# Nonsmooth modal analysis for a one-dimensional finite bar subject to unilateral contact using the Wave Finite Element Method

Carlos Yoong, Anders Thorin, Mathias Legrand

► To cite this version:

Carlos Yoong, Anders Thorin, Mathias Legrand. Nonsmooth modal analysis for a one-dimensional finite bar subject to unilateral contact using the Wave Finite Element Method. The Fourth Joint International Conference on Multibody Systems Dynamics, May 2016, Montréal, Canada. hal-01408111

HAL Id: hal-01408111

https://hal.science/hal-01408111

Submitted on 3 Dec 2016

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Nonsmooth modal analysis for a one-dimensional finite bar subject to unilateral contact using the Wave Finite Element Method

Carlos Yoong¹, Anders Thorin¹ and Mathias Legrand¹

¹Department of Mechanical Engineering, McGill University,
carlos.yoong@mail.mcgill.ca, anders.thorin@mcgill.ca, mathias.legrand@mcgill.ca

This contribution suggests a numerical procedure, involving the Wave Finite Element Method (WFEM), able to perform nonsmooth modal analysis [1] of a frictionless unilateral contact problem defined on a finite one-dimensional domain, as depicted in Fig 1.

The WFEM is a shock-fitting method which consists in locating and tracking shock waves propagating in a mechanical system [2]. In this approach, the space domain is discretized into cells of length $\Delta x = x_{i+1} - x_i$ and time is discretized into intervals of length $\Delta t = t_{n+1} - t_n$, where $i$ is the index of the cell interface and $n$ is the index of the time step. Then, a system of equations is derived for the calculation of the state (defined here by the stress $\sigma$ and velocity $v$) of the cells at each time step $t_n$. The simulation of an elastic wave propagating with a given finite speed $c$ is performed by iterating these relations in discrete time with a prescribed time interval $\Delta t = \Delta x/c$. This method accurately captures potentially discontinuous wave fronts propagating at a finite speed. For an unforced system without unilateral constraints, the WFEM yields the typical system of equations

$$Q^{(n+1)} = AQ^{(n)} \quad (1)$$

where $Q^{(n)} = [\sigma^{(n)} v^{(n)}]^\top$ is the state of the system at time step $t_n$. The matrix $A \in \mathbb{R}^{2N \times 2N}$, where $N$ is the number of cells, embeds the dynamics of the system. Using Eq. (1), $Q^{(n)}$ can be directly expressed in terms of the initial conditions $Q^{(0)}$

$$Q^{(n)} = A^n Q^{(0)} \quad (2)$$

where $A^n$ is the matrix $A$ to the power of $n$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Elastodynamic problem with unilateral contact constraints and free-fixed boundary conditions</figcaption></figure>

Consider the free-fixed bar in Fig. 1: it is subject to a unilateral contact constraint. Once the WFEM discretization is performed, the contact constraints are enforced using the concept of floating boundary conditions [2], which loosely speaking, can be regarded as a conditional switch between free and fixed boundary conditions when a penetration is detected during the iterative process. From Eq. (2), two types of matrices $A$ shall then be distinguished: $A_f$ for the free-fixed condition (no contact) and $A_c$ for fixed-fixed condition (contact). $T$-periodic solutions are sought using the condition $Q^{(n_T)} = Q^{(0)}$ where $n_T \Delta t = T$. The solution is assumed to be composed by $r$ consecutive steps in the free phase and $p$ consecutive steps in the contact phase, leading to the following periodicity condition

$$Q^{(n_T)} = A_c^p A_f^r Q^{(0)} \quad (3)$$

where $n_T = r + p$. The durations of free and contact phases corresponding to periodic motions are not known a priori; therefore, the values of $r$ and $p$ leading to solutions are unknowns. Also, note that periodic solutions with
more than one contact phase per period are not targeted with the proposed formulation. Periodicity condition in Eq. (3) simplifies to

$$ (A_c^p A_f^r - I) Q^{(0)} = 0. \quad (4) $$

The initial condition $Q^{(0)}$ satisfying Eq. (4) is called a “potential solution”. Potential solutions are admissible solutions of the formulation if they satisfy the following additional conditions:

COND1: Point of contact must not penetrate the wall during periodic motion: $x_1^{(n)} \le d$, $n = 0, 1, \dots, r$

COND2: Point of contact cannot separate from the wall during contact: $x_1^{(n)} = d$, $n = r + 1, r + 2, \dots, r + p$

COND3: Contact interaction must be compressive: $\lambda^{(n)} \le 0$, $n = r + 1, r + 2, \dots, r + p$

In the above conditions, $x_1^{(n)}$ is the displacement of the contact node at time step $t_n$, $d$ is the initial gap between the bar tip and the rigid wall, and $\lambda$ is the contact force.

As seen from Eq. (4), a periodic solution is necessarily an element of the kernel of the matrix $S_T = A_c^p A_f^r - I$, i.e. $Q^{(0)} \in \ker S_T$. The dimension $m$ of $\ker S_T$ depends on the combinations of $r$ and $p$. A non-trivial solution may exist only if $\det(S_T) = 0$, hence only the combinations of $r$ and $p$ that correspond to $m \ge 1$ are of interest. Given a basis $\{e_1, e_2, \dots, e_m\}$ of $\ker S_T$, $Q^{(0)}$ reads

$$ Q^{(0)} = \alpha_1 e_1 + \alpha_2 e_2 + \dots + \alpha_m e_m \quad (5) $$

Initial state $Q^{(0)}$ is completely determined by the unknown coefficients $\alpha_1, \alpha_2, \dots, \alpha_m$. The state $Q^{(n)}$ is calculated at each time step through Eq. (2); hence this state can be expressed in terms of the coefficients $\alpha_1, \alpha_2, \dots, \alpha_m$. A solution is obtained by calculating the coefficients that satisfy the above necessary conditions.

It has been observed that when $\dim(\ker S_T) = 1$, only one solution satisfies all the conditions. Additionally, by changing $r$ and $p$, it is possible to find other admissible solutions in the vicinity of an existing solution. A continuum of periodic orbits then emerges: it defines a nonsmooth mode of vibration (NSM) [3]. The displacement $x_1$ is represented in Fig. 2(a) for three solutions belonging to the first NSM (continuation of the first linear modeshape of the bar) whose Frequency–Energy branch is shown in Fig. 2(b). The periodic solutions are traveling waves with discontinuous wave fronts interacting with each other and the boundaries.

<figure><img src="image_3.png" /><figcaption>Fig. 2: First nonsmooth mode of the free–fixed bar subjected to unilateral contact constraints. Energy is normalized with respect to the energy of the grazing solution</figcaption></figure>

## References

[1] D. Laxalde and M. Legrand, “Nonlinear modal analysis of mechanical systems with frictionless contact interfaces,” *Computational Mechanics*, vol. 47, no. 4, pp. 469–478, 2011. [hal-00492775].

[2] B. Shorr, *The Wave Finite Element Method*. Springer, 2004.

[3] A. Thorin, M. Legrand, and S. Junca, “Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law,” in *International Design Engineering Technical Conferences & Computers and Information in Engineering Conference*, no. DETC2015-46352, (Boston), ASME, 2015. [hal-01185973].