Prizmatik mafsal içinden kayan elastik çubukların dinamiği
The Dynamics of elastic beams sliding through a prismatic joint
- Tez No: 46560
- Danışmanlar: PROF.DR. METİN GÜRGÖZE
- Tez Türü: Doktora
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Mafsal, Çubuk elemanlar, Joint, Bar elements
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Günümüzde; biomekanik uygulamaları, uzay araçları ve özellikle de robot manipülatörlerde kullanılan çubukların matematik modellerinde rijit yerine elastik çubuk kabulü kaçınılmaz hale gelmiştir. Kullanılan çubukların olabildiğince hafif olmaları isteği, doğal olarak esnek olmalarına yol açmaktadır. Diğer taraftan, yüksek hızlarda çalışılması atalet kuvvetlerinin büyümesine neden olmaktadır. Titreşimleri artırıcı yönde etkiyen bu durum ise performansı düşürmektedir. Bu nedenlerle, herhangi bir mekanik sistemde sözkonusu olan çubuk şeklindeki uzuvların gerçekçi bir matematik modelinin elde edilmesi, atılacak önemli adımlardan biridir. Bu çalışmada, uzay araçlarındaki antenler ve bazı manipülatör tiplerinde sıkça kullanılmakta olan, sabit bir eksen takımına göre üç boyutlu hareketi önceden verilen bir kayar mafsal içinden kayan elastik bir çubuk ele alınmıştır. Çubuğun bir ucunda bir noktasal kütle olduğu kabul edilmiştir. Bu hareket durumuna sahip elastik çubuğun, iki eksende eğilmeye ve kendi ekseni etrafında burulmaya maruz olduğu varsayılmıştır. Çubuktaki elastik yerdeğiştirmelerin küçük olduğu ve çubuk malzemesinin lineer elastik bir yapıya sahip olduğu kabul edilmiştir. Ayrıca; dönme ataleti, eksenel kuvvet ve ağırlık kuvvetlerinin etkisi gözönüne alınmış, buna karşılık kesme şekil değiştirmesi, eksenel yerdeğiştirme ve bu arada sönüm tesirleri de ihmal edilmiştir. Bu kabuller altında, elastik çubuğun hareket denklemleri, Hamilton prensibi ile elde edilmiş kısmi türevli diferansiyel denklem sistemleridir. Bazı özel hareket durumları için bütünlük açısından yeniden türetilmiş olan bu denklemler, Galerkin ve Kabul Edilmiş Modlar metodlanrı ile genelleştirilmiş koordinatlar cinsinden adi diferansiyel denklem takımlarına dönüştürülmüştür. Daha sonra bu denklem takımları, 4. mertebeden Runge- Kutta integrasyon metodu kullanılarak Mathematica bilgisayar dilinde hazırlanmış programlar ile sayısal olarak çözülmüştür. Elde edilen titreşim biçimleri ve değerleri grafikler halinde sunulmuş ve sonuçlar irdelenmiştir. vn
Özet (Çeviri)
In present, there is substantial room for improvement in the design of mechanical systems that include flexible beams, such as manipulators and spacecraft appendages. In an attempt to improve the structural performance of these systems, their loads and operating speeds are kept fairly low if they are designed under the assumption of rigid body dynamics. Thus, over the last two decades there has been a growing interest in obtaining new methods for the design, analysis and control of these systems under the consideration of flexibility of the beams. In the literature, some researchers have investigated such flexible beams with the assumption of infinitesimal strains [l]-[3], although some others have used finite strain assumption [4]-[6]. The majority of the studies on flexible manipulators has focused on the manipulators with only revolute joints [7]-[13]. On the other hand, manipulators and spacecraft appendages with prismatic joints have received considerable attention lately. However, there is a problem increasing the complexity of the dynamic models of these systems, since the length of the flexible beam is varying with time, and thus the coefficients of the equations of motion and the corresponding boundary conditions are time-dependent. As one of the earliest works, Taborrok et.al. [14] studied a clamped-free beam whose length varies with time. In the following studies on this subject [15]-[18], they have investigated the flexible beam by using an assumption familiar from the classical vibrations literature, the assumption that the beam is similar to a clamped-free beam at any instant. In a similar manner, Buffinton and Kane [19] have analyzed a free-free beam over supports. As a more complex problem, an axially moving beam attached to a moving base has been investigated in references [20]-[23]. Considering a particular beam having not only translational motion but also rotational motion or combinations of translational and rotational motions, they have formulated the vibration problem and presented certain properties of beam mode shapes. In almost all the studies so far, Euler-Bernoulli beam assumption that neglects the effects of rotary inertia and shear deformation has been vmused to investigate the vibration of axially moving flexible beams. In some of the studies, the effect of axial force,or structural damping has also been considered. Morever, some researchers have analyzed the beam with a concentrated end-mass while others with an end-force and moment. Furthermore, the effects of the tail section of the beam has been considered [24]. In this study, a more general problem including most of the effects mentioned above is discussed. Considering not only the effects of the axial force due to inertial forces, end-mass and the tail section of the beam, but also the effects of gravitation and rotary inertia, the flexural and torsional vibrations of a flexible beam sliding through a prismatic joint are investigated. As an important difference between this study and others, the prismatic joint here has an arbitrarily motion in 3D-space instead of a particular motion. Here, it is also assumed that the effects of shear deformation, axial deformation, and structural damping are negligible. The dynamic system under consideration is depicted in Figure 1. It consists of an elastic beam with a concentrated end-mass, undergoing flexural (in two planes) and torsional elastic deflections. It is also sliding through a prismatic joint according to a prescribed motion u(t). The motion of the joint is to be given arbitrarily by the Kardan angles a(t), y(t), (3(t) and the position vector rA(t) of its central point A. Figure 1: Mathematical model of the axially moving flexible beam. IXAs it can be seen in Figure 1. three coordinate systems are used in the mathematical analysis of the dynamic model. One of these is an inertial coordinate system denoted by XYZ. The other two are the moving coordinate systems denoted by xAyAzA attached at point A to the prismatic joint, and xEyEzE attached to a typical element at point E on the elastic beam. It is assumed that a typical element on the beam can elastically displace in the directions of the axes xA and zA and rotate about the axis yA. These elastic displacements are denoted by ux(y,t), uz(y,t) and 0(y,t) while y is the spatial coordinate showing the position of a typical element at point E on the beam with respect to the mass center S of the beam (Figure 1). Considering these displacements, one can show that twisting and bending moments are zero at the ends of the beam. However, at the joint, the translational and angular displacements and the slopes of the deflection curves are zero. The equations of motion of the dynamic system have been derived by applying the extended Hamilton's principle }[5(T-V) + 8'A]dt = 0 (1) '1 where T denotes, as usual, the kinetic energy, V denotes the potential energy, and S 'A represents the virtual work done by the nonconservative forces acting on the system. Here, however, virtual work of the driving force F(t) vanishes (8'A=0) since the sliding motion u(t) is a prescribed motion and therefore its variation is zero (5u=0). The kinetic energy consists of two parts which are due to translational and rotational motions of each element of the beam. The rotational kinetic energy can be written as Td6n = ^û,taI0<da +f î(2ö>ASAIEBV+ VTBTIEBV)dy (2) where p and L are mass density and length of the beam respectively, and SA is the transformation matrix relating the coordinate systems XYZ and xaYaza- The angular velocity vector with respect to the inertial frame is presented by <da. The matrices I0 and IE contain the moments of inertia, V contains the elastic displacements, and B contains transformed angular velocity components of the joint. In a similar fashion, the translational kinetic energy can be obtained asTdof =\) [PA + M5(y-£)] (VATVA + 2VATSAÛ“ + 2VXA.JW 2-^ 2 (3) + 2û üö u +ûT ü - uT coT u )dy xuz Aa xuyz xuz xuz xbjz A/i xtjv ' Here, A denotes the cross sectional area of the beam and M denotes the concentrated end-mass. The translational velocity of the central point A of the prismatic joint is given by the matrix VA, expressed in the inertial coordinate system. The other matrices uxnyz and uxuz consists of the rigid and elastic displacements. The matrix <5Aa expresses the angular velocity components of the joint in the moving coordinate system xAyAzA. Additionally in equation (3), a spatial Dirac delta function defined by 5(y-f)=0, y*f, * L (4) /6(y--)dy=l is utilized. On the other hand, the potential energy consists of three parts. These are the strain energy due to the bending, the potential energy associated with the axial force arising due to inertial forces in connection with the so called axial foreshortening, and the potential energy due to the gravitation. Firstly, let the potential energy due to the bending be written in the form V, =4 Lf[EIX2 +EIX2 +GIJT ]dy (5) where EIX and EIZ denote flexural rigidities and GIb is the torsional rigidity. Secondly, the potential energy due to the axial force associated with foreshortening can be written as V. =| J tt'+u?) Yndy+ff tt,+0 Y-dy (6> wrhere Y,(y,t) and Y”(y,t) are the axial forces which have different expressions for the front and tail sections of the beam. For the front section, the axial force can be shown to be XIYI=M(-YMy+gS32) + pA[(ü + Y2+SH2-gS32-<Au)(y--) (7) and for the tail section Yn = pA[(ü+y2 +SH2 -gS32 -co^^Cy+y) (8) where g represents the gravitational acceleration, yM gives the translational acceleration of the end-mass, and the other terms are related to translational and rotational motions of the joint. Furthermore, the terms associated with the elastic displacements are neglected in the derivation of the last two expressions. Finally, the gravitational potential energy can be expressed as u U V =g J[pA+M8(y--)] (r^+SJu.J dy (9) where rA shows the vertical position of the central point A of the joint in the inertial frame, and S3 represents the third row of the transformation matrix SA. Consequently, summing the kinetic and potential energy terms as T=Tdto+T^, V=VeI+Ve2+VB (10) then dividing the beam into two parts as first (I) and second (II) sections, and denoting the elastic displacements in the following forms ux(y,t)=uX2(y,t), u.Cy.t^Cy.t), P(y,t)=P2(y,t) ; -L/2<y<-u u*(y>t)=uxi(y,t), uz(y,t)=uzi(y,t), P(y,t)=P1(y5t) ; -u<y<L/2 and finally making use of Hamilton's principle given in(l), one can obtain after lengthy calculations the equations of motion for the front section (-u<y<L/2) as xnp[-b>ı; +kx +<& -b>i; hk -kk +&< -b3.)p: +(b* -Km M L -(Y,+SHl)-gS31]+pA[{-- (yMy-gS32)+(ü-(ö2XZAu+Y2+SH2-gS32)(y--) ^(y,^)K + {ü-CD^(u+y)+y,+S%-gS1IK]-EIXs=0, (12) p[-b;u;;+b;5üi;+(b;5-b;ju:;+(b;5-b;2)û:;+(b;5-b;2)p;+(b;5-b;2)p;] -[PA+M5(y-|)][2(G,XAû-tDYAûIı)+üzı+(caXA + Sffi23)(u+y)+S0JMu2]-(6YA-Stûı3)uxı +(Y3 + S“3)+gS33]+pA[{-^-(yMy-gS32)+(ü-G)LAu+y2+SH2-gS32)(y-|) »2 ok,, L2 (y2- j)K + (ü-o^u+y1+Slft-gS11-a)^y)u;]-m,ıÇ =0, (13) and rib^-b^+co^-cb^ +(b;3-b;6Kı+(b*43-b;6K] +(b;-b;6Kl+(b;-b;)û;]+GibpI”=o, and for the tail section (-L/2<y<-u) as, ri-bx + bX + (b;4 - b>;; + (b;4 - b;,)< + (t>;4 - b*31)p2 + (b;4 - b;,)p2] + pA[2(©^ü-<DYAü1B)-üllî + (<ö^-S(1M)(u+y)-SB11uIB-(6^+S^)u>B -(y1+s“1)-gS31+{(ü-öJXZAu+y2+sH2-gs32xy+-)-^(y2-j)K; + (ü-co2XZAu+y2 + SH2-gS32-(D2XZAyK2]-EIzu;=0, (15) pH>X +b;5üi; +(b;5 -b;2x; +(b;-b;)û;; +(b;3 -b'j p; +04 -vjp;] -PA[2((öXaû-coYaûX2)+ü22 +((üXa +SB23)(u+y)+SB33uZ2 -(Ö)Ya -SB13K2 +(Y3+SH3)+gS33 -{(ü-<Au+y2 +SH2 -gS32)(y-|)-^(y2 -£)}< -(ü-cD^u+y1+Slh-gSll-0)^yK]-m,ıı;=O, (16) and p[b;p2 -bj2 +a>SB3 -cbSI^ +o>; -b;6)<2 +(b; -b;x2 +(b;-b;K+(b;-b;)û;]+Gibp;'=o. xraThese six equations of motion of the system are subject to following boundary conditions; for the first section, ux|(-u,t) = 0, uzi(-u,t) = 0, p,(-u,t) = 0, u;(-u,t)=0, u;(-u,t) = 0, u:;(L/2,t) = 0, u:;(L/2,t) = 05 p;(L/2,t) = 0, p[coSIBl -cbSIB4 +b1*)u; +(b;, -b;4)u; +(b;, -b;)p, +(b; -b;4)û; +(b; -b;4)p-b;ü; +^(yMy-gs32) < +^u^ p p y=I72 = 0 +pK,B2 -<% +(K -K)K +KK +(K -t>;5)P, +(b; -b;5)u; M EI +(b;-b;)ft-b;ü; +-(yMy-gs33) <+^<i y=IV2 = 0 (18) and for the second section, uX2(-u,t) = 0, u22(-u,t) = 0, p2(-u,t) = 0, u;(-u,t)=0, u;2(-u,t)=0, u;;(L/25t) = 0, u:;(L/2,t) = o, p2(L/2,t) = 0, pKBi -<b< +KK HK -KK +(K ~K)& +(b; -b;>; EI +(b;1-b;4)p-b;ü;+-^u:;]|y__U2-=o, +pKB2 -av HK -KK2 +K<2 HK -K)h HK -K)K +(b;2-b;5)p2-b;5üi2+^u:;ı p y=-U2 =0 (19) The equations (12) to (17) and the associated boundary conditions given in (18) and (19) form a very complex boundary value problem for which an exact solution is not possible. Consequently, the partial differential equations of motion of the dynamic system will be solved by using approximate methods. In view of this, here, Galerkin and assumed modes methods have been used to discretize the equations of motion of the system for two special cases. Assuming that the torsional displacements can be neglected to simplify the problem, the methods consist of assuming solutions only for the bending displacements in the form of the infinite series as XTVu.(y,t)=Z Pjl(t) e”(y,t).“',1 = 1,2 (20) ua(y,t)=t qjl(t) ejj(y,t) where e”(y,t) and ej2(y,t) are properly selected admissible functions that depend on the spatial coordinate y and time t, and p;i(t) and qjs(t) are the time-dependent generalized coordinates to be determined. The eigenfunctions of a clamped-free beam have been chosen as the admissible functions since they satisfy all of the geometric boundary conditions of the system. They are ejl(y,t) = cosh^-^) - cos^^) - r^sinh^-^) - sin^-^)], u+- u+- u+- u+- 2 2 2 2 (21) ej2(y,t) = cosh(^ f-) - eos^ f-) - ^[sinh^ f~) - sin(Xj j-)] u-- u - u u - 2 2 2 2 where _coshA.j+cos?ij sinh^+sinXj and the dimensionless frequency parameters Xi satisfy the well known characteristic equation of a clamped-free beam cosbAjCOsV^-l, j=l,...,oo (23) In Galerkin method, substituting these admissible functions through the infinite series given in (20) into the equations of motion given in (12)- (13) and (15)-(16), a system of second order ordinary differantial equations of motion can be obtained in the form of M(y,t)p(t) + C(y, t)p(t) +K(y,t)p(t) = F(y,t) (24) This system of differential equations is to be solved numerically by means of a suitable computer code. However, the results concerning the elastic displacements should satisfy the boundary condition terms given in the last two rows of (18) and (19), in which the first three rows are satisfied with the admissible functions automatically. The satisfaction of these boundary conditions ensures that the solution of the differential equations is unique XV[28]. This event increases the complexity of the numerical solution of the problem. In order to avoid this consequence, the series given in (20) have been substituted into Hamilton's principle directly. Then, the equations of motion have been obtained in the same form as equation (24). As a result, the boundary conditions have been considered in the final form of equations of motion. On the other hand, in the assumed modes method, introducing these infinite series with the admissible functions into the kinetic and potential energy expressions and utilizing Lagrange's equations, dt dt ( m\.dp* 87 dp, dph (25), i = l,2,k = l,...,co İAJ dqK 3qB one can obtain the equations of motion in the same form of equation (24). Consequently, it is confirmed that both methods give the same results as expected. In the first special case, it is assumed that the elements of the system are constrained to move on the horizontal x-y plane. While the joint has any arbitrary motion on the plane, the beam is sliding through the joint according to the prescribed motion u(t). The beam has only bending displacements in the xA direction. For this dynamic system, the partial differential equations of motion and associated boundary conditions have been rewritten. Then, they have been obtained as systems of ordinary differential equations by using both methods. Then, these systems of equations have been solved by using a the fourth-order Runge-Kutta method associated with a suitable computer code. The results have been plotted for typical kinematical inputs. In the second special case, the same procedure has been followed for a different beam in flexure along the two axes. In other words, the partial differential equations of motion and associated boundary conditions have been rewritten for the beam in this special case. In contrast to the preceding case here, the joint has a rotational and also a translational motion. Both of the approximate methods have been used to discretize the equations of motion of this new system. The final forms of the differential equations were in the form of equation (24). Hence, this system of equations has been solved by means of the fourth-order Runge-Kutta method and a similar computer code. Finally, the results have been plotted for various kinematical inputs. XVI
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