Harmonic balance-based nonsmooth modal analysis of unilaterally constrained discrete systems

Abstract

Nonsmooth modal analysis of a unilaterally constrained one-dimensional bar with constant cross-sectional area was recently proposed. The corresponding formulation took advantage of the d’Alembert solution available for such systems and does not require any space semi-discretization of the governing equations. However, it is unable to cope with non-constant cross-sectional area bars, for instance. The present work suggests a formulation relying on various space semi-discretization methodologies (such as finite elements, Rayleigh–Ritz techniques, component mode synthesis, modal superposition and other reduced-order models) where the complementarity Signorini condition, reflecting the unilateral contact constraint, is enforced in a weighted residual sense in time through the harmonic balance method. Importantly, an impact law, such as Newton’s impact law, for instance, classically required for uniqueness purposes in a space semi-discrete dynamical framework, is here explicitly ignored in the proposed formulation and is, instead, implicitly satisfied in a weighted residual sense. As required for the existence of periodic solutions, the predicted vibratory responses would then correspond to an energy-preserving impact law in case the latter had to be explicitly implemented. Periodic responses are investigated in the form of classical energy-frequency backbone curves along with the associated displacement fields. It is found that for the constant cross-section benchmark system, the results compare well with existing works and the proposed methodology stands as a viable option in the field of interest when semi-discretization in space is required.

Appendices

A Periodic solutions of one-dof systems

This appendix briefly discusses the proposed weak enforcement of the Signorini boundary condition for two academic one-dof systems featuring exact solutions in the sense of distributions.

1.1 A.1 Forced one-dof system: the bouncing ball

A rigid mass m subject to gravity, illustrated in Fig. 22, is considered. The (partial) governing equations are:

$$\begin{aligned} m \ddot{u}(t)= & {} -mg +p(t),\quad u(t)\ge 0,\quad p(t)\ge 0,\nonumber \\{} & {} \qquad u(t)p(t)=0, \quad u(0) = u_0, \quad v(0) = 0\nonumber \\ \end{aligned}$$

(A.1)

Fig. 22

System of interest: bouncing ball

where u(t) is the displacement of the ball; g stands for gravity and p(t) is the contact force. The unilateral contact condition is supplemented with an impact law reading: for all t such that $u(t)=0$ with ${\dot{u}}(t-)<0$, then ${\dot{u}}(t+)=-e {\dot{u}}(t-)$ to guarantee solution uniqueness [7]. The restitution coefficient is set to $e = 1$ to preserve energy and thus existence of periodic solution. Normalization $\bar{u} = u/2u_0$, $\bar{t} = t/\sqrt{2u_0/g}$ and $\bar{p} = p/mg$ yields $\ddot{u}(t) = -g+p(t)$, $u(0) = 0.5$, $v(0) = 0$ and $g = 1$. The exact solution, shown in Fig. 23a, is periodic with period $T = 2$ and the solution in the range of $t\in [0,T]$ reads

$$\begin{aligned} u(t) = \left\{ \begin{aligned}&-0.5t^2+0.5{} & {} 0\le t\le 1\\&-0.5t^2+2t-1.5{} & {} 1< t\le 2 \end{aligned}\right. \end{aligned}$$

(A.2)

along with $p(t) = \delta (t-1)$ where $\delta (\cdot )$ is the Dirac delta distribution. The corresponding Fourier series are

$$\begin{aligned}{} & {} u(t) = \frac{1}{3}-\frac{2}{\pi ^2}\sum ^{\infty }_{m=1}\frac{(-1)^m}{m^2}\cos (m\pi t)\nonumber \\{} & {} \quad \text {and}\quad p(t) =1+2\sum ^{\infty }_{m = 1}(-1)^m\cos (m\pi t) \end{aligned}$$

(A.3)

where p(t) is also the time derivative of u(t) twice, termwise. Pointwise convergence is achieved for the displacement and distributional convergence for the contact force even away from the delta.

Fig. 23

Time-domain system response: a displacement, b contact force. Exact solution (black lines) and approximation with $M = 10$ (blue lines), 20 (red lines) and 40 (yellow lines). (Color figure online)

Fig. 24

Frequency-domain system response: a displacement, b contact force. Exact solution (black lines) and approximation with $M = 10$ (blue lines), 20 (red lines) and 40 (yellow lines). (Color figure online)

The methodology detailed in Sects. 3.2 and 3.3 is now used with

$$\begin{aligned}{} & {} u(t) \approx u_{\!M}(t)=\sum ^M_{m = 0}{\hat{u}}_m\cos (m\Omega t),\nonumber \\{} & {} \quad p(t)\approx p_{\!M}(t) =\sum ^M_{m = 0}{\hat{p}}_m\cos (m\Omega t). \end{aligned}$$

(A.4)

The unknowns of the problem are the $2M+3$ quantities $\{{\hat{u}}_m,{\hat{p}}_m,\Omega \}$, $m=0,\ldots ,M$. A Fourier transform on Eq. A.1 leads to the $M+1$ equations ${\hat{p}}_m = -(m\Omega )^2{\hat{u}}_m$, $m=1,\ldots ,M$ along with ${\hat{p}}_0 = g$. The weak enforcement of the Signorini condition as developed in the paper leads to the $M+1$ additional equations

$$\begin{aligned}{} & {} g_m(\hat{\textbf{u}},\hat{\textbf{p}},\Omega )\nonumber \\{} & {} \quad \equiv \frac{1}{T}\int _0^T \cos (m\Omega t)(-p_{\!M}(t)\nonumber \\{} & {} \qquad +\max [-\rho u_{\!M}(t)+p_{\!M}(t),0])\textrm{d} t=0,\nonumber \\{} & {} \quad \qquad m=0,1,\ldots , M, \end{aligned}$$

(A.5)

where $\hat{\textbf{u}}=({\hat{u}}_0,{\hat{u}}_1,\ldots ,{\hat{u}}_{\!M})$ and $\hat{\textbf{p}}=({\hat{p}}_0,{\hat{p}}_1,\ldots ,{\hat{p}}_{\!M})$. The last equation is $\sum ^M_{m = 0}{\hat{u}}_m = u_0$ since the initial displacement of the bouncing ball is prescribed and the frequency of the response is unknown. The initial velocity $v_0 = 0$ is implicitly enforced since Fourier series are limited to $\cos$ terms only. The above equations are collectively solved using a nonlinear solver. In practice $\{\Omega ,\hat{\textbf{u}}\}$ are the sole independent unknowns and Eq. A.5 only should now be solved. We set $\rho =10$.

The resulting numerically approximated contact force and displacement are shown in Fig. 23. For a low $M\approx 10$, the approximation exhibits a slightly longer period compared to the exact solution and the instantaneous impact is not accurately captured. For a higher $M\approx 40$, the approximate contact force converges to Dirac delta function (in a distributional sense). Convergence on the displacement and period is also achieved. Frequency-domain convergence analysis is illustrated in Fig. 24. It is self-explanatory and it is concluded that the proposed solution strategy to enforce a Signorini boundary condition via HBM is able to reproduces, in a weak sense, an energy-preserving impact law.

Fig. 25

1-dof spring-mass system with unilateral contact

1.2 A.2 Autonomous one-dof system

Another simple spring-mass example, illustrated in Fig. 25, closer in spirit to the one considered in the present paper is briefly explored. A mass m is attached to a spring k and subject to a unilateral condition with initial gap $g_0$. Normalized quantities are: time $\bar{t} = \sqrt{k/m}~t$, displacement $\bar{u} = u/g_0$ and impact force $\bar{p} = p/(g_0k)$. The corresponding normalized governing equation reads $u({t}) + \ddot{u}({t}) = {p}({t})$ where the upper bar notation is dropped. Unilateral contact is governed by the complementarity Signorini condition along with the Newton impact law with $e = 1$. The main difference with the previous system is the absence of external forcing. Accordingly, the exact solution feature a continuum of (nonlinear) natural frequency $\Omega \in ]1,2[$ and reads

$$\begin{aligned} u(t) = \frac{1}{\cos (T/2)}\left\{ \begin{aligned}&\cos (t){} & {} 0\le n\le T/2\\&\cos (t-T){} & {} T/2< n\le T \end{aligned}\right. \nonumber \\ \end{aligned}$$

(A.6)

with $T=2\pi /\Omega$ and $p(t) = 2\tan (T/2)\delta (T/2)$. The exact solution for $T = 4.5$ is shown in Fig. 26. The corresponding Fourier series read

$$\begin{aligned} u(t)= & {} \frac{2\tan (T/2)}{T}\Bigl (1 +\sum ^{\infty }_{m = 1}\frac{2(-1)^m}{1-m^2\Omega ^2}\cos (m\Omega t)\Bigr )\quad \text {and}\nonumber \\ p(t)= & {} \frac{2\tan (T/2)}{T}\Bigl (1 +\sum ^{\infty }_{m = 1}2(-1)^m\cos (m\Omega t)\Bigr ). \end{aligned}$$

(A.7)

Fig. 26

Time-domain system response: a displacement, b contact force. Exact solution (black lines) and approximation with $M = 10$ (blue lines), 20 (red lines), and 40 (yellow lines). (Color figure online)

Fig. 27

Frequency-domain system response: a displacement, b contact force. Exact solution (black lines) and approximation with $M = 10$ (blue lines), 20 (red lines), and 40 (yellow lines). (Color figure online)

Fig. 28

Convergence analysis: a displacement, b strain. Exact solution (black lines) and LM solution with $N = 5$ (blue lines) and $N = 20$ (red lines). (Color figure online)

The governing equations are again solved using the proposed solution strategy with $T = 4.5$. A total of $2M+2$ unknowns $\{\hat{\textbf{u}},\hat{\textbf{p}}\}$ are to be found by solving the $M+1$ equations $((1-(m\Omega )^2){\hat{u}}_m = {\hat{p}}_m$, $m=0,1,2,\ldots ,M$ and the $M+1$ additional Signorini conditions in the form (A.5) where $\Omega$, and thus T, is specified.

The displacement and contact force are shown in Fig. 26 for various M (assuming a discretization of the type (A.4)). Pointwise convergence is achieved for the displacement and distributional convergence for the contact force with a rate slightly slower than O(1/M) (not shown). Frequency-domain convergence analysis is illustrated in Fig. 27. Again, the proposed scheme finds approximate periodic solutions satisfying the impact law $e = 1$ in a weak sense, in agreement with results exposed in [26].

B Linear mode expansion convergence analysis

In this paper, the LM method was developed by considering a family of clamped-free (homogeneous Dirichlet-homogeneous Neumann) modeshapes. It is here shown how the method can generate an approximate solution to a bar problem with homogeneous Dirichlet-non-homogeneous Neumann boundary condition. To this end, we reduce the Helmholtz equation to a time-independent configuration. It the becomes $u_{,xx}(x) = 0$, $x\in ]0,1[$, with $u(0)=0$ and $u_{x}(1)=1$ (non-homogeneous Neumann BC). The exact solution is $u(x) = x$, $x\in [0,1]$. The clamp-free modes of the bar are listed in Eq. 14. Using them to expand the approximation leads to the weak form

$$\begin{aligned} \int _0^1\bigl (\varvec{\phi }_{x}(x)^{\top } \varvec{\phi }_{x}(x)\bigr ) \textrm{d} x ~\textbf{u}=\varvec{\phi }(1) \end{aligned}$$

(B.1)

with⁴$\textbf{u}\equiv \left[ a_1, \ldots , a_{\!N}\right] ^{\top }$ storing the contributions of the shape functions (ie, the linear modes) collected in $\varvec{\phi }(x) \equiv \left[ \phi _1(x), \ldots , \phi _{\!N}(x)\right] ^{\top }$. The system of linear equations simplifies to $\omega _i^2 a_i = (-1)^{i-1}\sqrt{2}$, $i=1,\ldots ,N$ so that $a_i = (-1)^{i-1}\sqrt{2}/\omega _i^2$ and the approximated displacement solution of the system reads

$$\begin{aligned} u_{\!N}(x) = \sum _{i = 1}^{N}(-1)^{i-1}\frac{2}{\omega _i^2}\sin (\omega _i x),\quad \omega _i = \frac{2i-1}{2}\pi ,\nonumber \\ \end{aligned}$$

(B.2)

which converges pointwise to the exact solution $u(x) = x$ as $N\rightarrow \infty$.

The approximate transfer function defined in the paper reduces here to $G_{\!N}=u_{\!N}(1)$ since it is assumed that $p=1$.Accordingly, the convergence of $u_{\!N}(1)$ as $N\rightarrow \infty$ is clarified. Equation B.2 yields

$$\begin{aligned} \lim _{N\rightarrow \infty }u_{\!N}(1) =\frac{8}{\pi ^2}\sum _{i=1}^\infty \frac{1}{(2i-1)^2}=1=u(1). \end{aligned}$$

(B.3)

The error caused by truncation at order N is

$$\begin{aligned} E_{\!N} =|u(1)-u_{\!N}(1)|= \frac{8}{\pi ^2}\sum _{i=N+1}^\infty \frac{1}{(2i-1)^2} \end{aligned}$$

(B.4)

with the following bounds:

$$\begin{aligned} \int _{N+1}^\infty \frac{1}{(2x-1)^2}\mathop {}\!\textrm{d}x\le & {} \sum _{i=N+1}^\infty \frac{1}{(2i-1)^2}\nonumber \\\le & {} \int _{N}^\infty \frac{1}{(2x-1)^2}\mathop {}\!\textrm{d}x. \end{aligned}$$

(B.5)

Further evaluation of the upper and lower bounds leads to

$$\begin{aligned} \frac{8}{\pi ^2}\frac{1}{4N+2}<E_{\!N}<\frac{8}{\pi ^2}\frac{1}{4N-2} \end{aligned}$$

(B.6)

indicating that the sequence $\{u_{\!N}(1)\}$ for N sufficiently large has the rate of convergence O(1/N) and so has $G_{\!N}$, which matches the derivations reported in Sect. 4.1.

Although $u_{\!N}(x)$ converges pointwise to the exact solution, see Fig. 28a, the approximated strain

$$\begin{aligned} u_{\!N,x}(x) = 2\sum _{i = 1}^{N}\frac{\cos (\omega _i x)}{\omega _i},\quad \omega _i = \frac{2i-1}{2}\pi \end{aligned}$$

(B.7)

only converges in $L^2$ sense to $u_x$. Obviously, the approximated strain vanishes at $x=1$, that is $u_{\!N,x}(1)=0$, which is independent of N, and does not converge to the exact strain $u_{x}(1)=1$. A Gibbs phenomenon is thus generated (here in space) around $x = 1$. The conclusion is the following: in the approximate solution, the LM method generates a Gibbs phenomenon in space which adds to the Gibbs phenomenon in time induced by HBM. The associated expansion family is clearly not optimal for the problem at hand but convergence is still achieved in a weak sense.