An efficient Galerkin averaging-incremental harmonic balance method for nonlinear dynamic analysis of rigid multibody systems governed by differential–algebraic equations

Abstract

An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed for steady-state nonlinear dynamic analysis of index-3 differential algebraic equations (DAEs) for general rigid multibody systems. The multibody dynamic modeling theory has made significant advances in generality and simplicity, and multibody systems are usually governed by DAEs. The bridge between the multibody dynamic modeling theory and nonlinear dynamic analysis theory is built for the first time in this work, and the EGA-IHB method can be used as a universal solver for obtaining steady-state periodic responses of DAEs for general multibody systems. Since the fast Fourier transform and EGA are used, the EGA-IHB method has excellent robustness. Since the Floquet theory cannot be directly used for stability analysis of periodic responses of DAEs, a new stability analysis procedure is developed, where perturbed, linearized DAEs are reduced to ordinary differential equations with use of independent generalized coordinates. A modified arc-length continuation method with a scaling strategy is proposed for calculating response curves and conducting parameter studies. Several examples are used to show the performance and capability of the current method. Periodic solutions of DAEs from the EGA-IHB method show excellent agreement with those from numerical integration methods. Amplitude–frequency and amplitude–parameter response curves are generated, and stability and period-doubling bifurcations are analyzed. The current method shows excellent computational efficiency and robustness in solving high-dimensional DAEs.

Appendices

Appendix A: EGA Procedure

To calculate coefficient vectors related to ${\mathbf{A}}_{i}^{{r_{1} }}$, ${\mathbf{A}}_{i}^{{r_{2} }}$, ${{\varvec{\upxi}}}_{M}^{ij}$, ${{\varvec{\upxi}}}_{\varepsilon }^{ij}$, ${{\varvec{\upxi}}}_{\kappa }^{ij}$, and ${{\varvec{\upxi}}}_{{\Phi_{q} }}^{ij}$ in Eqs. (21) and (25) using the FFT, the first step is to conduct time-domain sampling, which can be effectively done by a series of matrix operations. Define a time-series matrix

$$ {\mathbf{T}} = \left[ {\begin{array}{*{20}c} 1 & 1 & \cdots & 1 & 0 & \cdots & 0 \\ 1 & {\cos \left( {1\Delta \tau } \right)} & \cdots & {\cos \left( {H\Delta \tau } \right)} & {\sin \left( {1\Delta \tau } \right)} & \cdots & {\sin \left( {H\Delta \tau } \right)} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 1 & {\cos \left( {N\Delta \tau } \right)} & \cdots & {\cos \left( {NH\Delta \tau } \right)} & {\sin \left( {N\Delta \tau } \right)} & \cdots & {\sin \left( {NH\Delta \tau } \right)} \\ \end{array} } \right] $$

(56)

where $\Delta \tau = 2\pi /N$, in which N is the sampling rate. Note that N should be sufficiently large to avoid spectrum aliasing. Hence, $\tilde{q}_{i}$ and $\tilde{\lambda }_{i}$ in Eqs. (7) and (10) can be expanded in the time domain following

$$ \begin{gathered} {\hat{\mathbf{q}}}^{i} = {\mathbf{T}} \cdot {\mathbf{A}}_{i} ,\;\;i = 1,2, \cdots ,n \hfill \\ {\hat{\varvec{\uplambda }}}^{i} = {\mathbf{T}} \cdot {\mathbf{B}}_{i} ,\;\;i = 1,2, \cdots ,m \hfill \\ \end{gathered} $$

(57)

where ${\hat{\mathbf{q}}}^{i}$ and ${\hat{\varvec{\uplambda }}}^{i}$ are N-dimensional time-series vectors related to $\tilde{q}_{i}$ and $\tilde{\lambda }_{i}$, respectively. There are ${\hat{\mathbf{q}}}^{i\prime} = \frac{{d{\hat{\mathbf{q}}}^{i} }}{d\tau } = \frac{{d{\mathbf{\mathbf T}}}}{d\tau } \cdot {\mathbf{A}}^{i}$ and ${\hat{\mathbf{q}}}^{i\prime \prime} = \frac{{d^{2} {\hat{\mathbf{q}}}^{i} }}{{d\tau^{{2}} }} = \frac{{d^{{2}} {\mathbf{\mathbf T}}}}{{d\tau^{{2}} }} \cdot {\mathbf{A}}^{i}$ related to $\tilde{q}_{i}^{\prime}$ and $\tilde{q}_{i}^{\prime \prime }$, respectively. Substituting Eq. (57) into Eqs. (5) and (6), where $\tilde{q}_{i}$ and $\tilde{\varvec{\uplambda }}_{i}$ are replaced by ${\hat{\mathbf{q}}}^{i}$ and $\;{\hat{\varvec{\lambda }}}^{i}$, respectively, and scalar multiplications are replaced by elementary dots, one has the following time-domain sampling vectors:

$$ \begin{gathered} {\varvec{\hat{\mathbf{\kappa }}}^{ij}} = \frac{{\partial f_{i} \left( {{\hat{\mathbf{q}}}^{1} ,{\hat{\mathbf{q}}}^{2} , \cdots ,{\hat{\mathbf{q}}}^{n} ,\omega {\hat{\mathbf{q}}}^{{1}{\prime}} , \cdots ,\omega {\hat{\mathbf{q}}}^{{n}{\prime} }} \right)}}{{\partial q_{j} }} - \sum\limits_{k = 1}^{m} {\frac{{\partial^2 \Phi_{k} }}{{\partial q_{i} q_{j} }}\left( {{\hat{\mathbf{q}}}^{1} ,{\hat{\mathbf{q}}}^{2} , \cdots ,{\hat{\mathbf{q}}}^{n} } \right){\varvec{\hat{\mathbf{\lambda }}}}^{k} } \hfill \\ {\varvec{\hat{\mathbf{\varepsilon }}}^{ij}} = {\frac{1}{\omega}}\frac{{\partial f_{i} \left( {{\hat{\mathbf{q}}}^{1} ,{\hat{\mathbf{q}}}^{2} , \cdots ,{\hat{\mathbf{q}}}^{n} , \omega{\hat{\mathbf{q}}}^{{1}{\prime}} , \cdots ,\omega {\hat{\mathbf{q}}}^{{n}{\prime}} } \right)}}{{\partial q_{j}^{\prime } }} \hfill \\ {\hat{\varvec{\Phi }}}_{q}^{ij} = \frac{{\partial \Phi_{i} }}{{\partial q_{j} }}\left( {{\hat{\mathbf{q}}}^{1} ,{\hat{\mathbf{q}}}^{2} , \cdots,{\hat{\mathbf{q}}}^{n} } \right) \hfill \\ \end{gathered} $$

(58)

and

$$ \begin{aligned} &{\hat{\mathbf{r}}}^{1i} = - \omega^{2} \sum\limits_{j = 1}^{n} {M_{ij} {\hat{\mathbf{q}}}^{j\prime \prime} } - \sum\limits_{j = 1}^{m} {\frac{{\partial \Phi_{i} \left( {{\hat{\mathbf{q}}}^{1} , {{\hat{\mathbf{q}}}}^{2}\cdots ,{\hat{\mathbf{q}}}^{n} } \right)}}{{\partial q_{j} }}{\hat{\varvec {\lambda }}}^{j} } \\& \quad{\text{ + f}}_{i} \left( {\omega {\hat{\mathbf{q}}}}^{1\prime } , {\omega {\hat{\mathbf{q}}}}^{2\prime }\cdots, {\omega {\hat{\mathbf{q}}}}^{n\prime }, {\hat{\mathbf{q}}}^{1}, {\hat{\mathbf{q}}}^{2}\cdots, {\hat{\mathbf{q}}}^{n}, \tau \right) \hfill \\& {\hat{\mathbf{r}}}^{2i} { = } - \Phi_{i} \left( {{\hat{\mathbf{q}}}^{1} , {\hat{\mathbf{q}}}^{2} \cdots ,{\hat{\mathbf{q}}}^{n} }, \tau \right) \hfill \\ \end{aligned} $$

(59)

The FFT can then be used to generate ${\mathbf{A}}_{i}^{{r_{1} }} ,{\mathbf{A}}_{i}^{{r_{2} }} ,{{\varvec{\upxi}}}_{M}^{ij} ,{{\varvec{\upxi}}}_{\varepsilon }^{ij} ,{{\varvec{\upxi}}}_{\kappa }^{ij} ,$ and ${{\varvec{\upxi}}}_{{\Phi_{q} }}^{ij}$. The procedure to generate ${\mathbf{A}}_{i}^{{r_{1} }}$ is shown as an example here. According to the discrete Fourier transform (DFT) theory, one has

$$ \begin{gathered} A_{i}^{r1} = {\text{Re}} \left( {\sum\limits_{q = 1}^{N} {\hat{r}_{q}^{1i} e^{{ - \iota p\tau_{q} }} } } \right)\hat{r}_{{}}^{1i} ,\;\;i = 0,1, \cdots H, \hfill \\ A_{i}^{r1} = - {\text{Im}} \left( {\sum\limits_{q = 1}^{N} {\hat{r}_{q}^{1i} e^{{ - \iota \left( {p - H} \right)\tau_{q} }} } } \right)\hat{r}_{{}}^{1i} ,\;\;i = H + 1,H + 2, \cdots 2H + 1 \hfill \\ \end{gathered} $$

(60)

where $\iota = \sqrt { - 1}$. In practice, the FFT is used to efficiently calculate the DFT in Eq. (60). One can similarly calculate ${\mathbf{A}}_{i}^{{r_{2} }} ,{{\varvec{\upxi}}}_{M}^{ij} ,{{\varvec{\upxi}}}_{\varepsilon }^{ij} ,{{\varvec{\upxi}}}_{\kappa }^{ij}$, and ${{\varvec{\upxi}}}_{{\Phi_{q} }}^{ij}$.

Appendix B: Governing DAEs for Numerical Examples in Sect. 5

Governing DAEs for numerical examples in Sect. 5 are in the standard form in Eq. (1) with their system matrices and vectors given below. For the first numerical example, one has

$$ \begin{gathered} {\mathbf{M}} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{1} } & {m_{1} } & {m_{1} l^{2} /12} \\ \end{array} } \right)\; \hfill \\ {\mathbf{q}} = \left( {\begin{array}{*{20}c} {x_{1} } & {y_{1} } & {\varphi_{1} } \\ \end{array} } \right)^{T} \hfill \\ {\mathbf{f}} = \left( {\begin{array}{*{20}c} 0 & { - m_{1} g} & { - \varepsilon_{1} \dot{\varphi } + f\cos \omega t} \\ \end{array} } \right)^{T} \hfill \\ \end{gathered} $$

(61)

$$ {{\varvec{\Phi}}} = \left( {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ \end{array} } \right) + \left[ {\begin{array}{*{20}c} {\cos \varphi_{1} } & { - \sin \varphi_{1} } \\ {\sin \varphi_{1} } & {\cos \varphi_{1} } \\ \end{array} } \right]\left( {\begin{array}{*{20}c} { - l/2} \\ 0 \\ \end{array} } \right) $$

(62)

The system matrix and vectors for the inverted rod pendulum in the second example are

$$ \begin{gathered} {\mathbf{M}} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{1} } & {m_{1} } & {m_{1} l^{2} /12} \\ \end{array} } \right)\; \hfill \\ {\mathbf{q}} = \left( {\begin{array}{*{20}c} {x_{1} } & {y_{1} } & {\varphi_{1} } \\ \end{array} } \right)^{T} \hfill \\ {\mathbf{f}} = \left( {\begin{array}{*{20}c} 0 & { - m_{1} g} & { - \varepsilon \dot{\varphi }_{1} + \kappa ({\frac{\pi}{2}}-{\varphi }_{1} })^3 \\ \end{array} } \right) \hfill \\ \end{gathered} $$

(63)

$$ {{\varvec{\Phi}}} = \left( {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ \end{array} } \right) + \left[ {\begin{array}{*{20}c} {\cos \varphi_{1} } & { - \sin \varphi_{1} } \\ {\sin \varphi_{1} } & {\cos \varphi_{1} } \\ \end{array} } \right]\left( {\begin{array}{*{20}c} { - l/2} \\ 0 \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} 0 \\ {p\sin \omega t} \\ \end{array} } \right) $$

(64)

For the slider-rod pendulum in the third example, mass matrices related to the slider and rod are

$$ \begin{aligned} & {\mathbf{M}}_{1} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{1} } & {m_{1} } \\ \end{array} } \right) \hfill \\ & {\mathbf{M}}_{2} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{2} } & {m_{2} } & {m_{2} l^{2} /12} \\ \end{array} } \right) \hfill \\ \end{aligned} $$

(65)

respectively, constrained generalized coordinates related to the slider and rod are

$$ \begin{aligned} & {\mathbf{q}}_{1} = \left( {\begin{array}{*{20}c} {x_{1} } & {y_{1} } \\ \end{array} } \right)^{T} \hfill \\ & {\mathbf{q}}_{2} = \left( {\begin{array}{*{20}c} {x_{2} } & {y_{2} } & {\varphi_{2} } \\ \end{array} } \right)^{T} \hfill \\ \end{aligned} $$

(66)

respectively. Force vectors related to the slider and rod are

$$ \begin{aligned} & {\mathbf{f}}_{1} = \left( {\begin{array}{*{20}c} {f\cos \omega t - c_{1} \dot{x}_{1}^{{}} - k_{1} x_{1} } & { - m_{1} g} \\ \end{array} } \right)^{T} \hfill \\ & {\mathbf{f}}_{2} = \left( {\begin{array}{*{20}c} 0 & { - m_{2} g} & { - c_{2} \dot{\varphi }_{2} - k_{2} \left( {\varphi_{2} + \frac{\pi }{2}} \right)} \\ \end{array} } \right)^{T} \hfill \\ \end{aligned} $$

(67)

respectively. Constraint equations are

$$ \begin{gathered} \Phi_{1} = y_{1} \hfill \\ {{\varvec{\Phi}}}_{2} = \left( {\begin{array}{*{20}c} {x_{2} } \\ {y_{2} } \\ \end{array} } \right) + \left[ {\begin{array}{*{20}c} {\cos \varphi_{2} } & { - \sin \varphi_{2} } \\ {\sin \varphi_{2} } & {\cos \varphi_{2} } \\ \end{array} } \right]\left( {\begin{array}{*{20}c} { - l/2} \\ 0 \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

(68)

By combining Eqs. (65)–(68), the system matrix and vectors are

$$ {\mathbf{M = }}\left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{1} } & {} \\ {} & {{\mathbf{M}}_{2} } \\ \end{array} } \right]{\mathbf{,}}\;\;{\mathbf{q = }}\left( {\begin{array}{*{20}c} {{\mathbf{q}}_{1} } \\ {{\mathbf{q}}_{2} } \\ \end{array} } \right){\mathbf{,}}\;\;{\mathbf{f = }}\left( {\begin{array}{*{20}c} {{\mathbf{f}}_{1} } \\ {{\mathbf{f}}_{2} } \\ \end{array} } \right){\mathbf{,}}\;\;{\mathbf{\Phi = }}\left( {\begin{array}{*{20}c} {\Phi_{1} } \\ {{{\varvec{\Phi}}}_{2} } \\ \end{array} } \right) $$

(69)

The procedure to generate the governing DAEs for the slider-chain pendulum in the fourth example is illustrated in detail here. Constrained generalized coordinates for the slider and rods are

$$ \begin{aligned} & {\mathbf{q}}_{1} = \left( {\begin{array}{*{20}c} {x_{1} } & {y_{1} } \\ \end{array} } \right)^{T} \hfill \\ & {\mathbf{q}}_{i} = \left( {\begin{array}{*{20}c} {x_{i} } & {y_{i} } & {\varphi_{i} } \\ \end{array} } \right)^{T} \hfill \\ \end{aligned} $$

(70)

where $\;i = 2,3, \cdots ,41$. Kinetic energies of the slider and chain rods are

$$ \begin{aligned} & K_{1} = m_{1} \left( {\dot{x}_{1}^{2} + \dot{y}_{1}^{2} } \right)/2 \hfill \\ & K_{i} = m_{i} \left( {\dot{x}_{i}^{2} + \dot{y}_{i}^{2} } \right)/2 + m_{i} l^{2} \dot{\varphi }_{i}^{2} /24 \hfill \\ \end{aligned} $$

(71)

respectively, where $i = 2,3, \cdots ,41$, and their potential energies are

$$ \begin{aligned} & V_{1} = m_{1} gy_{1} + \left( {\kappa_{1} x_{1}^{2} } \right)/2 \hfill \\ & V_{2} = m_{2} gy_{2} + \hat{\kappa }\left( { - \pi /2 - \varphi_{2} } \right)^{2} /2, \hfill \\ & V_{i} = m_{i} gy_{i} + \hat{\kappa }\left( {\varphi_{i - 1} - \varphi_{i} } \right)^{2} /2 \hfill \\ \end{aligned} $$

(72)

where $i = 3,4, \cdots ,41$. Non-conservative generalized forces on the slider and rods are

$$ \begin{aligned} & Q_{1} = p\cos \omega t - \varepsilon_{1} \dot{x}_{1} \hfill \\ & Q_{2} = - \hat{\varepsilon }\left( {2\dot{\varphi }_{2} - \dot{\varphi }_{3} } \right) \hfill \\ & Q_{i} = - \hat{\varepsilon }\left( {\dot{\varphi }_{i - 1} - 2\dot{\varphi }_{i} + \dot{\varphi }_{i + 1} } \right) \hfill \\ & Q_{41} = - \hat{\varepsilon }\left( {\dot{\varphi }_{40} - \dot{\varphi }_{41} } \right) \hfill \\ \end{aligned} $$

(73)

respectively, where $i = 3,4, \cdots ,40$. Substituting Eqs. (71), (72), and (73) into the first kind of Lagrange’s equations

$$ \begin{aligned} & \frac{d}{dt}\frac{{\partial \left( {\sum\limits_{i = 1}^{41} {K_{i} } } \right)}}{{\partial {\dot{\mathbf{q}}}}} - \frac{{\partial \left( {\sum\limits_{i = 1}^{41} {K_{i} } } \right)}}{{\partial {\mathbf{q}}}} + \frac{{\partial \left( {\sum\limits_{i = 1}^{41} {V_{i} } } \right)}}{{\partial {\mathbf{q}}}} + {{\varvec{\Phi}}}_{{\mathbf{q}}}^{T} {{\varvec{\uplambda}}} = {\mathbf{Q}} \\ & {{\varvec{\Phi}}} = {\mathbf{0}} \\ \end{aligned} $$

(74)

one has mass matrices for all the bodies:

$$ \begin{aligned} & {\mathbf{M}}_{1} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{1} } & {m_{1} } \\ \end{array} } \right) \hfill \\ & {\mathbf{M}}_{i} = {\text{diag}}\left( {\begin{array}{*{20}c} {m_{i} } & {m_{i} } & {m_{i} l^{2} /12} \\ \end{array} } \right) \hfill \\ \end{aligned} $$

(75)

where ${\mathbf{M}}_{1}$ is related to the slider and ${\mathbf{M}}_{i}$ is related to the ith chain rod; force vectors for all the bodies are

$$ \begin{gathered} {\mathbf{f}}_{1} = \left( {\begin{array}{*{20}c} { - \varepsilon_{1} \dot{x}_{1} - \kappa_{1} x_{1} + p\cos \omega t} & { - m_{1} g} \\ \end{array} } \right)^{T} \hfill \\ \hfill \\ \end{gathered} $$

(76)

$$ {\mathbf{f}}_{2} = \left( {\begin{array}{*{20}c} 0 & { - m_{2} g} & { - \hat{\varepsilon }\left( {2\dot{\varphi }_{2} - \dot{\varphi }_{3} } \right) - \hat{\kappa }\left( { - \frac{\pi }{2} - 2\varphi_{2} + \varphi_{3} } \right)} \\ \end{array} } \right)^{T} $$

(77)

$$ \begin{gathered} {\mathbf{f}}_{i} = \left( {\begin{array}{*{20}c} 0 & { - m_{i} g} & { - \hat{\varepsilon }\left( {\dot{\varphi }_{i - 1} - 2\dot{\varphi }_{i} + \dot{\varphi }_{i + 1} } \right) - \hat{\kappa }\left( {\varphi_{i - 1} - 2\varphi_{i} + \varphi_{i + 1} } \right)} \\ \end{array} } \right)^{T} \hfill \\ \hfill \\ \end{gathered} $$

(78)

where $i = 2,_{{}} 3,_{{}} \cdots ,_{{}} 40$, and

$$ {\mathbf{f}}_{41} = \left( {\begin{array}{*{20}c} 0 & { - m_{2} g} & { - \hat{\varepsilon }} \\ \end{array} \left( {\dot{\varphi }_{41} - \dot{\varphi }_{40} } \right) - \hat{\kappa }\left( {\varphi_{41} - \varphi_{40} } \right)} \right)^{T} $$

(79)

The bodies are attached to each other using rotating joints; hence, constraint equations are

$$ \begin{aligned} \Phi_{1} & = y_{1} \\ {{\varvec{\Phi}}}_{i} & = \left[ {\begin{array}{*{20}c} {x_{i + 1} } \\ {y_{i + 1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\cos \varphi_{i + 1} } & { - \sin \varphi_{i + 1} } \\ {\sin \varphi_{i + 1} } & {\cos \varphi_{i + 1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} { - l/2} \\ 0 \\ \end{array} } \right] \\ & \quad - \left[ {\begin{array}{*{20}c} {x_{i} } \\ {y_{i} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {\cos \varphi_{i} } & { - \sin \varphi_{i} } \\ {\sin \varphi_{i} } & {\cos \varphi_{i} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {l/2} \\ 0 \\ \end{array} } \right] \\ \end{aligned} $$

(80)

By combining Eqs. (75)–(80), the system matrix and vectors are

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{1} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & \ddots & \vdots \\ {\mathbf{0}} & \cdots & {{\mathbf{M}}_{41} } \\ \end{array} } \right],\;\;{\mathbf{f}} = \left( {\begin{array}{*{20}c} {{\mathbf{f}}_{1}^{{}} } & {{\mathbf{f}}_{2}^{{}} } & \cdots & {{\mathbf{f}}_{{{41}}}^{{}} } \\ \end{array} } \right)^{T} $$

(81)

$$ {{\varvec{\Phi}}} = \left( {\begin{array}{*{20}c} {{{\varvec{\Phi}}}_{1}^{T} } & {{{\varvec{\Phi}}}_{2}^{T} } & \cdots & {{{\varvec{\Phi}}}_{40}^{T} } \\ \end{array} } \right)^{T} $$

(82)