Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

Abstract

A numerical scheme grounded on the Boundary Element Method expressed in the Frequency Domain is proposed to perform Nonsmooth Modal Analysis of one-dimensional bar systems. The latter aims at finding continuous families of periodic orbits of mechanical components featuring unilateral contact constraints. The proposed formulation does not assume a semi-discretization in space of the governing Partial Differential Equations, as achieved in the Finite Element Method, and so mitigates a few associated numerical difficulties, such as chattering at the contact interface, or the questionable approximation of internal resonance conditions. The nonsmooth Signorini condition stemming from the unilateral contact constraint is enforced in a weighted residual sense via the Harmonic Balance Method. Periodic responses are investigated in the form of energy-frequency backbone curves along with the associated displacement fields. It is found that for the one-bar systems, the results compare well with existing works and the proposed methodology stands as a viable option in the field of interest. The two-bar system, for which no known results are reported in the literature, exhibits very rich nonsmooth modal dynamics with entangled nonsmooth modal motions combining hardening and softening effects via the intricate interaction of various, possibly subharmonic and internally resonant, nonsmooth modes of the two bars.

Appendices

Appendix 1: Alternate equivalent frequency-domain formulation

It is possible to retrieve Expression (18) without relying on the developed FD-BEM formulation. The Fourier Transform (6) of the displacement and the resulting Helmholtz Eq. (7) are considered. The general solution to the Helmholtz Eq. (7) is $\hat{u}(x,\omega )=A\cos \omega x+B\sin \omega x$ which induces $\hat{u}_{x}(x,\omega )=-\omega A\sin \omega x+\omega B\cos \omega x$. Reading the two previous identities on the boundary $\{0\}\cup \{1\}$ leads to

$$\begin{aligned}&\hat{u}(0,\omega )=A\nonumber \\&\hat{u}(1,\omega )=A\cos \omega +B\sin \omega \nonumber \\&\hat{u}_{x}(0,\omega )=\omega B=-\hat{p}(0,\omega )\nonumber \\&\hat{u}_{x}(1,\omega )=-\omega A\sin \omega +\omega B\cos \omega =\hat{p}(1,\omega ) \end{aligned}$$

(33)

which is strictly equivalent to (18). The rest of the procedure follows. However, the extension of the FD-BEM to higher dimensions in space is more straightforward for the enforcement of the boundary conditions.

Appendix 2: Separation of variables

It seems appropriate to highlight a major difference between the proposed approach based on direct and inverse Fourier Transforms and the classical separation of variables sometimes used in solving the wave equation via the superposition principle. The technique can be summarized as follows:

• Consider an ansatz solution of the form $u(x,t)=\Phi (x)\exp (j\omega t)$ and plug it into the wave Eq. (1). This implies that the function $\Phi (x)$ is solution to the Helmholtz equation $\Phi _{xx}+\omega ^2\Phi =0$, identical to Eq. (7) stemming from a Fourier Transform.

• The general solution is $\Phi (x)=A\cos \omega x+B\sin \omega x$.

• Enforce the boundary conditions. Let us consider the Dirichlet–Signorini bar for simplicity. Accordingly, $\Phi (0)=0$, that is $A=0$.

• From the above, the solution now reads $u(x,t)=B\sin \omega x\exp (j\omega t)$. However, the Signorini condition cannot be properly enforced at this stage, as achieved in (29). The usual approach would be to consider an homogeneous Neumann condition at $x=1$ in order to generate a family of eigenfunctions $\Phi _k(x)=\sin (k\pi x/2)$, $k=1,3,\ldots$ so that the sought solution is now expressed as the infinite sum

  $$\begin{aligned} u(x,t)=\mathfrak {R}\Bigl (\sum _{k,\mathrm{odd}} B_k\sin (k\pi x/2) \exp (jk\pi t/2)\Bigr ) \end{aligned}$$

  (34)

  and then try to enforce the Signorini condition. However, this would require either a penalization or an impact law, with the corresponding questions on the values of the companion parameters (penalization coefficient or impact law restitution coefficient). Note that a finite-element-based approach in space would not help either.

The Frequency-Domain procedure exposed in the current work overcomes the above difficulties by handling the Signorini condition directly in the frequency domain and never assumes a solution in the form (34), or similar.

Appendix 3: D’Alembert solution and Fourier transform

The general solution to the wave Eq. (1) is D’Alembert solution $u(x,t)=f(x+t)+g(x-t)$. The Dirichlet condition at $x=0$ implies $f(t)+g(-t)=0$, that is $u(x,t)=f(x+t)-f(t-x)$. The corresponding strain field is $u_{x}(x,t)=f'(x+t)+f'(t-x)$. At $x=1$, both equations yield $u(1,t)=f(t+1)-f(t-1)$ and $u_{x}(1,t)=f'(t+1)+f'(t-1)$. Applying a Fourier Transform to each quantity along t leads to $\hat{u}(1,\omega )=2j\sin \omega \hat{f}(\omega )$ and $\hat{u}_{x}(1,\omega ) =2j\omega \cos \omega \hat{f}(\omega )$, expressions which agree with (23). The same procedure applies to the Robin–Signorini system and Expression (27) would be retrieved. Again, this shows that the proposed approach is a Frequency-Domain procedure based on a Fourier Transform of the D’Alembert solution.