Design methodology for the development of variable stiffness devices based on layer jamming transition

Figure 2 reports the general scheme of the proposed generalized design methodology that are described with four main steps, and which are discussed below.

Step 1. The first step concerns the selection of differential pressure acting on the system. The main idea underneath the choice is that, even if the pressure input may vary during activation, the maximum range must be taken into account as reference value for the dimensioning. Thus, in the governing equations, the pressure is always a known term.

Step 2. The second step regards the selection of the main load case condition. The choice must fall between the following alternatives: axial forces, transversal forces, bending moments and torsional moments (figure 2). Among the possible combinations between these load conditions, we also take into account the case in which both bending moments and transversal forces are applied. This combination is very common and used in several applications in which layer jamming systems are exploited to make variable stiffness actuators. However, as later discussed, this case does not need a separate discussion and can be treated as the case where only transversal forces are applied, simply introducing a different meaning of some parameters.

Step 3. Once the load case has been selected, two alternatives are possible. If the selected load case regards axial forces or transversal forces it is possible to choose between (i) a bulk configuration, in which the overall length of the system is always the same before slipping condition, or (ii) a contact area variation, in which the jamming transition can be activated at different lengths of the system between the rest and the maximum one. If either bending moments or transversal moments have been selected at the previous step, it is possible to move directly to the next one.

Step 4. At this step, the design mode can be selected and there are two alternatives: material selection or sizing. The first case can be used if the size of the system is known a priori (e.g. for space limitations) and the layer material has to fit some specifications in terms of slip threshold and stiffness. The second one is the right choice if slip threshold, stiffness and layer material are known, so the sizing of the layers has to be evaluated in order to fit with the required specifications. In both cases, the number of layers is not known beforehand.

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Once the correct configuration has been initialized through the selections at steps 1–4, the input parameters have to be defined and the output is evaluated through an iterative and interactive process. Both input and output parameters and the governing equations are different according to the configurations selected in the previous steps: axial forces, transversal forces, bending moments and torsional moments load cases are reported in figures 3, 4, 6 and 7, respectively. In order to better clarify how the methodology works, contour graphs are inserted in the detailed schemes. According to the selected case, these graphs show how the unknown terms depend on the known parameters through the model equations that are reported in the following subsections. The methodology becomes more interactive after the selection at step 4, where the user is asked to make use of external sources iteratively. If the case of imposed size has been selected (material selection), the methodology returns the numerical ranges of the material properties that the user should use to find viable products (for example on an external material library) whose characteristics lie in the provided ranges. On contrary, if the case of imposed material has been chosen (sizing), the methodology returns the ranges of functions describing both dimensions and number of layers. In this latter case, the user has to select the values in accordance with the given ranges.

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In all cases, the methodology ends with a post-process step in which the user is asked if the performances of the designed layer jamming system have to be evaluated with respect to the remaining load cases. If the answer is yes, some additional parameters are required and must be specified according to the selected load case. The possible load cases are reported and discussed below.

2.2.1. Axial forces

For a system made of n layers, with friction coefficient μ on which a differential pressure P is applied (figure 1), according to the equation reported in the study of Kim et al [16] the slip force between layers is given by the product between the tangential force induced by the pressure lwμP on a single layer and the number of layers (also expressible as the ratio between h and t):

$$\begin{eqnarray}&&\begin{array}{c}F=nlw\mu P=\displaystyle \frac{h}{t}lw\mu P\end{array}\end{eqnarray} \tag{ 1 }$$

where h is the total height of the system.

At first, the slip forces F_min and F_max, which define the lower and upper boundary of a range of acceptable performances respectively, must be selected by the user.

In case of material selection at step 4, the inputs are the layer dimensions l and w, that can be used to evaluate the ratio F/h as:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{F}{h}=\displaystyle \frac{\mu }{t}lwP\end{array}\end{eqnarray} \tag{ 2 }$$

For a guess value of h, the definition of the variation range of this ratio is equivalent to defining a range of acceptable value of the t/μ ratio, which corresponds to a family of materials that fit with the given requirements in terms of both friction coefficient and available layer height. The initial guess of h can be selected as the strictest condition, then the process can be reiterated with different value of h until one or more suitable materials are found. Once the selection process ends, by inserting the specification of the selected material (which also implies a specific number of layers) all the systems parameters are known, and the effective slip force can be evaluated.

If the case of contact area variation has been selected at step 3, a lower contact area should be taken into account which is given by the product between w and l_min, the useful length at which a maximum length of the system L specified by the user corresponds by the relation l_min = 2l − L. The same modelling equation reported above can be also used to evaluate the slip force at maximum length F_L . The user can accept this result or can reiterate the overall process with a different choice of h to obtain better performances.

In case of sizing selection at step 4, the inputs are the friction coefficient μ and the height t of the layers. In addition, if the layer dimensions are not known, it is reasonable to impose a specific shape factor l/w of the layers a priori, obtained from the optimization of material consumption, for example. In this case, by expressing the length l as function of w by means of the shape factor, the range of acceptable slip forces corresponds to a range of hw² values in which a restrictive choice on h (which for a given layer height correspond to a lower number of layers) brings to the need for wider layers. The slip forces are expressed as follow:

$$\begin{eqnarray}&&\begin{array}{c}F=h{w}^{2}\displaystyle \frac{\mu P}{t}\displaystyle \frac{l}{w}\end{array}\end{eqnarray} \tag{ 3 }$$

The user can balance these two design parameters as necessary to fit with some specific requirements, and once they have been chosen, they must be specified to proceed to the next steps.

If the contact area variation has been selected at step 3, since the size of the system plays a secondary role with respect to the previous case, the force at maximum length F_L is required as input value in the adoption of the governing equation, as showed before. According to this, the output of the calculation is the maximum length L which fits with the specified input and then, as in the previous case, the design process can end or be entirely repeated.

Finally, if the performances of the designed system have to be evaluated in other load cases, the additional information of Young's modulus and Poisson ratio of the selected layer material have to be specified.

2.4.1. Transversal forces

In case of transversal forces, the differential pressure P prevents the layers from slipping on each other so that the system can be considered as single monolithic block. As a consequence, the unjammed and the jammed state present different area moment of inertia, which are respectively I0 and Ij (see equations (4) and (5)) [20, 22]. This also leads to an increase of the stiffness from k0 to kj, defined as the ratio between applied forces and displacements of the free end (see equations (6) and (7)). The increase of area moment of inertia and stiffness can be expressed as follow:

$$\begin{eqnarray}&&\begin{array}{c}{I}_{0}=\,\displaystyle \frac{wn{t}^{3}}{12}\end{array}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\begin{array}{c}{I}_{j}=\displaystyle \frac{w{n}^{3}{t}^{3}}{12}\end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{c}{k}_{0}=\displaystyle \frac{3E{I}_{0}}{{l}^{3}}=\,\displaystyle \frac{Ewn{t}^{3}}{4{l}^{3}}\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{c}{k}_{j}=\displaystyle \frac{3E{I}_{j}}{{l}^{3}}=\displaystyle \frac{Ew{n}^{3}{t}^{3}}{4{l}^{3}}\end{array}\end{eqnarray} \tag{ 7 }$$

where E is the material Young's modulus. The ratio r between k_j and k₀ quantifies the stiffness increment between the jammed and unjammed configurations and it is equal to:

$$\begin{eqnarray}&&\begin{array}{c}r=\displaystyle \frac{{k}_{j}}{{k}_{0}}={n}^{2}\end{array}\end{eqnarray} \tag{ 8 }$$

In this configuration, it is also possible to identify a critical force above what the layers start to slip on each other. This behavior occurs when the shear stress in the structure equals the maximum shear stress (τ_max) that can be sustained by friction (μP):

$$\begin{eqnarray}&&\begin{array}{c}{\tau }_{{\rm{\max }}}=\mu P=\displaystyle \frac{3{V}_{{\rm{\max }}}}{2A}\end{array}\end{eqnarray} \tag{ 9 }$$

where V_max is the maximum shear force and A = ntw is the cross-section area. By substituting the value of the applied force to V_max, it is possible to obtain the slip force as follows:

$$\begin{eqnarray}&&\begin{array}{c}{V}_{{\rm{\max }}}=F\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\begin{array}{c}{F}_{slip}=\displaystyle \frac{2\mu PA}{3}=\displaystyle \frac{2}{3}\mu Phw=\displaystyle \frac{2}{3}\mu Pntw\end{array}\end{eqnarray} \tag{ 11 }$$

Several works [20, 21, 23] highlighted how once the critical conditions are exceeded, the layers slide progressively and the relationship between applied forces and displacements becomes nonlinear. Then, once all layers are sliding, they are fully decoupled and the overall behavior is again linear, similar to what happens below the critical value, but with a different stiffness (depending on the dynamic friction coefficient). Therefore, the relationship between forces and displacements of layer jamming systems is usually decomposed into three regions: (i) elastic, (ii) transition, and (iii) plastic.

Under the simplified hypotheses on which the methodology is based, it is not possible to capture the nonlinear behavior of the transition region. Therefore, the force versus displacement relationship is given by two regions: (i) the linear one, described by the stiffness k_j , and (ii) the plastic one, described by the stiffness k₀ . In this case, the user has to define the lower and upper boundaries of the range of acceptable stiffnesses, slip forces and jamming ratio, which are respectively: k_{j_min} and k_{j_max}, F_{slip_min} and F_{slip_max}, r_min and r_max.

The inputs are the layer dimensions l and w. The definition of the layer material features has to be done step by step. Once the guess value h has been defined, imposing the values of minimum and maximum slip forces, it is possible to identify a range of acceptable values of the ratio F_slip /h, which is equivalent to define a range of acceptable frictional coefficients μ. This ratio is expressed as follows:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{F}_{slip}}{h}=\mu \displaystyle \frac{2wP}{3}\end{array}\end{eqnarray} \tag{ 12 }$$

Then, imposing the minimum and maximum stiffnesses, it is possible to identify a range of acceptable values of the ratio k_j /h³ , which is equivalent to define a range of acceptable Young's moduli E, and it is defined as follows:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{k}_{j}}{{h}^{3}}=E\displaystyle \frac{w}{4{l}^{3}}\end{array}\end{eqnarray} \tag{ 13 }$$

It is possible to notice that these two steps lead to the definition of a rectangular area in the first plot reported in figure 4, where the choice of a greater value of h corresponds to both a reduction and a translation toward the axis origin of the area of acceptable materials. For each material compliant with the constraints imposed in the two previous steps, the minimum and the maximum values of jamming ratio have to be taken into account. The methodology returns the minimum and maximum number of layers, thanks to the relation expressed in equation (8). Consequently, for each acceptable material found, the user has to check the availability of layers with height t, which ensures to obtain the selected h with an acceptable number of layers.

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If the contact area variation has been selected at step 3, the user has to define the maximum length L. Then, it is possible to overlook the change of area moment of inertia due to the sliding among layers, and the methodology returns the corresponding stiffness k_L . According to the adopted theory, the slip threshold is not affected by the length change.

The procedure can be entirely repeated with different overall height values until the given results match with the user requirements. If no possible solutions are identified, a fast analysis of the results obtained at the single steps can be useful to find better alternatives in the adoption of non-uniform jamming structures. If the material selection has been restricted mostly because of the range of acceptable Young's modulus, non-uniform strategies such as the jamming sandwich structures (which are composed of layers of one type between two layers on another one) [24] or the use of parallel-guided structures [25] can be adopted in order to fit with the required stiffnesses. On contrary, if the friction coefficient range has mostly influenced the material selection, the threshold between non-slip and slip conditions can be incremented using a non-uniform strategy in which a layer of one type is placed between layers of another one.

The inputs are the friction coefficient μ, the Young's modulus E, the height t, and the shape factor l/w of the selected layers. By imposing the minimum and maximum of the slip force, a range of acceptable values for the product nw is identified, whereas by imposing the minimum and maximum of the stiffness, the range of acceptable values for the ratio n³/w² can be obtained, as expressed by the following equations:

$$\begin{eqnarray}&&\begin{array}{c}nw=\displaystyle \frac{3{F}_{slip}}{2t\mu P}\end{array}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{n}^{3}}{{w}^{2}}=\displaystyle \frac{4{k}_{j}{\displaystyle \frac{l}{w}}^{3}}{E{t}^{3}}\end{array}\end{eqnarray} \tag{ 15 }$$

Once the selected values of number and width of layers have been specified, if the contact area variation has been selected at step 3, the user needs to define the stiffness at maximum length k_L and the methodology returns the corresponding value of L. If the result is not acceptable, the procedure can be reiterated starting from the previous step. As in the previous case, the adoption of non-uniform jamming structure can be an alternative approach in case the output of the methodology does not match same specific user requirements.

Finally, the user is asked to specify the Poisson ratio of the selected layer material as additional information, in order to evaluate the performances of the designed system in other load cases.

The same approach reported in the previous section can be used if both bending moments and transversal forces are applied. In this case, considering both the reference system and the loads showed in figure 5, the following equations hold:

$$\begin{eqnarray}&&\begin{array}{c}d\varphi =\displaystyle \frac{{M}_{tot}\,}{EI}dx=\displaystyle \frac{F\left(l-x\right)-M\,}{EI}dx\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\begin{array}{c}df=xd\varphi \end{array}\end{eqnarray} \tag{ 17 }$$

where dφ is the angle between two sections of the beam at infinitesimal length dx in the deformed configuration. The relationship between loads and displacements of the free end f can be then expressed as:

$$\begin{eqnarray}&&\begin{array}{c}f=\displaystyle {\int }_{0}^{l}\displaystyle \frac{F\left(l-x\right)x-Mx\,}{EI}dx=\left(F-\displaystyle \frac{3M}{2l}\right)\displaystyle \frac{{l}^{3}}{3EI}\end{array}\end{eqnarray} \tag{ 18 }$$

Assuming the stiffness can be evaluated as follows:

$$\begin{eqnarray}&&\begin{array}{c}{k}_{j}=\displaystyle \frac{\left(F-\tfrac{3M}{2l}\right)}{f}=\displaystyle \frac{3EI}{{l}^{3}}=\displaystyle \frac{Ew{h}^{3}}{4{l}^{3}}\end{array}\end{eqnarray} \tag{ 19 }$$

it is possible to obtain the same results reported in equation (7). For what concerns the slip threshold, the presence of a bending moment does not affect the shear force, so equation (11) still holds. In summary, the design methodology is still valid, but the values entered as k_{j_min} and k_{j_max} must be considered as equivalent stiffnesses representing the combined load, as shown in figure 5.

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This configuration can be adopted when, for example, a layer jamming system is combined with an actuator and transversal forces are applied. In these cases, the actuator effect can be taken into account by means of a moment opposite to the external transversal forces. Several applications, such as variable stiffness grippers and manipulators can be modelled in this way.

This configuration can be adopted when, for example, a layer jamming system is combined with an actuator and transversal forces are applied. In these cases, the actuator effect can be taken into account by means of a moment opposite to the external transversal forces. Several applications, such as variable stiffness grippers and manipulators can be modelled in this way.

2.7.1. Bending moments

Under the same simplified hypotheses adopted for both axial and transversal forces, the modelling equations in case of bending moments are:

$$\begin{eqnarray}&&\begin{array}{c}{k}_{b}=\displaystyle \frac{Ew{h}^{3}}{12}\end{array}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\begin{array}{c}r=\displaystyle \frac{{k}_{b}}{{k}_{{b}_{0}}}={n}^{2}\end{array}\end{eqnarray} \tag{ 21 }$$

The value k_b represents the relation between the bending moment that acts on the system and the assumed curvature, thus it does not depend on the length of the structure. No shear forces act on the system, hence in this case no information about the slip and non-slip region can be obtained.

As a preliminary information, the user is required to set the lower and upper boundaries of the range of acceptable bending stiffnesses and jamming ratio, which are defined through k_{b_min}, k_{b_max}, r_min and r_max, respectively.

The user has to specify the width w of the layers as input. Then for a given guess value of the overall height h, imposing the minimum and maximum bending stiffnesses in equation (20), the range of acceptable Young's moduli E is obtained.

For the acceptable materials, the user has to select the layer height t that returns the specified overall height h, once multiplied for a number n in the range of acceptable values obtained by the conditions on the minimum and maximum jamming ratio (see equation (21)). As in previous cases, this process can be repeated iteratively until the obtained results fits with the user requirements.

The inputs are the Young's modulus E and the layer height t of the selected layer. In this case, the range of acceptable stiffnesses corresponds to a range of acceptable wn³ values through equation (20). The n value has to be selected in the range of acceptable values given by the conditions on the minimum and maximum jamming ratio, as in the previous case. Then the layer width w can be evaluated.

Finally, if the performances of the designed system have to be evaluated in other load cases, the user needs to define the additional inputs of selected material: the friction coefficient, the Poisson ratio, and the layers length.

2.9.1. Torsional moments

For a cantilever beam with rectangular section, considering the simplifying hypotheses adopted so far and the additional one of $w\gg h,$ the governing equations in torsional moments load case are:

$$\begin{eqnarray}&&\begin{array}{c}{k}_{{t}_{j}}=\,\displaystyle \frac{Gw{h}^{3}}{3}\end{array}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\begin{array}{c}{M}_{slip}=\displaystyle \frac{h{w}^{2}\mu P}{3}\end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\begin{array}{c}r=\displaystyle \frac{{k}_{{t}_{j}}}{{k}_{{t}_{0}}}={n}^{2}\end{array}\end{eqnarray} \tag{ 24 }$$

The value ${k}_{{t}_{j}}$ represents the relation between the torsional moment which acts on the system and the resulting rotation angle of the free end. As previously described in transversal forces load case, equation (23) can be obtained equating the shear force on the external layer and the maximum shear force that can be sustained by friction.

In this case, the user has to define the lower and upper boundaries of the range of acceptable torsional stiffnesses, slip moments and jamming ratio, which are respectively: ${k}_{{t}_{j}\,{\rm{\min }}}$ and ${k}_{{t}_{j}\,{\rm{\max }}},$ ${M}_{slip\_\,{\rm{\min }}}$ and ${M}_{\,slip\_\,{\rm{\max }}},$ ${r}_{{\rm{\min }}}$ and ${r}_{{\rm{\max }}}.$

This design process has to be done step by step. As proposed for the transversal moments load case, for a given guess value of the overall height h, imposing the minimum and maximum slip moments, the range of acceptable friction coefficients is obtained, as shown in the following equation:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{M}_{slip}}{h}=\mu \displaystyle \frac{{w}^{2}P}{3}\end{array}\end{eqnarray} \tag{ 25 }$$

Then, imposing the minimum and maximum torsional stiffnesses, it is possible to identify a range of acceptable ${k}_{{t}_{j}}/{h}^{3}$ ratios:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{k}_{{t}_{j}}}{{h}^{3}}=\,G\displaystyle \frac{w}{3}\end{array}\end{eqnarray} \tag{ 26 }$$

or equivalently a range for the shear modulus G = E[2(1 + ν)], where E is the Young's modulus and ν is the Poisson ratio.

As in the transversal forces load case, these two steps lead to the definition of a rectangular area (shown in the first graph reported in figure 7), in which a high value of h brings to both a reduction of the area of acceptable materials and its translation toward the axis origin. Then, through equation (24), the methodology returns the minimum and maximum number of layers, which set the range of acceptable values of the jamming ratio. With this result, the user can check the availability for each acceptable material found, considering the layer height t that ensures to obtain the selected h with an acceptable number of layers. As in previous cases, the procedure can be reiterated with different guess values of h, if obtained results are not acceptable.

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The friction coefficient μ, the shear modulus G and the layer height t of the selected layer have to be specified. A range of acceptable nw² can be identified imposing the minimum and maximum values for the slip moment:

$$\begin{eqnarray}&&\begin{array}{c}n{w}^{2}=\displaystyle \frac{3{M}_{slip}}{t\mu P}\end{array}\end{eqnarray} \tag{ 27 }$$

while the range of acceptable n³ w can be obtained imposing the minimum and maximum values for the torsional stiffnesses:

$$\begin{eqnarray}&&\begin{array}{c}{n}^{3}w\,=\displaystyle \frac{3{k}_{{t}_{j}}}{G{t}^{3}}\end{array}\end{eqnarray} \tag{ 28 }$$

Once the values of the selected n and w have been specified by the user, the Young's modulus of the selected layer material and the layers length are required as additional information, in order to evaluate the performances of the designed system in other load cases.