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# Flexible Joint - Measurement of the Bending Stiffness
Author: Dehaeze Thomas

## Table of Contents

- 1. Finite Element Model
- 2. Setup
- 3. Effect of Bending
- 4. Computation of the bending stiffness
- 5. Effect of Shear
- 6. Effect of Torsion
- 7. Full stroke measured displacement and applied force as a function of \(H\)
- 8. Negligible bending of the supporting bar
- 9. Conclusion

The goal is to design a test bench to measure the bending stiffness of a flexible joint with 1% accuracy.

## 1 Finite Element Model

From the Finite Element Model, the stiffnesses and strokes of the flexible joint have been computed.

    ka = 94e6; % Axial Stiffness [N/m]
    ks = 13e6; % Shear Stiffness [N/m]
    kb = 5;    % Bending Stiffness [Nm/rad]
    kt = 260;  % Torsional Stiffness [Nm/rad]

    Fa = 469;    % Axial Force before yield [N]
    Fs = 242;    % Shear Force before yield [N]
    Fb = 0.118;  % Bending Force before yield [Nm]
    Ft = 1.508; % Torsional Force before yield [Nm]

    Xa = Fa/ka; % Axial Stroke before yield [m]
    Xs = Fs/ks; % Shear Stroke before yield [m]
    Xb = Fb/kb; % Bending Stroke before yield [rad]
    Xt = Ft/kt; % Torsional Stroke before yield [rad]

| Stiffness [N/um] | Max Force [N] | Stroke [um]
--- | --- | --- | ---
Axial | 94 | 469 | 5
Shear | 13 | 242 | 19

| Stiffness [Nm/rad] | Max Torque [Nmm] | Stroke [mrad]
--- | --- | --- | ---
Bending | 5 | 118 | 24
Torsional | 260 | 1508 | 6

## 2 Setup

Let’s say a force is applied on top of the flexible joint with a distance \(H_F\) with the joint’s center.

The displacement of the flexible joint is also measure at an height \(H_D\) from the joint’s center.

Figure 1: Figure caption

## 3 Effect of Bending

The torque applied is:

\begin{equation} M_b = F \cdot H_F \end{equation}

The flexible joint is experiencing a rotation \(R_b\) due to the torque \(M_b\):

\begin{equation} R_b = \frac{M_b}{k_b} = \frac{F \cdot H_F}{k_b} \end{equation}

This rotation is then measured by the displacement sensor. The measured displacement is:

\begin{equation} D_b = H_D \tan(R_b) = H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \label{eq:bending_meaured_disp} \end{equation}

## 4 Computation of the bending stiffness

From equation \eqref{eq:bending_meaured_disp}, we can compute the bending stiffness:

\begin{equation} k_b = \frac{\tan^{-1}\left( \frac{D_b}{H_D} \right)}{F \cdot H_F} \end{equation}

And therefore, to precisely measure \(k_b\), we need to:

- precisely measure the motion \(D_b\)
- precisely measure the applied force \(F\)
- precisely know the height from the flexible joint’s center to the measurement point \(H_D\)
- precisely know the height from the flexible joint’s center to the force application point \(H_F\)

If there are estimation errors for \(H_D\) or \(H_F\) as shown in Figure 2, this will induce an error for the estimation of the stiffness.

For 1% accuracy estimation of \(k_b\), we can write the following approximate requirements:

| Accuracy
--- | ---
Force Measurement | 1%
Displacement Measurement | 1%
\(H_D\) | 1%
\(H_F\) | 1%

Figure 2: Error in the estimation of the height of the force sensor and displacement sensor

## 5 Effect of Shear

The effect of Shear on the measured displacement is simply:

\begin{equation} D_s = \frac{F}{k_s} \end{equation}

We would like to have this displacement much smaller than the displacement induced by the bending effects:

\begin{equation} D_b \gg D_s \end{equation}

Which is equivalent as to have:

\begin{equation} H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \frac{F}{k_s} \end{equation}

Here to simplify, we suppose \(FH_F/k_b \ll 1\) (which is the case in practice), and we suppose \(H_D = H_F = H\).

The obtained condition is then:

\begin{equation} H \gg \frac{k_b}{k_s} \end{equation}

Table 1: Height \(H\) to have less than certain amount of error due to shear effects
| 10% error | 1% error | 0.1% error
--- | --- | --- | ---
\(H\,[mm]\) | 6 | 62 | 620

In order to limit the effect of shear of less than 1%, the height from the joint’s center to the force application point and to the measurement point should be larger than 62mm.

    H = 62e-3; % [m]

## 6 Effect of Torsion

If the application force is not aligned with the vertical axis of the flexible joint, this will induce a torsion motion that will induce a measurement error (Figure 3).

Figure 3: Horizontal position error of the force sensor and displacement sensor

Let’s note the offset of the force sensor \(\epsilon_{F,y}\) and the offset of the measurement point \(\epsilon_{D,y}\). The vertical torque (torsion) will be equal to:

\begin{equation} M_t = F \cdot \epsilon_{F,y} \end{equation}

And the induced torsion:

\begin{equation} R_t = \frac{M_t}{k_t} = \frac{F \cdot \epsilon_{F,y}}{k_t} \end{equation}

The effect on the measured displacement is:

\begin{equation} D_t = \epsilon_{D,y} \tan \left( R_t \right) = \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{k_t} \right) \end{equation}

And we would like to have:

\begin{equation} D_b \gg D_t \end{equation}

Which is equivalent as to have:

\begin{equation} H_D \tan\left( \frac{F \cdot H_F}{k_b} \right) \gg \epsilon_{D,y} \tan\left( \frac{F \cdot \epsilon_{F,y}}{k_t} \right) \end{equation}

Supposing \(H_F = H_D = H\) and \(\epsilon_{F,y} = \epsilon_{D,y} = \epsilon_{y}\), the condition becomes:

\begin{equation} \epsilon_{y} \ll H \sqrt{\frac{k_t}{k_b}} \end{equation}

Table 2: Maximum lateral position error \(\epsilon_y\) to have less than certain amount of error due to torsion effects
| 10% error | 1% error | 0.1% error
--- | --- | --- | ---
\(\epsilon_y\,[mm]\) | 44.7 | 4.5 | 0.4

For 1% error, the lateral positioning errors \(\epsilon_y\) for both the force sensor and the displacement sensor should be less than 4.5mm.

## 7 Full stroke measured displacement and applied force as a function of \(H\)

Applying a force with a large height \(H\) means the induced rotation (for constant force) will be larger. This also means that the measured displacement \(D_b\) will also be larger.

Note that we here suppose the force axis is co-linear with the measurement axis (\(H_F = H_D = H\)).

Let’s compute:

- \(D_b\) as a function of \(H\) \[ D_b \approx H \tan (R_b) \]
- the applied force \(F_{\text{max}}\) to induce the maximum rotation as a function of \(H\) \[ F_{\text{max}} \approx \frac{X_b \cdot k_b}{H} \]

Figure 4: Applied force \(F_{\text{max}}\) and measured displacement \(D_b\) as a function of \(H\)

With an offset of 62mm, we obtained values shown in Table 3.

Table 3: Maximum displacement and maximum applied force for \(H = 62\,[mm]\)
\(D_b\,[mm]\) | \(F_m\,[N]\)
--- | ---
1.5 | 1.9

## 8 Negligible bending of the supporting bar

This should be confirmed with FEM.

## 9 Conclusion

Table 4: Conclusions in terms of forces and displacement measurements
| Range | Accuracy
--- | --- | ---
Force Measurement | 2 N | 1% = 0.02 N
Displacement Measurement | 1.5 mm | 1% = 15 um

Table 5: Conclusions in terms of required positioning accuracy
| Value | Precision | Comment
--- | --- | --- | ---
\(H_D\) | 62mm |  | For negligible Shear
\(H_F\) | 62mm |  | Same
\(\epsilon_y\) | 0 | 4.5mm | For negligible Torsion
\(\epsilon_z\) | 0 | 1% = 0.6mm | For torque estimation precision

Load cells:

- https://www.te.com/usa-en/product-CAT-FLS0020.html?q=&n=510872&d=681051%20564247&type=products&samples=N&inStoreWithoutPL=false&instock=N#mdp-tabs-content
- https://www.digikey.com/en/products/detail/honeywell-sensing-and-productivity-solutions/FSS005WNSB/6056404

Displacement sensors:

- http://www.vaco-france.com/outillage-de-metrologie/comparateur-a-cadran-de-precision.htm
- https://fr.rs-online.com/web/p/comparateurs/1940101/

Author: Dehaeze Thomas

Created: 2021-01-15 ven. 23:32