2007年多体动力学计算方法综述：应用科学中的计算机技术

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《多体动力学（2007）》是ComputationalMethodsinAppliedSciences系列中的一本专著，由E.Oñate担任系列编辑。本书主要探讨了多体系统动力学的计算方法及其在各领域的应用。作者包括Juan Carlos García Orden，来自西班牙马德里理工大学；José M. Goicolea，同样来自马德里理工大学；以及Javier Cuadrado，来自西班牙卢戈大学。该书的出版方是Springer出版社，位于荷兰多尔德recht。本书针对多体系统的复杂动态行为进行了深入研究，这些系统涉及机械工程、航空航天、生物力学、机器人技术等多个学科。书中涵盖了多体系统的基本原理，如刚体动力学、运动学、动力学方程的建立、接触与碰撞分析、动力学建模、数值积分方法等核心概念。它不仅提供了理论框架，还介绍了先进的计算机模拟工具和技术，使得读者能够理解和应用这些理论解决实际问题。多体动力学在现代工程实践中具有重要作用，例如在设计飞机结构、汽车悬挂系统、机械臂运动控制以及大规模结构动态分析时。书中强调了实验验证、数值解算的精度和稳定性，以及如何处理非线性动力学问题和不确定性因素。此外，书中的章节可能包含对多体系统动力学建模软件的介绍，如基于有限元法（FEM）、多体动力学软件包（如ADAMS、MATLAB/Simulink等）的应用示例。值得注意的是，本书在版权方面有严格的限制，未经Springer出版社许可，任何复制、存储或传输行为都必须得到书面授权。电子版书籍的ISBN号分别为ISBN-10 1-4020-5684-2和ISBN-13 978-1-4020-5684-0，购买者可以借此获取数字内容。《多体动力学（2007）》是一本实用的参考文献，对于理解多体系统动态行为，特别是在工程设计和仿真中的应用，具有很高的价值。对于学习者和研究人员来说，这本书提供了理论基础和实践经验，是深入研究多体动力学领域的宝贵资源。

G. Jeleni´c and J.J. Muñoz

Fig. 3. Scheme of the screw joint and the rack-and-pinion joint.

2.2.1 Kinematics of joints with dependent degrees of freedom

Setting the released displacement r

as the dependent variable and the released ro-

tation θ

as the independent variable, the relationship between the two of them may

be written through a function r

= f (θ

In the joints commonly considered, only one component of the released rotation

will be non-zero, in which case θ

and δβ

are coaxial. Therefore δθ

= δβ

,and

the variation of r

= f (θ

) reads δr

∂f (θ

)

∂θ

δβ

. Replacing δr

into (9) the

matrix N turns into

N =



0 

∂

f (θ

)

∂θ

I −





0 



. (16)

2.2.2 Screw joint and rack-and-pinion joint

These joints have important common features that enable deﬁning them in a unique

manner. In both cases there exists a released rotation around axis t

(see Figure 3),

and a released displacement with components along axes t

(attached to the master

node) and t

× t

. For the screw join t (sc) axes t

and t

are coincident, while in the

rack-and-pinion joint (rp) they are perpendicular to each other. The latter also has a

non zero initial displacement (the radius of the pinion) in a d irection p erpendicular

to t

and t

The particular form of r

= f (θ

) for these joints can be written u sing the

scalar parameters b and c as

= f (θ

)

= f (θ

)

= b(t

×t

)+c(θ

·t

= b



+c(t

⊗t

)θ

, (17)

where b is the radius of the pinion and c has a different meaning for the two joints.

In the rack-and-pinion joint c = b and in the screw joint it corresponds to the pitch

of the thread. Note that since t

and t

have the same direction in the screw joint, the

Modelling Multi-Body Systems Using the Master–Slave Approach

constant term b



in th is case vanishes. Differentiating the kinematic relation (17)

gives rise to the terms

∂f (θ

)

∂θ



∂f (θ

)

∂θ



= ct

⊗ t

(18)

needed in (16) and subsequently in (14) and (15).

2.2.3 Rigid segment

Within the master–slave approach, a rigid segment (rs) can be formulated using the

deﬁnition o f the rack-and- pinion joint in which the rack is ﬁxed to the pinion. This

is equivalent to setting c = 0 in (17), in which case the master and the slave node

remain at a ﬁxed distance b throughout the motion with the relative orientation of

one with respect to the other also ﬁxed. In this case

∂f (θ

)

∂θ



= 0, which inserted

in (1 6) deﬁnes the linear operator N at the node.

2.2.4 Cam joint

Let us consider a cam joint with a simple eccentric cam-lobe proﬁle as shown in

Figure 4. By setting the upper node of the rotating element B as the slave node

and the left-end node of the element A as the master node, the relation between the

released translational displacement r

and the released rotation θ

may be written

= f (θ

)

cam

= (R cos |θ

|−R − a)t

, (19)

where a and 2 R + a are the minimum and maximum released displacements of the

arm A and t

is the unit vector attached to the m aster node (direction in which the

relative translation takes place). The term

∂f (θ

)

∂θ

for the cam joint is computed as

∂f (θ

)

∂θ



cam

=−R sin |θ

⊗ t

, (20)

which is needed in (16) in order to deﬁne the linear operator N at the node.

2.3 Numerical stability and conservative integration

In order to avoid the problems with numerical stability in the standard end-poin t nu-

merical integration of equations of motion, a different, conservative, class of methods

for numerical integration can be applied to the problem. For a conservative force-free

class of dynamical problems, this procedure introduces an energy- and momentum-

conserving transformation which relates the ﬁnite changes of the slave and the mas-

ter/released variables over a time step rather than the variations of the slave and

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2007年多体动力学计算方法综述：应用科学中的计算机技术

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