Esnek uzuvlu bir manipulatörün dinamik davranışının analizi
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Date
2000-07-04
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Uludağ Üniversitesi
Abstract
Bu çalışmada, esnek uzuvlu bir manipülatörün dinamik analizinin aşamaları verilmiş, modellemede göz önünde tutulması gereken hususlar temel kavram ve yaklaşımlar irdelenerek belirlenmiştir. Daha sonra tek serbestlik dereceli bir manipülatörün farklı yöntem ve yaklaşımlarla matematiksel modeli oluşturulmuş, hareket denklemleri çıkarılmıştır. Bu hareket denklemlerinde geometrik yumuşama ve katılık etkileri, enine ve boyuna elastik deplasmanlar, kesit dönme etkisi ve kesme kuvveti etkisi çeşitli şekillerde hesaba katılmıştır. Sayısal çözümlerde kullanılmamakla birlikte ayrıca iki serbestlik dereceli düzlemsel bir manipülatörün hareket denklemleri de çıkarılmış ve sınır şartlan verilmiştir. Hareket denklemlerinin çıkartılmasında sistemin Lagrangianından yararlanılmıştır. Enerji ifadelerinde eksenel ve enine elastik deplasmanlar kullanılarak elastik harekete ait kısmi türevli diferansiyel denklemler elde edilmiştir. Ayrıca yaklaşık yöntemler olarak "Kabul Edilmiş Modlar" ve "Sonlu Elemanlar" yöntemleri kullanılmıştır. Bu yöntemler için hareket denklemleri seçilen mod veya eleman sayısıyla orantılı sayıda adi diferansiyel denklem takımlarından ibarettir. Elde edilen hareket denklemleri literatürde verilen bazı örneklere uygulanmış ve sonuçlar karşılaştırmalı olarak verilmiştir. Hareket denklemlerinin çözümünde katı (stiff) diferansiyel denklemlere uyarlanmış Runge-Kutta metodu kullanılmıştır. Hareket denklemlerinin çözümleri çok esnek sistemlerde nonlineer uzama oranlarının kullanılmasının gerekliliğini vurgulamaktadır. Ayrıca literatürde refere edilen bazı sonuçların dayandıkları teori itibariyle tutarsızlıkları ortaya konmuştur.
In this study, the stages of the dynamic analysis of a flexible manipulator are presented, and the matters to be considered in the modelling are determined by evaluating some basic concepts and approaches. Then, the mathematical model of a single-link flexible manipulator is established, and its equations of motion are derived using different methods and approximations. In these equations, the geometric softening and stiffening, the shear force and the rotary inertia effects are included completely and/or individually. Additionally, the equations of motion of a planar two-link manipulator are derived although they are not employed to obtain numerical results. In the derivations of the equations of motion, the system Lagrangian is utilized. By expressing the energy integrals in terms of elastic displacements and their time derivatives, the equations of motions are obtained as partial differential equations. Additionally, two approximate methods, i.e. the assumed modes method and finite elements methods, are utilized. In this case, the equations of motion consist of a set of ordinary differential equations the number of which depends on the number of modes or elements used in those methods. These equations of motion are applied to some examples given in the literature, and the results are plotted and interpreted comparatively. To solve these equations numerically, a Runge-Kutta algorithm modified for stiff systems has been used. It is concluded from the numerical results that the non-linear relation between curvature and bending moment along with the inclusion of non-linear strains must be utilized at the first step. The equations should carefully be simplified considering the operating conditions of the system. Furthermore, it is shown that some results presented in the literature contradict with the theory formulated by those authors.
In this study, the stages of the dynamic analysis of a flexible manipulator are presented, and the matters to be considered in the modelling are determined by evaluating some basic concepts and approaches. Then, the mathematical model of a single-link flexible manipulator is established, and its equations of motion are derived using different methods and approximations. In these equations, the geometric softening and stiffening, the shear force and the rotary inertia effects are included completely and/or individually. Additionally, the equations of motion of a planar two-link manipulator are derived although they are not employed to obtain numerical results. In the derivations of the equations of motion, the system Lagrangian is utilized. By expressing the energy integrals in terms of elastic displacements and their time derivatives, the equations of motions are obtained as partial differential equations. Additionally, two approximate methods, i.e. the assumed modes method and finite elements methods, are utilized. In this case, the equations of motion consist of a set of ordinary differential equations the number of which depends on the number of modes or elements used in those methods. These equations of motion are applied to some examples given in the literature, and the results are plotted and interpreted comparatively. To solve these equations numerically, a Runge-Kutta algorithm modified for stiff systems has been used. It is concluded from the numerical results that the non-linear relation between curvature and bending moment along with the inclusion of non-linear strains must be utilized at the first step. The equations should carefully be simplified considering the operating conditions of the system. Furthermore, it is shown that some results presented in the literature contradict with the theory formulated by those authors.
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Keywords
Esnek manipülatörler, Elastik sistemlerin titreşimi, Esnek uzuv, Dönen esnek kiriş, Flexible manipulators, Vibration of elastic systems, Flexible link, Rotating flexible beam
Citation
Telli, S. (2000). Esnek uzuvlu bir manipulatörün dinamik davranışının analizi. Yayınlanmamış doktora tezi. Uludağ Üniversitesi Fen Bilimleri Enstitüsü.