Scholarly article on topic 'The equations of Lagrange for a continuous deformable body with rigid body degrees of freedom, written in a momentum based formulation'

The equations of Lagrange for a continuous deformable body with rigid body degrees of freedom, written in a momentum based formulation Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — H. Irschik, M. Krommer, M. Nader, Y. Vetyukov, H.-G. von Garssen

Abstract The present paper is concerned with Lagrange׳s Equations, applied to a deformable body in the presence of rigid body degrees of freedom. The Lagrange description of Continuum Mechanics is used. An exact version of the Equations is derived first. This version, which represents an identical extension of the Fundamental Law of Dynamics, does involve the idea of virtual motions. The virtual motion is described in the framework of the Ritz-Ansatz, but our derivation does not make use of D׳Alemberts principle, the principle of virtual work, or variational principles. From the exact version, by involving arguments related to the Galerkin approximation technique, we derive an approximate Ritz type version of Lagrange׳s Equations. This approximate version coincides with the traditional one, which is based on the notion of kinetic energy. However, since our derivation stems from the Fundamental Law of Dynamics, we have at our disposal an alternative formulation, which is based on the notion of local momentum. This momentum based version, which is the main topic of the present contribution, can be used for the purpose of performing independent checks of the energy based version of Lagrange׳s Equations. The momentum based version also clarifies that and how certain terms in the energy based version do cancel out. The momentum based version is worked out in the framework of the Floating Frame of Reference Formulation of Multibody Dynamics. Explicit formulas for the single terms of Lagrange׳s Equations are derived for the translational, rotational and flexible degrees of freedom of the deformable body, respectively. Corresponding Lagrange׳s Equations are explained in the light of the relations of Balance of Total Momentum, Balance of Total Moment of Momentum, of the Mean Stress Theorem and the notion of Virial of Forces. An embedding into the literature is given.

Academic research paper on topic "The equations of Lagrange for a continuous deformable body with rigid body degrees of freedom, written in a momentum based formulation"

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Journal of Sound and Vibration

journal homepage: www.elsevier.com/locate/jsvi

The equations of Lagrange for a continuous deformable body with rigid body degrees of freedom, written in a momentum based formulation

H. Irschik3'*, M. Krommera, M. Naderb, Y. Vetyukova, H.-G. von Garssenc

a Institute of Technical Mechanics, Johannes Kepler University Linz, Austria b Linz Center of Mechatronics (LCM), Austria c Siemens Corporate Technology, Munich, Germany

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ARTICLE INFO

Article history: Received 24 January 2014 Received in revised form 25 July 2014

Accepted 18 August 2014 Handling Editor: W. Lacarbonara Available online 23 October 2014

ABSTRACT

The present paper is concerned with Lagrange's Equations, applied to a deformable body in the presence of rigid body degrees of freedom. The Lagrange description of Continuum Mechanics is used. An exact version of the Equations is derived first. This version, which represents an identical extension of the Fundamental Law of Dynamics, does involve the idea of virtual motions. The virtual motion is described in the framework of the RitzAnsatz, but our derivation does not make use of D'Alemberts principle, the principle of virtual work, or variational principles. From the exact version, by involving arguments related to the Galerkin approximation technique, we derive an approximate Ritz type version of Lagrange's Equations. This approximate version coincides with the traditional one, which is based on the notion of kinetic energy. However, since our derivation stems from the Fundamental Law of Dynamics, we have at our disposal an alternative formulation, which is based on the notion of local momentum. This momentum based version, which is the main topic of the present contribution, can be used for the purpose of performing independent checks of the energy based version of Lagrange's Equations. The momentum based version also clarifies that and how certain terms in the energy based version do cancel out. The momentum based version is worked out in the framework of the Floating Frame of Reference Formulation of Multibody Dynamics. Explicit formulas for the single terms of Lagrange's Equations are derived for the translational, rotational and flexible degrees of freedom of the deformable body, respectively. Corresponding Lagrange's Equations are explained in the light of the relations of Balance of Total Momentum, Balance of Total Moment of Momentum, of the Mean Stress Theorem and the notion of Virial of Forces. An embedding into the literature is given. © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

Since the discovery of Lagrange's Equations, see Dugas [1] for the history, these relations have appealed engineers and researchers from both, the application and the fundamental point of view. A collection of solved problems has been recently

* Corresponding author. E-mail address: hans.irschik@jku.at (H. Irschik).

http://dx.doi.org/10.1016/josv.2014.08.016

0022-460X/© 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

presented by Gignoux and Silvestre-Brac [2]. For a tutorial presentation, which has inspired the present contribution, see Ziegler [3] as well as Irschik et al. [4].

In the framework of mechanical systems that consist of a finite number of discrete material particles, a critical discussion on various methods of deriving Lagrange's Equations was given in a paper by Casey [5], who made the following classification: (a) derivations leading from Newton's law (formulated for the single particles in the physical space) by a formal manipulation of partial derivatives, (b) derivations resting on D'Alemberts principle and the principle of virtual work,

(c) derivations originating from variational principles, and (d) derivations employing differential geometry and tensor calculus. While in [5] Casey criticized methods (a)-(c) as unenlightening, obscure, or suffering from serious shortcomings, respectively, he succeeded in presenting a concise and elegant extension of the geometrical method (d), where he described the system of particles as a single fictitious particle that moves in a higher dimensional configuration space. From a generalized Newton's law formulated in the configuration space for the fictitious particle, he eventually derived Lagrange's Equations for the system of particles. This geometrical method of derivation, which is also very well suited for understanding the role of constraints, was extended by Casey [6] to a rigid continuum, and he presented a geometrical derivation of Lagrange's Equations for a pseudo-rigid body in [7]. For generally deformable (non-rigid and non-pseudo-rigid) continua, however, the geometrical method (d) for deriving Lagrange's Equations at present appears to be an open problem.

In the literature, the derivation of Lagrange's Equations for deformable bodies usually is performed utilizing Continuum Mechanics based extensions of the methods (a)-(c), performing a reduction of the continuous body to a system with a finite number of degrees of freedom via the Ritz approximation technique. A derivation in the framework of the Lagrange or material description of Continuum Mechanics was presented by Washizu [8], while a derivation of Lagrange's Equations in the context of the Euler or spatial description can be found in Ziegler [3]. The rigid body is included as a special case in these formulations. A historical exposition on the Rayleigh-Ritz technique was presented by Leissa [9]. In the framework of the Ritz approach, Lagrange's Equations also can be applied successfully to systems consisting of several continuous deformable bodies that are connected by joints, springs and dampers, denoted as multibody dynamic systems, see the books by Shabana [10] and Bremer [11].

The present paper is motivated by the previous work of our group on a deformable single body in a rapid motion, where we have described the motion of the deformable body by a problem-oriented combination of Lagrange's Equations and the Ritz approach, see Refs. [12-14]. In their traditional form, which we have utilized in the latter contributions, Lagrange's Equations are based on the notion of kinetic energy in the form of partial derivatives with respect to generalized coordinates and velocities, see [3,8]. It however appears to be desirable to have an alternative method of description at disposal, which does not start from the notion of kinetic energy, in order to perform an independent check of the resulting equations of motion. Moreover, it is to be required that the mathematical form of the equations of motion should be as compact as possible, in order that these equations are accessible to modern model based control methods utilizing symbolic computation. In this respect, terms that are annihilating one another should be avoided in the formulation from the onset.

It is the scope of the present contribution to derive an alternative form of Lagrange's Equations, which satisfies the above requirements. It will turn out that this formulation is governed by the notions of momentum, of moment of momentum, mean stress and an extended virial of forces. We therefore will subsequently talk about a momentum based derivation, instead of the traditional kinetic energy based derivation. Notwithstanding the conceptual merits of the geometrical method

(d) in the configuration space, the considerations in the present paper will be based on an extension of the method (a) mentioned above, where we start from the local relation of balance of momentum in physical space, also denoted as the Fundamental Law of Mechanics, see Ziegler [3]. In the present paper, we utilize the Lagrange description of nonlinear Continuum Mechanics. In the Lagrange description, the physical entities associated with the material particles of a continuous body are referred to the place of these particles in a reference configuration. In the Euler formulation, which we do not utilize here, the actual place of the material particles is used for the sake of description, where a reference configuration comes into play just for describing the elastic properties of the continuum and its volumetric density. For detailed discussions on the Lagrange and the Euler descriptions of Continuum Mechanics, and for their history, see Truesdell and Toupin [15]. A kinetic energy based derivation of Lagrange's Equations involving both, the Lagrange and the Euler descriptions of Continuum Mechanics, has been presented in [16].

Our paper is organized as follows. In Section 2 below, Lagrange's Equations are first derived as an exact consequence of the Fundamental Law of Dynamics. In order to do so, we mathematically manipulate the latter vectorial relation, which describes our actual problem at hand, by a scalar multiplication with certain vectorial entities that belong to some virtual motion. This virtual motion is imagined to be described by a finite set of time-dependent generalized coordinates in the framework of a Galerkin method. For a general methodology of mathematically manipulating the fundamental relations of balance and jump, see Ref. [17]. From a systematic point of view, the present contribution can be understood as an extension of the results in the latter paper with respect to Lagrange's Equations, which were not addressed there. Integrating over the body under consideration, a form of Lagrange's Equations is first obtained, which must be satisfied identically by the exact solution of the actual problem at hand. This form, which however is nothing else than an identical transformation of the Fundamental Law of Dynamics, does not involve a Ritz approach for the actual motion, but only for the virtual one; it however gives a firm justification for using the notion of virtual displacements in a nonlinear Continuum Mechanics setting, without involving the principles needed for methods (b) and (c) mentioned above. A related preliminary version of the derivations in Section 2 has been recently stated in a contribution by Irschik and Holl [18] on Lagrange's Equations for

mechanical systems with a flow of mass through the boundary, for so-called open systems, in which latter contribution the possibility of obtaining a momentum based formulation of Lagrange's Equations however was not stressed. In the present paper, we assume that mass is conserved for the deformable body under consideration. For energy based derivations of Lagrange's Equations for systems in which mass is not conserved, see, Ref. [18], and Cveticanin [19], Irschik and Holl [20] and Pesce [21].

In Section 3, which deals with an approximate form of Lagrange's Equations, we assume that the virtual motion represents a candidate for obtaining a sufficiently accurate approximation of the actual one. In the Lagrange description of Continuum Mechanics, we eventually obtain a form of Lagrange's Equations, which was derived before by Washizu [8], who involved a Ritz-approach for the actual motion from the onset of his considerations. Thanks to the momentum based derivation in Section 2, we are however in the position to replace the kinetic energy based terms in Lagrange's Equations by momentum based relations. In the framework of this momentum formulation, we can show that and how certain terms must cancel out in the traditional kinetic energy based version of the formulation, and that this computationally unappealing behavior is absent in the present momentum based formulation of Lagrange's Equations; for an analogous study in the dynamics of rigid bodies see Irschik et al. [4].

In Section 4, the momentum based approximate formulation is adapted in order that the Floating Frame of Reference Formulation can be applied, the use of which is frequent in the field of Multibody Dynamics, see Shabana [10]. Accordingly, the momentum based formulation of Lagrange's Equations is explicitly worked out in Sections 5-7 below for the translational, the rotational and the flexible generalized coordinates, respectively. Particularly, we present a relation to a practically valuable formulation for the global relations of Balance of Total Momentum and Balance of Total Moment of Momentum presented by Ziegler [3]. With respect to the flexile coordinates, we interpret the results as a special case of the Mean Stress Theorem discussed by Truesdell and Toupin [15], using an extension of the notion of Virial of Forces. As a main result of the present paper, in Sections 5-7 and in Appendices A-C below we present explicit expressions for the dynamic terms in Lagrange's Equations in the momentum based formulation, separately for the translational, rotational and flexible generalized coordinates. These explicit expressions are given in condensed form, and should therefore be valuable for programming purposes. In Appendix D, further computationally advantageous simplifications are noted, and it is shown that for the special case of vanishing of the flexible coordinates, i.e. for a rigid body, the present formulation in physical space coincides with results on a rigid continuum found earlier by Casey [6] in the configuration space, see Casey and O'Reilly [22] for systems of rigid bodies. Our paper ends with some concluding remarks, in which we extend the interpretation concerning the Mean Stress Theorem to the total of Lagrange's Equations, and present additional relations to Analytical Mechanics.

2. Lagrange's Equations as an exact consequence of the fundamental law of dynamics

In the following, we consider the motion of a continuous deformable material body, which is described by a certain initial-boundary value problem at hand. Since the solution of this problem is our actual concern, we subsequently will denote this motion as the actual motion of the body. The actual motion must satisfy the local relation of balance of linear momentum, also denoted as the Fundamental Law of Dynamics, Ziegler [3]. In the Lagrange or material description of Continuum Mechanics, the Fundamental Law of Dynamics can be written as

b0 + Div n^j =0 (1)

The linear momentum associated with a material particle in the actual position, the local linear momentum J 0, taken per unit volume in the reference configuration, is given by

J' = P0 v (2)

The above relations hold with respect to an inertial frame. In the following, a certain inertial frame is fixed and denoted as the global inertial frame. In the Lagrange description, every entity is described as a function of the place of the material particles in a reference configuration and of time t. In order to label the place in the reference configuration, we subsequently use the time-independent vector-type entity P. When the reference configuration is taken at rest with respect to the global inertial frame, the position vector of the particle in the reference configuration can be used for P. A proper definition of P in the case of an arbitrarily moving rigid reference configuration will be given detail in the next section, where we will deal with a so-called Floating Frame of Reference Formulation. In any case, the field of the actual positions of the particles from the origin of the global inertial frame at time t in the Lagrangian description obeys a functional dependence written as r = r (P, t). For the sake of brevity, we do not distinguish between an entity and its functional representation. The absolute velocit-y in the actua l position is formed by the material time derivative with respect to the global inertial frame, v = d r =dt =v (P, t), where P is kept fixed. The mass per unit volume in the reference configuration is p0 = p0(P), which is assumed to bes constant with respect to time in the following, but it may vary throughout the reference configuration. Moreover, b0 = b0(P, t) is the body force per unit volume in the reference configuration, the first Piola-Kirchhoff stress tensor is denoted as n = n(P, t), and the symbol Div stands for the divergence operator with respect to the place of the particles in the reference configuration.

The Fundamental Law of Dynamics, Eq. (1), represents a single vector partial differential equation in space and time. In order to relate it to a system of scalar ordinary differential equations of the type of Lagrange*'s Equations, we consider some virtual motion of the body. The virtual motion is described by a field of position vectors r , where it is imagined that the functional dependency upon time can be further parameterized by means of a set of l time-dependent generalized coordinates qk=qk(t), k=1,...,/, such that we may write

— * r =

P, q\t),

, qk(t),

, q'(t)

The superscript (*) will be used for indicating entities that belong to the virtual motion, in contrast to the actual motion. Note, however, that we utilize the same reference configuration for both, the actual and the virtual motion, the place of the particles in the reference configuration being labeled by the entity P, which shall be considered as an identifier of a material particle at Lagrangian description, see Ziegler [3], Vetyukov [23]. The use of upper indices for numbering the generalized coordinates qk allows us to remain compatible with the conventional summation rules in the following. For the sake of simplicity, no explicit time-dependence is considered in Eq. (3), and qk are taken as independent from one another. Thus, holonomic constraints, if any, are already taken into account in the description of the virtual motion stated in Eq. (3). The absolute velocity of the particles in the virtual motion is obtained as the material time derivative of r , where Eq. (3) yields

V = dtr (P'

, qk(tt),

kè, W

= V*( P,...,qk(t),...,...,qk(t),...), k = 1,...,'

which brings into the play the generalized velocities

qk = dqk/dt = qk(t).

Recall that for performing the material time

derivative d/dt, it is taken into account that qk and qk in general do depend on time, but P is kept fixed. It is assumed that no non-integrable constraints are to be considered for the generalized velocities qk in the virtual motion. Note, however, that the presence of such non-holonomic constraints would not decrease the number l of the generalized coordinates considered in the virtual motion, see Casey [5]. For methodologies of using a smaller set of so-called quasi-coordinates in the non-holonomic case, the interested reader is referred to the Multibody Dynamics formulation of Bremer [11].

The following useful relations can be derived as consequences of the functional dependencies stated in Eqs. (3) and (4), see e.g. Ziegler [3] and Lurie [28]:

dr* dv*

w=~¡q

d id r *\ dv * dtl dqk = dqk

Moreover, we recall the definition of the kinetic energy per unit volume in the reference configuration, associated with a particle in the virtual motion, as

1 r n r n

T* =1 Po v V

The scalar product is indicated by a dot. Introducing the local linear momentum in the virtual motion

J = Po v

the following two relations follow by partial differentiation:

* * d v d v

df=Po v ■ df=J

Hence, using Eqs. (5) and (6), we find that

w=Po v U W=J

d J' dv_ 3( d v \ J,* did v \

dt dq k dn J

ät\ dqk ät\ dq k dqk

which holds for every k=1,.,/.

In order to develop the desired system of scalar ordinary differential equations in the form of Lagrange's Equations, we let ourselves be motivated by Eq. (11), and thus perform a scalar multiplication of the Fundamental Law of Dynamics, Eq. (1),

— * k

by means of the vectors dv /dq , for k-1,...,/. This results in the following set of l relations:

— — * — *

J . Êq~(b.+Div n ■ dqk - . k _ 1.....l (12)

Subtracting Eq. (12) from Eq. (11), the following system of scalar ordinary differential equations in time is obtained:

AÎÊH)_ OH-o ,(a-*>- k-1 I (13)

dt{dqk) dqk -0k ; k - 1 l (13)

where the generalized force, taken per unit volume in the reference configuration, follows to:

or"-@ b .+Div n - ¡Afj. % (,4)

The difference in the local linear momentum between actual and virtual position of the particle is

— — — — — *

AJ'- J'- J '* - Po ( V - v ) (15)

It is understood that the entities in Eqs. (11) and (12) are taken for the same instant of time, and for the same material particle. Note that we have used the superscript (a, *) in order to indicate that entities from both, the virtual and the actual motion are involved in Eq. (14). Integrating Eq. (13) over the volume V0 of the reference configuration, and introducing the total kinetic energy associated with the virtual motion

T* - f T'* dV0 (16)

the following global relations, which have the well-known kinetic energy based form of Lagrange's Equations, are obtained:

1 - dH = Qian). k = 1 l (17)

dt^qV dqk =Qk ' k =1' (17)

The total differential with respect to time is used, since Eq. (17) refers to the entire body, in contrast to the local formulation in Eq. (13). The total generalized force is

Q(a,n) = r dv0

/ s \ s n s n s n

= 1. (b0-+№) %dS°-L? G"d%dv» (18)

where the divergence theorem has been used, and n 0 denotes the unit outer normal vector at the surface S0 in the reference configuration. The gradient operator with respect to the place in the reference configuration is abbreviated by Grad.

Concerning Eqs. (13) and (17), the following remarks seem to be in order. First, recall that the derivation of Eq. (13) has involved entities associated with the same material particle in both, the virtual and the actual motion, and at the same instant of time. Thus, it has become possible to perform a unifying integration of the local relations in Eq. (13) over the volume of this reference configuration. Since the latter has been taken as rigid, where the surface of the reference volume does not move relatively to the particles located on it, we have been allowed to interchange the integration and the differentiation, which leads to the global relations in Eq. (17). Note that this operation of interchanging rests upon the assumption that the mass of the system is conserved. It is to be moreover noticed that Eqs. (13) and (17) do represent exact relations, in the sense that they must be satisfied exactly, if the exact solutions of the actual problem at hand for v and n are substituted into the generalized force Q'(a'*) in Eq. (14). However, this is of little wonder, since Eqs. (13) and (17) have been obtained by an identical transformation of the Fundamental Law of Dynamics, see Eq. (12). Tacitly, it has been only assumed above that the actual and virtual motions under considerations are nice enough in order that the differentials and integrals in the above derivations make sense. From a systematic point of view, it might be noticed that we have not considered variational techniques or virtual work principles. Instead, we have derived Eqs. (13) and (17) as consequences of the Fundamental Law of Dynamics, which has made the derivation short-handed and free from the need of introducing certain smallness assumptions for the virtual motion under consideration.

3. Approximate form of Lagrange's Equations

In principle, the exact solutions for v and n for the actual problem at hand need not to be related to the generalized coordinates qk, which have been used above for defining the virtual motion r , see Eq. (3). On the other hand side, Lagrange's Equations in the form of Eq. (17) generally are not sufficient for obtaining the exact solution for the actual motion, even when constitutive relations for the stress n and suitable initial and boundary conditions have been suitably stated. In order to approximately close the problem, i.e.ein order to obtain approximate solutions, we now assume that the virtual

placement in Eq. (3) and the virtual velocity in Eq. (4) represent suitable candidates for sufficiently close approximations of the actual motion. We thus split the exact solution in the virtual one and corresponding error terms

^n — — — n

r = r + rE, v = v + ve, J' = J'+ Je, Ji = Po vE, n = 77* +77 (19)

where the error terms are assigned with an index e, and where we have introduced a local error momentum J 'E. The error terms are again functions of the place Pand the time t, e.g. v e = v e(P, t), etc. In the framework of the formulation stated in

Eq. (19), with Eq. (11) we are allowed to write, instead of Eqs. (13) and (14),

* * * *

dJ' dv dJ' d r d 13T*\ dT* ' ~dif = ~dt ' = ~ ~dqk

J _ DiV n) U ^

dt ~e dqk

= Qk _ Rf _ Div nj k = 1,I (20)

with the approximate generalized force

dv* dr*

Qn = (bo + Div n*) ■ dqr =(bo + Div 77*) • (21)

Note that Eqs. (20) and (21) represent only a re-formulation of the exact formulations given in Eqs. (13) and (14). In order to obtain their global forms, Eqs. (17) and (18), we need to integrate Eqs. (20) and (21) over the volume in the reference configuration . This is the point, where we leave the exact formulation in order to obtain an approximation. We interpret the expression dJ 'E =dt - Div nE in Eq. (20) as an error body force per unit volume in the reference configuration. Motivated by the Galerkin method, see Ziegler [3], Bremer [11], Vetyukov [23], we require that the weighted integral of this error force over the volume in the reference configuration must vanish, where we use d r =dqk as weighting function. Completeness of the approximation Eq. (3) with increasing number of degrees of freedom in the model l shall then guarantee the convergence of the resulting approximate solutions towards the exact one. Indeed, the equivalence of the Galerkin orthogonality condition and Lagrange's Equations in the framework of the Ritz approach has been shown by Chiroiu and Nicolae [24] for non-homogeneous viscoelastic solid dynamics.

Now, setting the weighted volume integrals of the error force to zero, the superscripts (*) can be omitted for the sake of brevity, since no distinction between actual and virtual motion is necessary any longer. This abbreviation will be used from now on, also when referring to the formulas in Section 2 above, where we no longer will distinguish between the entities r and r , J' and J' , etc.

We thus can write

Xo l|dV 0 - ¡VoJ • % dV 0

4(1) -W -- k -- 1.....l (22)

with the generalized forces, see Eqs. (18) and (5)

Qk={ QkdVo

- i b0 • |rkdV0 +i t„0 • |r,dS0 -f n • Gradl^dV) (23)

Jv0 dqk Js0 0 dqk .¡V0 ~ dqk

where we have introduced the surface traction t„0 — nn0. In order to evaluate the last volume integral in Eq. (23), expressions for the approximate stress n must be computed by substituting r and its derivatives into the actual constitutive relations of the problem at hand, using the approximation stated in Eq. (3). The surface integral in Eq. (23) follows from the boundary conditions at the surface S0.

Again, some remarks seem to be in order. First, it might be noticed that the above sketched approximation procedure conceptually differs from the usual approximate procedures of deriving Lagrange's Equations for a continuous body, which do assume dependencies of the Ritz type in Eqs. (3) and (4) for the actual motion from the onset, and in which virtual changes are studied afterwards, see Ziegler [3] and Washizu [8]. In Eq. (22), we indeed have obtained the kinetic energy based formulation stated in the latter references; however we now have at our disposal also an alternative momentum based version.

It is also to be mentioned that the quality of the approximation will depend on the appropriateness of the functional dependency in Eqs. (3) and (4) for the problem at hand. Particularly, in the following we must accept also for the actual motion the independence conditions for the generalized coordinates and velocities that have been introduced above for the virtual motion. Additional constraint force terms involving Lagrange multipliers however could be added to Eq. (22) in order to overcome this deficiency, analogous to procedures presented in the book by Shabana [10]. See Bremer [11] for the

additional constraint force terms that are needed for classical non-holonomic constraints, as well as for a valuable alternative formulation using quasi-coordinates and the so-called projection equation. A rather general class of constraint forces has been successfully treated by O'Reilly and Srinivasa [25]. For the sake of brevity, however, in the following we remain in the framework of independent generalized coordinates and velocities.

As a main consequence of Eqs. (22) and (23), we can say that we are now in the position to relate Lagrange's Equations to the notion of local linear momentum per unit volume in the reference configuration, J0 , see Eq. (2), in a straight-forward manner. Hence, the terms containing J0 in Eq. (22) subsequently will be used to derive alternative formulas for explicitly computing the single terms in Lagrange's Equations, replacing the notion of local kinetic energy by the notion of local linear momentum. In order to have a motivation for this procedure, note from Eqs. (5), (6), (9) and (10) that

dv\ dJ' dv

(Sr.) \dqkJ

dT' _ j, dv _ J, d ( d r'

dqk J dqk J dt\ dqk

It is thus seen that the term dT0/dqk, and hence also the total dT/dqk, indeed do cancel out in the energy based formulation of Lagrange's Equations, Eqs. (20) and (22), a fact, which one can easily observe in practice when performing computations by hand, e.g. for working out some tutorial problems [4].

4. Lagrange's Equations adapted for the floating frame of reference formulation

The approximate relations stated in Section 3 above now will be applied in the framework of the Floating Frame of Reference Formulation, which is suitable for describing the motion a deformable body having additional rigid body degrees-of-freedom, and as such forms an important basis of the Multibody Dynamics technique, see Shabana [10]. In the framework of the Floating Frame of Reference Formulation, see Fig. 1, we describe the deformation of the body with respect to a rigid reference configuration, which moves together with the actual motion of the deformable body, and which is called the floating reference configuration.

We rigidly attach to the floating reference configuration a Cartesian coordinate system with base vectors Ex, a _ 1,2,3, denoted as the floating coordinate system, since it moves together with the floating reference configuration. The actual position vector of the origin A of the floating coordinate system, taken from the origin O of the global inertial frame, is expressed as

rA _ 2 xact) ea (26)

For the global inertial frame, a space-fixed inertial Cartesian coordinate system with base-vectors ea, a _ 1,2,3, and with origin in point O is used in Eq. (26). This system is denoted as global coordinate system in the following. The three

floating reference configuration

coordinates of rA in the global coordinate system are used as the first three of the generalized coordinates in Eq. (3)

q1 (t) _ x\(t), q2(t) _ xiit), q3(t) _ x3(t) (27)

Subsequently, we denote these as the translational coordinates. Since the base vectors e a do not change with time, we obtain the following functional representation:

rA _ 2 qa(t) ea _rA(q\t), q2(t), q3(t)) (28)

In the floating coordinate system, the directed vector from the origin A to the place of the particle in the reference configuration can be written as

p _ 2 Pa Ea (29)

where the components Pa, the Cartesian coordinates of particles in the reference configuration, do not depend on time, while the base vectors Ea do move with the floating reference configuration. The components P shall be used for labeling the place of the particles in the moving reference configuration. In Eq. (4) and in the following, this is performed by introducing the time-independent vector, see Fig. 1:

- 3 . 3 dn - dr 3 d2 r ->

P _ „?!P e * Div n _ a21 dpaGraddq. _ B?1 WW.® etc (30)

see Eqs. (1) and (23).

Now, each of the moving base vectors E in Eq. (29) can be obtained from the global base vectors ea via a single time-dependent rotation tensor R, which describes the rotation of the co-moving rigid reference configuration with respect to the global coordinate system. As is well known, the rotation tensor is orthogonal, R ~1 _ RT, and it can be completely described by three time-dependent rotation angles Ê*1, d2 and 03, Shabana [10]. Subsequently, these rotation angles will be used as the next three of the generalized coordinates qk in Eq. (3), and will be called the rotational coordinates:

q4(t) _ 61 (t), q5(t) _ 02(t), q6(t) _ 03(t) (31)

The global base vectors e do not change with time, when seen from the inertial frame. We thus can write

Ea _ Re _ R(q4(t), q5 (t), q6(t))e _E (q4(t), q5(t), q6(t)) (32)

Hence we obtain from Eqs. (29) and (32) that

p _ R 2 Pa ea _ R P _p( P, q4(t), q5(t), q6(t)) (33)

a_ 1 v '

The remaining generalized coordinates in Eq. (3) are used to describe the deformation of the body with respect to the co-moving reference configuration. For the total position vector, we write

r _ rA + p + u (34)

with the field of displacement from the floating rigid reference configuration

u _ 2 q<pj, (35)

In Eq. (35), the generalized coordinates qj(t), j_7,...,/, represent the so-called flexible coordinates, and the corresponding are Ritz Ansatz-vectors, also denoted as deformation modes, which approximate the displacement from the floating reference configuration, and which are to be selected in a problem-oriented manner. We utilize the following functional representation for the deformation modes:

I._ 2,<P)E(q4(t),q5(t),q6(t))_-k{P,q4(t),q5(t),q6(t)) (36)

and hence

l 3 - , - ,

u _ 2 qj 2 Ea _u(P, q4(t), q5(t), q6 (t), q7(t),..., ql(t)) (37)

j _ 7 _ 1

Note that the components of the deformation modes in the floating coordinate system are taken as time-independent,

4a _ 4k (P) in Eq. (36). Also, a mathematically more complex Ritz Ansatz, not assuming a series that consists of terms each of which is separable in space and time, could have been associated with Eq. (3), instead of Eq. (35). Nevertheless, in the following we shall utilize the traditional separable form, Eqs. (35)-(37).

In the subsequent sections, the functional dependencies stated in Eqs. (26)-(37) will be used in order to conveniently perform the partial derivatives necessary for computing Lagrange's Equations in the momentum-based formulations of Eqs. (20) and (22). For the local linear momentum in Eqs. (20) and (22), we also need the absolute velocity of the particles in the framework of the Floating Frame of Reference Formulation, see Eq. (2). From Eq. (34), we obtain the total velocity as the material time derivative

where Eqs. (28), (29), (35) and (36) yield

- d rA d

v = St + TTÀp + u

§= 1" = ji

i(p + u) = Z^P. + ji Wffi + ji7 to,

Using the fact that unit vectors of the floating coordinate system can be obtained from the unit vectors in the global system by means of a rotation with matrix R, it can be shown that

dEa - -

ST = w x E

where the vector of angular velocity of the floating coordinate system, and thus of the floating rigid reference configuration, can be expressed as

- = i q gj

The three vectors g{ — g{(t), k — 4,5,6, form a skew covariant basis; they are the axial vectors belonging to the skew-symmetric matrices

C, = -^RT = -R

~k qk~ ~ qk With Eq. (41), it is found that Eq. (40) can be written as

k = 4,5,6

dt(p+u) = - x ( p+u) + x^fy

Using Eqs. (39)-(44), we obtain for the velocity in Eq. (38) that

- 3 . - - / - - x l -

v = z qqe+wx [p+u) + x q<P,

j = 1 V 7 j = 7

For the momentum-based formulation of Lagrange's Equations in Eqs. (20) and (22), we need the time rate of the local momentum in Eq. (2). Since the mass per unit volume in the reference configuration is time-independent, we have

dJ' _ dv

St = Po St

From Eq. (45), we obtain the absolute acceleration again by performing the material time-derivative. Using Eq. (44), this yields

dv = ^ qq ej+x( p + u) + w x (w x( p + + 2w x i q-j + i q'"Î>j

The relation in Eq. (47) coincides with the results of the kinematics of single mass-points, see Ziegler [3]. The first term at the right hand side of Eq. (47) is the translational part, the second and third terms form the rotational part, the fourth term is the Coriolis part, and the last term is the relative part of the absolute acceleration.

5. Lagrange's Equations for the translational generalized coordinates, k = 1,2,3

For the translational coordinates, Eqs. (26)-(37) yield

dv d r drA __ .-.To

—r = tt = TT = e<; k = 1,2,3 dqk dqk dqk ' '

Based on Eq. (48), we first present relations of Lagrange's Equations and Balance of Total Momentum. The total linear momentum is found by integration of J' over the body in the reference configuration (see Ziegler [3]):

J = J' dVo = A, v dVo

■JV o JVo

Hence, in Eq. (22) we obtain for the translational coordinates

f dJ' dr dJ _ ■ ^-rdV0 = -f- ■ ek Jv0 dt dqk 0 dt k

Lagrange's Equations for the translational coordinates thus are found to be the projections of the rate of total momentum upon the unit vectors of the global coordinate system, see Eq. (22):

dJ _ d /ôt\ dT n .

dîFUek =dt(dqk)~W<= ; k =12'3

The coordinates of the rate of linear momentum in the global coordinate system do represent the dynamic terms in three Lagrange's Equations for the translational coordinates

dL = £ fdfdr) _dT

dt = j hi\dt\ dqj dqj e

Using Eqs. (21) and (23), together with Eq. (38) and the divergence theorem, the generalized force in Eq. (51) can be written as

The resultant force exerted upon the body is

/ b o dVo +/ tno dSo ■ e,c = F ■ e,c

'Vo JSo

F = b odVo + tno dSo

JVo JS0

In other words, the generalized forces here represent the components of F in the global coordinate system:

F = 1 Qj ej

Hence, E__s. (51) and (53) are in direct coincidence with the global relation of Balance of Total Momentum, which reads dJ /dt = F, see, Ziegler [3].

Explicit expressions for the translational dynamic terms in Lagrange's equations are presented in Appendix A. These explicit formulations have the advantage that the volume integrals can be performed conveniently in the reference configuration, before the time integration of Lagrange's equations is performed. Further computational simplifications that result from a suitable choice of the deformation modes are presented in Appendix D.

6. Lagrange's Equations for the rotational generalized coordinates, k=4,5,6

For the rotational coordinates, Eqs. (26)-(37) yield

%=5=&p+=)=i (- j7 q*) # k=4,5,6

Similar to Eq. (41), it can be shown that

d F _> —

dEk =Sk x F ; k = 4; 5; 6

with the covariant base vectors gk introduced above in Eq. (42), see fundamental considerations on the dual Euler basis by O'Reilly [26] for the case of a rigid body. Hence, with Eqs. (29) and (37), it is found that Eq. (56) becomes

fk — fk — & x (p -u); k — 4,5,6 (58)

It follows that:

I % — ((? -) x J U i (59)

Based on Eq. (59), we first present relations to the law of Balance of Total Moment of Momentum, see Ziegler [3]. Introducing the relative moment of momentum per unit volume in the reference configuration

da — (p -u) x ( f-p0 v^ (60)

a short computation involving the product-rule of differentiation, and noting the properties of the vector cross-product, it is found that Eq. (59) can be replaced by

$ ■ % -( P+u) x■ g. (6,)

The total relative moment of momentum with respect to the moving origin A, see Ziegler [3], is

da — f da dV0 (62)

In Eq. (22), we hence find for the rotational coordinates that

0V<V0 — (f-f x /jp + U>0dV0) ■ i (63)

With Eq. (62), Lagrange's Equations, Eq. (22), for the rotational coordinates can be interpreted as being the projections of the rate of the relative moment of momentum terms upon the vectors gk:

(f -t x V( (P - U) ■ g, — - £ — Qk; k — 4,5,6 (64)

We now introduce the dual or contravariant basis with the properties

gj gk — Sf; j — 4,5,6; k — 4,5,6 (65)

where S denotes the Kronecker delta. Multiplying Eq. (64) by the contravariant base vectors gj, j — 4,5,6, and summing up, it is found that the covariant components of the moment of momentum terms are formed by the dynamic terms in Lagrange's Equations for the rotational coordinates:

x l(pp-Uu)p0dV0 — jf4(i(!) j (66)

Using Eqs. (21) and (23), together with Eq. (58) and the divergence theorem, and introducing the cross-product of two second-order tensors, which is a vector, the generalized force in Eq. (64) can be written as

Qk — (m a - Jv (l+ Grad u) x 77 dV^ ■ gk — Ma ■ gk (67)

The resultant moment with respect to the point A, exerted by the body forces and the forces acting upon the surface S0, is given by

ma — Jv ( P + u) x b0dV0 + ^ (p + u} x dS0 (68)

The cross product in Eq. (67) represents two times the axial vector belonging to the skew part of the second-order tensor h+ Grad u \nT, and thus is omitted as being zero due to the symmetry conditions, which are known to hold for the product of the first Piola-Kirchhoff stress tensor n with the deformation gradient, see Truesdell and Toupin [15] for the latter. From Eqs . (65) and (67), it is seen that the generalized forces for the rotational coordinates represent the covariant components of Ma:

ma — Z Qjgj (69)

j — 1

Hence, Eqs. (64) and (67) are in direct coincidence with the global relation of Balance of Total Moment of Momentum, which, in a problem-oriented formulation, with a moving point of reference A, can be written as (dDA/dt)-(d VA/dt) x /v0(p + u)po dVo = Ma, see Ziegler [3].

Explicit expressions for the rotatory dynamic terms in Lagrange's equations are derived in Appendix B. Again, these explicit formulations have the advantage that the volume integrals can be performed conveniently in the reference configuration, before the time integration of Lagrange's equations is performed. For further simplifications that result from a proper choice of the deformation modes, see Appendix D.

7. Lagrange's Equations for the flexible generalized coordinates, k=7,...,l

For the flexible coordinates, Eqs. (26)-(37) yield

dV dr du

W=dq'=

It thus follows that:

^ = = = k = 7, ■ ,l (70)

d-fpT = J _k (71)

dt dqk dt K '

from which Lagrange's Equations can be obtained according to Eq. (22).

In order to assign a mechanical meaning to the corresponding procedure, in the present section we start with the generalized forces. Using Eqs. (21) and (23), together with Eq. (70) and the divergence theorem, the total generalized forces for the flexible coordinates can be written as

Qk = Vk - i n • Grad__ k dVo (72)

The kth modal virial of the forces per unit volume in the reference configuration is

Vk = f bo • Ik dVo +i tno • Ik dSo (73)

JVo J So

For the notion of the virial of forces, which dates back to Finger, and for the notion of a displacement virial, the reader is referred to Irschik [27], where a principle of virtual displacement virials has been presented. Since we perform the scalar product of the forces with the mode <k in Eq. (73), we here talk about a modal virial of the forces. Using Eq. (22), we hence find for the flexible coordinates that

X.dJU*kdVo = dt(|k)-§ = Vk-Xon• Grad^kdVo; k = 7,l (74)

Note that, when is replaced by the position vector p in Eq. (73), then in the static case the original virial formulation of Finger, see Truesdell and Toupin [15] and Irschik [27], is obtained. More generally, Eq. (74) also can be interpreted as a special case of the Mean Stress Theorem, see Sections 216 and 219 of Truesdell and Toupin [15], in which the auxiliary function, denoted by in Ref. [15], is to be replaced by the mode <pk. The notion of a mean or weighted stress then comes into the play through the last term in Eq. (74).

Explicit expressions for the flexible dynamic terms in Lagrange's equations are stated in Appendix C, these explicit formulations having the advantage that the volume integrals can be performed conveniently in the reference configuration, before the time integration of Lagrange's equations is performed. For further simplifications that result from a proper choice of the deformation modes, see Appendix D.

8. Concluding remarks

In the present paper, we have laid emphasis upon deriving a momentum based formulation of Lagrange's Equations, which traditionally are based on the notion of kinetic energy. Using the Lagrange description of Continuum Mechanics, and involving a virtual motion prescribed in the framework of the Ritz approximation technique, we first have derived an exact version of Lagrange's Equations as a consequence of the Fundamental Law of Dynamics for the actual problem at hand. This version, stated in Eqs. (17) and (18), must be satisfied exactly, when the exact solutions of the problem at hand are substituted. The corresponding derivations, particularly the formulations given in Eqs. (9)-(11), have enabled us to derive an approximate (Ritz-type) version of Lagrange's Equations for a deformable body, which is based on the notion of local linear momentum per unit volume in the reference configuration. From Eq. (22), which also contains the traditional kinetic energy based formulation, the momentum based version reads,

/Jt • dqdVo=Qk; k=1 l (75)

with the following formulation for the generalized forces, see Eq. (23):

Qk—L} 0 ■ -.fSo tn0 ■ -.fVo n ■ GradfkdV0 (76)

Eq. (76) is the form of Lagrange's Equations, which we have proposed in the present paper. The momentum based version has enabled us to show, which terms do cancel out in the kinetic energy based version, see Eqs. (24) and (25).

In Section 5, we have worked out Eqs. (75) and (76) for the Floating Frame of Reference Formulation of Multibody Dynamics. Correspondingly, relations to Balance of Total Momentum have been stated in Eqs. (51)-(55). A computationally advantageous explicit momentum based expression for the translational coordinates is provided in Appendix A, see Eqs. (79) and (80) below. Relations to the equation of Balance of Total Moment of Momentum have been recorded in Eqs. (60)-(69). Corresponding explicit momentum based expressions for the rotational coordinates are given in Appendix B, see Eqs. (93)-(94) below. Relations to the Theorem of Stress Means and the notion of Virial of Forces have been discussed in Eqs. (72)-(74). The flexible coordinates are treated Appendix C in explicit form, see Eqs. (97)-(101) below. Situations, for which certain terms in Lagrange's Equations do vanish, are addressed in Appendix D below. The explicit formulations presented in the appendices have the advantage that the volume integrals can be performed conveniently in the reference configuration, before the time integration of Lagrange's equations is performed.

In the light of the remarks given in Section 7 for the flexible coordinates, it is now easy to see that the total of the momentum based formulation of Lagrange's Equations, Eqs. (75) and (76), concerning each of the generalized coordinates, can be interpreted as a special case of the Mean Stress Theorem, see Section 216 of Truesdell and Toupin [15], where the auxiliary function, denoted by in Ref. [15] is to be replaced by dr /dqk.

The resulting system of equations for the translational degrees of freedom Eq. (80), rotational degrees of freedom Eq. (94) and flexible ones Eq. (101) has been validated against the equations of motion, which formally follow from the traditional Lagrange's Equations, based on the kinetic energy. Featuring a system of computer algebra, the analysis has proven equivalence of the two systems of equations in the general case irrespective from the particular geometry of the body, its inertial properties or chosen set of deformation modes in Eq. (36). This check has been performed for some practical examples concerning the modeling of high-speed rotors presented in Refs. [12-16].

We finally note that relations equivalent to the momentum based Continuum Mechanics formulation given in Eqs. (75) and (76) are to be found in the literature on Analytical Mechanics, for systems consisting of single particles, where the volume integrals are replaced by finite sums, and corresponding formulations for the generalized forces are used, see Eq. (6.3.8) of the book by Lurie [29]. The extended case of considering redundant and non-holonomic constraints was presented in Eq. (6.3.7) of the latter reference. These relations were used by Lurie [29] only afterwards to derive the energy-based version of Lagrange's Equations, see his Eq. (7.1.7). The present work may be understood as a Continuum Mechanics based extension, however using a different method of derivation, which is not based on methods (b) and (c) mentioned in the above Introduction. It is remarked that the book of Lurie [29] also contains comprehensive discussions on the geometrical description of Analytical Mechanics, mentioned as method (d) in our Introduction. It is hoped that the present paper, which has been formulated in physical space, will stimulate further research on the geometrical method in the configuration space. In physical space, based on the above momentum based formulation, a main topic of the present contribution has been to provide explicit formulations for the dynamic terms in Lagrange's Equations, which are compact enough in order to enable an efficient symbolic computation, and which can serve as convenient mathematical models for applying methods of nonlinear automatic control.

Acknowledgment

This work has been partially supported by the Austrian COMET-K2 programme of the Linz Center of Mechatronics (LCM), and was funded by the Austrian federal government and the federal state of Upper Austria.

Appendix A. Explicit expression for the translational terms in Lagrange's Equations

In the momentum based version of Eq. (22), the expressions for the local momentum in Eqs. (46) and (47), are used. First, we note the identity (see [23])

W x (w x ^p + — (w ® w- (w■ ( P + ^ (77)

The unit tensor is written as I . Introducing the total mass as

m0 — [ P0 dV0 (78)

we find that the momentum type expression for the dynamic terms in Lagrange's Equations becomes, using Eq. (37):

/, J - g-. - qk- + ( * - #) - (/, ? p"dv»-, I7 *

Alternatively, we may write

X. ^odV o)

+ ^ (w ® w - (w - w ¿J '^fvP Po d Vo + 2 qj jv 4jPo dVo j

+ 2(ek x w) -(.I^l 4jPo dV^ + ek - (¿j ^ dV^ (79)

dJ0 dv am €k , / dw

A. at ¿tdVo = qkmo x{l;ppodVo+j2<jdVo) + ^ __ ® __ - (w • w l) ^ X p Po dVo + 2 q?f WjPo dVo j + 2w x ^<jPo dV^ + WjPo dV^ • (8o)

This second formulation, Eq. (8o), is written in the light of Eq. (51), as a projection upon ek, while the foregoing result, Eq. (79), which separates the integrals from terms containing _ek and w, appears to be somewhat more appealing from a programming point of view. The integrals in Eqs. ( ) and (8o) can be conveniently computed in the floating coordinate system, substituting Eq. (29) for p, and Eq. (36) for <. When the description is made in the global coordinate system, the transformation of the base-vectors of the floating system, Ea, a = 4,5,6, is to be performed by using Eq. (32).

Appendix B. Explicit expressions for the rotatory dynamic terms in Lagrange's Equations

For the rotational coordinates, an explicit expression for the dynamic terms in Lagrange's Equations, Eq. (22), using the local momentum expressions in Eqs. (46) and (47), is presented. The momentum type expression for the dynamic terms in Lagrange's Equations becomes

^- dk- po{i1 *e+i x(p+u) +w x (w x(p+u))

dJ' dv /3 ..i- dw - ~c - Po 2 €" 1 •

qk v -1

+2w x I q fy + I * j - (gk x (p+u\) (8i)

j - 7 j - 7

It is worth to study the single terms at the right hand side of Eq. (81) in more detail. In the following, their integrals over the volume in the reference configuration are written in a form, which allows to separate the factors containing ej, gk and w from the integral terms, this form being even more appealing from a programming point of view; see Eq. (79) for the translational coordinates. Alternatively, we give a formulation with reference to Eq. (64). Using Eq. (37), we obtain for the first two terms in Eq. (81) that

X tj- ( gk x ( P + u))PodVo -(ej x gk)- ^ pPodVo + .I^X &Po dVoj

i pPo dVo + itfi wiPo dVo j x ej) - gk (82)

'Vo i - 7 jvo j j

X x ( P + u) - ( gk x ( p + u))Po dVo - ^ ® g,^ - ¿A - ^¿A^pJ - gk (83)

Note, that a dot between two tensors has a meaning of "full contraction", or of a trace of a conventional scalar product. The moment of inertia tensor, formulated with respect to the moving origin A, and taking into account the displacement from the reference configuration, but computed by integration over the latter, is

Ya - ftr ea\-ea (84)

The trace of a tensor is abbreviated by the symbol tr. The symmetric Euler tensor EA in Eq. (84) consists of three parts:

Ea = Eao + 2 2 q sym Ea + 2 2 q'q> sym Eij (85)

j = 7 i = 7 j = 7

with the Euler tensor for the undeformed reference configuration

Eao = P 0 PPo dVo

and the generalized Euler tensors, which contain the deformation modes:

Ea,= p 0 (pjPo dVo

Eij = 0 $jPo dVo

Furthermore, in Eq. (81) there is

/ w x (w x (p + u

gk x (P + u) )PodVo = (w 0 (w x gk) ) • Ea

= - I W x ( E.W

We also find that

w x z qfy ] • ( gk x(p+u) )Po dVo=(w 0 gk) • z qTa,

j = 7 J

= ( x q Zaw ) • gk

The generalized inertia tensors TAj, which in general are not symmetric, are given by

Ta,= (tr( EAj+ qiEj) JI-( EAj+

The last term in Eq. (69) yields

jv .Z^fy u(gk x ( P + u))PodVo =-2 Z^q axiaU EAj+ Z^Eij ) • gk

The axial vector of the skew part of a second-order tensor is abbreviated by the abbreviation axial. The integrals in Eqs. (82) and (86)-(88) can be conveniently computed in the floating coordinate system, by substituting Eq. (29) for P, and Eq. (36)

for When the description is made in the global coordinate system, the transformation of the base-vectors of the former system can be performed afterwards, using Eq. (32). Substituting Eqs. (82)-(92) into Eq. (64) yields the momentum-oriented formulation of dynamic terms in Lagrange's Equations for the rotational coordinates. On the one hand side, we have

L J • I* dV o = Z W, x g,}

fv PPo dVo + .Z^^ $iPo dVo J + (dw 0 gk ) • Ta

(w 0 (w x gk)) • Ea + 2(w 0 gk) • Z^Ta,-2 Z^q axia^EAj+ Z^Ev^ • gk (93)

Alternatively, we can write

fß • i dVo=(j i q{L0pPodVo+i Z7 qjVo ^o)x *+^

Eaw) + 2 Z^qTa,w-2 Z^ axial ^Ea,+ Z^Eiil ) ' gk

Appendix C. Explicit expressions for the flexible dynamic terms in Lagrange's Equations

We next present a direct computation of Lagrange's Equations for the flexible coordinates, Eq. (74), using the local momentum expressions in Eqs. (46) and (47). Eventually, we find that the momentum type expression for the dynamic terms in Lagrange's Equations become

sr ■ jqk—*>(, |q e+W x(P+ U)+Wx (W x(P+u))

+2W x z q ft + z q j ■ .k (95)

j — 7 j — 7 J

Again, the single terms in Eq. (95) are re-written in a form, which allows integrating them over the volume in the reference configuration, such that factors containing ej and w can be separated from the terms containing the integrals, see also Eq. (79) for the translational and Eq. (93) for the rotational coordinates. Since <k is a function of the place in the reference configuration, a description analogous to Eqs. (80) and (94) is not suitable here. We obtain

/ tj ■ <kPodVo =ejU <kPodVo (96)

JVo JVo

JvlÊ x ( P + u) ■ <kPo dV0 = -2dw ■ axial ^ÊAk + qEji^ (97)

J w x (w x( p + u)) ■ <kp0dVo = —w 0 w) ■ TAk (98)

Jv(w x £ q j ■ <kPo dVo = - 2w ■ £ q axial j (99)

i Îq4>j ■ <kPo dVo = k) (100)

■JV o j = 7 j = 7 V /

Substituting Eqs. (96)-(100) into Eq. (95), yields the dynamic terms in Lagrange's Equations for the flexible coordinates in the momentum-based formulation

l^T' dV dV 0 = .2 q Cj Jv - kPo d V0 - • axia^ EAk + ^ £

- (w ® w) • rAk-4w • q'axia^E^ + ^tr(§fc) (101)

Appendix D. Some simplifications of Lagrange's Equations

The resulting form of Lagrange's Equations in the Floating Frame of Reference Formulation can be considerably simplified by selecting the floating coordinate system and the modes in the Ritz approximation such that certain terms do vanish. In the present appendix, we first assume that the floating coordinate system is formed by the principal axes of inertia of the floating rigid reference configuration, the center of mass of the latter being used as origin A. In the above formulas, we then can use the following relations:

pP0 dV0 = 0 (102)

EA0 = if/" (P" )2P0 dV^ E a 0 Ea (103)

a = 1 V JV0 J

When normalized small-deformation elastic free vibration modes are used as Ritz Ansatz-functions, then there is

jvP0^Ik )dV0 = Sj (104)

Moreover, when free-surface small-deformation elastic modes are taken for the Ritz deformation modes, then there is

i p0-k dV0 = 0 (105)

axial (gj = 0 (106)

since Balance of Total Momentum and Moment of Momentum must hold for the free vibration modes. This fact was observed for the first time by Korn [28]. It turns out to be advantageous to utilize these simplifications in a numerical time-stepping routine for solving Lagrange's Equations, see the results provided by our research group in Refs. [12-14] for applications to high-speed deformable rotors.

Now assume that the body under consideration is rigid, such that only the translational and rotational generalized coordinates are present, 1= 6. Utilizing Eq. (102), the Lagrange Equations reduce to, see Eqs. (53) and (79), as well as Eqs. (67) and (94)

qkm0 = F • Ck, k = 1,2,3 (107)

TW+w x k = MA • gk, k = 4,5,6

These results are in coincidence with the fundamental paper by Casey [5] on the geometric derivation of Lagrange's Equations for a rigid continuum. For Eq. (108), see Eqs. (4.49), (4.50b) and (2.31) of the latter reference. Note from Eq. (84) above that there is

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[15] C. Truesdell, R.A. Toupin, The classical field theories, in: S. Flügge (Ed.), Handbuch der Physik, Vol. II/l: Prinzipien der klassischen Mechanik und Feldtheorie, Springer, Berlin196o, pp. 226-793.

[16] H. Irschik, H.J. Holl, F. Hammelmüller, The Rayleigh-Ritz technique and the Lagrange equations in continuum mechanics: formulations for material and non-material volumes, in: H. Irschik, K. Schlacher (Eds.), Advanced Dynamics and Control of Structures and Machines, CISM Courses and Lectures No. 444, Springer-Verlag, Wien-New York2oo4, pp. 21 -36.

[17] H. Irschik, On rational treatments of the general laws of balance and jump, with emphasis on configurational formulations, Acta Mechanica 194 (2oo7) 11-32.

[18] H. Irschik, H.J. Holl, Lagrange's equations for open systems, derived via the method of fictitious particles, and written in the Lagrange description of continuum mechanics, Acta Mechanica (2o14), http://dx.doi.org/1o.1oo7/soo7o7-o14-1147-8, in press.

[19] L Cveticanin, Dynamics of Machines with Variable Mass, Gordon and Breach Science, London, http://dx.doi.org/1o.1oo7/soo7o7-o14-1147-8.

[20] H. Irschik, H.J. Holl, The equations of Lagrange written for a non-material volume, Acta Mechanica 153 (2oo2) 231-248.

[21] C.P. Pesce, The application of Lagrange equations to mechanical systems with mass explicitly dependent on position, ASME Journal of Applied Mechanics 7o (2oo3) 751-756.

[22] J. Casey, O.M. O'Reilly, Geometrical derivation of Lagrange's equations for a system of rigid bodies, Mathematics and Mechanics of Solids 11 (2oo6) 4o1-422.

[23] Y. Vetyukov, Nonlinear Mechanics of Thin-walled Structures. Asymptotics, Direct Approach and Numerical Analysis, Springer, Wien, 2o14.

[24] V. Chiroiu, V. Nicolae, Lagrange equations in the non-homogeneous viscoelastic solid dynamics, Revue roumaine des sciences techniques. Série de mécanique appliquée (Romanian journal of technical sciences. Applied mechanics) 24 (1979) 855-86o.

[25] O.M. O'Reilly, A.R. Srinivasa, On a decomposition of generalized constraint forces, Proceedings of the Royal Society of London. Series A 457 (2oo1) 13o7-1313.

[26] O.M. O'Reilly, The dual Euler basis: constraints, potentials and Lagrange's equations in rigid-body dynamics, Transactions of ASME 74 (2oo7) 256-258.

[27] H. Irschik, Zum Fingerschen Virial verformbarer Körper in der nichtlinearen Statik (in German), Sitzungsberichte der österreichischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung II 2o9 (2ooo) 47-66.

[28] A. Korn, Über die Eigenschwingungen eines elastischen Körpers bei verschwindenden Druckkomponenten an der Oberfläche (in German), Palermo Rend 3o (191o) 153-184.

[29] A.I. Lurie, Analytical Mechanics, Springer, Berlin, 2oo2 (translated by A.K. Belyaev).

References