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Static condensation method for the reduced dynamic modeling of mechanisms and structures

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Abstract

In this paper, a novel static condensation method is extended to mechanisms and structures with internal joints. The formulation is framed inside the static reduction techniques in a similar way to the classic Guyan–Iron reduction. A new class of joint nodes is added to the existing classes of inner and boundary nodes to consider joint constraints inside the compatibility conditions. Within the set of joint nodes, a reduced subset of independent nodes can be individuated to perform the static reduction. A simple connection rule is provided to determine the independent nodes of a mechanical system composed of rigid bodies, flexible components and joints. The inner and boundary nodes are then expressed in terms of the independent joint nodes to form a new transformation matrix able to reduce the stiffness and mass matrices of the original system without connections. Joint stiffness and external forces are also included in the final reduction process. Finally, two examples prove the efficiency and demonstrate the equivalence and compatibility of the proposed method with other static reduction techniques in order to create a unique substructuring framework.

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Acknowledgements

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Dipartimento Ingegneria Civile e Architettura, University of Catania, Via Santa Sofia, 54, 95125, Catania, Italy

    Alessandro Cammarata, Rosario Sinatra & Pietro Davide Maddìo

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  1. Alessandro Cammarata
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  2. Rosario Sinatra
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  3. Pietro Davide Maddìo
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Correspondence to Alessandro Cammarata.

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Appendix

Appendix

In Sect. 3.3, the following system (20) has been derived:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf{K}_\mathrm{II}\mathbf{u}_\mathrm{I} + \mathbf{K}_\mathrm{IB}\mathbf{u}_\mathrm{B} = \mathbf{0}\\ \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BI}\mathbf{u}_\mathrm{I} + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{u}_\mathrm{B} = \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{f}_\mathrm{B}\\ \mathbf{u}_\mathrm{B} = \mathbf{G}_\mathrm{B}\mathbf{u}_O + \mathbf{H}_\mathrm{B}{\varvec{\theta }}_M \end{array}\right. } \end{aligned}$$
(35)

It can be recognized that from the first equation of this system, the Guyan–Iron transformation is recovered. Then, after substituting the third equation into the second one, we derive

$$\begin{aligned} \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BI}\mathbf{u}_\mathrm{I} + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{G}_\mathrm{B}\mathbf{u}_O + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{H}_\mathrm{B}{\varvec{\theta }}_M = \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{f}_\mathrm{B} \end{aligned}$$
(36)

After computation, \({{\varvec{\theta }}}_M\) is obtained, i.e.,

$$\begin{aligned} {{\varvec{\theta }}}_M = -({\mathbf{H}_\mathrm{B}}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{H}_{B})^{-1}({\mathbf{H}_\mathrm{B}}^\mathrm{T}\mathbf{K}_\mathrm{BI}\mathbf{u}_\mathrm{I} + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{G}_{B}\mathbf{u}_O - \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{f}_{B}). \end{aligned}$$
(37)

Substituting \({{\varvec{\theta }}}_M\) into the third equation of the system, we finally obtain

$$\begin{aligned} \mathbf{u}_\mathrm{B} = \mathbf{L}_\mathrm{B} \mathbf{u}_O + \mathbf{L}_F \mathbf{f}_\mathrm{B} \end{aligned}$$
(38)

where

$$\begin{aligned} \mathbf{L}_F= & {} {\varvec{\varGamma }}_\mathrm{B}^{-1}(\mathbf{H}_{B}{\varDelta _\mathrm{B}}^{-1}{\mathbf{H}_{B}}^\mathrm{T}) \end{aligned}$$
(39a)
$$\begin{aligned} \mathbf{L}_\mathrm{B}= & {} {\varvec{\varGamma }}_\mathrm{B}^{-1}(\mathbf{H}_{B}{\varDelta _\mathrm{B}}^{-1}{\mathbf{H}_{B}}^\mathrm{T} \mathbf{K}_\mathrm{BB}\mathbf{G}_\mathrm{B} - \mathbf{G}_\mathrm{B}) \end{aligned}$$
(39b)
$$\begin{aligned} {\varvec{\varGamma }}_\mathrm{B}= & {} \mathbf{H}_{B}{\varDelta _\mathrm{B}}^{-1}{\mathbf{H}_{B}}^\mathrm{T} \mathbf{K}_\mathrm{BI}{\mathbf{K}_\mathrm{II}}^{-1}\mathbf{K}_\mathrm{IB} - \mathbf{1}_\mathrm{B} \end{aligned}$$
(39c)

and \(\varDelta _\mathrm{B} = {\mathbf{H}_\mathrm{B}}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{H}_{B}\). Without changing the sense of the explanation, let us impose \(\mathbf{f}_\mathrm{B}=\mathbf{0}\), and then, the expression between \(\mathbf{u}_\mathrm{B}\) and \(\mathbf{u}_O\) is reduced to

$$\begin{aligned} \mathbf{u}_\mathrm{B} = \mathbf{L}_\mathrm{B} \mathbf{u}_O \end{aligned}$$
(40)

where \(\mathbf{L}_\mathrm{B}\) is used to create a new transformation matrix \(\varPsi _L\), defined as

$$\begin{aligned} \left[ \begin{array}{c} \mathbf{u}_\mathrm{I}\\ \mathbf{u}_\mathrm{B} \end{array} \right] = \varPsi _L \mathbf{u}_O \Rightarrow \varPsi _L = \varPsi \mathbf{L}_\mathrm{B} \end{aligned}$$
(41)

where \(\varPsi \) has been defined in Eq. (4) for the Guyan–Iron condensation. It can be verified by direct substitution that when all joints are locked, that is when \(\mathbf{H}_{B} \equiv \mathbf{O}\), it follows that \(\mathbf{L}_\mathrm{B} \equiv \mathbf{G}_\mathrm{B}\) and \(\varPsi _L \equiv \varPsi \mathbf{G}_\mathrm{B}\). The latter expresses that the Guyan–Iron transformation can be recovered simply by blocking the joints. Considering \(\varPsi _L\), the following reduced system is written

$$\begin{aligned} \mathbf{M}_L \ddot{\mathbf{u}}_b + \mathbf{K}_L \mathbf{u}_b = \mathbf{F}_L \end{aligned}$$
(42)

where \(\mathbf{M}_L\), \(\mathbf{K}_L\), respectively, are the reduced mass and stiffness matrices, defined as

$$\begin{aligned} \mathbf{M}_L = \varPsi _L^\mathrm{T} \mathbf{M}\varPsi _L,\quad \mathbf{K}_L = \varPsi _L^\mathrm{T} \mathbf{K}\varPsi _L \end{aligned}$$
(43)

and \(\mathbf{F}_L\) is the array of reduced forces. The expressions of \(\mathbf{M}_L\), \(\mathbf{K}_L\) and \(\mathbf{F}_L\) are not reported for brevity. Compared to the previous formulations proposed by the authors in [4, 6, 7], the new approach is global and does not require different cases of coupling between bodies.

In particular cases, the vector \(\mathbf{f}_\mathrm{B}\) may include elastic forces. Therefore, suppose that an elastic force/torque is applied to the boundary node of a generic joint, the corresponding generalized force vector \(\mathbf{f}_b\) becomes

$$\begin{aligned} \mathbf{f}_b = -k_e \mathbf{h}_b \theta _m \end{aligned}$$
(44)

where \(k_e\) is the stiffness at the joint location. This expression can be extended to the global vector \(\mathbf{f}_\mathrm{B}\), i.e.,

$$\begin{aligned} \mathbf{f}_\mathrm{B} = -\mathbf{K}_E \mathbf{H}_\mathrm{B} {\varvec{\theta }}_M,\quad \mathbf{K}_E = \textsf {diag}(\mathbf{O}_6,\ldots ,k_e\mathbf{1}_6,\ldots ,\mathbf{O}_6) \end{aligned}$$
(45)

where \(\mathbf{K}_E\) is the generalized stiffness matrix containing all joint stiffness. By substituting this expression into the right-side term \(\mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{f}_\mathrm{B}\) of Eq. (36), we obtain

$$\begin{aligned} \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BI}\mathbf{u}_\mathrm{I} + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{G}_\mathrm{B}\mathbf{u}_O + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{H}_\mathrm{B}{\varvec{\theta }}_M = -\mathbf{H}_\mathrm{B}^\mathrm{T} \mathbf{K}_E \mathbf{H}_\mathrm{B} {\varvec{\theta }}_M \end{aligned}$$
(46)

from which it can be obtained that \({{\varvec{\theta }}}_M\) becomes

$$\begin{aligned} {{\varvec{\theta }}}_M = -[{\mathbf{H}_\mathrm{B}}^\mathrm{T}(\mathbf{K}_\mathrm{BB} + \mathbf{K}_E)\mathbf{H}_{B}]^{-1}({\mathbf{H}_\mathrm{B}}^\mathrm{T}\mathbf{K}_\mathrm{BI}\mathbf{u}_\mathrm{I} + \mathbf{H}_\mathrm{B}^\mathrm{T}\mathbf{K}_\mathrm{BB}\mathbf{G}_{B}\mathbf{u}_O) \end{aligned}$$
(47)

This expression replaces Eq. (37) and allows for obtaining the transformation matrix \(\varPsi _L\) in a manner similar to what has been done previously.

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Cammarata, A., Sinatra, R. & Maddìo, P.D. Static condensation method for the reduced dynamic modeling of mechanisms and structures. Arch Appl Mech 89, 2033–2051 (2019). https://doi.org/10.1007/s00419-019-01560-x

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