation Ü-JŞL.- -*L.+ j£L- Qi C4.66D dt 62i. ÖZi 6Zi in which, T represents kinetic energy, U represents potential energy and Q represents generalized force which is determined from the work done by applied force pCx.tDdw in the virtual displacement <5q, the equation for ZiCtD is found as ZlCtD= ZL(0)cos w. t +.? sin cot 1. CO. L t 6>. Ct-rD fCrDdr C4.8aD co~ Ml ”J " q~ Dynamic factor for l'th mode is t y.CtD» «. f sin w. Ct-rD fCrDdr C4. 833 O In smokestack problem dynamic factor for i'th mode i: found as V. <t)= oj. f ygCT) Sin to.Ct-T3 dr (4.90) xtit