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: Belirtilmemiş.
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 90
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 = (1- cos tot) tr for second phase“.. r, 2 Sin co 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 = 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 = 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 Ft 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, h 1 O d CO d f K + 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 - - ? 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 ], x a~y(.K,t),.“, EIz 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 . 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.