剛性異方性構造の低剛性化と課題

\(\begin{bmatrix}  d_x \\ d_y \\ \theta_z \\\end{bmatrix}
=　\begin{bmatrix}
S_{11} & S_{12} & S_{16} \\  & S_{22} & S_{26} \\ \text{sym.} &  & S_{66} \end{bmatrix}
\begin{bmatrix}  F_x \\ F_y  \\ M_z \\\end{bmatrix}\)

\(\boldsymbol{T} = [S]\boldsymbol{W}\\
\begin{bmatrix}  d_x \\ d_y \\ d_z \\ \theta_x \\ \theta_y \\ \theta_z \\\end{bmatrix}
=　\begin{bmatrix} 
\frac{l^3}{3EI_x} & 0 & 0   & 0 & 0 & \frac{l^2}{2EI_x}\\
0 & \frac{l}{Ebh} & 0       & 0 & 0 & 0\\
0 & 0 & \frac{l^3}{3EI_z}   & -\frac{l^2}{2EI_z} & 0 & 0 \\
0 & 0 & -\frac{l^2}{2EI_z} & \frac{l}{EI_z} & 0 & 0  \\
0 & 0 & 0                  & 0 & \frac{l}{GJ} & 0     \\
\frac{l^2}{2EI_x} & 0 & 0  & 0 & 0 & \frac{l}{EI_x}  \end{bmatrix}
\begin{bmatrix}  F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \\\end{bmatrix}\)

\(J = \frac{bh^3}{3} \left(1-\frac{192h}{\pi^5b} \sum_{n=1}^{\infty} {1 \over 2n-1}\tanh{(2n-1)\pi b\over 2h}\right)\)

\(\boldsymbol{W} = [N] \boldsymbol{W'}\\
\begin{bmatrix}  F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \\\end{bmatrix} =　\begin{bmatrix}
\boldsymbol{n_1} &
\boldsymbol{n_2} &
\boldsymbol{n_3} &
\boldsymbol{0} &
\boldsymbol{0} &
\boldsymbol{0} \\
\boldsymbol{L} \times \boldsymbol{n_1} &
\boldsymbol{L} \times \boldsymbol{n_2} &
\boldsymbol{L} \times \boldsymbol{n_3} &
\boldsymbol{n_1} &
\boldsymbol{n_2} &
\boldsymbol{n_3}
                \end{bmatrix}
\begin{bmatrix} 
F_x' \\ F_y' \\ F_z' \\ M_x' \\ M_y' \\ M_z' \\\end{bmatrix}\)

\([Q] = [N][Q'][N]^\boldsymbol{T}\)

\([M]\frac{d^2 \boldsymbol{T}}{dt^2} + [Q]\boldsymbol{T} = \boldsymbol{W}\)

\([M]^{-1}[Q]\boldsymbol{T} = \omega_n^2\boldsymbol{T}\)

Introduction

　In the previous article, we computed the behavior of a slider structure in a simplified manner as a structure that is less stiff in only one direction. It was also noted that the slider structure has higher stiffness in the direction of the beam thickness in which the deformation is to be induced, compared to a simple cantilever beam.
In this article, we discuss how to achieve lower stiffness and the challenges faced in doing so.

To achieve lower stiffness and challenges

For lower stiffness in X direction

　If the deformation is to be induced in the x-direction, we can make \(S_{11}\), the component of the compliance matrix with respect to the x-direction, larger than the others.

\(\begin{bmatrix}  d_x \\ d_y \\ \theta_z \\\end{bmatrix}
=　\begin{bmatrix}
S_{11} & S_{12} & S_{16} \\  & S_{22} & S_{26} \\ \text{sym.} &  & S_{66} \end{bmatrix}
\begin{bmatrix}  F_x \\ F_y  \\ M_z \\\end{bmatrix}\)

In the previous article, it was mentioned that the behavior of beams connected in series can be obtained by adding the compliance matrices, and that the deformation can be increased by connecting beams in series.

  To increase the ratio of compliance matrix components (such as \(S_{11}/S_{22}\) or \(S_{11}/S_{66}\)), structures with larger ratios must be combined. If \(S_{11}\) is made larger, other deformations must be allowed to increase (\(S_{11}/S_{22}\) and \(S_{11}/S_{66}\) become smaller).

  On the contrary, structures with more pronounced stiffness anisotropy cannot be realized by simply changing the beam arrangement, but only by reducing the beam thickness or increasing the beam length. However, there is a limit to how thin the beams can be made, given the buckling strength and manufacturing constraints. （If the structure is allowed to deform under load after fabrication, the QZS structure may provide a solution.)

Connect slider structures in series

　To reduce the stiffness in the X-direction without significantly compromising rigidity in the Y-direction or against rotation, modify the geometry as follows

• Increase the design area by changing the Y-directional dimension from 20 mm to 40 mm
• Change the beam thickness from 1 mm to 0.5 mm
• Connect the slider structure in series
  

The behavior when a directional load of 1 N is applied is shown below. The thin dotted line indicates a cantilevered beam, and the solid line indicates the series structure shown above.

Compared to the cantilever beam, the deformation in the X-direction is larger and the rotation angle is smaller. The displacement in the y-direction is larger, but negligible compared to the displacement in the X-direction.

  By allowing a small amount of Y-directional displacement and rotation, the structure can be connected in series to obtain a structure with lower stiffness in the X-direction.

Torsional deformation increases

　This is not a happily-ever-after situation. The lower stiffness may cause problems with out-of-plane stiffness that have not been taken into account.

3-dimensional deformation behavior

　To account for out-of-plane deformation, it is necessary to consider the behavior in three dimensions. Using the spatial mechanism approach, the calculation can be performed as simply as in the 2D case.

  If out-of-plane deformation is also considered, the compliance matrix \([S]\) of a uniform beam becomes

\(\boldsymbol{T} = [S]\boldsymbol{W}\\
\begin{bmatrix}  d_x \\ d_y \\ d_z \\ \theta_x \\ \theta_y \\ \theta_z \\\end{bmatrix}
=　\begin{bmatrix} 
\frac{l^3}{3EI_x} & 0 & 0   & 0 & 0 & \frac{l^2}{2EI_x}\\
0 & \frac{l}{Ebh} & 0       & 0 & 0 & 0\\
0 & 0 & \frac{l^3}{3EI_z}   & -\frac{l^2}{2EI_z} & 0 & 0 \\
0 & 0 & -\frac{l^2}{2EI_z} & \frac{l}{EI_z} & 0 & 0  \\
0 & 0 & 0                  & 0 & \frac{l}{GJ} & 0     \\
\frac{l^2}{2EI_x} & 0 & 0  & 0 & 0 & \frac{l}{EI_x}  \end{bmatrix}
\begin{bmatrix}  F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \\\end{bmatrix}\)

where J is the torsional constant of the beam, and in this case the following equation was used

\(J = \frac{bh^3}{3} \left(1-\frac{192h}{\pi^5b} \sum_{n=1}^{\infty} {1 \over 2n-1}\tanh{(2n-1)\pi b\over 2h}\right)\)

Let \([N]\) denote the coordinate transformation matrix from \(\boldsymbol{W'}\) to\(\boldsymbol{W}\).

\(\boldsymbol{W} = [N] \boldsymbol{W'}\\
\begin{bmatrix}  F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \\\end{bmatrix} =　\begin{bmatrix}
\boldsymbol{n_1} &
\boldsymbol{n_2} &
\boldsymbol{n_3} &
\boldsymbol{0} &
\boldsymbol{0} &
\boldsymbol{0} \\
\boldsymbol{L} \times \boldsymbol{n_1} &
\boldsymbol{L} \times \boldsymbol{n_2} &
\boldsymbol{L} \times \boldsymbol{n_3} &
\boldsymbol{n_1} &
\boldsymbol{n_2} &
\boldsymbol{n_3}
                \end{bmatrix}
\begin{bmatrix} 
F_x' \\ F_y' \\ F_z' \\ M_x' \\ M_y' \\ M_z' \\\end{bmatrix}\)

The stiffness matrix when the beam is moved is expressed as

\([Q] = [N][Q'][N]^\boldsymbol{T}\)

where \(\boldsymbol{n_1}\),\(\boldsymbol{n_2}\),\(\boldsymbol{n_3}\) in the transformation matrix \([N]\) are the unit vectors of X'Y'Z' in each axis direction and \(\boldsymbol{L}\) is the position vector represents the position vector of the tip of the beam.

  The behavior of beams connected in series or parallel can also be obtained by adding the compliance and stiffness matrices like in the two-dimensional case.
  The behavior of beams connected in series or parallel can also be obtained by adding the compliance and stiffness matrices like in the two-dimensional case.

Natural Vibration Modes

　Stiffness anisotropy of a structure can also be understood by looking at its natural vibration modes (depending on the application, this may be easier to consider as a real problem). Lower natural frequency modes indicate lower stiffness.

  The natural frequency values and modes can be obtained using the equations of motion with inertia taken into account.

\([M]\frac{d^2 \boldsymbol{T}}{dt^2} + [Q]\boldsymbol{T} = \boldsymbol{W}\)

where `\([M]\) is the matrix constructed by the mass and inertia tensors (\(diag(m,m,m,I_x,I_y,I_z)\)) transformed in coordinates. When the load is zero, the following equation holds for the natural frequency \(\omega_n\).

\([M]^{-1}[Q]\boldsymbol{T} = \omega_n^2\boldsymbol{T}\)

Therefore, by calculating the eigenvalues and eigenvectors of \([M]^{-1}[Q]\), the eigenfrequencies and eigenmodes of 6 degrees of freedom can be obtained.

  The table below compares the natural frequencies and eigenvectors for the slider structure alone with a beam thickness of 1 mm treated in the previous article and for the above series structure. The upper part of the structure is 1 mm thick steel (density 7850 kg/m^3). Each eigenvector is normalized to have a norm of 1, so the absolute values of the components are meaningless.

Slider structure

	1st	2nd	3rd	4th	5th	6th
Natural frequency [Hz]	284	2073	3481	4922	5680	7772
dx [m]	0.97	0	0	0	0	0
dy [m]	0	0	0	0	1	0
dz [m]	0	-0.01	0	0	0	0.01
θx [rad]	0	1.00	0	0	0	1.00
θy [rad]	0	0	1	0	0	0
θz [rad]	0.24	0	0	1	0	0

Serial structure

	1st	2nd	3rd	4th	5th	6th
Natural frequency [Hz]	23	168	309	426	758	1154
dx [m]	0.90	0	0	0	0	0
dy [m]	0	0	0	0	1	0
dz [m]	0	0	-0.04	0	0	0.00
θx [rad]	0	0	1.00	0	0	1.00
θy [rad]	0	1	0	0	0	0
θz [rad]	0.43	0	0	1	0	0

A comparison of the natural frequencies shows that the series structure is less rigid for each mode. Although Z-axis rotation appears as a first-order mode in both structures, the center of rotation is calculated to be y = 4.0 m and y = 2.1 m, which is quite far from the overall dimensions, indicating that it is almost a translational motion in the X direction. The deformation mode close to the rotation around the Y-axis is a cubic mode for the single structure, whereas it is a quadratic mode for the series structure.

  As shown above, when the stiffness is reduced in the X direction, the stiffness with respect to the out-of-plane XY direction is also reduced, especially with respect to rotation around the longitudinal axis.

Torsional properties of slider structure

　The reduced stiffness against torsion is due to the characteristics of the slider structure. Compare the torsional properties of a simple cantilevered beam and a slider structure when varying the dimensions. When the thickness or length is changed, the compliance matrix diagonal component \(S_{55}\) with respect to \(\theta_y\) changes as follows

Both structures are shown based on a width of 20 mm, a thickness of 0.5 mm, and a length of 20 mm. The beams of the slider structure were fixed at ±10 mm. When the beam length is increased and the beam thickness is decreased, the slider structure shown by the solid line shows a larger change in \(S_{55}\) than the cantilever beam shown by the dotted line. In other words, when the stiffness in the x-direction is adjusted to be small, the slider structure tends to have larger torsional deformation.

  The smaller beam-to-beam distance is also a significant factor in this series structure. The figure below shows the variation of \(S_{55}\) with distance between beams.

The smaller the beam spacing, the closer to a cantilevered beam structure, and the more prone to twisting. To maintain a large beam-to-beam distance, extrusion constraints must also be eliminated.

Higher rigidity against torsion

　A slider structure connected in series has low stiffness against torsion, so try to find a way to increase the stiffness against torsion without significantly increasing the stiffness against displacement in the x-direction.

  Try applying a structure that is prone to only bending deformation to each beam, as shown in the figure.

The natural frequencies were calculated using finite element analysis, in addition to the structure before the improvement.

Before improvement

	1st	2nd	3rd	4th	5th	6th
Vibration mode
Natural frequency [Hz]	25	170	200	583	645	1426

After Improvements

	1st	2nd	3rd	4th	5th	6th
Vibration mode
Natural frequency [Hz]	35	252	279	598	660	1780

The simplified calculation results did not take into account the mass of the resin section or the deformation of the thick-walled section, resulting in differences from the finite element analysis results, especially in the higher-order modes.

  Comparing the natural frequencies before and after the improvement, the natural frequency for translational motion in the X direction increased by a factor of 1.40. However, the natural frequency for Y-axis rotational motion is 1.64 times higher, indicating that the stiffness is efficiently improved in this direction.

  As shown above, the stiffness against rotation can also be improved by applying a shape that reflects the deformation mode. Since this shape was created on a first thought, it is possible that a better structure can be obtained by adjusting the parameters in various places.

Summary and challenges

　In the slider structure, we explained that in order to reduce the stiffness only for the deformation to be induced, the beam must be thinned or lengthened, and that in doing so, the beam is greatly reduced in stiffness against torsion. In addition, as a way to improve the stiffness against torsion, we introduced a method of applying a structure where only bending deformation is likely to occur in the beam section.

  Although we introduced stiffness anisotropic structures in this article, it is necessary to consider the following issues when applying them to actual products.

• The difficulty of manufacturing may have raised due to the more complex geometry.
• The unit cost and weight will increase due to the increased amount of materials.
• Various strength evaluations should be conducted.
• Since micro-deformation is assumed, rigidity anisotropy may no longer be valid with large deformation.
• Robustness against geometric tolerances must be considered.

  I hope to be able to introduce these measures sometime in the future. Thank you for your patience until the end.