Simulation and stability analysis of periodic flexible multibody systems

Abstract

The dynamic response of many flexible multibody systems of practical interest is periodic. The investigation of such problems involves two intertwined tasks: first, the determination of the periodic response of the system and second, the analysis of the stability of this periodic solution. Starting from Hamilton’s principle, a unified solution procedure for continuous and discontinuous Galerkin methods is developed for these two tasks. In the proposed finite element formulation, the unknowns consist of the displacement and rotation components at the nodes, which are interpolated via the dual spherical linear interpolation technique. Periodic solutions are obtained by solving the discrete nonlinear equations resulting from continuous and discontinuous Galerkin methods. The monodromy matrix required for stability analysis is constructed directly from the Jacobian matrix of the solution process. Numerical examples are presented to validate the proposed approaches.

Appendices

Appendix A: Discrete Fourier transforms

Let $n_{T} > 1$, $n_{H} = \left \lceil (n_{T} - 1)/2 \right \rceil$, and $\tau _{k} = 2 k \pi /n_{T}$, $k = 0,1,\ldots ,n_{T}-1$, where notation $\left \lceil \cdot \right \rceil$ indicates the least integer greater than or equal to $(\cdot )$. The operator for the discrete Fourier transform and its inverse, both of size $n_{T}\times n_{T}$, are defined as

$$ \begin{aligned} {{F}}_{0,n_{T}} &= \frac{2}{n_{T}} \begin{bmatrix} 1/2 & 1/2 & \cdots & 1/2 \\ \cos \tau _{0} & \cos \tau _{1} & \cdots & \cos \tau _{n_{T} - 1} \\ \sin \tau _{0} & \sin \tau _{1} & \cdots & \sin \tau _{n_{T} - 1} \\ \vdots & \vdots & \ddots & \vdots \\ \cos (n_{H} \tau _{0}) & \cos (n_{H} \tau _{1}) & \cdots & \cos (n_{H} \tau _{n_{T} - 1}) \\ \sin (n_{H} \tau _{0}) & \sin (n_{H} \tau _{1}) & \cdots & \sin (n_{H} \tau _{n_{T} - 1}) \end{bmatrix} , \\ {{F}}_{0,n_{T}}^{-1} &= \begin{bmatrix} 1 & \cos \tau _{0} & \sin \tau _{0} & \cdots & \cos (n_{H} \tau _{0}) & \sin (n_{H}\tau _{0}) \\ 1 & \cos \tau _{1} & \sin \tau _{1} & \cdots & \cos (n_{H} \tau _{1}) & \sin (n_{H} \tau _{1}) \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cos \tau _{n_{T} - 1} & \sin \tau _{n_{T} - 1} & \cdots & \cos (n_{H} \tau _{n_{T} - 1}) & \sin (n_{H} \tau _{n_{T} - 1}) \end{bmatrix} , \end{aligned} $$

(57)

for odd values of $n_{T}$, and

$$ \begin{aligned} {{F}}_{0,n_{T}} &= \frac{2}{n_{T}} \begin{bmatrix} 1/2 & 1/2 & \cdots & 1/2 \\ \cos \tau _{0} & \cos \tau _{1} & \cdots & \cos \tau _{n_{T} - 1} \\ \sin \tau _{0} & \sin \tau _{1} & \cdots & \sin \tau _{n_{T} - 1} \\ \vdots & \vdots & \ddots & \vdots \\ \cos (n_{H} \tau _{0})/2 & \cos (n_{H} \tau _{1})/2 & \cdots & \cos (n_{H} \tau _{n_{T} - 1})/2\end{bmatrix} , \\ {{F}}_{0,n_{T}}^{-1} &= \begin{bmatrix} 1 & \cos \tau _{0} & \sin \tau _{0} & \cdots & \cos (n_{H} \tau _{0}) \\ 1 & \cos \tau _{1} & \sin \tau _{1} & \cdots & \cos (n_{H} \tau _{1}) \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cos \tau _{n_{T} - 1} & \sin \tau _{n_{T} - 1} & \cdots & \cos (n_{H} \tau _{n_{T} - 1})\end{bmatrix} , \end{aligned} $$

(58)

for even values of $n_{T}$. These discrete operators are found by approximating the continuous Fourier transform and its inverse through a trapezoidal quadrature rule [43].

Appendix B: Radau polynomials

Legendre’s polynomials of degree $k$, denoted as $P_{k}(\xi )$, $\xi \in [-1, 1]$, are orthogonal polynomials generated by the following recurrence relationship:

$$ (k+1) P_{k + 1}(\xi ) = (2k + 1) \xi P_{k - 1}(\xi ) - k P_{k - 2} ( \xi ),\quad k \ge 2. $$

(59)

The few lowest-order polynomials are $P_{0} (\xi ) = 1$, $P_{1} (\xi ) = \xi$, $P_{2} (\xi ) = (3 \xi ^{2} - 1) / 2$, $P_{3} (\xi ) = (5 \xi ^{3} - 3 \xi ) / 2$. The set of polynomials of degree less or equal to $N$ forms a vector space of dimension $N + 1$, denoted as ${\mathbf{P}}_{N}$. Clearly, the set of Legendre’s polynomials up to the $N$th degree, $\{P_{0}, \ldots , P_{N}\}$, forms an orthogonal basis of ${\mathbf{{P}}}_{N}$.

The left and right Radau polynomials [44] of degree $k$ are defined as

$$\begin{aligned} \bar{\ell }_{k}&= \displaystyle \frac{1}{2} (P_{k} + P_{k-1}), \end{aligned}$$

(60a)

$$\begin{aligned} \bar{r}_{k} &= \displaystyle \frac{(-1)^{k}}{2} (P_{k} - P_{k-1}). \end{aligned}$$

(60b)

The left and right Radau points are zeros of polynomials $\bar{\ell }_{k}$ and $\bar{r}_{k}$, respectively. Clearly, the right Radau polynomial $\bar{l}_{k}$ is orthogonal to any polynomial $p\in {\mathbf{{P}}}_{k-2}$. It is verified easily that

$$\begin{aligned} &\bar{\ell }_{k}(-1) = 0, \qquad \bar{\ell }_{k}(1) = 1, \end{aligned}$$

(61)

$$\begin{aligned} &\bar{r}_{k}(-1) = 1, \qquad \bar{r}_{k}(1) = 0. \end{aligned}$$

(62)

Considering a polynomial $p \in {\mathbf{{P}}}_{k-1}$, integration by parts leads to

$$\begin{aligned} \int _{-1}^{1} p \bar{\ell }_{k}^{\,\prime } \, \mathrm{d}t &= \left . p \bar{\ell }_{k} \right \vert _{-1}^{1} - \int _{-1}^{1} p' \bar{\ell }_{k} \, \mathrm{d}t = p(1), \end{aligned}$$

(63a)

$$\begin{aligned} \int _{-1}^{1} p \bar{r}_{k}^{\,\prime } \, \mathrm{d}t &= \left . p \bar{r}_{k} \right \vert _{-1}^{1} - \int _{-1}^{1} p' \bar{r}_{k} \, \mathrm{d}t = - p(-1), \end{aligned}$$

(63b)

because of identities (61) and $p' \in {\mathbf{{P}}}_{k-2}$.