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Free vibration analysis and mode management of bistable composite laminates using deep learning


In this paper, for the first time, the deep learning technique of the artificial neural network method is used to determine the free vibration parameters of the rectangular bistable composite plates. For this purpose, first, the static and free vibration behaviours of a cross-ply bistable composite plate are studied using analytical, finite element and experimental methods. By comparing them, it is turned out that there is a considerable difference among obtained natural frequencies so that the analytical method is only able to determine the fourth natural frequency and cannot estimate the first three natural frequencies. To solve this problem, the deep neural network is employed to model the modal parameters of the bistable laminate as an explicit mathematical relationship that can be generalized to the other bistable composite plates. This mathematical relation makes it possible to obtain the natural frequencies in each of the stable configurations based on the geometric dimensions of the plate. In the following that, the inverse problem method is considered and the mode management capability is investigated. A fast swarm intelligence algorithm called firefly algorithm is used to optimize the optimization function of the mode management problem. Mode management using the evolutionary algorithm provides the appropriate physical dimensions of the plate according to the scenarios for natural frequencies arrangement. The results are validated by comparing them with those obtained by the finite element method and experiment test, which results show that this method estimates the modal parameters with high accuracy. The method used in this paper can also be applied to determine the modal parameters of other morphing structures.

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The authors acknowledge the reviewers for the valuable criticism and suggestions which helped to improve the manuscript.

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Correspondence to S. Saberi.

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The weigh and bias matrices of trained DNNs which are used in this paper are represented here. As stated, a separate network is considered to estimate each natural frequency. Since three natural frequencies of both stable states have been considered, then six separate networks have been designed and trained. So, there are six sets of weight and bias matrices which each of them is related to a network. The network topology has three layers and is shown in Fig. 

Fig. 11

Topology of three layered network


In the following, the weight and bias matrix of each network is presented.

First natural frequency of the first stable state (Table 6):

Table 6 Weight and bias matrices for the first natural frequency of the first stable state

Second natural frequency of the first stable state (Table 7):

Table 7 Weight and bias matrices for the second natural frequency of the first stable state

Third natural frequency of the first stable state (Table 8):

Table 8 Weight and bias matrices for the third natural frequency of the first stable state

First natural frequency of the second stable state (Table 9):

Table 9 Weight and bias matrices for the first natural frequency of the second stable state

Second natural frequency of the second stable state (Table 10):

Table 10 Weight and bias matrices for the second natural frequency of the second stable state

Third natural frequency of the second stable state (Table 11)

Table 11 Weight and bias matrices for the third natural frequency of the second stable state


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Saberi, S., Ghayour, M., Mirdamadi, H.R. et al. Free vibration analysis and mode management of bistable composite laminates using deep learning. Arch Appl Mech 91, 2795–2816 (2021).

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  • Bistable composites
  • Deep learning
  • Rayleigh Ritz
  • Free vibration
  • Artificial neural network
  • Firefly algorithm