Shabana AA (1998) Dynamics of Multibody Systems. Cambridge University Press, Cambridge UK  Home/Search   Document Not in Database   Summary   Related Articles   Check

....dynamics of the hydraulic control system to the rotorcraft model to verify the actual aeroservoelastic properties of the rotor. A multibody multidisciplinary approach has been undertaken to exploit the capability of modern multibody formulations to account for exact kinematics description [2, 3], coupled to sophisticated structural exibility models such as those commonly addressed, in the literature, as exact or intrinsic [4] The multibody approach proved fundamental in analyzing sophisticated aeroelastic rotorcraft models in complex and unconventional operating conditions with no ....

A. A. Shabana, Dynamics of Multibody Systems. Cambridge, MA: Cambridge University Press, 1998.

....of the object in Euclidean coordinates. 3. 3 Equations of Motion We model the dynamics of the deformable body as a system of second order ordinary differential equations that is obtained by applying the finite element method to the Lagrangian formulation of the equations of elasticity (see [29][22] 25] We represent the state of the body at time t as a column vector of generalized coordinates q = q(t) whose a th component qa (t) is a 3 dimensional vector. Due to equation (4) we can express both kinetic energy T and potential energy V in the form T = T ( q) and V = V (q) where q ....

....large deformations are required and significant error is unacceptable, then the full non linear formulation is necessary. Our approach to the large rotation small deformation scenario deserves further comment. Terzopoulos et al. 33] and similar formulations in the engineering literature, e.g. [29]) integrate a moving frame of reference into the dynamic equations, adding greatly to the complexity of the exposition and implementation. The frame of reference attempts to track the configuration of the object as if it were a rigid body. Besides the added complexity, another problem is that over ....

A. Shabana. Dynamics of Multibody Systems. Cambridge University Press, 1998.

....e.g. Simeon [11] For the modeling of the deformation of a multibody system, different approaches exist. The floating frame of reference method uses a reference frame for the description of the deformation of the body while inertial forces lead to the nonlinear coupling, for details see Shabana [10]. The finite segment method splits the single bodies into small segments connected with springs and dampers. It therefore can be interpreted as a nonlinear finite element formulation using piecewise constant shape functions. Simo and Vu Quoc [12] introduced the large rotation vector formulation to ....

....to efficiently describe the motion and large deformation of beam, plate and shell structures. The absolute nodal formulation uses finite elements and constraints without a reference frame. While it naturally leads to problems in the higher frequencies and even to instabilities, see also Shabana [10], stabilization techniques exist which add an artificial numerical damping, like the HHT method, see e.g. Hairer et al. 6, 7] Alternatively, Gonzalez and Simo [5] derived stable energy momentum methods for Hamiltonian systems and it has been extended to contact and impact problems by Demkowicz ....

A.A. Shabana, Dynamics of multibody systems, John Wiley & Sons (1989).

....our framework for computing the elastic dynamics of an elastic body. We model the dynamics of the deformable body as a system of second order ordinary differential equations obtained by applying the finite element method (FEM) to the Lagrangian formulation of the equations of elasticity (see [39, 30, 34]) To establish notation, we begin with a quick review of the method. Consider a body whose rest state is defined by a domain . The trajectory of the body over time is represented by a function p :# : x, t) p(x, t) 1) It is convenient to decompose p(x, t) as the sum of the rest ....

....large deformations are required and significant error is unacceptable, then the full non linear formulation is necessary. Our approach to the large rotation small deformation scenario deserves further comment. Terzopoulos et al. 44] and similar formulations in the engineering literature, e.g. [39]) integrate a moving frame of reference into the dynamic equations, adding greatly to the complexity of the exposition and implementation. The frame of reference attempts to track the configuration of the object as if it were a rigid body. Besides the added complexity, another problem is that over ....

A. Shabana. Dynamics of Multibody Systems. Cambridge University Press, 1998.

....and 1. The position of P within the element can therefore be measured in a regular (and easy to interpolate) way using parametric coordinates. The physical and parametric coordinates are linked by the so called shape functions. For a hexahedral element, these functions are of the following form [5, 6, 7, 8]: N 0 = 1 Gamma p) 1 q) 1 r) 8 N 1 = 1 Gamma p) 1 q) 1 Gamma r) 8 N 2 = 1 p) 1 q) 1 Gamma r) 8 N 3 = 1 p) 1 q) 1 r) 8 (1) N 4 = 1 Gamma p) 1 Gamma q) 1 r) 8 N 5 = 1 Gamma p) 1 Gamma q) 1 Gamma r) 8 N 6 = 1 p) 1 Gamma q) 1 Gamma r) 8 N 7 = 1 p) 1 Gamma ....

A. Shabana, Dynamics of Multibody Systems, Cambridge University Press, 1998, pp. 171-178.

....the flexibility effects are treated quite rigorously, the linear graph of the system is only used to generate kinematic constraint equations for the absolute coordinates employed by the author. The dynamic equations are generated using an assembly procedure very similar to that proposed by Shabana [10], which does not require (or exploit) the topological equations available from the GT model. As a result, a very large system of differential algebraic equations (DAEs) are obtained for relatively simple systems. In a previous paper [11] we have proposed and validated the use of virtual work as ....

....the symbolic form of the equations of motion. Our GT formulation also allows the analyst to retain non working constraint reactions in the dynamic equations via Lagrange multipliers, or to eliminate these reactions by exploiting the principle of virtual work. These two approaches have been called [10, 14] the augmented formulation and the embedding technique , respectively, although other names have been used in the literature. Both approaches have been included in our GT formulation. In the embedding technique, it is necessary to identify a subset of fqg that are independent. For a system with ....

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A.A. Shabana, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, 1998.

....also serve to further validate the virtual work graph theoretic approach. To clearly define the number of monomials of the same type used to discretize the deformation variables of a beam, the notation uabcd is used to denote the discretization scheme that uses a monomials (which start with x [23]) for the axial deformation, b monomials (which start with x 2 ) for the deflection in the Y direction of the beam s body frame, c monomials (which start with x 2 ) for the deflection in the Z direction and d monomials (which start with x) for the angle of twist. The u in the denotation serves ....

....in their initial conditions) the O(1) results do not. This set of results for both O(1) and O(2) formulations has also been supported by using finer discretization schemes, other types of polynomials, and second order kinetic equations. In short, it is seen that when the floating frame approach [23] is adopted in dealing with EulerBernoulli beams, a complete second order deformation field is of paramount importance for some cases. As mentioned earlier in the paper, failure to use a complete second order deformation approximation results in the loss of some first order terms in the ....

A.A. Shabana, Dynamics of multibody systems, John Wiley & Sons, Inc. 1989.

....the components of q, where f (u; t) is the force distribution applied to the model. See [13, 7] for explicit formulas for the above matrices and vectors. 4 Constrained Nonrigid Motion We can extend (11) to account for the motions of composite models with interconnected deformable parts. Shabana [11] describes the well known Lagrange multiplier method for multibody objects. We form a composite generalized coordinate vector q and force vectors g q and f p for an n part model by concatenating the q i , g q i , and f p i associated with each part i = 1; n. Similarly, the composite ....

Shabana, A., (1989) Dynamics of multibody systems, Wiley, New York.

....To start, image velocities in the optical flow constraint equation are interpreted as projections of the model s three dimensional velocities; this produces a system of optical flow equations that constrain the velocities of the motion parameters of the model. In the theory of dynamic systems [Sha89] velocity constraints such as these are called non holonomic. The velocities of the motion parameters are already accounted for as resulting from the application of edge based forces; finding the equilibrium resulting from these forces amounts to a straightforward optimization problem (which ....

....constraint equation, which was discussed in Section 2.2.2. Instead of solving it on its own, however, it is used as a constraint on the motion of the model. 5.2.2 Solving the dynamic system The constrained system of equations (2.8) and (5. 6) are solved using the method of Lagrange multipliers [Sha89, Str88] The Lagrange multiplier technique adds additional degrees of freedom (one for each degree of constraint) to form a larger, unconstrained system. The initial dynamic equation of motion (2.8) now split into two parts corresponding to q b and qm , is modified by adding the constraint ....

A. Shabana. Dynamics of Multibody Systems. Wiley, 1989.

....being robust to noisy constraints. Our approach can be summarized as follows. Within a deformable model framework, we start with a modelbased version of the optical flow constraint equation, which constrains the velocities of the motion parameters of the model. In the theory of dynamic systems [41], velocity constraints such as these are called non holonomic. The velocities of the motion parameters are already accounted for as resulting from the application of edge based forces; finding the equilibrium resulting from these forces amounts to a straightforward optimization problem. With the ....

....is used as a hard constraint on the motion of the model. 4. 2 Solving the dynamic system Constraining the equations of motion with the model based flow equation results in the constrained system: q = f q subject to B qm I t = 0 (15) This is solved using the method of Lagrange multipliers [41, 45]. The Lagrange multiplier technique adds additional degrees of freedom (one for each degree of constraint) to form a larger, unconstrained system (with the constraints built in ) The initial dynamic equation of motion (5) now split into two parts corresponding to q b and qm , is modified by ....

A. Shabana. Dynamics of Multibody Systems. Wiley, 1989.

....Many mechanical systems such as vehicles, space structures, robotics and aircrafts consist of interconnected rigid and deformable components that undergo large translation and rotation displacements. Such complex mechanical systems, called multibody systems, require specific tools to be simulated (Ahmed A. Shabana 1989). Several software packages (W.O. Schielen 1990) exist for the design and analysis of such systems. But these systems only deal with specific aspects of the design process (K. P. Shah et al. 1994) In addition, incompatibilities between the physical models make intractable any data base transfert ....

....express explicit and implicit (i.e. constraint) energy exchanges. Algorithm The algorithm automatically and symbolically builds the motion equations from the low level description of the scene by applying a lagrangian formalism which can deal with general multibody systems (W.O. Schielen 1984) (Ahmed A. Shabana 1989). The following list describes the computation scheme : firstly, the algorithm traverses the motion frame graph and creates the kinematic graph ; secondly, it computes the kinetic energies C of each element using the kinematic tree ; these elements are either mass and inertial matrices ....

Ahmed A. Shabana. 1989. Dynamics of Multibody Systems. John Wiley and sons.

....attributes interpolation functions attributes Table 1: High level and low level descriptions 2. 2 Algorithm The algorithm automatically builds the motion equations from the low level description of the scene by applying a lagrangian formalism which can deal with general multibody systems [6, 7]. The following list describes the computation scheme : firstly, the algorithm traverses the motion frame graph and creates the kinematic tree ; secondly, it computes the kinetic energies C of each element using the kinematic tree ; these elements are either the mass and inertial matrices ....

Ahmed A. Shabana. Dynamics of Multibody Systems. Wiley Interscience, 1989.

....is possible (integration of angular velocity is meaningless) Define the local contact frame through the rotation matrix R loc = # xg yg zg # T . 39) Let G f = G(q f,rot ) Gg = G(q g,rot) w hereG(qrot) is the matrix operator mapping quaternion velocities to angular velocities [Haug, 1992, Shabana, 1998] given by G(qrot) 2 # qrot 2 qrot 1 qrot 4 qrot 3 qrot 3 qrot 4 qrot 1 qrot 2 qrot 4 qrot 3 qrot 2 qrot 1 # . 40) The velocity [v T # T ] T is the motion of surface g relative to surface f . We write our world space velocities in terms of the local frame. It can be ....

Shabana, A., Dynamics of Multibody Systems, Cambridge University Press, 1998.

....notation such as S is used to indicate the local body frame. Use of minimal coordinate representations such as DenavitHartenburg or Hayati parameters, would need to be extended to include surface interactions. We use the augmented generalized coordinate representation of mechanical systems [14, 32] to incorporate both simple mechanical joints and complex surface constraints. The position and quaternion orientation coordinates for body i are denoted by q i . The parametric contact coordinates u = u i v i u j v j ] T for two surfaces i and j in contact are completely dependent on ....

.... for virtual mechanical assemblies previously developed by the authors [29] Force in configuration space of the mechanism Wc , such as in the Cartesian quaternion set of coordinates, may be expressed in terms of body space force and torque W through standard transformation operators, denoted by G [32, 14], W = 1 4 G(q)Wc (5) This force and torque may be projected onto joint axes to provide the amount of torque required by a controls system to produce a motion. For our interactive haptics force feedback application, W can be transmitted to the user. 7 3 and 6 Axes of Force Feedback Our system ....

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Shabana, A., Dynamics of Multibody Systems, Cambridge University Press, 1998.

....minimization of a cost function when dealing with several dioeerent tasks, while Section 5 details the concept of a smooth switch between tasks. A sensor based control law is presented in Section 6 the performances of which are illustrated in Section 7. Section 8 then concludes. 1 According to [Shabana, 1989], a mechanical system is said to be holonomic when all the kinematic constraints are fully integrable. All other systems are obviously nonholonomic. INRIA Commande R#f#renc#e Capteurs des AUVs Holonomes 5 2 The task function concept 2.1 Some mathematical notations Consider E , the ....

Shabana, A. (1989). Dynamics of multibody systems. John Wiley and Sons, Ltd.

....then uses the summed up force and torque vectors to update the body position and orientation respectively. In our model, we used the generic rigid body motion simulator described in [BW97] We do not present the basic mechanics equations here. The reader can refer to [BW97] Haug92] and [Shabana89] for a complete description of the rigid body dynamics. In order to solve the governing ODEs, a desirable integration method can be chosen. We opted for Euler s method, the simplest one, because we wanted to see how the model behaves in the basic case. Using quaternions to represent rotation ....

Shabana A. A., Dynamics of Multibody Systems, John Wiley & Sons, 1989.

....discussion, and explicitly define the modular relationship between bodies, constraints, and the computation of the Lagrange multipliers. A more basic introduction, including information on reduced cordinate methods, multiplier approaches, and various numerical methods can be found in Shabana [14]. 4.1 Notation With the above in mind, we introduce a small amount of notation. The dimension of the ith body is denoted dim(i) and is the number of d.o.f. s the body has when unconstrained. We describe the ith body s velocity as a vector v i # IR dim(i) a force F i actingontheith body is ....

A. Shabana. Dynamics of Multibody Systems. Wiley, 1989.

.... (integration of angular velocity is meaningless) Define the local contact frame through the rotation matrix R loc = Theta xg yg zg T : 39) Let G f = G(q f;rot ) Gg = G(qg;rot ) where G(qrot) is the matrix operator mapping quaternion velocities to angular velocities [Haug, 1992, Shabana, 1998] given by G(qrot) 2 Gammaq rot 2 qrot 1 qrot 4 Gammaq rot 3 Gammaq rot 3 Gammaq rot 4 qrot 1 qrot 2 Gammaq rot 4 qrot 3 Gammaq rot 2 qrot 1 # : 40) The velocity [v T T ] T is the motion of surface g relative to surface f . We write our world space velocities in terms of ....

Shabana, A., Dynamics of Multibody Systems, Cambridge University Press, 1998.

....objects. Therefore, the object is first manually decomposed into simple articulated subparts. Each subpart contains a small number of object components. Components conforming a subpart are confined to motion within a plane (coplanar motion) and connected to each other by revolute joints[10] i.e. the spatial constraints between two components are modeled by one revolute joint. A revolute joint allows only relative angular rotation between components about the revolute joint axis which is perpendicular to the motion plane and does not allow that components may rotate themselves. ....

....constraints. For motion estimation, the object is first automatically articulated into flexibly connected object components using the method for object articulation proposed in [5] 7] Each subpart contains a single object component. Object components are connected together by spherical joints[10] instead of revolute joints. A spherical joint allows non restricted relative angular rotations between two object components and that object components may rotate themselves. Motion estimation determines first the motion of the largest object component without considering spatial constraints. ....

A. Shabana, Dynamics of multibody systems, John Wiley & Sons, Inc., USA, 1989, Cap. 1 and 2, pp. 1--116. 0 2 4 6 8 0 10 20 30 40 Fig. 1 Simplified stick model of an articulated object m 3 m 1 root object--component object--component (link) spherical joint Fig. 3 Simplified stick model of the articulaed

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Shabana AA (1998) Dynamics of Multibody Systems. Cambridge University Press, Cambridge UK

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Shabana, A. A., Dynamics of Multibody Systems , Wiley, New York, 1989.

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Shabana, A.A., Dynamics of Multibody Systems, Cambridge University Press, 1998.

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A.A. Shabana, Dynamics of Multibody Systems, Cambridge, 1998.

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A.A. Shabana. Dynamics of Multibody Systems. John Wiley & Sons, New York, USA, 1989.

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A.A. Shabana, "Dynamics of Multibody Systems", John Wiley and Sons, New York, 1989.

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