# Dynamics of Quasiperiodic Beams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Continuous Beam with Stiffeners

^{3}. A unit cell of such a structure is shown in Figure 1.

#### 2.1. The Periodic Case: Unit Cell Analysis

#### 2.2. Spectral Properties—Bulk and Finite Domains

#### 2.3. Experimental Results on a Finite Beam

## 3. Sandwich Quasiperiodic Beams

#### 3.1. Dynamics of Sandwich Beams

#### 3.1.1. Geometric and Material Properties

^{3}. Using the work of Gibson and Ashby [46], the shear modulus for the regular honeycomb (material A) is obtained to be $G=17.9$ MPa and density $\rho =48$ kg/m

^{3}as opposed to the shear modulus of the auxetic core (material B) whose $G=95.7$ MPa and $\rho =148$ kg/m

^{3}. Furthermore, the distributed parameter model to analyze the behavior of such sandwich structures is also taken from Ruzzene et al. [43].

#### 3.1.2. Harmonic Response

#### 3.2. Numerical Results: Frequency Response Function

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

3-D | Three-dimensional |

FEM | Finite Element Method |

FRF | Frequency Response Function |

IDS | Integrated Density of States |

QP | Quasi-Periodic |

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**Figure 3.**(

**a**) dispersion behavior of the unit cell polarized to the out-of-plane bending mode, i.e., stronger red color represents the more contribution from the out-of-plane bending mode; (

**b**) dispersion behavior of the unit cell polarized to the in-plane bending mode, i.e., red color represents in-plane bending mode. Figures (

**c**–

**i**) are the mode shapes of the unit cell for the respective branch. Purple–white–green colormap is used to represent the deformation in the mode shapes with green representing the lowest and purple representing the height amplitude of deformation. Roman numerals I–VII are used to label the modes such that (

**c**) I—out-of-plane bending (first), (

**d**) II—torsion (first), (

**e**) III—out-of-plane bending (second), (

**f**) IV—in-plane bending (first), (

**g**) V—torsion (second), (

**h**) VI—in-plane bending (second), (

**i**) VII—axial (first).

**Figure 4.**Bulk frequency spectrum for all beam modes. (

**a**) raw frequency spectrum; (

**b**) frequency spectrum polarized with polarization factor ${\mathcal{P}}_{b}^{o}$, revealing butterflies associated with various modes.

**Figure 5.**For the first in-plane bending mode, (

**a**–

**c**) represent the bulk spectrum obtained from a commensurate structure of 100 unit cells, finite spectrum (in red) of the cantilever beam made of 20 unit cells overlapped with the the bulk spectrum (in black), and the representative mode shapes for given frequencies and $\theta =0.89$, respectively. Similar information is provided for the first torsional mode in (

**d**–

**f**) as well as the out-of-plane bending modes in (

**g**–

**i**).

**Figure 6.**(

**a**–

**c**) Integrated Density of States (IDS) as a function of $\theta $ exhibits a sharp jump in color level at the bandgaps for (

**a**) first in-plane bending mode, (

**b**) first torsional mode, and (

**c**) first out-of-plane bending mode; (

**d**) IDS for the out-of-plane bending mode at the projection parameter value of $\theta =0.89$ that is chosen in Section 2.3 for further theoretical and experimental studies.

**Figure 7.**Bulk (black dots) and finite (red dots) spectrum for the out-of-plane bending mode (

**a**). FRFs in out-of-plane bending, spatially averaged over a portion of the beam length (

**b**–

**d**): $x\in [0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}L]$ (

**b**), $x\in [0.2L,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.4L]$ (

**c**), and $x\in [0.75L,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.9L]$ (

**d**).

**Figure 8.**Experimental set-up: (

**a**) stiffened beam with parameter $\theta =0.89$, (

**b**) cantilever beam’s free end equipped with disc-shaped piezoelectric actuator, (

**c**) mounts representing the cantilever boundary.

**Figure 9.**FRF of the beam subject to tip harmonic excitation: (

**a**) numerically obtained FRF; (

**b**) experimentally obtained FRF.

**Figure 10.**(

**a**) Overlapping bulk spectrum obtained from a commensurate structure of 100 unit cells for both torsional (blue) and out-of-plane bending (black) modes; (

**b**) bulk spectrum of out-of-plane bending mode overlapped with the finite spectrum of cantilever beam made of 20 unit cells associated with the out-of-plane bending mode in red and the torsional mode in blue.

**Figure 11.**(

**a**) Numerical results with solid blue line representing case with $\theta =0.89$; (

**b**–

**d**) experimental frequency response for the case with $\theta =0.89$ at the spatial location: (

**b**) 0.20L; (

**c**) 0.25L; (

**d**) averaged over the range. The red and green regions represent the numerically obtained non-trivial bandgap, whereas the blue region represents the trivial bandgap. (

**e**) Experimentally obtained mode shapes for the peaks in the experimental frequency response plots lying in the theoretical bandgaps are actually the torsional modes due to contamination of the out-of-plane bending mode’s frequency response. The points shown are the locations of the experimentally obtained wavefield data. A jet colormap is overlapped with the deformation to provide a reference.

**Figure 15.**Numerically obtained deformed shapes of the periodic sandwich beam at frequencies: (

**a**) 300 Hz, (

**b**) 600 Hz, and (

**c**) 900 Hz.

**Figure 16.**Numerical frequency-response function using dynamic stiffness matrix under base excitation while the response is evaluated at (

**a**) middle node, (

**b**) tip of the sandwich beam.

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**MDPI and ACS Style**

Gupta, M.; Ruzzene, M.
Dynamics of Quasiperiodic Beams. *Crystals* **2020**, *10*, 1144.
https://doi.org/10.3390/cryst10121144

**AMA Style**

Gupta M, Ruzzene M.
Dynamics of Quasiperiodic Beams. *Crystals*. 2020; 10(12):1144.
https://doi.org/10.3390/cryst10121144

**Chicago/Turabian Style**

Gupta, Mohit, and Massimo Ruzzene.
2020. "Dynamics of Quasiperiodic Beams" *Crystals* 10, no. 12: 1144.
https://doi.org/10.3390/cryst10121144