Forward Dynamics of Flexible Multibody Systems

Transcription

1 11 Forward Dynamics of Flexile Multiody Systems So far, several approaches to the solution of the kinematics and dynamics of multiody systems have een presented. It has een assumed in these approaches that all the odies satisfy the rigid ody condition. A ody is assumed to e rigid if any pair of its material points do not present relative displacements. In practice, odies suffer some degree of deformation; so this assumption does not hold in the strict sense. However, in the majority of the cases the relative displacements are so small that they do not affect the system's ehavior. Therefore, they can e neglected without committing an appreciale error. There are some important cases, however, in which deformation plays an important role. This is the case of lightweight spatial structures and manipulators or high-speed machinery. The dynamics of those systems is influenced y the deformation; thus the formulation of the preceding chapters cannot e applied. The complexity of the equations of motion considering deformation grows consideraly. So does its size, since all the variales defining the deformation must also e considered. In this chapter some of the methods that have een presented in the literature for the dynamics of flexile multiodies will e reviewed. Next a general method ased on the moving frame approach will e descried with natural coordinates that can e used when the elastic displacements are small. A formulation for eam-like elements ased on the large displacement theory will e presented, and expressions for a nonlinear finite element that uses the same kind of Cartesian variales such as coordinates of points and components of unit vectors used in the previous chapters will e developed. Finally, some practical examples will e shown An Overview In this section a quick overview will e given on some of the methods presented in the literature for the analysis of flexile multiody systems. Some of the work in the field was aimed at developing formulations suitale for particular 375

2 Forward Dynamics of Flexile Multiody Systems mechanisms such as the four-ar linkage or the crank and rocker mechanism. These approaches are not reviewed here, ut the reader is referred to the papers y Lowen and Jandrasits (1972), Lowen and Chassapis (1986), Erdman and Sandor (1972), and Erdman and Sung (1986). All the methods currently availale may e divided into three main groups: a) the simplified methods ased on elasto-dynamics, ) the methods ased on defining the deformation with respect to a moving reference frame, and c) the methods ased on defining the overall motion plus deformation with respect to an inertial frame. In the simplified elasto-dynamic methods the deformation is considered uncoupled from the rigid ody motion which is considered known y means of rigid ody dynamics and is called the nominal motion. The main assumption is that the nominal motion induces deformations which are considered small, ut that the deformations do not affect the nominal motion. This approach originally proposed y Winfrey (1971) was later expanded y Midha et al. (1978) and Sunada and Duowsky (1981) to include inertial and centrifugal effects in the elastic equations. Naganathan and Soni (1987) proposed, for the case of openchain flexile manipulators with independent coordinates (no constraint conditions), the use of elasto-dynamics with an iterative procedure that couples the elastic deformations with the nominal motion. For more general applications, the simplified approaches ased on elasto-dynamics cannot e accepted since the coupling terms may strongly influence the solution. In such cases, one of the other two families of methods must e used. The methods in the second group include all the nonlinear coupling terms in the formulation. Two kinds of variales are considered: first, the rigid ody variales, that express the large nonlinear overall motion and characterize the moving frame of each ody; second, the deformation variales, that express the state of deformation with respect to the moving frames. Both the relative displacements and the gradient of the displacements are assumed to e small, in order that the linear theory of elasticity holds. Some authors take as deformation variales the nodal variales resulting from a finite element discretization of the flexile ody (Song and Haug (1980) and Serna and Bayo (1989)). Since this may lead to a large numer of unknowns, one way of reducing the size of the prolem consists in assuming that during the motion only a few modes will e excited and in taking the amplitude of such modes as unknowns. Shaana and Wehage (1983) used a popular sustructuring technique called component mode synthesis (Hurty (1965)) to reduce the numer of unknowns in each ody. Other ways of selecting the most convenient assumed modes may e found in Craig (1981). Other authors (Book (1980), Kim and Haug (1988 and 1989), Changizi and Shaana (1988)) have developed recursive formulations that are ased on the same approach for the definition of the deformation. A major advantage of the moving frame approach is that it makes use of the classical linear finite element theory to introduce either the nodal variales or the assumed mode shapes. Since there are a large numer of reliale finite element codes well-known y engineers, this method has a special attractiveness. Some of the limitations of this

3 11.1 An Overview 377 method have een pointed out y Kane et al. (1987), who showed that the moving frame approach with linear elasticity fails to consider the rotational stiffening effects that appear at very fast speeds of operation and which ecome important. As pointed out y Simo and Vu-Quoc (1987), second-order strain measures are necessary to capture these centrifugal stiffening effects through the geometric stiffness. The third group encompasses a series of more recent methods, introduced first y Simo and Vu-Quoc (1986), that are ased on the large rotation theory. Its main purpose is to develop nonlinear finite elements to e emedded in the multiody formalism. There is only one kind of variales, which are the gloal positions and orientations of the nodes referred either to an initial undeformed state (total Lagrangian formulation) or to a previously known state of deformation (updated Lagrangian formulation). These variales define at the same time the large translations, rotations, and deformations of the ody. This method allows for the existence of aritrarily large relative displacements and displacement gradients. However, since elastic constitutive relations are most commonly used, the assumption of small strains is often made. Unlike the moving frame approach, this method incorporates automatically the correct rotational stiffening terms and is well suited to study instailities and uckling. Its main drawack is that the size of the prolem cannot e reduced as in the moving frame approach. Therefore it is usually large. Furthermore, this formulation is limited to flexile odies that can e modeled using eam and shell elements. After considering this overview of all the methods availale so far for the dynamic analysis of flexile multiodies, we are going to concentrate in this chapter: first, in the formulation of the classical approach of the moving frame with natural coordinates (Section 11.2) and secondly on a new formulation for eamlike elements ased on the large displacement theory that is also ased on the use of natural coordinates (Section 11.3). Note that oth formulations are non-exclusive in the sense that one may e preferred over the other depending on the type of applications. Since oth are ased on the natural coordinates, they can perfectly coexist with the rigid-ody formulation of the previous chapters, in a general purpose simulation package. In those cases in which the geometry is complicated and only small deformations are expected, the moving frame approach with assumed modes will e the est choice. Conversely, with simplified geometry and appearance of nonlinear effects the second approach will e the way to go The Classical Moving Frame Approach In this section the moving frame method using the natural coordinates will e descried (Vukasovic et al. (1993)). A complete formulation of the moving frame approach within the setting provided y the reference point coordinates has een descried y Shaana (1989)). By means of the classical formulation, we use the natural coordinates of the ody (or element) to unequivocally define the

4 Forward Dynamics of Flexile Multiody Systems Z Z Y u j i j X u i Y X Figure Flexile ody with natural coordinates and the moving frame. moving frame that moves with the large overall rigid ody motion and to which the elastic deformation variales are referred. The natural coordinates of the ody do not include relative translations or rotations and are sujected to the corresponding rigid ody constraints. The formulation of the joint constraints is different than in the rigid ody, ecause now points and vectors cannot e shared at the joints and the elastic deformations at those points need to e included Kinematics of a Flexile Body Figure 11.1 shows a flexile ody which will e denoted as and which is defined y the two points i and j and y the two vectors u i and u j. Note that points i and j are not material points ut just two points chosen for the definition of the moving frame. We assume that r j Ðr i, u i, and u j are not coplanar. If a ody has more than two points and two vectors, the formulation can readily e modified to accommodate the additional coordinates y simply adding additional rigid ody constraints (See Chapter 2). Consider the moving reference frame (X, Y, Z ) attached to the ody. Let A e the orthogonal rotation matrix that relates the inertial frame (X, Y, Z) to the ody moving frame. We can write X = A X (11.1) where X represents a (3 3) matrix whose columns are, respectively, r j Ðr i, u i, and u j. The upper ar denotes vectors referred to the moving frame. Similarly (See Chapter 4), X is a (3 3) constant matrix, whose columns are r j Ðr i, u i, and u j. Matrix A can e otained from equation (11.1) y simple inversion: A = X (X ) Ð1 (11.2)

5 11.2 The Classical Moving Frame Approach 379 Figure Cantilever modes of a eam-like flexile ody. Consider the ody and in it a material point P. Consider also a material segment through P, with its direction defined y a unit vector u P. The deformed position of P and u P can e written as: r P = r i + A (r P Ð r i ) + A d r P (11.3) u P = A ( u P + d u P ) (11.4) where d r P represents the elastic displacement P, and d u P the elastic incremental rotation of u P, oth expressed in the moving reference frame. We can now proceed with the spatial discretization of the elastic displacement y defining a set of N R Ritz vectors, such as finite elements or assumed modes for ody, namely f Pk, k = 1,..., N R. These vectors are functions of the material coordinates r P of the point. The set f Pk contains the assumed displacement field corresponding to the assumed modes or finite elements with the rotations defined y the derivatives f' Pk. Using this set of Ritz vectors, the displacements and rotations at point P can e expressed as: N R d r P = h k (t) f Pk = F P h (11.5) å k=1 N R d u P = h k (t) f'pk = F' P h (11.6) å k=1 where h k t are the time-dependent amplitude factors of the Ritz vectors (assumed modes or finite elements). At this point, the analyst has two choices: a) consider a finite element model from which one can extract a reduced set of assumed modes using, for instance, component mode synthesis; or ) otain a set of such modes experimentally through a viration analyzer. Note that f Pk does not depend on time; thus it will not e differentiated. That the rigid ody modes must also e eliminated from the

6 Forward Dynamics of Flexile Multiody Systems Ritz vectors, since the rigid ody motion is already taken into account y the natural coordinates of the moving frame. One of several possile ways of imposing this condition is to select all the Ritz vectors with a clamped end at the origin of the moving frame. In this case, the moving frame attached to the elastic modes will e defined y (r j Ðr i ) and u i. Note that r j is not a material point of the elastic ody. Figure 11.2 shows the two first cantilever modes for a eam-like ody Derivation of the Kinetic Energy In order to otain the expression for the inertia forces we first derive the expression for the kinetic energy of the ody in the form T = 1 2 T r P rp dm (11.7) v Then, sustituting equation (11.5) in (11.3) one can otain r P = r i + A (r P Ð r i ) + A F P h ) (11.8) Sustituting equations (4.49) and (4.50) into (11.8) one otains r P = C P q + A F P h (11.9) where (See Chapter 4) C P is a time invariant matrix that depends on the location of P, and q is the vector that contains the Cartesian coordinates of oth the points i and j and the unit vectors u i and u j. The velocity of P is otained y the differentiation of equation (11.9): r P = C P q + A F P h + A F P h (11.10) Matrix A may e expressed in terms of q in the following manner: first, differentiate (11.2) to otain A = X (X ) Ð1 (11.11) then, sustitute this result into (11.10) to otain r P = C P q + X (X ) Ð1 FP h + A F P h (11.12) Now the second term on the RHS of (11.12) can e modified y defining a modal transformation matrix Y P, such that Y P º (X ) Ð1 FP The mentioned term may e expressed as (11.13)

7 11.2 The Classical Moving Frame Approach 381 X (X ) Ð1 F P h = X Y P h = X NR å k=1 t Pk hk may e ex- where t Pk are the transformed modal vectors. The product of X t Pk pressed as: X t Pk = r j Ð r i u i u j t 1 t 2 t 3 r i Pk = (11.14) (11.15) = Ðt 1 I 3 t 1 I 3 t 2 I 3 t 3 I 3 Pk r j u i u j º T Pk q where T Pk is a matrix that plays the same role of C P ut now is applied to the coordinates of the point P. Sustituting (11.14) into (11.10) a final expression for the velocity of P is otained: r P = C P q + NR å k=1 hk T Pk q + A F P h (11.16) Finally, y sustituting (11.18) into the expression of the kinetic energy (11.7), we arrive at the following final expression: where T = 1 2 qt h T M rr M fr M rf M ff q h (11.17) T M rr = C P C P dm Vol NR NR å å k=1 l=1 NR å k=1 + h k + h k h l Vol T T (C P T Pk + T Pk T T Pk TPl Vol dm C P) dm + (11.18) M rf = C P T A F P dm Vol NR å + h k k=1 Vol T T Pk A FP dm (11.19) M ff = F P T A T A F P dm Vol = F P T F P dm Vol (11.20) The su indexes (-) r and (-) t have een used to differentiate the terms corresponding to the rigid ody motion characterized y q from those that correspond to elastic deformations characterized y h. It may e oserved from those expres-

8 Forward Dynamics of Flexile Multiody Systems sions how oth q and h are coupled in two different ways: a) through the coupling matrices M rf y Mfr; ) and y means of the h dependent terms that are in- cluded in M rr and which appear in equation (11.20). The first term on the RHS of equation (11.20) coincides with the mass matrix of the rigid element as developed in Section 4.2. The second term contains a summation term that is linear in the elastic deformations h. The last term contains a doule summation that depends on the square of the elastic deformations. If one is consistent with the assumption of small deformations, these square terms may e neglected for all practical purposes. However, the second linear term in h may not e neglected as a general rule. Only after a careful comparison of the magnitude of these terms with those corresponding to the rigid case, may they e neglected. It may e finally oserved that M ff does not depend either on the deformation or the rigid ody coordinates. It is the constant mass matrix usually considered in structural dynamics (Craig (1981)). Equation (11.18) contains integrals that are independent of oth position as well as time. These can e computed only once prior to the numerical integration of the equations of motion. The integral of the first term of this equation was seen in detail in Section 4.2. In the following exercise, it will e shown how to compute in an efficient manner the second term of this equation. Example 11.1 Assuming that the Ritz vectors have een computed using finite elements, develop a procedure to integrate V T T (C P TPk + T Pk CP) dm Solution: Recall that the expressions for C P and T Pk are: y cp have een defined in equations (11.16) and (4.55), re- where the vectors t Pk spectively, as: T Pk = Ðt 1 I 3 t 1 I 3 t 2 I 3 t 3 I 3 (ii) C P = (1Ðc 1 ) I 3 c 1 I 3 c 2 I 3 c 3 I 3 (i) (iii) t Pk = X Ð1 f Pk (iv) c P = X Ð1 (r P Ð r i ) It ecomes ovious that one does not need to integrate the matrix products as they appear in (i). It is sufficient to compute the following integral V c T P t Pk dm = V c 1 c 2 c 3 P t 1 t 2 t 3 Pk dm (v) (vi)

9 11.2 The Classical Moving Frame Approach 383 Equation (vi) contains all the terms necessary to uild equation (i). If finite elements are used to compute (vi), one may proceed in the following manner. First, sustitute in (vi) the results of (iv) and (v): V c T P t Pk dm = X Ð1 T (r P Ð r i ) f Pk dm V X ÐT (vii) Now, interpolate the spatial variales (geometry as well as deformed shapes) appearing in the integrand of (vii) using the finite element deformed shapes (r P Ð r i ) = N (r P Ð r i ) x (viii) f Pk = N (r P Ð r i ) p k where N (r P Ð r i ) are the finite element functions that are used to interpolate oth the geometry as well as the gloal deformed shapes of the ody. The vector x represents the coordinates of the finite element mesh, and p k are the values of k mode in the nodes of the finite element mesh. Sustituting the equations (viii) and (ix) in the integral (vii), one can otain V c T P t Pk dm = X Ð1 N (r P ) (x p k T ) N (r P ) T dm V X ÐT Only interpolation functions are part of the integrand, since the expression inside the parenthesis in the middle of the integral does not depend on the spatial coordinates. The integral (x) may e calculated in a ody-y-ody asis. (ix) (x) The final step is to express the kinetic energy of the whole multiody system as an addition of the energies of each individual ody: T = å T = 1 2 qt M q (11.21) where M and q have een otained y assemling the sumatrices M, and vectors q and h, respectively Derivation of the Elastic Potential Energy The expression for the elastic potential energy takes a very simple form with the moving frame approach, since it is only due to the contriution of the elastic displacement. The overall rigid ody motion does not contriute to the potential energy. Consequently, the potential energy of a ody is given y P = 1 2 ht K ff h (11.22) where K s = K ff, if the stiffness has een otained y using finite elements or K ff = F T K s F (11.23) y using assumed modes defined in the local frame.

10 Forward Dynamics of Flexile Multiody Systems The elastic potential energy of the multiody system is otained y assemling the potential energies of each of its odies as: P = å P = 1 2 å q T h T K ff q h = 1 2 qt K q (11.24) where K is the resulting gloal stiffness matrix that only affects the suset of q that corresponds to the elastic displacements h Potential of External Forces Only the case of a concentrated external force f that is applied at a point P of the ody will e considered in this section. Assuming that the force f p is defined in gloal coordinates and making use of the equation (11.9), the virtual work of this force may e written as: dw = dr P T f P = (dq T C P T + dh T F P T A T ) f P = dq T dh T C P T F P T A T f P (11.25) from which one may find the generalized force Q P = C P T F P T A T f P (11.26) Consequently, the potential may e easily calculated as V = q qo dq T Q P (11.27) The same procedure is used to calculate the generalized force and potential of any other type of loading Constraint Equations The constraint equations for flexile odies modeled with natural coordinates also come from two different sources, namely, rigid ody constraints and joint constraints. The rigid ody constraints now correspond to the definition of the moving frame and are derived and formulated as descried in Chapter 2. However, the joint constraints must e modified, since these joints also include elastic deformations. As a consequence, variales such as points and unit vectors can no longer e shared at the joints.

11 11.2 The Classical Moving Frame Approach 385 Z u j u m i u i j m n X Y u n Figure Revolute joint etween two flexile odies. The reason why the variales cannot e shared at the joints is explained with a simple example. Consider the two contiguous odies ij and mn shown in figure 11.3, with each of them defined y two points and two vectors. Consider for the moment that the flexiility is modeled y taking for each of them the first cantilever mode. In addition, assume that there is a revolute joint etween the two odies. Now consider that instead of taking two points j and m, only one point, say i, that is shared etween the two odies is taken. Further consider that there is only one vector in the joint, say u j, which is shared. Note that in the flexile case the natural coordinates do not necessarily coincide with material points or directions, ut they are only a mathematical tool to descrie the overall motion. Since a joint constraint must e imposed etween material points and directions, the sharing of variales does not in this case enforce the revolute joint constraint. Rather, the condition that the deformed end of the ody ij is coincident with the deformed origin of the ody mn must e imposed. Since mn is clamped at the origin, the previous condition means that the ody ij cannot deform. This oviously is unacceptale. This reasoning can e extended to more than one mode, ut the conclusion is always that the sharing of variales limits the deformation modes in an unacceptale way. After this consideration, one can now formulate the constraint equations for the revolute joint shown in Figure First, the deformed positions of j and m must coincide. Similarly, the deformed unit vectors u j and u m must also coincide. Those conditions can e written as: r j + d r j Ð r m + d r m = 0 (11.28) u j + d u j Ð u m + d u m = 0 (11.29) Using expressions (11.5) and (11.6), the displacements dr j, dr m, du j, and du m can e expressed as a linear comination of the Ritz vectors as:

12 Forward Dynamics of Flexile Multiody Systems d r j = A d r j = NR å k=1 h k (t) A f jk = A F j h (11.30) d u j = A d u P = NR å k=1 h k (t) A f 'Pk = A F' P h (11.31) and analogously dr m and du m. The constraint equations for the whole multiody system that can include the definition of relative coordinates at the joints are otained y putting together all the rigid ody and joint constraints in the form F q, t = 0 (11.32) Governing Equations of Motion Several methods for deriving the equations of motion have een presented in Chapter 4 for rigid ody dynamics. The equations of motion in the flexile case are derived in an analogous way, and, therefore, the final form is identical. Having developed expressions for the potential and kinetic energies, the most reasonale way of otaining the equations of motion is through the Lagrange's equations, which leads to the following result: M q + K q + F q T l = Qex Ð M q + T q (11.33) The last two terms on the RHS of this equation are velocity-dependent in oth the rigid and the elastic coordinates. If the second derivative of the constraints is appended, the final form of the equations is M F q T F q 0 q l = Q ex Ð M q + T q Ð K q Ð F q q Ð F t (11.34) However, the Lagrange multiplier approach is generally not the est way of integrating these equations of motion. This is due mainly to two reasons: First, the numer of constraints and therefore Lagrange multipliers is now much larger than in the rigid ody case, since there is not sharing of variales at the joints. Secondly, the elastic terms in the RHS of (11.22) induce a considerale amount of numerical stiffness to the integration process, particularly if high frequency modes of virations such as axial modes are present in the formulation. The first prolem can e remedied y the use of the penalty formulation as done in Bayo and Serna (1989) which eliminates the multipliers from the equations of motion; thus reducing consideraly the size of the system of equations. Other approaches, descried in Chapters 5 and 8 to formulate the equations of motion in independent coordinates using the projection matrix R, can also e applied for flexile multiody dynamics.

13 11.2 The Classical Moving Frame Approach 387 Figure Flexile space root in reorientation maneuver. It is easy to verify that using the penalty formulation, the equations of motion ecome: T (M + F qa Fq ) q + Kq = T = Q ex Ð Mq + T q Ð F qa (F q q + 2m W F + W 2 F ) (11.35) The second prolem, which is related to the stiffness of the resulting equations, may e solved y using the A-stale numerical algorithms presented in Chapter 7. The most appropriate implementation of these algorithms for the case at hand is that explained in Chapter 8, Section 8.5. In this explanation, the difference equations of the integrator are sustituted into the equations of motion. The resulting set of nonlinear equations is solved using Newton-Raphson iteration Numerical Example Using the method presented aove, a satellite deployment maneuver has een simulated, where a flexile root turns and repositions a satellite. Figure 11.4 shows the complete system, and Figure 11.5 shows the set of points and vectors used to define the mathematical model. The main links of the root (odies 3 and 4) are assumed to e flexile and have een modeled using six eam elements, while the other links of the root (end-effector, wrist) are assumed rigid. The main ody of the satellite is supposed rigid, while the solar arrays are flexile and modeled with eleven eam elements of equivalent stiffness and mass. An inertial reference frame is located at the ase of the manipulator on the shuttle ay.

14 Forward Dynamics of Flexile Multiody Systems Figure Natural coordinates model of a flexile root. Figure Deviation of the x-coordinate of the root end-effector. The input to the system is a known variation of the angles driving the manipulator that leads to the desired motion of the system. This motion is a 90 degrees rotation around the Z axis, and, simultaneously, a 180 degrees rotation around the Y axis, in order to re-orientate the satellite. Using this input, the dynamic response of the system has een calculated. Figure 11.6 shows the deviation of the position of the root end-effector (x-coordinate) relative to the rigid ody motion that represents the elastic viration response superimposed on the large rigid ody motion. Once the driving input is finished at time 600 sec., a residual viration remains in the system due to the asence of damping. Calculations have een carried out on a Silicon Graphics Power Iris 4D/240 computer, using only one processor. The required CPU time has een aout 2000 sec. for a maneuver that lasts 800 sec. in real time.

15 11.3 Gloal Method Based on Large Rotation Theory Gloal Method Based on Large Rotation Theory The methods in this section are contriutions from Avello (1990). As mentioned in the overview of the different methods, the classical moving frame approach is ased on the assumption of small displacements and equilirium in the undeformed configuration. Kane, Ryan, and Banerjee (1987) showed that these assumptions lead to a spurious loss of stiffness, when the rotational velocities are large. Moreover, the method seen in the previous section cannot handle larger displacements than those for which the linear finite element method yields accurate results. When oth the elastic displacements are small and the stiffening effects are not important, the classical method yields sufficiently accurate results. It is usually preferred ecause of the reduced numer of equations and the possiility of using either assumed or experimentally found modes of viration. When the stiffening effects ecome important and/or displacements ecome finite, the gloal or asolute method descried in this section can e applied. It is called gloal or asolute ecause the entire motion of the ody (finite rotation plus deformation) is all referred to a fixed frame. This produces a shifting of non-linearity from the inertia terms in the moving frame approach to the deformation terms in this new approach. A formulation of this type was first presented y Simo and Vu-Quoc (1986 and 1988) for multiodies modeled as planar and threedimensional eams, respectively. In this section it will e assumed that the flexile odies are long and slender and that they can e correctly modeled as eams. TimoshenkoÕs eam theory will e used, under the asic assumption that plane sections initially normal to the centroidal line remain plane after gloal deformation has taken place. With these asic assumption, one will derive expressions for a simple nonlinear finite element method that can e used to model flexile odies in a multiody formalism. The most attractive features of this formulation are its simplicity and the compatiility with the natural coordinates so far used in this ook, since the nodal variales of the new eam element are also Cartesian points and unit vectors Kinematics of the Beam Figure 11.7 shows an initially straight prismatic eam of length L and constant cross section A. One can define a fixed reference frame (X 1, X 2, X 3 ), with the X 1 axis coincident with the centroidal line, and axes X 2 and X 3 coincident with the principal axes of inertia. Any cross section of the eam can e descried in this initial state y the coordinates (X 1, 0, 0) of the intersection point etween the cross section and the centroidal line, and y two mutually orthogonal vectors M and N parallel to the X 2 and X 3 axes. Vectors M and N can e considered as corotational vectors that move rigidly attached to the cross section to which they elong.

16 Forward Dynamics of Flexile Multiody Systems n m p X 3 r X 1 N X 2 P X 2 M X 3 X 1 Figure Deformed and undeformed prismatic 3-D eam. After the eam has undergone finite displacements, the position of its cross sections can e defined with the coordinates of the intersection point r and with the components of the co-rotational vectors m and n, as shown in Figure Upper-case letters will e used for the undeformed positions (material coordinates) and lower-case letters for deformed positions (spatial coordinates). If a Lagrangian formulation is used, one can write the deformed positions as a function of the undeformed ones. Since the initially straight prismatic eam is characterized in its undeformed position y just the X 1 coordinate, vectors r, m, and n can e written as a function of X 1, and the time t, as r=r(x 1, t), m=m(x 1, t), and n=n(x 1, t). The deformed coordinates x=(x 1, x 2, x 3 ) of a particle whose material coordinates are X=(X 1, X 2, X 3 ) can e written as x X, t = r X 1, t + X 2 m X 1, t + X 3 n X 1, t (11.36) where X 1 is not a function of time A Nonlinear Beam Finite Element Formulation In the finite element method, a proper inter-element continuity for the interpolated function and its derivatives must e assured y the shape functions. Typical Timoshenko eam elements require continuity only in the displacements and rotations ut not in their derivatives (C 0 continuity). This is achieved y interpolating independently the displacements and rotations inside each element. In this section, an independent interpolation will e assumed for the nodal variales. However, the nodal variales that will e used in this section are different, in na-

17 11.3 Gloal Method Based on Large Rotation Theory 391 n i mi X 3 ri X 1 X 2 Figure Cartesian dependent coordinates for a eam section. ture and numer, from the classical nodal variales used in linear eam elements. The nodal variales in the classical eam elements are composed of three displacements ui and three rotations qi. Instead, the nodal variales used here are composed of the three coordinates of the position r i and the six components of the two orthogonal unit vectors m i and n i, as shown in Figure The nine nodal variales (r i, m i, n i ) are redundant, ecause only three of the six components of m i and n i are independent. In fact, there are three constraint equations that m i and n i must satisfy, two corresponding to the unit norm condition and the third corresponding to the orthogonality condition etween them. Redundant variales have een extensively used in the kinematic and dynamic analysis of multiody systems, as has een seen throughout this ook, ut seldom in the finite element method. The main advantage of using redundant variales is that the overall complexity of the formulation is reduced. The degree of non-linearity of the prolem is reduced as the numer of variales is increased. The cost that one has to pay is the introduction of constraint equations to enforce the satisfaction of the constraints at the nodes. Let (r i, m i, n i ), i = 1,..., p e e the values of (r, m, n) in the p e nodes that elong to the finite element e. The values of (r, m, n) inside each finite element are otained through the following interpolation scheme: p e p e p e r e = N i r i, m e = N i m i, n e = N i n i (11.37) å i=1 å i=1 where N i are the shape functions that can e found in any standard ook in the finite element method (See Bathe (1982)). Note that in expression (11.37) unit vectors are eing interpolated. Since the shape functions are not required to preserve the norm, vectors m e and n e have no longer a unit module. In the same way, the interpolated values m e and n e are not orthogonal. This interpolation inconsistency adds a new source of numerical er- å i=1

18 Forward Dynamics of Flexile Multiody Systems ror that is added to the gloal error of the finite element method. A full discussion on how this error affects the accuracy of the solution goes eyond the scope of this chapter, ut the following points may give more insight. Ð First, the magnitude of the interpolation errors depends on the relative rotation among the nodes of a single finite element. This means that small relative rotations imply small interpolation errors. For example, one can interpolate the two vectors: m 1 = 0 1 with the linear shape functions: N 1 = L Ð x L m 2 = cos j sin j (11.38) N 1 = x (11.39) L The resulting interpolated vector inside the element can e otained as m e = N 1 m 1 + N 2 m 2 (11.40) The module of this vector can e easily calculated as: m et m e 2 2 = N 1 + N2 + 2 N1 N 2 m 1T m 2 = 2 x 2 Ð x 1 Ð cos j + 1 (11.41) L 2 L The maximum constraint violation is otained in the middle of the element. Take j=10 degrees and compute the module of m e y taking the square root of equation (11.41). The resulting value is m e = , which represents an error elow the 0.4%. Since one does not expect to handle rotations larger than 10 degrees among the nodes of the same finite element, the approximation seems quite reasonale. Ð Secondly, the convergence of the finite element method is guaranteed, ecause as the numer of elements increases the error due to the interpolation decreases. In the limit, no error is otained. Ð Finally, the results otained with this formulation are similar to the ones otained with other nonlinear formulations Derivation of the Kinetic Energy In order to otain the inertia forces, one must first develop the expression for the kinetic energy, which can e otained from the integral: T e = 1 2 x e T x e dm V e (11.42) The velocity of a material point x e is otained y differentiating expression (11.36) particularized for element e, and y sustituting the interpolation scheme given in (11.37). This leads to

19 11.3 Gloal Method Based on Large Rotation Theory 393 x e p e = N i (r i + X 2 m i + X 3 n i ) (11.43) å i=1 Sustituting equation (11.43) into (11.42) yields T e = 1 2 p e p e å å i=1 j=1 V e N i N j r it r j + X 2 2 m it m j + X3 2 n it n j + 2 X 2 r it m j + 2 X 3 r it n j + X 2 X 3 m it n j dm + (11.44) where the only terms that depend on the variales of the integral are X 2 and X 3. Since X 2 and X 3 are principal axes of inertia and recalling that X 1 coincides with the center of gravity of the cross section, the three last terms in the integral vanish, ecause they represent two static moments of first order and an inertia product. After reordering equation (11.44), the standard form of the kinetic energy is otained as T e = 1 2 qet M e q e (11.45) where q e is a vector that contains the nodal variales of element e as q et = r 1T m 1T n 1T (r pe ) T (m pe ) T (n pe ) T (11.46) The matrix M e is constant, symmetric, and is composed of sparse sumatrices M ij of size (9x9). In an homogeneous eam, M e takes the form: M e = M 11 M 12 M 1p e M 21 M 22 M 2p e M p e 1 M p e 2 M p e p e (11.47) M ij = r A c ij I I 2 c ij I (11.48) I 3 c ij I 3 where r is the volumetric density, I 3 the (3 3) unit matrix, and c ij the integral over the length of the element of the product of shape functions (N i N j ). Compare this simple and constant expression for the mass matrix with the highly nonlinear matrix otained in Section Although the mass matrix is simpler, the elastic potential energy in the next section is more complicated than with the moving frame method.

20 Forward Dynamics of Flexile Multiody Systems X 3 X x X 1 X 2 Figure Undeformed and deformed position vectors of a point Derivation of the Elastic Potential Energy One of the asic assumptions often made in structural theory is that displacements and displacement gradients are small. When this assumption holds, the Cauchy strain tensor can e used and accurate results are given. However, the Cauchy strain tensor does not work for large displacements, since it does not exhiit the proper invariance under rigid ody rotations of the displacement field. Therefore, when large rotations are considered, a different measure of the strain must e used. Several different kinds of strain measures have een proposed when displacements, displacement gradients, or oth are finite (Malvern (1969)). These strain measures can e included in two major groups. Eulerian formulations formulate the prolem in the deformed configuration, while Lagrangian formulations formulate it in the undeformed configuration. Eulerian formulations are used in applications where an undeformed or initial state does not exist or is unknown, as in fluid mechanics. In elasticity, however, it seems more useful to use a Lagrangian formulation, since an undeformed configuration is always assumed to exist and is taken as a reference state. The Green strain tensor has typically een used in nonlinear elasticity to characterize the deformation field of odies undergoing large displacements. As the displacements and displacement gradients get smaller, the Green tensor tends to the Cauchy tensor and, in the limit, they are identical. Consider a continuous ody and a fixed reference frame (X 1, X 2, X 3 ), as can e seen in Figure Capital letters X=(X 1, X 2, X 3 ) will e used to refer to the coordinates of a particle in an initially undeformed position, and lower-case letters x=(x 1, x 2, x 3 ) will e used for the currently deformed position. In a Lagrangian formulation, x is taken as a function of X and time, and therefore can e written

21 11.3 Gloal Method Based on Large Rotation Theory 395 x = x(x, t) (11.49) The deformation gradient F is defined as the matrix that contains the partial derivatives of x with respect to X. An infinitesimal vector in the deformed position dx can e expressed in terms of the deformation gradient and of its undeformed position dx as dx = x dx = F dx (11.50) X The Green deformation tensor C is defined as the one that gives the new squared length (ds) 2 of vector dx, into which the given vector dx has deformed. Thus, ds 2 = dx T C dx (11.51) The Green strain tensor E gives, y definition, the change in squared length etween the deformed and the undeformed state of a vector dx ds 2 Ð ds 2 = 2 dx T E dx (11.52) where (ds) 2 is the original length of vector dx. Comparing equations (11.50), (11.51), and (11.52), the two following relations can easily e found: C = F T F (11.53) E = C Ð I 3 (11.54) 2 where I 3 is the (3 3) identity matrix. The potential energy for a linearly elastic homogeneous material can e written in terms of the strain vector E = {E 11 E 22 E 33 E 12 E 13 E 23 } T as V = 1 E T D EdV 2 V e (11.55) where the integral is extended to the ody in the undeformed configuration. D represents the matrix of elastic constants, which is defined in terms of LameÕs constants l and G as l +2G l l l l +2G l D = l l l +2G G 0 0 (11.56) G G The values of l and G in terms of the Young modulus E and the Poisson ratio n are: l = E n 1+n 1Ð2n G = E 2 1+n (11.57)

22 Forward Dynamics of Flexile Multiody Systems As seen in equation (11.36), the deformed coordinates of any point of the eam can e written as x X, t = r X 1, t + X 2 m X 1, t + X 3 n X 1, t (11.58) The deformation gradient F can e easily computed as F = x X = x,1 x,2 x,3 = r,1 + X 2 m,1 + X 3 n,1 m n (11.59) where the vertical ars in equation indicate the separation etween columns. The notation (Ð),i is used to represent (Ð)/ X i. The Green strain tensor can e otained y sustituting equation (11.59) into equations (11.53) and (11.54) as E = 1 2 x T,1 x,1 Ð 1 x T,1 m x T,1 n x T,1 m 0 0 x T,1 n 0 0 (11.60) with x,1 = r,1 + X 2 m,1 + X 3 n,1 (11.61) Sustituting equation (11.61) into (11.60) and operating, the following expression is otained for the components of the strain vector E: E 11 = 1 2 x T,1 x,1 Ð 1 = 1 2 r T 2,1 r,1 + X T 2 2 m,1 m,1 + X T 3 n,1 n, X 2 r T,1 m,1 + 2 X 3 r T,1 n,1 + 2 X 2 X 3 m T,1 n,1 Ð 1 E 22 = E 33 = 0 E 12 = 1 2 x,1 T m = 1 2 r T,1 m + X 2 m T,1 m + X 3 n T,1 m (11.62a) (11.62) (11.62c) E 13 = 1 2 x,1 T n = 1 2 r T,1 n + X 2 m T,1 n + X 3 n T,1 n (11.62d) E 23 = 0 (11.62e) If it is assumed that the strains are sufficiently small (note, however, that finite elastic displacements and rotations are still eing considered), the products (m T,1 m,1 ), (n T,1 n,1 ), and (m T,1 n,1 ) in E 1 are second-order terms that can e neglected. Furthermore, the products (m T,1 m) and (n T,1 n) are zero, as can easily e seen y differentiating with respect to X 1 the two unit norm conditions (m T m Ð 1 = 0) and (n T n Ð 1 = 0), respectively. With these simplifications, the strain measures can finally e written as:

23 11.3 Gloal Method Based on Large Rotation Theory 397 T r r T Axial:,1-1 T,1 Bending: r,1 n,1 X 3 Bending: r,1 m,1 X 2 2 T T T T Torsion: n m,1 X 2 Torsion: mn,1 X 3 Shear: n r,1 Shear: m r,1 Figure Axial, ending, torsion, and shear strains in a eam. E 11 = 1 2 T T T r,1 r,1 Ð 1+ 2 X 2 r,1 m,1 + 2 X 3 r,1 n,1 E 22 = E 33 = E 23 =0 E 12 = 1 2 r T,1 m + X 3 n T,1 m (11.63a) (11.63) (11.63c) E 13 = 1 2 r T,1 n + X 2 m T,1 n (11.63d) which is in accordance with the strain distriution predicted y the elemental theory of strength of materials for a prismatic eam under axial, shearing, ending, and torsion loads, as illustrated in Figure For example, the term T (r,1 r,1 Ð 1)/2 in E 11 represents a constant strain distriution corresponding to a pure axial load. Analogously, the term (X 2 r T,1 m,1 ) in E 11 represents a strain distriution that varies linearly with X 2, with a zero value at the centroid and extreme values at the edges, as corresponds to a pure ending load. The two constant shear strains predicted y the Timoshenko eam theory are known to e incorrect, and a paraolic strain distriution should appear. This has typically een corrected y multiplying the area of the cross section y a factor (5/6 for rectangular sections) which gives the correct shear strain energy of the eam. The potential energy of a single element can now e written as P e = 1 E E G E G E 13 dv 2 V Sustituting equation (11.63) and operating, one can otain (11.64)

24 Forward Dynamics of Flexile Multiody Systems P e = 1 2 V E A G E I2 G E I3 G G As2 G G A s3 G G Ip G 6 2 dx1 (11.65) where G 1 represents the axial strain, G 2 and G 3 are the ending unit rotations per unit length, G 4 and G 5 are the shearing strains, and G 6 is the torsion rotation per unit length. Their expressions are: G 1 = r T,1 r,1 Ð 1 2 G 2 = r T,1 n,1 G 3 = r T,1 m,1 G 4 = r T,1 m G 5 = r T,1 n G 6 = n,1 T m (11.66) where A s2 and A s3 are the equivalent shear areas, and I 2, I 3, and I p have the following meaning: 2 2 I 2 = X 3 da I 3 = X 2 A A da I p = (X X3 2 ) A da (11.67) The finite element interpolation given in equation (11.37) can e introduced into equations (11.65) and (11.66). After some algeraic manipulations and rearrangements, the following expressions for the strains G i can e otained: G i = 1 2 qet G i q e Ð i, i = 1,, 6 (11.68) with 1 = 1/2, i = 0, i = 2,..., 6, and where q e was defined in (11.46). The matrices G i are symmetric, sparse, and depend only on the shape functions and their derivatives with respect to X 1. Their expressions are as follows: G 1 = N i,1 N j,1 I (11.69a) G 2 = N i,1 N j,1 I (11.69) N i,1 N j,1 I G 3 = 0 3 N i,1 N j,1 I N i,1 N j,1 I (11.69c) G 4 = 0 3 N i,1 N j I N j,1 N i I (11.69d)

25 11.3 Gloal Method Based on Large Rotation Theory 399 u a P Figure Definition of a revolute joint. G 5 = N i,1 N j I (11.69e) N j,1 N i I G 6 = (11.69f) N j,1 N i I N i,1 N j I The total potential energy for the eam is otained y adding the potential energy of all the elements given y expressions (11.65), (11.68), and (11.69) as P = å P e (11.70) e Oserve that in this eam element the potential energy is otained as a polynomial of order 4 in the position variales ecause P e depends on the square of G i, and G i depends on the square of q e. It is unlike the classical moving frame formulation of Section 11.2, in which the potential energy is a quadratic function of the position variales. This complicates the implementation of the elastic forces, ut the mass matrix otained in Section is constant and can e computed only once. Therefore, the complexity is transferred from the inertia forces to the elastic forces, ut the overall complexity remains similar to the moving frame method's complexity Constraint Equations Since the position variales are not independent, constraints must e introduced at the finite element nodes and at the joints. The constraints at the nodes account for the unit norm and orthogonality conditions that the unit vectors must satisfy.

26 Forward Dynamics of Flexile Multiody Systems The constraints at the joints restrict the relative motion of adjacent odies to the rotations or translations allowed y the kinematic joints. Each node introduces six constraints of the form l i T l i Ð 1 l i T mi m T i m i Ð 1 m T i n i n T i n i Ð 1 = 0 (11.71) n i T l i The constraint equations at the joints can e written in terms of the nodal variales of the nodes next to the joint. The constraint equations for a revolute joint are presented elow as an example. Figure shows two eam-like odies linked at point P y a revolute joint of axis u. Let a and e the two nodes next to the joint, each of them elonging to one of the eams. The revolute joint constraints must enforce oth the condition that point P as attached to the frame in a and as attached to the frame in coincides, and the condition that the vector u as attached to a and as attached to also coincide. Both conditions can e written through the following two vector equations equivalent to six scalar equations: f = r a + A a A a a r P Ð r Ð A r P a u Ð A = 0 (11.72) u where only two of the last three equations are independent. The matrices A a and A are (3x3) orthogonal rotation matrices given y A a = m a Ù n a m a n a and A = m Ù n m n (11.73) where the vertical ars denote the separation etween columns. The values of a r P, r P, a u, and u are the coordinates of point P and the components of vector u expressed in frames a and, respectively. In the previous example, the joint is linking two eams, ut the joint could also e thought as linking a eam and a rigid ody or a eam and a flexile ody with assumed deformation modes. The joint constraints would e developed in the same way, ut different position variales would e used for one of the odies Governing Equations of Motion Once more the equations of motion can e derived using any of the methods seen in Chapter 4. Here, the Lagrange multipliers method will e used again. The Lagrangian function L can e written as

27 11.3 Gloal Method Based on Large Rotation Theory 401 Clamped 45 R = 100 cm Figure Cantilever eam 45-degree end. L = T Ð P + F T l (11.74) where F contains the constraints that arise from the unit norm and orthogonality condition that the nodal variales have to satisfy at the nodes and from the kinematic constraints imposed at the joints. Vector l contains the Lagrange multipliers corresponding to the constraints. The application of the LagrangeÕs equations leads to M q + F q T l = Q Ð F (11.75) where M is the mass matrix otained y assemling the mass matrices M e of each element, F q the Jacoian matrix of the constraint equations, Q the vector of generalized external forces, and F the elastic forces. The elastic forces are otained y differentiating equation (11.65) with respect to q e, giving F e L = E A G 1 G 1 + E I 2 G 2 G 2 + E I 3 G 3 G 3 + G A s2 G 4 G (11.76) + G A s3 G 5 G 5 + G I p G 6 G 6 dx 1 q e The matrices G i are very sparse, and consequently the multiplications y q e can e carried out analytically with very few arithmetic operations Numerical Examples In this section, the results otained in three examples are presented in order to test oth the accuracy of the present eam finite element and the numerical inte-

28 Forward Dynamics of Flexile Multiody Systems Tale Tip displacement (cm) in the cantilever 45-degree end. f = 300 Kg f = 450 Kg f = 600 Kg x1 x2 x3 x1 x2 x3 x1 x2 x3 Present method Cardona (1989) Simo (1986) Bathe and B. (1979) Crisfield (1990) gration procedure. In all cases, the penalty matrix a was taken as ai, and the value of the penalty factor a was taken as 10 6 times the largest term appearing in the tangent stiffness matrix H qf otained through differentiation of equation (11.76). No attempt was made to optimize the value of the penalty factor. It was found that the iteration process converged in few iterations and that the constraint violation was kept small, roughly f < a Ð1. The calculations were performed in a Silicon Graphics 4D/240 using only one processor. To avoid the shear-locking, reduced integration has een used for the shear terms. Example 11.2 The 45-degree end cantilever eam shown in Figure of radius equal to 100 cm shall e considered. It is located in a horizontal plane and a vertical static load f will e considered acting at the tip. The eam has a unit square cross section and E = G = 10 7 Kg/cm 2. It is discretized using eight linear straight elements. In Tale 11.1, the three coordinates of the tip in the deformed position are presented for three different values of the force. The values otained with the finite element developed in this chapter are compared to the values otained previously y Cardona (1989), Simo (1986), Bathe and Bolourchi (1979), and Crisfield (1990). The total load is applied in six equally-spaced load increments. It is worth noting that the tip displacements are of the same order of magnitude as the length of the eam, which is cm. Therefore, the ehavior of the eam is totally nonlinear, with finite displacements which could not e studied using a mode superposition method. As the value of the load increases the solution provided y the proposed method gives larger displacements than the other formulations. This is ecause the interpolation in each finite element, as discussed in Section , does not satisfy the orthogonality condition for variales (m, n). This static example is an extreme one, and it has een presented to prove that this assumption is valid. In fact, the maximum discrepancy etween the results presented in Tale 11.1 for the different methods is aout two per cent.

29 11.3 Gloal Method Based on Large Rotation Theory 403 X3 F t X 2 X 1 F Figure Right-angle cantilever eam. Example 11.3 This example, a right-angle cantilever eam, was first proposed y Simo and Vu- Quoc (1988) and solved with quadratic elements. Later, it was solved y Cardona (1989) using linear elements with a very similar formulation as the previous one. The prolem consists of a right-angle cantilever eam composed of two straight parts of length L=10, each, as shown in Figure The physical characteristics of the eam are not realistic, ut they are useful to test the accuracy of the method in a dynamic simulation when large relative displacements appear. Their values, using the notation in Simo and Vu-Quoc (1988), are given elow: GA = EA = 10 6 EI 2 = EI 3 = GI p = 10 3 A r = 1 I r1 = 2I r2 = 2I r3 = 20 There is a dynamic vertical load F acting at the elow with a triangular variation law. The load acts for 2 sec and reaches a peak of F max =50 at t=1 sec, as can e seen in Figure The prolem has een solved with two different discretizations using four and eight linear elements. The total simulation time is 30 sec. In Figures and 11.15, the vertical displacements of the elow and the tip otained with four and eight elements are plotted. The agreement of this dynamic response compared to Simo and Vu-Quoc (1988) and Cardona (1989) is poor for the four elements discretization, ut it is good when eight elements are used. The results were otained using a constant step size of sec. The average numer of iterations in the Newton-Raphson procedure was three. The CPU times were 20.6 sec for the four elements discretization and 44.4 sec for the eight elements discretization.

30 Forward Dynamics of Flexile Multiody Systems 5.00 Elow vertical displacement elements 8 elements Time (s) Figure Elow vertical displacement using four and eight finite elements Tip vertical displacement Time (s) Figure Tip vertical displacement using four and eight finite elements.

31 11.3 Gloal Method Based on Large Rotation Theory 405 e 2 e 2 q 1 e 3 e 1 e 1 e 3 L 1 q 2 L 2 q 3 q 4 L 3 L 4 Figure Spatial manipulator with two flexile links. Example 11.4 A flexile spatial manipulator composed of two rigid and two flexile links is presented in Figure Links 2 and 3 are flexile eams of tuular section. Each link is connected to the previous one through a revolute joint. At the midpoint of link 4 a lumped mass of 200 Kg has een attached to represent a load. The geometric and material properties of the links are: L1 = 0.3 m Inner radius of the cross section for links 2 and 3. L2 = 4.0 m ri = 0.04 m L3 = 5.0 m L4 = 0.5 m Outer radius of the cross section for links 2 and 3. E = N/m 2 ro= 0.05 m r = 2699 Kg/m 3 Links 1 and 4 have een modeled, respectively, with a single finite element of high-elasticity modulus, and links 2 and 3 have een modeled with four elements each. The simulation that has een carried out is ased on a prescried motion in each revolute joint that moves the manipulator from the initial configuration to the final one, oth shown in Figure The prescried motion is such that

32 Forward Dynamics of Flexile Multiody Systems X Y Z Deviation (m) Time (s) 25 Figure Tip deviations in the X, Y, and Z directions with respect to the rigid ody trajectory. there is a rotation of 90 degrees in joints 1 and 4 and a rotation of 45 degrees in joints 2 and 3. The variation law of each joint is the following: q 1 = q 4 = p 2T s t Ð T s sin 2pt 2p p 2 T s 0 t T s t ³ T s q 2 = q 3 = p 4T s t Ð T s sin 2pt 2p p 4 T s 0 t T s t ³ T s The total simulation time is 25 sec, and Ts was taken to e 15 sec. Figure illustrates the three (X,Y,Z) components of the tip deviation with respect to the nominal motion (that is, the trajectory otained with all the links considered as rigid) as a function of time. The CPU time is 43.2 sec with a fixed step size of 0.2 sec and an average of 3.3 Newton-Raphson iterations per step.

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