Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints

Abstract

In this paper, a new analytical approach suitable for the stability analysis of multibody mechanical systems is introduced in the framework of Lagrangian mechanics. The approach developed in this work is based on the direct linearization of the index-three form of the differential-algebraic dynamic equations that describe the motion of mechanical systems subjected to nonlinear constraints. One of the distinguishing features of the proposed method is that it can handle general sets of nonlinear holonomic and/or nonholonomic constraints without altering the original mathematical structure of the equations of motion. While the typical state-space dynamic description associated with multibody systems leads to the definition of a standard eigenproblem, which is impractical, if not impossible, to implement in the case of complex systems, the method developed in this paper involves a generalized state-space representation of the dynamic equations and allows for the formulation of a generalized eigenvalue problem that extends the scope of applicability of the stability analysis to complex mechanical systems. As demonstrated in this investigation employing simple numerical examples, the proposed methodology can be readily implemented in general-purpose multibody computer programs and compares favorably with several other reference computational approaches already available in the multibody literature.

Appendix A

In this appendix, the main equations describing four alternative computational algorithms for the determination of the generalized acceleration vector of a multibody system are synthetically reported.

1.1 A.1 Embedding technique

$$\begin{aligned} {{\mathbf{q}}_i}= & {} {{\mathbf{B}}_i}{\mathbf{q}} \end{aligned}$$

(154)

$$\begin{aligned} {\tilde{\mathbf{I}}}= & {} \left[ {\begin{array}{*{20}{c}} {\mathbf{O}}\\ {\mathbf{I}} \end{array}} \right] ,\quad {{\mathbf{J}}_c} = \left[ {\begin{array}{*{20}{c}} {\mathbf{J}}\\ {{{\mathbf{B}}_i}} \end{array}} \right] ,\quad {{\mathbf{w}}_c} = \left[ {\begin{array}{*{20}{c}} { - {{\mathbf{Q}}_d}}\\ {\mathbf{0}} \end{array}} \right] \end{aligned}$$

(155)

$$\begin{aligned} {{\bar{\mathbf{J}}}_c}= & {} {\mathbf{J}}_c^{ - 1}{\tilde{\mathbf{I}}},\quad {{\bar{\mathbf{w}}}_c} = {\mathbf{J}}_c^{ - 1}{{\mathbf{w}}_c} \end{aligned}$$

(156)

$$\begin{aligned} {{\mathbf{M}}_{i,i}}= & {} {\bar{\mathbf{J}}}_c^\mathrm{T}{\mathbf{M}}{{\bar{\mathbf{J}}}_c},\quad {{\mathbf{Q}}_i} = {\bar{\mathbf{J}}}_c^\mathrm{T}\left( {{{\mathbf{Q}}_b} + {\mathbf{M}}{{{\bar{\mathbf{w}}}}_c}} \right) \end{aligned}$$

(157)

$$\begin{aligned} {{\mathbf{M}}_{i,i}}{{\ddot{\mathbf{q}}}_i}= & {} {{\mathbf{Q}}_i} \end{aligned}$$

(158)

$$\begin{aligned} {\ddot{\mathbf{q}}}= & {} {{\bar{\mathbf{J}}}_c}{{\ddot{\mathbf{q}}}_i} - {{\bar{\mathbf{w}}}_c} \end{aligned}$$

(159)

1.2 A.2 Amalgamated formulation

$$\begin{aligned} {{\mathbf{q}}_i}= & {} {{\mathbf{B}}_i}{\mathbf{q}} \end{aligned}$$

(160)

$$\begin{aligned} {\tilde{\mathbf{I}}}= & {} \left[ {\begin{array}{*{20}{c}} {\mathbf{O}}\\ {\mathbf{I}} \end{array}} \right] ,\quad {{\mathbf{J}}_c} = \left[ {\begin{array}{*{20}{c}} {\mathbf{J}}\\ {{{\mathbf{B}}_i}} \end{array}} \right] ,\quad {{\mathbf{w}}_c} = \left[ {\begin{array}{*{20}{c}} { - {{\mathbf{Q}}_d}}\\ {\mathbf{0}} \end{array}} \right] \end{aligned}$$

(161)

$$\begin{aligned} {{\bar{\mathbf{J}}}_c}= & {} {\mathbf{J}}_c^{ - 1}{\tilde{\mathbf{I}}},\quad {{\bar{\mathbf{w}}}_c} = {\mathbf{J}}_c^{ - 1}{{\mathbf{w}}_c} \end{aligned}$$

(162)

$$\begin{aligned} {{\mathbf{M}}_{am}}= & {} \left[ {\begin{array}{*{20}{c}} {\mathbf{M}}&{}{\mathbf{I}}&{}{\mathbf{O}}\\ {\mathbf{I}}&{}{\mathbf{O}}&{}{ - {{{\bar{\mathbf{J}}}}_c}}\\ {\mathbf{O}}&{}{ - {\bar{\mathbf{J}}}_c^\mathrm{T}}&{}{\mathbf{O}} \end{array}} \right] ,\quad {{\mathbf{q}}_{am}} = \left[ {\begin{array}{*{20}{c}} {{\ddot{\mathbf{q}}}}\\ { - {{\mathbf{Q}}_c}}\\ {{{{\ddot{\mathbf{q}}}}_i}} \end{array}} \right] ,\quad {{\mathbf{Q}}_{am}} = \left[ {\begin{array}{*{20}{c}} {{{\mathbf{Q}}_b}}\\ { - {{{\bar{\mathbf{w}}}}_c}}\\ {\mathbf{0}} \end{array}} \right] \end{aligned}$$

(163)

$$\begin{aligned} {{\mathbf{M}}_{am}}{{\mathbf{q}}_{am}}= & {} {{\mathbf{Q}}_{am}} \end{aligned}$$

(164)

$$\begin{aligned} {\ddot{\mathbf{q}}}= & {} {{\mathbf{B}}_q}{{\mathbf{q}}_{am}} \end{aligned}$$

(165)

1.3 A.3 Projection method

$$\begin{aligned} {{\mathbf{q}}_i}= & {} {{\mathbf{B}}_i}{\mathbf{q}} \end{aligned}$$

(166)

$$\begin{aligned} {\tilde{\mathbf{I}}}= & {} \left[ {\begin{array}{*{20}{c}} {\mathbf{O}}\\ {\mathbf{I}} \end{array}} \right] ,\quad {{\mathbf{J}}_c} = \left[ {\begin{array}{*{20}{c}} {\mathbf{J}}\\ {{{\mathbf{B}}_i}} \end{array}} \right] \end{aligned}$$

(167)

$$\begin{aligned} {{\bar{\mathbf{J}}}_c}= & {} {\mathbf{J}}_c^{ - 1}{\tilde{\mathbf{I}}} \end{aligned}$$

(168)

$$\begin{aligned} {{\mathbf{M}}_{pr}}= & {} \left[ {\begin{array}{*{20}{c}} {{\bar{\mathbf{J}}}_c^\mathrm{T}{\mathbf{M}}}\\ {\mathbf{J}} \end{array}} \right] ,\quad {{\mathbf{Q}}_{pr}} = \left[ {\begin{array}{*{20}{c}} {{\bar{\mathbf{J}}}_c^\mathrm{T}{{\mathbf{Q}}_b}}\\ {{{\mathbf{Q}}_d}} \end{array}} \right] \end{aligned}$$

(169)

$$\begin{aligned} {{\mathbf{M}}_{pr}}{\ddot{\mathbf{q}}}= & {} {{\mathbf{Q}}_{pr}} \end{aligned}$$

(170)

1.4 A.4 Fundamental equations of constrained motion: Udwadia–Kalaba equations

$$\begin{aligned} {{\mathbf{M}}^ * }= & {} {\mathbf{M}} + {{\mathbf{J}}^\mathrm{T}}{\mathbf{J}},\quad {\mathbf{Q}}_b^ * = {{\mathbf{Q}}_b} + {{\mathbf{J}}^\mathrm{T}}{{\mathbf{Q}}_d} \end{aligned}$$

(171)

$$\begin{aligned} {\ddot{\mathbf{q}}}= & {} {\left( {{{\mathbf{M}}^ * }} \right) ^{ - 1}}{\mathbf{Q}}_b^ * + {\left( {{{\mathbf{M}}^ * }} \right) ^{ - 1}}{{\mathbf{J}}^\mathrm{T}}{\left( {{\mathbf{J}}{{\mathbf{M}}^ * }{{\mathbf{J}}^\mathrm{T}}} \right) ^ + }\left( {{{\mathbf{Q}}_d} - {\mathbf{J}}{{\left( {{{\mathbf{M}}^ * }} \right) }^{ - 1}}{\mathbf{Q}}_b^ * } \right) \end{aligned}$$

(172)