• Influence of the aspect ratio of the sheet for an electric generator utilizing the rotation of a flapping sheet, Mechanical Engineering Journal, Vol. 8, No. 1 (2021).
  https://doi.org/10.1299/mej.20-00459

@article{Akio YAMANO202120-00459,
    title={Influence of the aspect ratio of the sheet for an electric generator utilizing the rotation of a flapping sheet},
    author={Akio YAMANO and Hiroshi IJIMA and Atsuhiko SHINTANI and Chihiro NAKAGAWA and Tomohiro ITO},
    journal={Mechanical Engineering Journal},
    volume={8},
    number={1},
    pages={20-00459-20-00459},
    year={2021},
    doi={10.1299/mej.20-00459}
}

• Flow-induced vibration and energy-harvesting performance analysis for parallelized two flutter-mills considering span-wise plate deformation with geometrical nonlinearity and three-dimensional flow, International Journal of Structural Stability and Dynamics, Vol. 22, No. 14, (2022).
  https://doi.org/10.1142/S0219455422501632

@article{doi:10.1142/S0219455422501632,
    author = {Yamano, Akio and Chiba, Masakatsu},
    title = {Flow-Induced Vibration and Energy-Harvesting Performance Analysis for Parallelized Two Flutter-Mills Considering Span-Wise Plate Deformation with Geometrical Nonlinearity and Three-Dimensional Flow},
    journal = {International Journal of Structural Stability and Dynamics},
    volume = {22},
    number = {14},
    pages = {2250163},
    year = {2022},
    doi = {10.1142/S0219455422501632}
}

• Influence of boundary conditions on a flutter-mill, Journal of Sound and Vibration, Vol. 478, No. 21 (2020).
  https://doi.org/10.1016/j.jsv.2020.115359

@article{YAMANO2020115359,
    title = {Influence of boundary conditions on a flutter-mill},
    journal = {Journal of Sound and Vibration},
    volume = {478},
    pages = {115359},
    year = {2020},
    doi = {https://doi.org/10.1016/j.jsv.2020.115359},
    author = {A. Yamano and A. Shintani and T. Ito and C. Nakagawa and H. Ijima}
}

├─double_sheets
│  ├─cores
│  │  ├─functions
│  │  │  ├─fluid
│  │  │  └─structure
│  │  └─solver
│  │      ├─fluid
│  │      └─structure
│  ├─save
│  │  └─fig
│  │      └─modes
│  └─ToolBoxes
└─single_sheet
    ├─functions
    │  ├─fluid
    │  └─structure
    ├─save
    │  └─fig
    │      └─modes
    ├─solver
    │  ├─fluid
    │  └─structure
    └─ToolBoxes

[Step 1] Install the ToolBoxes

The following ToolBoxes in “./XXXX/ToolBoxes/” ("XXXX" is "double_sheets" and "single_sheet") are required,

For numerical analysis:

• “Meshing a plate using four noded elements” by KSSV:
  https://jp.mathworks.com/matlabcentral/fileexchange/33731-meshing-a-plate-using-four-noded-elements

• “Sparse sub access” by Bruno Luong:
  https://jp.mathworks.com/matlabcentral/fileexchange/23488-sparse-sub-access

• “Vectorized Multi-Dimensional Matrix Multiplication” by Darin Koblick:
  https://jp.mathworks.com/matlabcentral/fileexchange/47092-vectorized-multi-dimensional-matrix-multiplication?s_tid=prof_contriblnk

For plotting results:

• “TriStream” by Matthew Wolinsky:
  https://jp.mathworks.com/matlabcentral/fileexchange/11278-tristream

• “mmwrite” by Micah Richert:
  https://jp.mathworks.com/matlabcentral/fileexchange/15881-mmwrite

• “Quiver 5” by Bertrand Dano:
  https://jp.mathworks.com/matlabcentral/fileexchange/22351-quiver-5?s_tid=FX_rc3_behav

[Step 1.2] Add path to installed ToolBoxes

Modify "add_pathes.m" to add path to abovementined installed ToolBoxes as follows,

addpath ./ToolBoxes/XX;

where XX is the name of folder of the installed ToolBox.

[Step 1.3] Modify the source code in the “TriStream” ToolBox

"TriStream.m" must be modified to plot the stram line Line.

Line 49: x=x(:)'; y=y(:)'; x0=x0(:)'; y0=y0(:)'; u=u(:)'; v=v(:)';

→

 x=x(:)'; y=y(:)'; x0=x0(:)'; y0=y0(:)'; u=double(u(:)'); v=double(v(:)');

and

line 63: TRI = tsearch(x,y,tri',Xbeg,Ybeg);

→

TRI = tsearchn([x.' y.'], tri',[Xbeg.' Ybeg.']);

[Step 2] Start GUI form

Open the “GUI.fig” from MATLAB.

[Step 2.1] Pre-setting

Push the "Parameters" buttun and edit parameters.

[Step 3] Start analysis

Push the “exe” button and wait until the finish of the analysis.

[Step 4] Plot results

Push the “plot” button.

[Step 5] View plotted results

Results (figures and movie) plotted by [Step 4] are in "./XXXX/save" directory.

Analytical condisions are in "./XXXX/save/param_setting.m"

%% Analytical conditions
End_Time = 20;                                          %% Nondimensional analysis time [-]
d_t = 1.0e-3;                                           %% Nondimensional step time [-]
core_num = 6;                                           %% Core number [-]
speed_check = 0;                                        %% 1:ON, 0:OFF [-]
alpha_v = 0.5;                                          %% 1:implicit solver，0:explicit solver [-]

Ma = 1.0;                                              %% Mass ratio [-]
Ua = 15.0;                                            	%% Nondimensional flow velocity [-]
theta_a_vec = 0e-1*[ 0 10];                           	%% Nondimensional material damping [-]

• Mass ratio $M^*$ is the density ratio of the fluid and sheet, which is defined by,

$$M^* := \dfrac{1}{\mu} = \dfrac{\rho_f L}{\rho_m h},$$

where $\rho_f$ and $\rho_m$ are the density of fluid and sheet, respectively. $L$ is the total length of sheet, and $h$ is the thickness of sheet.

• Nondimensional flow velocity $U^*$ represents the free-stream velocity nondimensionalized by the flag rigidity and inertia [^1][^2][^3],

$$U^* := \sqrt{ \dfrac{\mu}{\eta}} = \sqrt{ \dfrac{ \rho_m h L^2 H U^2_\infty }{ EI }}, $$

where $H$ is the width, $E$ is the Young's modulus, and $I := Hh^3/12$ is second moment of area of the sheet, respectively.

Initial disturbances on two sheets to break the trivial equilibrium are written as,

q_in_norm = @( time)( 0.5*sin( pi*time/0.2).*( time < 0.2 ) );              %% Initial disturbance (upper sheet)
q_in_norm_1 = @( time)( 0.0*sin( pi*time/0.2).*( time < 0.2 ) );        	%% Initial disturbance (lower sheet)
q_in_vec = [ 0 0 1].';                                                      %% Force direction [-]  

Dimensions of sheets are defined by,

Length = 1.0;                                   %% (Nondimensional) length [-]
Width = 1.0;                                    %% aspect ratio [-]
Height = 2.0;                              	    %% distance between two sheets [-]
thick = 1e-3;                                   %% thickness [-]

where the aspect ratio Width is $H^* := H/L$, the nondimensional thickness thick is $h^* := h/L$, and the nondimensional distance between two sheets Height is $D^* := D/L$.

Boundary conditions for two sheets are written as,

• Clamped at the leading-edge

%% Boundary conditions for two sheets

%%[0] Upper sheet
node_r_0 = [ 1:Ny+1 ];                                                      %% Node number giving the displacement constraint [-]
node_dxr_0 = [ 1:Ny+1 ];                                                    %% Node number giving x-directional gradient constraint [-]
node_dyr_0 = [ 1:Ny+1 ];                                                    %% Node number giving y-directional gradient constraint [-]

%%[1] Lower sheet
node_r_0_1 = [ 1:Ny+1 ];                                                    %% Node number giving the displacement constraint [-]
node_dxr_0_1 = [ 1:Ny+1 ];                                                  %% Node number giving x-directional gradient constraint [-]
node_dyr_0_1 = [ 1:Ny+1 ];                                                  %% Node number giving y-directional gradient constraint [-]

• Pinned at the leading-edge

%% Boundary conditions for two sheets

%%[0] Upper sheet
node_r_0 = [ 1:Ny+1 ];                                                      %% Node number giving the displacement constraint [-]
node_dxr_0 = [ ];                                                           %% Node number giving x-directional gradient constraint [-]
node_dyr_0 = [ 1:Ny+1 ];                                                    %% Node number giving y-directional gradient constraint [-]

%%[1] Lower sheet
node_r_0_1 = [ 1:Ny+1 ];                                                    %% Node number giving the displacement constraint [-]
node_dxr_0_1 = [ ];                                                         %% Node number giving x-directional gradient constraint [-]
node_dyr_0_1 = [ 1:Ny+1 ];                                                  %% Node number giving y-directional gradient constraint [-]

where index in vector shows the node index around a plate element to apply boundary conditions.

(b) Shell elements on a plate. A flexible sheet is partitioned by shell elements when the number of elements is $N_{elem} := N_x N_y$ [^2].