4.1 Introduction Issues of applied dynamics CHAPTER 4. DYNAMICS 191

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1 Chapter 4 Dynamics Dynamics is the branch of mechanics that is concerned with the study of motion and the relation between the forces and motion. The central focus of our study is the dynamics of systems of rigid bodies and its application to technical problems. Furthermore, we are basically concerned with the computer aided dynamics of rigid bodies to give an insight into the contemporary classical dynamics from the computational point of view. This should familiarise the reader (and user of this book) with the basic concepts of today s computational dynamics realized in various program packages [4], [5]. The motivation for this approach stems from the fact that in the contemporary engineering praxis a lot of dynamical problems arise but only very few of them can be solved in the analytical form by following classical calculation by hand approaches. For the majority of problems (large-scale problems, analytically nonsolvable differential equations, non-linear tasks, coupled problems etc.) computational methods have to be applied. This fact gives rise to many open questions concerning the optimal use of the computational tools available within the various program packages [17]. The experience shows that for an accurate and effective computation, the mechanical and mathematical models of the given engineering problem have to be properly established. The computational model should contain all the necessary pieces of information considering the mechanical phenomena under investigation. It also should be formulated properly to suit the computational method that is intended to be utilized to obtain the final solution. On the other hand, many computational methods for the various kinds of problems are at the user s disposal today. Among them the appropriate ones for the problem at hand should be chosen and applied. In this chapter our main goal is to provide the basic principles of the contemporary computational dynamics of rigid body systems as well as the necessary theoretical background. The starting point is the question: What should be considered 190

2 CHAPTER 4. DYNAMICS 191 in establishing a proper computational model that should be successfully solved? 4.1 Introduction Issues of applied dynamics Dynamics can be classified into the several sub-domains. Each of them has its own modelling assumptions and procedures. In most of the cases, the computational methods are also different. According to the characteristics of the problem and the focus of the intended dynamical analysis, the sub-domain whose approach is best suited to the problem at hand should be chosen. In the sequel of the chapter, an overview of the characteristics of the sub-domains and problems of the contemporary dynamics is given. Multibody dynamics Multibody dynamics deals with the mechanical systems of interconnected rigid bodies that undergo large displacements and rotations [4], [5], [19]. The bodies are interconnected by kinematical constraint elements and coupling elements. Both visco-elastic and inertia properties of the real technical system are discretised during the process of shaping of the system s mechanical model [18]. The mathematical modelling of the established discretised mechanical model leads to the ordinary differential equations (ODE) (minimal form mathematical models) or to the differential-algebraic equations (DAE) (mathematical models in descriptor form) [5]. The concepts of the multibody dynamics can be successfully utilized within the framework of the following technical applications: vehicle systems, aircraft subsystems, robotic systems, various kind of mechanisms, biomechanical systems, mechatronics. Structural dynamics Structural dynamics deals with the deformable mechanical structures whose segments generally do not undergo large displacements and rotations (not kinematical chains). The mass and visco-elastic properties of the system are distributed along the structure [18]. The basic mathematical modelling generally leads to partial differential equations (PDE). The discretisation of the system that is usually performed in the sequel of the mathematical modelling procedure yields a mathematical model in the form of ODE. By using finite element approach [1], very powerful computational proce-

3 CHAPTER 4. DYNAMICS 192 dures are available for tackling the problems of structural dynamics, see also chapter 5. Typical structural dynamics applications are: plates, shells, aircraft structures, trusses, civil engineering structures. Flexible multibody dynamics In the framework of flexible multibody dynamics, segments of a system are considered to be flexible [19]. Flexible multibody dynamics typically deals with non-linear structures whose segments undergo large rigid body motion superposed by flexible deformations [10]. Modelling and computational procedures of the multibody dynamics and structural dynamics are being combined in order to formulate efficient procedures for problems of this kind. The methods of the flexible multibody dynamics are subjects of extensive ongoing research activities [5]. The applications of flexible multibody dynamics systems can be found in various multibody systems with connected rigid and flexible segments like aircraft rotary wings, flexible robots, biomechanical systems, high-speed mechanisms. Problems of dynamics In dynamics various classes of problems can be distinguished. ffl Inverse dynamics The inverse dynamics deals with the determination of the applied and constraint forces and torques for a mechanical system whose motion is prescribed [16]. Beside the full dynamic approach, in which all the forces of the system are considered in the computation, the quasi-static approach of the inverse dynamics can be applied. Within the framework of the quasi-static approach, the inertia forces of the system are neglected. In most of the cases, an inverse dynamics problem leads to a set of algebraic equations. ffl Forward dynamics The forward dynamics deals with the determination of the motion of the system that is subjected to prescribed applied forces and torques [4]. In most of the technical applications (system s bodies undergo large displacements and rotations, coupling elements of the system possess non-linear characteristics), a forward dynamics problem leads to solving non-linear ordinary

4 CHAPTER 4. DYNAMICS 193 Dynamics Dynamics of MBS Structural dynamics Dynamial behaviour (stability tests) Inverse dynamics Forward dynamics Optimisation Dynamical modelling Deriving dynamical equations Newton Euler approach Lagrange approach, Jourdain s principle etc. Inverse dynamics (Solving of linear algebraic equations) Descriptor form formulation Reduction before integration Minimal form formulation Stability criteria Stability analysis Forward dynamics DAE system Reduction during integration Forward dynamics ODE system Linear analysis Linearization of the equations Linear forward dynamics Linear ODE (vibration analysis) Integration of DAE Integration of ODE Figure 4.1: Issues of applied dynamics differential equations. Depending on the formulation of the mathematical model, additional algebraic equations may be imposed on the system. ffl Vibrations In most of the cases in linear domain [13] [15], solving the vibrational problem leads to the determination of the system eigenvalues and modes (Chapter 5). The system stability problem can also be mentioned in this context.

5 CHAPTER 4. DYNAMICS 194 In the framework of the some very important industrial applications (nonlinear vibration within the vehicle sub-systems, acoustical problems etc.) nonlinear vibrational problems have to be tackled. ffl Optimisation The problem of the optimisation of the mechanical systems (weight, costs, deformation and stresses, dynamical trajectories etc.) is very important in engineering and lies far out of the scope of this book [2], [6]. However, it can be stated that specialised methods that allow for the optimisation of the mechanical systems according to the specified criteria may be applied. Another possibility is to look for an improved design via repeated simulations and variations of the system s parameters. In Figure 4.1 issues of applied mechanics, problems and solving methods of solution are depicted schematically Modelling of Mechanical Systems The modelling of mechanical systems has two major steps that are illustrated in Figure 4.2. Figure 4.2: Steps of modelling The first step is the mapping of the reality (technical system) into a set of simplified entities in order to establish a mechanical model [18]. The mechanical model should include at least the effects under consideration, but not more, i.e. the models should be as complex as necessary but as simple as possible (A. Einstein: Everything should be made as simple as possible, but not simpler ). The mechanical modelling is not an unique but an iterative process. It needs a lot of engineering

6 CHAPTER 4. DYNAMICS 195 experience since proper analogies between the reality and the model, dependent on the goals of the analysis, have to be established [5]. Once the mechanical model is built, in the second step a mathematical model, i. e. a set of the governing equations which describe the model s dynamical behavior, has to be formulated [7]. The mathematical modelling is also not an unique process. It depends on the goals of the analysis and the computational procedures and tools as well as the computer hardware that are intended to be used. Mechanical modelling Mechanical modelling is a process that is affected by the character of the problem and focus of the intended analysis in the first place. Second, the characteristics of the real objects are important, but only within the scope of the given task and intended analysis. A real object can be modelled using different mechanical elements: an aircraft can be considered as a rigid body within the scope of flight mechanics, but it has to be modelled as a system of elastic bodies to analyse the landing dynamics phenomena. If its space trajectory is under investigation, a large space station can be modelled as a particle, but on the other hand, a tennis ball has to be considered to be an elastic body in the case of its impact analysis. The crucial modelling criterion is that the mechanical model should be able to describe (take into account) those mechanical properties of the real system that are under the consideration with the desired accuracy [16], [18]. Mathematical modelling Mathematical modelling is a process of formulating a mathematical text (a set of the equations of motion, for example), referred to the established mechanical model by following physical laws and principles (Newtonian classical mechanics, smooth or non-smooth theory). A good and effective mathematical model has to reflect the type and character of the analysis that is to be performed (linear or non-linear analysis, for example), but also has to be properly formulated to suit the computational procedures and algorithms that are intended to be used for the manipulation and evaluation of the generated equations [7]. In some special cases, the solution of the established mathematical model may be found analytically, where the obtained solution is exact under the assumptions made during mechanical and mathematical modelling. Nevertheless, in most of the cases computational procedures have to be utilized to find the numerical solutions (Chapter 4.3.1). In the past three decades numerous computational techniques and algorithms

7 CHAPTER 4. DYNAMICS 196 have been established to generate the governing equations for the various classes of problems and specific kinds of the analysis (multibody systems, structural systems, systems with the unilateral or variable constraints etc.). These algorithms are the core of the various program packages that are offered in the market today [18], [19]. Although very often the intended mechanical analysis can be carried out by starting initially from different mathematical models, an appropriate mathematical modelling can influence the computational procedure itself to a great extent (reducing the computation time or gaining more accurate results). 4.2 Mechanical Modelling As it was mentioned in Chapter 4.1.2, the mechanical modelling is a process of the mapping of the reality to a set of simplified elements. The established set of the elements (mechanical model) has to be able to describe those mechanical properties of the real system which influence the dynamical phenomena under consideration. Given the goals of the analysis and characteristics of the real system whose dynamical behaviour has to be investigated, a first step toward establishing a proper mechanical model is a decision whether the system is to be modelled as the multibody system or the modelling principles of structural dynamics are to be applied [18], [10]. Many technical systems consist of the large number of bodies interconnected by the constraint elements such as the joints, bearings, springs, dampers or actuators. These systems can be successfully modelled as multibody systems. It can be stated generally that if the bodies in the system undergo large motion and small vibrations, a very powerful tool is the modelling using the multibody system approach [19]. If the multibody system concept is adopted for modelling purposes, a real system will be discretised by means of the elements that will be reviewed in the sequel of the chapter. The discussion will be confined to the modelling principles of the classical multibody dynamics (the models are established as the systems of interconnected rigid bodies) and flexibility of the segments is not considered. Some phenomena of the dynamic behaviour of the elastic bodies are addressed in Chapter 5. Since the kinematical structure of the system determines its characteristics to a great extent, the types and the character of the kinematical constraints and the way they determine the behaviour of the system will be discussed in detail. The classification of the forces that appear in the multibody systems will be also overviewed. Another part of the mechanical modelling is the idealised description of the real load. It may be introduced in the model as concentrated forces and moments as well as forces and moments distributed over line, surface or volume [18], [1]. An appropriate modelling of the load is also dependent on the particular task and the

8 CHAPTER 4. DYNAMICS 197 established model itself. Once the mechanical model is established, the corresponding mathematical model has to be formulated Elements of Multibody Systems As it is depicted in the Fig. 4.3, multibody systems consist of elements with inertia and constraint elements and coupling elements without inertia [18]. Inertia is represented by a rigid body or, as a special case, a particle. Therefore, systems of particles and lumped mass systems may be regarded as special cases of multibody systems. Within the coupling elements two types of actuators can be distinguished: ffl Actuators that prescribe the particular applied forces as the functions of time ( force actuators ). The motion of the system caused by this type of the actuators is generally not known. It is a subject of the forward dynamic analysis of the system. ffl Actuators that prescribe the motion of the system, i.e. prescribe the particular displacements or rotations of the system s bodies as the functions of time ( displacement actuators ). The forces imposed by the actuators of this type are generally not known. These forces are the subject of the inverse dynamic analysis of the system [16]. Since these actuators prescribe the system s motion (the system is constrained to evolve in time in the specific way), the actuators of this type can be considered as the kinematical constraints. Consequently, the forces imposed by them are classified as the constraint forces (see classification of forces and kinematical constraints in the sequel of the chapter). The actuators of this type are also called kinematical drivers System forces The forces that appear in the multibody systems can be classified into the categories as discussed in the sequel [12], [18]. The classification of the sytem forces is important due to the fact that the different types of forces play a different role in the process of establishing of the mathematical model of the system (see Chapter 4.3.2).

9 CHAPTER 4. DYNAMICS 198 Passive elements rigid body with nodal points P i and center of gravity C mass point P Coupling elements spring Constraint elements rod damper support, bearings, joints ÓÓ Active elements Coupling elements actuator (force / moment) Constraint elements actuator (displacement / rotation) (kinematical driver) Figure 4.3: Elements of multibody system ffl External and internal forces This classification is based on the individual choice of the system s boundary. The external forces act from the outside of the boundary. The internal forces act inside the boundary of the system. The internal forces always appear in pairs. ffl Applied and constraint forces The applied forces are forces imposed on the system by the coupling elements as well as forces which can be described by physical laws. They influence the way how the system evolves in time (as well as the system constraint forces). Some examples of applied forces are: gravity force, force actuators, springs, dampers, forces due to the magnetic field, etc.

10 CHAPTER 4. DYNAMICS 199 The constraint forces are imposed to the system by the kinematical constraint elements (the joints, bearings, actuators that prescribe motion of the system). In the case of the ideal constraints, these forces are collinear to the direction of the restricted motion (see discussion on the kinematical constraints below). They influence the possible motion of the system. In Figure 4.5 system forces that appear in double pendulum are analysed and classified by deriving the free body diagram of the system Kinematical constraints The kinematical constraints are the mechanical entities that are imposed by the joints, bearings and the system prescribed motions (kinematical drivers) [21]. They restrict the system motion and reduce its degrees of freedom and are represented by the equations that describe the kinematical restrictions imposed on the system. The kinematical constraints are independent if these equations are linearly independent [16], [21] (the number of the independent kinematical constraints is equal to the number of the linear independent equations between the constraint forces, rank of the matrix in the equation (4.82)). The kinematical constraints can be independent of time (scleronomic constraints) or can prescribe the motion of the system as a function of time (rheonomic constraints). If the kinematical constraints are represented by equations comprising only displacements and rotations i.e. the constraints are at the position level since there are no velocities or accelerations in the equations, the constraints are called holonomic constraints. If the constraint equations are at the velocity level (containing time derivatives of position coordinates) which can be directly transformed by integration into the position level, they are also holonomic. If the constraint equations are at the velocity level and they can not be directly transformed into the position level, they are called non-holonomic constraints [18]. In the case of ideal kinematical constraints (the joints and bearings as well as the kinematical drivers are assumed to be rigid and frictionless), the direction of the constraint force that is imposed by the particular kinematical constraint is directed along the direction of the constraint itself. The considerations in this chapter are restricted to ideal and holonomic constraints. System degree of freedom In the case of a totally unconstrained free system of p rigid bodies, the degrees of freedom (DOF) of the system are 6p. It stems from the fact that 6p independent coordinates are necessary to describe the kinematical configuration (the position

11 CHAPTER 4. DYNAMICS 200 and orientation of the system s bodies) uniquely [4]. In Table 4.1 typical constraint elements and their characteristics are depicted. Table 4.1 Types and valency of bearings If q holonomic constraints are added to the system, its degree of freedom is reduced. ffl If all q constraints are independent, the degrees of freedom of the system are f =6p q. ffl If only r of the q constraints are independent, the degree of freedom of the system is f = 6p r. The number r of independent constraints is equal to the rank of matrix Q in equation (4.82). If the system possesses f DOF, there are f independent coordinates necessary to describe the configuration of the system uniquely. These coordinates are called generalised coordinates and can be choosen in different ways appropriate to the particular problem. The choice of a set of generalised coordinates may strongly influence the process of mathematical modelling as well as the process of solving the equations (see Chapter 4.3). Degree of freedom of double pendulum is determined in Figure 4.4. F Figure 4.4: Degree of freedom of mechanical system: double pendulum Types of mechanical systems The mechanical systems can be classified in terms of the number of its DOF and how the imposed kinematical constraints are arranged (Figure 4.6 and Figure 4.7).

12 CHAPTER 4. DYNAMICS 201 ÓÓ system s boundary Force external internal applied constraint Figure 4.5: Forces in mechanical system: a double pendulum ffl Statical determination If all q constraints are independent, the system is a statically determined. On the other hand, if only r of the q constraints are independent, than n = q r constraints are superflous. The system is statically n times overdetermined. In this case the constraint forces can not be calculated without introducing further modelling assumptions (elastic properties). In Figure 4.6 and Figure 4.7, the systems 4.6 d) and 4.7 d) are statically overdetermined.

13 CHAPTER 4. DYNAMICS 202 a) b) c) (Slider) d) e) Ó ÓÓ y (Pendulum) x Figure 4.6: Systems with various constraints and DOF: beams with various supports ffl Kinematical determination A system is kinematically determined, f =0, if the displacements and rotations of all its members are completely determined by the constraints. If all kinematical constraints do not depend on time, the system is a statical one. The kinematical constraints that do not depend on time are called scleronomic constraints [3]. In Figure 4.7, the systems c) and d) are kinematically determined (statical systems). Otherwise, if at least one constraint is dependent on time, the system does not have a fixed configuration but evolves in time and can be considered as a kinematical or dynamical system. The kinematical constraints that depend on

14 CHAPTER 4. DYNAMICS 203 a) b) Ó c) Ó ÓÓ statically and kinematically determined support Ó d) Ó ÓÓ ÓÓ statically overdetermined support y Ó ÓÓ x Figure 4.7: Systems with various constraints and DOF: structures and mechanisms time are called rheonomic constraints [3]. If the kinematical configuration of the system is not fully constrained by the kinematical constraints, i.e. f =6p r > 0, the system has f degrees of freedom. All examples presented in Fig. 4.6 as well as the examples a) and b) in Fig. 4.7, are kinematically undetermined. Note: mechanisms are kinematically undetermined f => 0 as long as their motion is not prescribed. For example, the well known four-bar linkage posses 1 DOF if there is no rheonomic constraint which determines its motion

15 CHAPTER 4. DYNAMICS 204 (if so,f =0). Types of statical and dynamical analysis Depending on the kinematical structure of the mechanical system (Figure 4.6 and Figure 4.7), different kinds of analysis can be undertaken. ffl Kinematically determined system In the case of the kinematically determined system, a static analysis (Chapter 3) or an inverse dynamic analysis can be performed (it depends on whether the system is a statical one or its structure evolves in time). In both cases the geometrical configuration of the system is not dependent on the applied forces that are imposed on the system. The motion of the system is completely defined by the kinematical constraints [16]. ffl Kinematically undetermined system If the system is not fully kinematically constrained but posseses f degree of freedom, the time evolution of the system s kinematical configuration is not fully determined by the kinematical constraints and it is dependent on the system s applied forces [4]. To determine the system s motion, the forward dynamical analysis must be performed. If the system s constraint forces are of the interest, they can be calculated during forward dynamic analysis or subsequently after the motion of the system is determined (depending on the formulation of the system s governing equations). 4.3 Mathematical modelling Introduction to mathematical modelling Before formulating of the governing equations of multibody systems, we survey dynamics of particles and rigid bodies based on the laws of classical mechanics. The vectorial entities like displacements, velocities, forces and torques, possesing a magnitude and direction, come into account represented by vectorial components or by the scalar magnitudes (coordinates) related to the vector bases. In computational mechanics vector entities are represented by arranging the coordinates in one-dimensional arrays called matrices or in a sloppy manner vectors. Similarly, tensors are arranged in multi-dimensional arrays. The aim of adopting matrix representations is to allow for performing required vector/tensor

16 CHAPTER 4. DYNAMICS 205 operations by using operations with matrices that can be easily utilised in computer applications. Starting from the classical vector representation, the matrix equations will be derived. Dynamics of particles x z O r y f v Figure 4.8: Motion of particle By applying Newton s second law the equation of motion of a particle depicted in Figure 4.8 can be written [20] f = m r = m _v = ma = d (mv) ; (4.1) dt where mv is the linear momentum of a particle. The angular momentum of a particle with respect to O is h O = r mv : (4.2) EQUATIONS OF MOTION OF SYSTEM OF PARTICLES If the system of p particles shown in Figure 4.9 is considered, the Newton s law for the i-th particle yields f i = f e i + f i i = m ia i : (4.3) where f ie denotes the resultant of the external forces acting on the i-th particle and the resultant of the internal forces f i i is given by the equation j=1 f ij = f i i : (4.4)

17 CHAPTER 4. DYNAMICS 206 z f 1 e e f 2 r 2 f i 12 f i 21 f i 1p f i 2p f i p1 f i p2 x O r 1 y r p f p e Figure 4.9: System of p particles In the equation (4.4), f ij are the system internal forces which acts between the bodies i and j and according to the Newton s third law [12] it is f ij = f ji : (4.5) When summing up over the entire system of p particles, it can be written i=1 f e i + i=1 f i i = i=1 i=1 f e i = m i a i ; i=1 i=1 f i i =0; (4.6) m i a i : (4.7) The equation (4.7) can be elaborated by utilizing the relations ( ). The position r C of the system mass centre C is defined by mr C = i=1 m i r i : (4.8) where m = P p i=1 m i is the total mass of the system. Differentiation of (4.8) with respect to time leads to the linear momentum of the system of particles mv C = The second differentiation of (4.8) yields ma C = i=1 i=1 m i v i : (4.9) m i a i ; (4.10)

18 CHAPTER 4. DYNAMICS 207 which can be introduced into (4.7) i=1 f e i = ma C : (4.11) According to (4.11), the mass centre of the system of particles moves as if the entire mass of the system were concentrated at that point and all the external forces were applied there [20]. ANGULAR MOMENTUM OF SYSTEM OF PARTICLES i th particle, z r Ai f i x O y v A v i Figure 4.10: Angular momentum of i-th particle The angular momentum of the i-th particle about an arbitrary moving point A (Figure 4.10) is h Ai = r Ai m i v i (4.12) and differentiation leads to _h Ai = _r Ai m i v i + r Ai m i _v i : (4.13) If A coincides with the fixed point O ( _r Ai v i = v i v i = 0), then it can be written _h Ai = r Ai m i v_ i = r Ai f i ; (4.14) or _h Ai = l Ai ; (4.15) where l Ai is the resultant torque with respect to A (Figure 4.10). The angular momentum of the system of particles about the moving point A is h A = i=1 r Ai m i v i : (4.16)

19 CHAPTER 4. DYNAMICS 208 After differentiation of (4.16) with respect to time and some algebraic operations and substitutions, it can be shown that equation (4.13) for the system of particles has the form _h A = _r AC mv C + i=1 r Ai f e i ; (4.17) where r AC is the position vector of the mass centre of the system of particles with respect to point A. When point A coincides with the fixed origin O ( _r AC = _r C = v C, v C mv C = 0 ) or point A coincides with the centre of mass C, equation (4.17) reduces to _h A = i=1 r Ai f e i : (4.18) So, if A coincides with C, equation (4.17) can also be written as or where _h C = l C = i=1 r Ci f i e ; (4.19) _h C = l C ; (4.20) i=1 r Ci f i e (4.21) is the resultant moment of all external forces acting on the system of particles about the mass centre C. It should be mentioned that dynamics of particle is uniquely described by the Newton s equation (4.3). The introduction of an angular momentum (Eq. 4.13) does not bring any new information into account. It has been introduced here as a pre-stage to dynamics of rigid body, where the consideration of angular momentum leads to the essential Euler s equation. Dynamics of rigid body Prior to deriving governing equation of rigid body dynamics, in the next section some basic kinematical relations will be repeated. BASIC KINEMATICAL RELATIONS In Figure 4.11 the following notation is used:! angular velocity of a body, r A position vector of the point A (body-fixed reference point), (x; y; z) inertial coordinate system k, (x 0 ;y 0 ;z 0 ) coordinate system fixed to the body k. 0

20 CHAPTER 4. DYNAMICS 209 z z r A v A y r C r C r x O y x r Figure 4.11: Rigid body Since the body (Figure 4.11) is rigid, r is a body-fixed position vector and does 0 not change its magnitude but only its orientation due to the rotation of body. Its time derivative with respect to the inertial system k can be expressed as _r 0 =! r 0 : (4.22) With the previously introduced notation, r can be written as and the velocity is obtained as r = r A + r 0 v = v A + _ r 0 = v A +! r 0 : (4.24) The orientation of a body in the inertial coordinate system can be determined via the Euler angles '; #; ψ that specify the orientation of the fixed body system k with 0 respect to the inertial system k. Other possibilities to describe the orientation of the body-fixed system include Bryant (cardan) angles, Euler parameters, Rodriguez parameters etc. [3], [16]. The relation between! and the derivatives of the Euler angles _x R =[_' # _ ψ] _ T can be expressed in matrix form! = H R _x R : (4.25) LINEAR MOMENTUM The mass of a body is given by m = Z dm ; (4.26) m

21 CHAPTER 4. DYNAMICS 210 and the position vector of the body centre of mass C in the system k is r C = 1 m Z r dm : (4.27) The position of C with respect to the body-fixed point A is given by 0 r AC = r C = 1 m Z r 0 dm : (4.28) From (4.23), the linear momentum of the body can be written in the form Z Z Z (v A +! r 0 ) dm = v A r 0 dm ; (4.29) dm +! or m Z m m (v A +! r 0 ) dm = v A m +! mr C 0 m = m(v A +! r C 0) = mv C = m _r C : (4.30) ANGULAR MOMENTUM The absolute angular momentum of a body with respect to O (origin of the inertial coordinate system) is determined by [3] or after introducing h O = Z m The term R m h O = (r A + r 0 ) (v A +! r 0 ) dm = r A (v A +! r C 0)m + r C 0 va m + r 0 Z m r _r dm (4.31) r = r A + r 0 ; (4.32) (! r 0 ) dm can be written in the form Z r 0 (! r 0 ) dm = Z Z m r 0 (! r 0 ) dm : (4.33) (r 0 2 E r 0 r 0 ) dm! m m = I A! ; (4.34)

22 CHAPTER 4. DYNAMICS 211 where I A is the inertia tensor of the body with respect to A I A = Z m (r 0 2 E r 0 r 0 ) dm ; (4.35) E is the unit vector and r 0 r 0 denotes dyadic product. Finally, if the body s centre of mass C is chosen as the reference point A (r C 0 = 0 ; v A! v C ; r A! r C ) and equation (4.34) is taken into account, (4.33) becomes h O = r C v C m + I C! : (4.36) EQUATIONS OF MOTION OF RIGID BODY The Newton s equation determines dynamics of the body s translational motion [12] d dt (mv C)=f ; (4.37) where: mv C is the linear momentum of the rigid body, f is the resultant of all forces acting on the body. If the mass of the body is constant (dm=dt =0), the equation (4.37) becomes ma C = f : (4.38) The Euler s equation determines the dynamics of the body s rotational motion _h O = l O ; (4.39) where: h O is the absolute angular momentum with respect to the fixed reference point O in the inertial space, l O is the resultant torque with respect to O. The result of the derivative of equation (4.36) with respect to time is _h O = v C v C m + r C ma C + I C _! +! I C! ; (4.40) and since v C v C m = 0, _h O = r C ma C + I C _! +! I C!: (4.41) By substituting equation (4.41) into equation (4.39) and by considering equation (4.38), the Euler s equation can be written in form I C _! +! I C! = l O r C f = l C ; (4.42)

23 CHAPTER 4. DYNAMICS 212 or in short I C _! +! I C! = l C : (4.43) NEWTON-EULER EQUATIONS IN MATRIX FORM By following the rules of matrix algebra, the vector-valued equations (4.37) and (4.42) can be written in the matrix form [16] ma C = f ; (4.44) I C ff + ~!I C! = l C : (4.45) The matrix ff stands for the body s angular acceleration ff = _! (4.46) and the vector product is performed using the skew-symetric matrix ~!. The matrix equation (4.44) is derived from the non-coordinate expression (invariant form) (4.37) by using the inertial coordinate system k. On the other hand, the matrix equation (4.45) is derived from invariant form (4.43) by using the body-fixed coordinate system k 0. By using a body-fixed coordinate system the components of the inertia tensor of the body with respect to C remain constant. This is very convenient from the computational point of view Mathematical models and procedures Mathematical models As a result of the mathematical modelling via different methods for the formulation of the governing equations, the two basic forms of the mathematical models can be distinguished: descriptor form and minimal form. Each of these forms possesses specific characteristics, being more or less appropriate for a particular dynamic analysis. Once the model is established, these characteristics determine to a great extent the computational procedures that are to be used in the subsequent computational process [5]. ffl Descriptor form characteristics number of coordinates and differential equations are larger than the number of DOF type of differential equations: DAE lower degree of non-linearity of the differential equations ffl Minimal form characteristics

24 CHAPTER 4. DYNAMICS 213 number of coordinates and differential equations is equal to the number of DOF type of differential equations: ODE highly non-linear differential equations Governing equation (holonomic system) p...bodies, q...constraints, f...dof Full descriptor form 6p dynamical equations of the free body diagram, 6p cartesian coordinates x q kinematical constraints equations Minimal form f equations of motion, f generalised coordinates y q kinematical constraint equations Figure 4.12: Forms of mathematical model Approaches to computational procedures Independent on the form of the established mathematical model, two forms of obtaining a solution of the governing equation can be distinguished: closed form solution and numerical (approximate) solution. If the methods for obtaining numerical solution have to be applied (this is the case for the most technical applications), this can be done using either symbolic or numerical approach to computational procedures [4]. ffl Closed form solution Searching for the closed form solution pays off if there are indications that a solution of the established mathematical model can be found by using pure analytical methods (the result is expressed in the form of functions). This solution is exact under the assumptions that have been made during the mechanical and mathematical modelling of the system (the obtained solution would be free of numerical errors of any kind).

25 CHAPTER 4. DYNAMICS 214 Unfortunately, in most of the cases (except for some linear models and the simpler tasks of small dimensionality), it is not possible to find the closed form solution and a numerical procedure has to be applied to obtain the solution of the model (the numerical procedure may be launched immediately after the mathematical model is established, or some symbolic manipulations and simplifications can be performed prior to the numerical calculations). ffl Symbolic approach The symbolic mathematical operations consist of the manipulations with the mathematical entities without assigning their numerical values. If a computational tool gives possibilities for the symbolic calculations, sometimes a more efficient computational procedure can be achieved by simplifying the established mathematical model before an iterative numerical procedure is launched. Once the mathematical model using symbolic formalisms is established, it can be used for repeated numerical calculations e.g. in numerical integration schemes [11]. However, the extent to which the efficiency may be improved using the symbolic tools is dependent on the task (mathematical model) at hand. The symbolic manipulations are often computationally more costly than the numerical procedures and for some types of problems very efficient numerical procedures can be utilized e.g. the sparse matrices techniques). Although symbolic procedures are much in use in today s computation, the design and implementation of symbolic algorithms are the topics of the ongoing research activities. ffl Numerical approach By using this approach, the numerical values are assigned to the symbolic items as soon as the mathematical model is established and the whole computational process deals with the numerical values. The majority of the computational packages on today s market are numerically oriented, especially packages and tools that are designed for a general use [4] Formulation of governing equations of mechanical systems When performing dynamic analysis of a given mechanical system, the formulation of governing equations is the main part of mathematical modelling. It is the first stage of mathematical modelling independent of the dynamical task at hand (inverse dynamical problem, forward dynamics, optimization problems, etc.).

26 CHAPTER 4. DYNAMICS 215 The derived mathematical model serves as a basic set of equations by means of which the system s motion and constraint forces can be determined [18]. In the most of the cases, the basic set of equations will have to be manipulated further to suit the intended analysis and computational procedure. As it was already explained, an output of the different formalisms consists of mathematical models shaped in different forms which require different numerical procedures and algorithms in order to obtain the final solution [4]. In the sequel of the chapter the methods for formulating the governing equations of mechanical systems, that are most commonly used in the computational dynamics today, are given briefly. The main characteristics of each method as well as the application properties are provided concisely. The computational procedures, appropriate for handling specific tasks and based on the formulation methods given below, are discussed in Chapter 4.4. In Chapter 4.5 some illustrative examples are given. Multibody systems of free bodies Prior to the investigation of constrained mechanical system, a system of free bodies is considered to prescribe the nature of underlying dynamics. The multibody system of free bodies, shown in Figure 4.12, is a mechanical system of rigid bodies whose motion is not constrained by kinematical constraints of any kind. Therefore, if the system consists of p bodies, it posseses 6p degrees of freedom (DOF) [19]. The f 1 e l 1 1 f i 1p z v 1 y O f i 12 v p f i p1 f p e l p x l 2 f i 21 2 v 2 f i p2 p f 2 e f i 2p Figure 4.13: Free-body diagram of multibody system of free bodies determination of the absolute position and orientation of the i-th body of the system

27 CHAPTER 4. DYNAMICS 216 is given by the vector of the body mass centre e.g. expressed in the inertial Cartesian coordinate system (other coordinate systems can also be chosen) x Ti =[x i y i z i ] T ; (4.47) and e.g. the Euler angles of the body s absolute orientation x Ri =[' i # i ψ i ] T : (4.48) (Note: In Eq. (4.46), (4.47) and subsequent text the index C is omitted since centre of mass will always be reffered to describe a position of the body) By grouping equations (4.46) and (4.47) together, the body absolute position vector can be introduced in the form x i =[x T Ti x T Ri ]T : (4.49) Newton-Euler equations of i-th body The Newton-Euler equations are basic equations of rigid body dynamics, see Chapter 4.1. The Newton equation determines dynamics of the body s translational motion, while the body s rotational motion is determined by the Euler equation [16]. The Newton equation is given by m i a i = f i ; (4.50) or in the matrix form m i ẍ Ti = f i : (4.51) The Euler equation is expressed by or, following the rules of the matrix algebra, I i _! i +! i I i! i = l i ; (4.52) I i ff i + ~! i I i! i = l i ; (4.53) where the body s angular acceleration is given by the equation ff i = _! i : (4.54) The relation between the body s angular velocity! i and the time derivatives of Euler angles x Ri = [' i # i ψ i ] T, by means of which the absolute orientation of the body in the inertial coordinate system is specified, can be given in the form [3]! i = H Ri _x Ri ; (4.55) and the differentiation with respect to time using the chain rule yields ff i = H Ri ẍ Ri + μff i : (4.56)

28 CHAPTER 4. DYNAMICS 217 In equation (4.55), all terms in which the second derivative appear linearly are expressed in the product H Ri ẍ Ri and all others are grouped in μff i [18]. By taking into account equations (4.55), equation (4.52) can be written in the form I i H Ri ẍ Ri + I i μff i + ~! i I i! i = l i : (4.57) Futhermore, the equations (4.50) and (4.56) can be grouped together to form the Newton-Euler equations of the i-th body in the matrix form:»»»»» mi E 0 E 0 ẍti 0 fi + = ; (4.58) 0 I i 0 H Ri ẍ Ri I i μff i + ~! i I i! i l i or in short M i H i ẍ i + q v i = qa i : (4.59) The dimensions of the matrices in equation (4.58) are dim[m i ]=6 6; dim[h i ]=6 6; dim[ẍ i ]=6 1 ; dim[q v i ]=6 1; dim[qa i ]=6 1 : Newton-Euler equations of p bodies By formulating equation (4.58) for each body in the system (i = 1:::p), the Newton-Euler equations of the multibody system of free bodies [16] can be obtained in the form: MHẍ + q v = q a ; (4.62) where the matrices are specified as follows M = H = x =[x T 1 x T 2 ::: xt p ]T ; dim[x] =6p 1 ; (4.63) m 1 E I m 2 E I m p E I p E H R E H R E H Rp ; dim[m] =6p 6p ; (4.64) ; dim[h] =6p 6p ; (4.65)

29 CHAPTER 4. DYNAMICS 218 q v = I 1 μff 1 + ~! 1 I 1! 1 0 I 2 μff 2 + ~! 2 I 2! 2 0 I p μff p + ~! p I p! p q a = f 1 l1 f 2 l2 f p lp ; dim[q v ]=6p 1 ; (4.66) ; dim[q a ]=6p 1 : (4.67) In the case of the multibody system of free bodies, the Newton-Euler equations (4.59) are the equations of motion of the system. Equation (4.59) represents 6p dimensional ODE system. It can be integrated in time for specified initial conditions x 0 ; _x 0 to determine the system s motion (variables x ; _x ; ẍ). Some computational issues The inertia matrix MH in Eq.(4.59) has non-symetrical properties which may decrese significantly the efficiency of computation. In the framework of integration of governing equations, a non-symetric inertia matrix prevents use of very efficient numeric procedures which require its symetrical properties (e.g. Cholesky method). Therefore, to improve the efficiency of the procedure, it may be helpful to symmetrise the matrix MH. This can be done by premultiplying Eq.(4.59) by H T, which reads H T MHẍ + H T q v = H T q a : (4.68) In many applications the integration of (4.65) requires a less computer-power than integration of (4.59). Constrained multibody systems MATHEMATICAL MODEL IN FULL DESCRIPTOR FORM The constrained multibody system (Figure 4.13) is a mechanical system of rigid bodies whose motion is constrained by kinematical constraints. If the system consists of p bodies whose motion is constrained by q kinematical constraints, the sys-

30 CHAPTER 4. DYNAMICS 219 tem possess f =6p q DOF [21]. e 1 l2 e k f1 e 1 f1 e h l2 Ó Figure 4.14: Constrained multibody system (mechanical model) The following notation is used: f e j i :::j-th (j =1:::k) applied external force that acts on the i-th body (i =1:::p) l e j i :::j-th (j =1:::h) applied external torque that acts on the i-th body (i =1:::p). In Figure 4.13 this is ilustrated at the body i =1. Forces in constrained multibody system Applied forces In Figure 4.14, the free-body diagram of constrained multibody system is derived. The resultant applied force that acts on the i-th body is f i = f e i + f i i ; (4.69) where the resultant force of the external applied forces, reduced to the centre of mass C i,is f e i = kx j=1 f e j i (4.70) and the resultant force of the internal applied forces (internal springs, dampers, etc.), reduced to the centre of mass, is given by the sum of the internal applied forces between the bodies i and j f i i = j=1 f i ij : (4.71)

31 CHAPTER 4. DYNAMICS 220 c l1 c f21 e f 1 l1 c f12 i f1p i fp1 c lp e f p lp l2 e f 2 c l2 i f2p i fp2 c fp0 Figure 4.15: Free-body diagram of constrained multibody system The reduction of forces with different application points to one specific point implies the formulation of an equivalent couple of force and torque acting at this specific point. The resultant torque about the centre of mass C i of the applied forces and torques that act on the i-th body (Figure 4.14) is l i = hx j=1 l e j i + j=1 l i ij + l red i ; (4.72) where l i ij is an internal applied torque (internal torsional spring, for example) that acts between bodies i and j and l redi is the torque due to the reduction of the forces f e and i f i to the centre of mass i C i. Constraint forces The resultant constraint force that acts on the i-th body, reduced to the centre of mass C i (Figure 4.14), is given by the equation f c i = j=0 f c ij ; (4.73) where f c ij is a constraint force that acts between bodies i and j (i; j = 0:::p, the index 0 stands for the body of external world ).

32 CHAPTER 4. DYNAMICS 221 If an index i or j is equal to zero, the constraint force is of the external type (a force due to a kinematical constraint with the external world, a bearing for example). If none of indices i; j is zero, the constraint force is of the internal type (due to a kinematical constraint that restricts relative motion of the bodies, revolute joint for example). The resultant torque about the centre of mass C i of the constraint forces and torques that act on the i-th body can be given in the form l c i = j=0 l c ij + l c red i ; (4.74) where l c ij is a constraint torque that acts between bodies i and j or a body and the external world (i; j =0:::p) and l c red i is a reduction torque of the forces f ij. c Newton-Euler equations of i-th body As previously mentioned, the Newton equation determines the dynamics of a rigid body s translational motion, while the body s rotational motion is determined by the Euler equation. When a body is kinematically constrained, the constraint forces and torques also influence the motion of a body and have to be considered in the framework of the Newton-Euler equations together with the applied forces and torques [19]. The Newton equation is given by m i a i = f i + f c i ; (4.75) or in the matrix form m i ẍ Ti = f i + f c i : (4.76) The Euler equation is expressed by or following the rules of the matrix algebra I i _! i +! i I i! i = l i + l c i ; (4.77) I i ff i + ~! i I i! i = l i + l c i : (4.78) By considering (4.55), equation (4.74) can be expressed in the form I i H Ri ẍ Ri + I i μff i + ~! i I i! i = l i + l c i : (4.79) After introduction of the body absolute position vector (4.60), equations (4.73) and (4.76) can be grouped together to form the Newton-Euler equations of the i-th body»»»»»» mi E 0 E 0 ẍti 0 fi f c + = + i 0 I i 0 H Ri ẍ Ri I i μff i + ~! i I i! i l i l c ; i (4.80)

33 CHAPTER 4. DYNAMICS 222 or in short M i H i ẍ i + q v i = qa i + qc i : (4.81) The dimensions of the matrices in equation (4.78) are dim[m i ]=6 6; dim[h i ]=6 6; dim[ẍ i ]=6 1 ; dim[q v i ]=6 1; dim[qa i ]=6 1; dim[qc i ]=6 1 : Newton-Euler equations of constrained system of p bodies By formulating equation (4.78) for each body in the system (i = 1:::p), the Newton-Euler equations of the constrained multibody system can be arranged in the form MHẍ + q v = q a + q c : (4.84) As it will be described in the sequel, the system constraint forces q c in the equation (4.79) can be expressed via kinematical constraint equations and additional parameters. Governing equations of constrained multibody systems Kinematical constraints equations The Newton-Euler equations (4.79) are part of the governing equations of the constrained multibody system. Since motion of the system bodies is kinematically constrained, the components of the system position vector x are not independent but satisfy a set of q kinematical constraint equations, which can be put in the form [16] g(x;t)=0 ; dim[g] =q: (4.85) By differentiation of (4.80) with respect to time, the equation that expresses the relation between the system velocities is = 0 ; (4.86) or in the short form where the matrix Q is defined as Q ; (4.87) ; dim[q] =q 6p : If (4.80) is differentiated twice, the equation that expresses dependency between system accelerations can be formulated. After application of the chain rule of differentiation, the kinematical constraint equations at the level of acceleration can be written in the short form Q ẍ = μc : (4.89)