Within this work, the beam’s transverse elastic deformations
w(
x,
t) are described in terms of the
N lowest-frequency bending modes. Here
x is the spatial coordinate in the direction of the beam’s length and
t the time. The clamping and the support frame are assumed to be perfectly rigid. We assume the beam to be homogeneous, with constant cross section area
A and density
\(\rho \), and to behave linear elastically, with the Young’s modulus
E. As in the experiments, the beam’s length is assumed to be much larger than the thickness, such that Euler–Bernoulli theory is justified and, accordingly, we neglect shear, Poisson effect, rotational and longitudinal inertia. The vibrational amplitude is assumed to be small in comparison with the beam’s length; however, we consider nonlinear terms due to stretching enforced by the clamped–clamped boundary conditions. As shown e.g. in [
4,
15,
18], this leads to the following equation system (for the clamped–clamped beam without slider):
$$\begin{aligned}&w(x,t) \approx \sum _{n=1}^N \varphi _n\left( x\right) q_n(t), \end{aligned}$$
(1)
$$\begin{aligned}&\ddot{q}_n + 2D_n \omega _n^\mathrm {lin} \dot{q}_n + \left( \omega _n^\mathrm {lin} \right) ^2 q_n + \sum _{j=1}^N \sum _{k=1}^N \sum _{l=1}^N b_{j,k,l}^{(n)} q_j q_k q_l = \gamma _n \ddot{w}_0(t) \quad \quad \mathrm {with} \quad n=1,2,\ldots ,N , \end{aligned}$$
(2)
$$\begin{aligned}&b_{j,k,l}^{(n)} = - \frac{EA}{2L} \int _0^L \varphi '_j \varphi '_k \mathrm {d}x \int _0^L \varphi ''_l\varphi _n \mathrm {d}x . \end{aligned}$$
(3)
Differentiation with respect to
t and
x is denoted
\(\dot{()}\) and
\(()'\). Equation
1 represents the ansatz, which assumes that the beam’s deformation can be well approximated by a linear combination of the
N lowest-frequency mode shapes,
\(\varphi _n(x)\) (mass-normalized), of the underlying linear system (linear Euler-Bernoulli theory).
\(\omega _n^\mathrm {lin}\) are the associated natural frequencies,
\(D_n\) the modal damping ratios and
\(q_n\) the modal coordinates. Furthermore,
\(\gamma _n =-\,\rho A \int _{0}^{L} \varphi _n \mathrm {d}x\) and
\(\ddot{w}_0\) is the imposed base acceleration. Equation
2 is the equation of motion, condensed to the transverse direction. Equation
3 defines the coefficients of the cubic nonlinear polynomial that describes the beam’s geometric stiffness nonlinearity.
To underline the significance of the geometric nonlinearity, the lowest natural frequency is plotted in Fig.
3 as a function of vibrational amplitude at the beam’s center. The beam’s properties are set as specified in Sect.
3.1. The natural frequency is determined by nonlinear modal analysis (NMA) and compared to a single-mode single-term Harmonic Balance (HB) approximation. For the NMA, the extended periodic motion concept [
6] is used, which is also applicable to damped systems. To this end, a HB approach with modal truncation after
\(N=7\) modes is considered, using the free Matlab tool NLvib [
8]. The harmonic truncation order is set to
\(H=10\), such that the condition
\(\mathrm {max}(\omega _1){\cdot } H>\omega _N^\mathrm {lin}\) holds in the considered amplitude range. Increasing
N and
H does not change the picture in the considered amplitude range, but yields additional loops for higher amplitude levels which are not relevant for this work.
Herein
\({\hat{q}}_1\) denotes the amplitude of the first modal coordinate which is linked to the physical amplitude at the beam’s center via
\({{\hat{w}}_{L/2}=\varphi _1(L/2) {\cdot } {\hat{q}}_1}\). Both approaches match very well up to
\({\hat{w}}_{L/2}/h \approx 2.6\), where
h is the beam’s thickness. The deviations for larger amplitudes, including a loop in the NMA at
\({\hat{w}}_{L/2}/h \approx 4\), can be explained by modal interactions, see e.g. [
5], which cannot be captured with the single mode approximation in Eq.
4. However, the largest vibration levels encountered in the present study remain sufficiently small,
\({\hat{w}}_{L/2}/h\le 2.5\). It can thus be stated that the approximation in Eq.
4 is valid in the range of vibration levels relevant in the present study.