(Non-)linear stiffness customisation of metallic additive manufactured springs

ABSTRACT Additive manufacturing (AM) facilitates the fabrication of compliant mechanisms through its free-form and design customisation capabilities. Specifically, the properties of kinetic mechanisms such as springs can be extended with regards to their inherent (non-)linear stiffness functions. This allows for the customisation of AM springs according to user preferences. By combining the design synthesis approach of building blocks with the structural optimisation approach for AM, it is possible to define and customise spring stiffness functionalities. The optimisation process employs an automated computational framework based on a genetic algorithm scheme, which has been demonstrated through randomised and reference case studies. This framework enables the attainment of linear, progressive (stiffening), and degressive (softening) stiffness curves. The manufacturability of the springs has been validated through laser powder bed fusion using stainless-steel material 17–4 PH (H900). The springs have resulted in an accuracy error of maximum 6.48% and precision error of maximum 5% through compression testing.


Introduction
A compliant mechanism (CM) is a monolithic mechanical device that performs one or multiple functions by transferring force and/or motion through elastic bending of its flexible elements within the structure (Howell, Magleby, and Olsen 2013).These compliant elements are the key distinguishing feature of a CM, which utilises compliance to replace joints and hinges found in conventional rigid-body systems (Machekposhti, Tolou, and Herder 2015).As a result, a monolithic CM offers several advantages over conventional rigid-body assemblies, such as lower manufacturing costs, reduced assembly time, and no need to stock additional parts (Howell 2013).Depending on their dominant function, CMs can be categorised into different classes: kinematic CMs follow a specific motion, while kinetic CMs such as springs follow a force-displacement function when loaded (Howell, Magleby, and Olsen 2013;Christine 2008).Kinetic mechanical parts like springs are widely used in various fields such as medical devices (Hitt et al. 2010;Carrozza et al. 2004), rehabilitation devices (Vallery et al. 2013;Ham et al. 2009), and artificial implants (Block, O' Connell, and Lowe 2013;Dodgen et al. 2012).In robotics, they are applied as joints to achieve rotational and translational force-displacement (or stiffness) relations (Vanderborght et al. 2013;Negrello et al. 2017).Their designs are mainly determined by conventional manufacturing technologies (e.g.coil-spring, flat-spring, machined-spring) (Keith and Nisbett 2014;Kobelev 2018).However, these conventional designs have limitations in terms of their functionality such as stiffness non-linearity.Non-linear softening or stiffening stiffness curves can only be achieved through extra assembly effort, such as the combination of multiple Belleville springs (Keith and Nisbett 2014) or contact-aided mechanism designs that induce additional friction, wear, or stick-slip effects (Arredondo-Soto, Cuan-Urquizo, and Gómez-Espinosa 2021).This assembly effort increases the weight of the overall system and decreases the compactness.Furthermore, customising traditional springs to meet the individual user's needs can be time-and cost-consuming (Christine 2008;Kobelev 2018).
Advanced manufacturing techniques, such as additive manufacturing (AM), have the potential to overcome the limitations of traditional spring designs.AM enables the production of mechanical parts (e.g.gear wheels, screws, etc.) that are complicated or difficult to fabricate using conventional manufacturing techniques (Cuellar et al. 2018;Lussenburg, Sakes, and Breedveld 2021).This allows for freeform geometries, rapid prototyping and shorter lead time, providing more design flexibility and variability in the manufacturing process.The same part can be printed with slight variations at no additional cost (Gibson, Rosen, and Stucker 2015;Yang and Zhao 2015).By combining AM with CMs, it is possible to create compact and lightweight designs that can easily integrate multiple functions into a noncontact or non-assembly mechanism (Thompson et al. 2016;Gao et al. 2015).Furthermore, AM allows for the customisation of designs to meet user-defined requirements, such as specific spring stiffness or load capacity (Ngo et al. 2018;Rosen 2014).
The free-form potentials of AM can overcome the aforementioned compactness, assembly, and customisation efforts of conventional spring designs.Plastic 3D printing was shown to be an effective method for integrating multiple flexural elements in a systematic and controlled manner (He et al. 2019;Haq, Nazir, and Jeng 2021) to create highly compact and non-assembly mechanisms (Lin et al. 2021;Zhang, Guo, and Hu 2021).These AM parts targeted the customisation of force-displacement functions (Cheng and Savarimuthu 2021;Scarcia et al. 2016).The customised designs were synthesised using well-established methods for a systematic design approach such as structural optimisation or the building block (BB) approach (Gallego and Herder 2009;Yu et al. 2009).These methodologies aimed to make the complex and unintuitive CM design process more guided (Howell, Magleby, and Olsen 2013).Without these methodologies, the high design complexity of CMs can make the design process subject to a more arduous trial-and-error approach relying on the experience and intuition of the designer (Gallego and Herder 2009;Yu et al. 2009;Sigmund 1997).However, with the help of increased computational power and more sophisticated simulation software, this design challenge was addressed (Lu and Kota 2006;Bruns and Tortorelli 2001).For example, a structural optimisation approach heavily dependent on computation was used to define the optimal topology, shape, and size of the designed structure (Arredondo-Soto, Cuan-Urquizo, and Gómez-Espinosa 2021) and left most of the design choices to an algorithm (Gallego and Herder 2009).A BB approach defines a set of components with specific functions that can be composed to create a CM design (Hoetmer, Herder, and Kim 2009).By using the structural optimised approach to design BBs, the control of the BB approach and the computational design power were leveraged to provide the optimal solution (Bernardoni et al. 2004;Naves, Brouwer, and Aarts 2017).
The customisation of conventionally produced metal springs was demonstrated through the use of wire-electrical discharge machining.An optimisation framework was used to tailor the springs with regards to the deformation range, the load range, and the (non-)linear stiffness curve (Jutte and Kota 2008;Jutte and Kota 2010).However, the number of customised metal AM springs that have been produced is currently limited (Lussenburg, Sakes, and Breedveld 2021), despite the numerous customisable plastic AM mechanisms that were developed.Metal CMs have the advantage of being more durable and resistant to fragility and creeping, and can also be used in a wider range of operating temperatures (Ngo et al. 2018;Laban et al. 2021).By exploiting the potentials of metallic AM techniques such as laser powder bed fusion (LPBF), it becomes possible to 3D print metallic springs.
Prior work presented a BB approach where LPBF springs were synthesised in a self-supported manner within various print orientations (Danun et al. 2021).The BBs were self-supported as they took into account the design rules of LPBF (e.g.overhang and print orientation), and minimised the required support structure.This reduction decreased the post-processing effort (Han et al. 2018;Leutenecker-Twelsiek, Klahn, and Meboldt 2016).Post-processing of metallic AM is considered a design challenge as it is both cost-and timeconsuming (Feng et al. 2020;Calignano 2014).Additionally, the mechanical post-processing of support structures increases the risk of damaging thin-structured designs (Merriam et al. 2013;Coemert et al. 2017).The contribution of this work is to present a synthesis approach for metallic 3D-printed springs based on BBs that are customisable in their stiffness functions (Figure 1).Both linear and non-linear stiffness curves shall be customisable.The metallic AM springs can also be tailored in their force and deformation ranges.These goals are realised with the following contributions (Figure 2): (1) Synthesising linear and non-linear BBs (Sec.2.1) (2) Establishing a computational customisation framework for shaping the BBs to tailor stiffness characteristics (Sec.2.2) (3) Customising force-and displacement-range through serial and parallel aggregation (Sec.2.3) (4) Ensuring manufacturability and validation in terms of stiffness performance (Sec.3) (5) Demonstrating the applicability of the stiffness customisation framework through case studies (Sec.4) The discussion and outlook are described in Sec. 5, and the conclusion of this work is presented in Sec. 6.

Synthesis methodology of linear and nonlinear BBs
Single BBs are chosen as the guiding template for customising linear and non-linear stiffness curves.The 3D BB design is constrained to thin beam-based elements with two bases: a loading base and a rigid base.The structural non-linearity of the BBs relies on a combination of the compliant element beam geometry and the boundary conditions (BCs) of both bases.Figure 3 illustrates the relation for both linear and non-linear stiffness curves.The inherent mechanical characteristics of cantilever beam models are used as a simplified analogy to derive the type of stiffness curves (Figure 3(a)).
The bending of a cantilever beam with a fixed end behaves linearly within the elastic limit and allows the loading end to bend in the direction of the applied force.The principle for non-linear bending lies in adding a kinematic constraint to the tip of the beam where it is being deflected (by a force).The constraint induces an axial stretch-dominated stress which is gradually added to the initially loaded transverse bending stress, thus creating a non-linearity.Since a beam under axial forces is stiffer than under bending stress, it results in a progressive (stiffening) curve or vice versa in a degressive (softening) curve (Arredondo-Soto, Cuan-Urquizo, and Gómez-Espinosa 2021).The applied force elongates or compresses the cantilever beam.This elongation or compression leads to axial stresses, as illustrated in Figure 3(a).
This simplified analogy of the cantilever beam models is transferred to synthesise the linear and non-linear BBs.The BC constraints of the BB are replaced with beam geometries that behave in an analogy to the cantilever beam models.The synthesised linear, progressive, and degressive BBs are shown in Figure 3(b-d).All three types of BBs contain two connecting beam elements, a top beam and a base beam, constrained by the loading and rigid bases.The rotational symmetric configuration of the single BBs constrains the translation of the loading base along the centreline (dashed black line in Figure 3(b-d)).
The linear stiffness is bending-dominated, without constraining the BC of both ends.Linearity is achieved with a thin, slender base and top beam (Figure 3(b)).This bending-dominating deformation of the base beam is additionally supported by halving the length of the rigid constraint.For the progressive BB, the ends of both beams are shaped and constrained geometrically so that they rotate and bend with a limited motion in the axial direction (Figure 3(c)).It has a stiff, thicker base beam paired with a thin, connecting top beam.The stiffer base beam allows rotation and bending and is not assumed to be infinitely stiff.The compliant top beam is tilted along the loading direction to induce the axial stresses (tension stress).The non-linearity analogy for the design of the degressive BB is reversed to the progressive design, specifically it goes from an initially stiffer axial stress state to a relatively softer bending stress state (Figure 3(d)).The degressive design uses tilting the top beam against the loading direction.The axial stress is in compression and not tension.The geometry of the base beam and the top beam are both slender and compliant, similar to the linear design.
The parametric model with nine design parameters that produces changes in the stiffness characteristic is visualised in Figure 4.The overall stiffness of the structure is adjusted by manipulating the in-plane thickness of the top beam and base beam by introducing thickness measures at either end of each beam.This is because structural stiffness is highly sensitive to the in-plane beam thicknesses, and is used to tune it accordingly.This relation corresponds well with the simple cantilever beam model's stiffness equation k (Eq. 1) of a cantilever beam model.The stiffness depends on the thickness (t) and length (l) with the power of three and linearly on the width (w).
The parameters T1, T2, T3, and T4 are implemented for changing or tapering the beam thickness.T1 is taken to enhance linearity and reduce stresses in the edge blend between the top beam and base beam.T4 is applied to increase the progressivity of the spring.The angle parameter α defines the angle of the top beam.Considering the cantilever beam model analogy of Figure 3(a), this parameter defines the graduation of the transition from bending to axial forces, which is necessary for the non-linearity of the BBs.It determines the shape of the stiffness characteristic, for example, how progressive (α < 1), linear (0 < α < 1), or degressive (α > 1) the spring will be.The height H of the BB is also parameterised.This parameter regulates the height difference between the top beam and the rigid base.Therefore, it regulates the amount of maximal spring deformation.A small height allows for small deformations before the top beam touches the base.Two radii (R1 and R2) are introduced to avoid sharp edges and stress concentrations.Large radii reduce stresses and make the model stiffer.Smaller radii increase  stresses but make the model more compliant.The overall length of the top beam is changeable using parameter L. A large L makes the spring softer and more linear.H and L specify the required design space volume for manufacturing the overall spring structure.

Manual customisation workflow
In the case of manual stiffness customisation, an iterative five-step design workflow is presented in Figure 5 using the aforementioned information about the nine design parameters.The width (w) (out-of-plane thickness) for all three BBs was kept constant.The three main parameters, L, α, and T4, caused the most sensitivity of the stiffness and its (non-)linearity.The remaining parameters were used for fine-tuning.The importance of each parameter determined in what order they were adjusted to facilitate the process of manually designing a spring.Springs are usually restricted by their volumetric design space and the amount of target displacement.Consequently, the maximum values for L and H were constrained by these specifications and were set first.Next, α was chosen as it influences the (non-)linearity of the stiffness characteristic.As a third step, T4 was defined.A larger T4 made the spring more non-linear.The parameters with the next biggest influence were the radii.Large radii reduced stresses and made the model stiffer.
Smaller radii increased stresses but made the model more compliant.In the last step, set the beam thicknesses T1, T2, and T3 to tune the overall stiffness.Once the parameters were set for the first iteration, the design was simulated, and the process was repeated.As the solution converged, parameters were fixed to restrict the design space increasingly.This five-step manual stiffness customisation was analysed by numerical finite element analysis (FEA) in ANSYS WORKBENCH R19.2.Details of the FEA setup are described in Appendix A.1.

Automated customisation framework
The automatically customised design of the linear and non-linear springs was generated with a genetic algorithm (gamultiobja variant of NSGA-II).The genetic algorithm (GA) optimisation framework was implemented by interfacing the parametric CAD model with the MATLAB optimisation toolbox and ANSYS FEA.The flowchart of this optimisation feedback loop is presented in Figure 6.The MATLAB optimisation toolbox had a standard implementation of a GA, which was capable of solving the prescribed stiffness problem.The integration of MATLAB and ANSYS allowed for the utilisation of MATLAB's algorithmic capabilities and the power of FEA to customise the stiffness characteristics.The optimisation feedback loop begun with an initial design population size of 20, where a random combination of all nine parameters was chosen within a user-defined bounded domain.The CAD parameters' domain defined the GA's design space.ANSYS conducted an FEA for the entire population.A crossover fraction of 70% was set.A stiffness characteristic, in the form of force-deformation design points (DPs), was then returned to the optimisation process.The target characteristic closely approximated the desired DPs.The DPs were used to create a discretized stiffness characteristic, which can be determined through single data points.Five DPs were used to accurately represent the spring characteristic.The GA used feedback from the initial population to create a second generation of fitter individuals using an elitism strategy.Fitness was measured using a loss function that calculates the error between the target and current characteristics.This feedback loop continued until the convergence criteria were met.The convergence criterion for stall generation was three, and the tolerances for the function were set at 1%.The GA terminated after a maximum of 10 generations.

Demonstration spring customisation framework
To demonstrate the effectiveness of the proposed customisation framework, an arbitrarily targeted single forcedisplacement point was selected (Figure 7).At this point, the linear, progressive, and degressive target curves intersected each other.Firstly, the linear spring design was computed within 121 design iterations, which shared the same maximal force-deflection point as the softening and stiffening spring.Each linear curve Figure 5.The springs' manual stiffness customisation followed a five-step manual design workflow that can be iteratively applied to define the design parameters of a single BB.
in Figure 7(a) represents a design iteration.The non-linear target curves were defined using a quadratic polynomial function.The degressive and progressive curves were approximated within 158 and 155 design iterations, respectively.These various design iterations of calculated (non-)linear stiffness curves are shown in Figure 7(b,c).The results of the stiffness curves that closely resemble the prescribed curves (solid black line) are highlighted with a blue dashed curve in Figure 7 and selected in Figure 8.
The performance of the linear and non-linear target curves crossing at a force-displacement point were evaluated with the accuracy forecasting error metric: symmetric mean absolute percentage error (SMAPE) (Flores 1986) for the linear, progressive, and degressive curves.The formula for SMAPE is given in Eq. 2.
This metric calculates the absolute differences between the forecasted design points (DP i forecast ) and actual design points (DP i actual ), and divides it by the sum of the absolute values of DP i forecast and DP i actual .The errors were then averaged over all the number of points N DP .The resulting value was multiplied by 100% to express the accuracy error as a percentage.The objective was to achieve a low SMAPE value, indicating that the deviation between the target and customised values was close.The SMAPE values between the target and customised curves for the linear spring was 0.77%, the progressive spring was 0.75%, and the degressive spring was 1.56%.
The design parameter values of the resulting customised BB models and their aggregation are overviewed in Figure 9.The spring was aggregated in series and parallel four times and was spaced to fit in three batches on the build chamber of the LPBF machine (see Sec. 2.3).
Figure 6.The input parameters were customised using an automated optimisation feedback loop with a genetic algorithm (GA).The initial population in FEA and CAD served as the entry point for the loop.The target characteristic and convergence criteria were set as user inputs.Once the optimisation process converged, the final customised result was outputted.

Serial-and parallel-aggregation
The potential of the BB approach allowed for the synthesis of (non-)linear BB stiffness (k i ), which was then aggregated in series and parallel (Figure 10).This aggregation enabled the stiffness function to be tailored to the desired force and displacement range.The displacement range was increased by stacking the BBs in series, resulting in an equivalent spring stiffness (k serial ) that was divided by the number of BBs.The serial aggregation consisted of pairing mirrored BBs in parallel with each other, with connections at the bases.Figure 10 shows the serial aggregation of four degressive BBs.Additionally, the parallel aggregation of the synthesised BBs, with four BBs arranged in a rotational symmetric configuration, determined the force range.The equivalent spring stiffness (k parallel ) was multiplied by the number of aggregated BBs, and the number of parallel aggregated BBs increased the force range.The combination of these two BB aggregations further customised the stiffness curve of the degressive spring (k total = k serial • k parallel ), resulting in a fourfold increase in force and displacement range.

Manufacturability and stiffness performance validation
The manufacturability and stiffness performance of the designed and as-fabricated springs were validated by testing four serial and parallel aggregated BBs.The customised BBs, with the design parameters shown in Figure 9, were chosen for this purpose.A designed offset of −0.02 mm in the beam thickness was integrated as a standard design parameter to account for potential stiffness discrepancies caused by manufacturing errors.The chosen offset value corresponded to both the manufacturing tolerance of ±0.02 mm and the surface roughness factor R z = 0.025 mm of the LPBF machine.This established method of incorporating designed offsets to compensate for manufacturing errors was proposed and implemented in literature (Wei et al. 2021;Pham et al. 2017;Fiaz, Settle, and Hoshino 2016).Consequently it was applied in the designed BB thickness to compensate potential deviations from the ideal design during the manufacturing process.
The fabrication of the linear, progressive, and degressive springs was carried out using a LPBF technique on a Concept Laser Mlab Cusing R machine.The slices were created with a layer thickness of 0.025 mm, utilising a laser spot beam diameter of 0.05 mm.A meander island scanning strategy with islands of 4 mm and a 90°scan angle was employed, with a hatch spacing of 0.091 mm and an island shift of 0.8 mm in both the x and y directions.The laser power and scan speed were set at 95 W and 700 mm/s, respectively.Additional information about the LPBF machine can be found in the manufacturer's data sheet (GE: Concept Laser GmbH 2023).The metallic powder used for fabrication was 17-4 PH, a precipitation-hardening stainless steel with a powder distribution of D 10 = 18.23 μm, D 50 = 28.87 μm, and D 90 = 44.52 μm.The composition of the powder is described in detail in (GE: Concept Laser GmbH 2017).The material properties of a 17-4 PH additive manufacture tensile probe are listed in Appendix A.2.The springs were then annealed and hardened using the H900 process, with an annealing temperature of 1050°C and a hardening temperature of 480°C to increase their yield strength.All three spring versions were sized in  their overall L and H to fit as a single batch in the LPBF machine building platform size of 90 × 90 × 80 mm, and the batch configuration was printed repeatedly five times.Figure 11 shows the metallic additive manufactured linear, progressive, and degressive springs, which were created by aggregating single BBs in series and parallel to scale the stiffness curve and adjust the force and displacement range.
A compression test was conducted to attain the effective spring stiffness using a Shimadzu AGS-X Series machine equipped with a 200 N load cell.The experimental setup, which utilised a tensile machine, is depicted in Figure 12.The spring was mounted for compressive loading, and the displacement was controlled to a predefined end position of 12 mm with a preload of 0.5 N (at a loading rate of 2 mm/min).All replicas of each spring configuration were tested, and the resulting averaged force-displacement curve is presented in Figure 13.Additionally, the target and customised curves are also complemented in Figure 13 for comparison.
The performance of the AM springs was validated through the precision error and accuracy error of the designed and fabricated springs.The accuracy error was measured via the SMAPE value.The precision error, represented by the ratio of the standard deviation to the mean, of the five tested replicas was limited to a maximum of 5% (Table 1).The as-tested stiffness curves of the (non-)linear springs exhibited higher stiffness when compared to the customised curves, as seen in Figure 13.Additionally, Table 2 provides an overview of the SMAPE values for the three variations: target and customised, customised and as-tested, and target and as-tested, for both the linear and non-linear stiffness curves.The highest SMAPE values of 2.0% or  2.1% were observed between the target and as-tested curves of the non-linear AM springs.In contrast, the linear spring shown low SMAPE values between 0.5% and 1.0% for all three variations.

(Non-)linear referenced case studies
In these referenced case studies, the BBs and automated customisation framework were utilised to match the stiffness curves of Jutta and Kota (Jutte and Kota 2008) as a concrete reference.Linear and non-linear examples of stiffness curves were selected, including a linear, progressive, and degressive target curve (Figure 14).The target curves were described using five DPs within the automated customisation framework (Sec.2.2).The design parameters of the BBs were bounded within a predefined design space that adhered to the build volume requirements of 90 × 90 × 80 mm and a minimum wall thickness of 0.4 mm of the LPBF machine.To maintain consistency with Sec.2.2 and Sec.2.3, the case studies were customised at the level of individual BBs.The customised BBs were then configured in series and parallel to match the referenced stiffness curves.
For the linear stiffness curve (Figure 14(a)), 103 design iterations were calculated to match the target curve.The spring stiffness achieved a SMAPE of 0.2%, but only 38% of the force and displacement range of the reference curve were considered.This limitation was due to the design parameters of height H = 7 mm and length L = 34 mm of the BBs, which have a significant impact on overall stiffness (see Eq. 1).Increasing these parameters would result in greater structural compliance and a larger displacement range.However, these limitations were necessary to adhere to the design space constraints, which encompass a space of 85 × 74 × 74 mm.The design parameters of the customised linear spring are listed in Figure 15, and the series and parallel arrangement is also visualised.
The customisation of the progressive target curve (Figure 14(b)) was achieved through 189 design iterations.The SMAPE yielded a value of 4.5% and indicated the highest value between the target and customised curves.The progressive spring's stiffness curve and its corresponding force-displacement range was customised without exceeding any space requirements.The design space for the progressive spring was 85 × 44 × 52 mm.The displacement range of the progressive curve was realised through the series connection Figure 13.All five replicas of each spring configuration were tested and the results were averaged.The stiffness curves of the target, customised, and as-tested linear, progressive and degressive AM springs were related.
Table 1.The AM spring configurations with five replicas had a precision error of maximum 5%.The degressive spring had the highest precision error among the three configurations.However, the linear and progressive springs had a repeatability error of maximum 3%.

Maximum Precision Error [%]
5 Replicas Linear 3 Progressive 3 Degressive 5 Table 2.The SMAPE values for the linear, degressive, and progressive stiffness curves are overviewed.The SMAPE values were compared as following: target and customised, customised and as-tested, and target and as-tested.In the case of the target and customised degressive spring, a fit of SMAPE = 1.1% was reached.The curve and resulting force range were customised in accordance to the reference curve, which required 163 design iterations.However, the displacement range was limited, as the degressive spring did not consider constant force with horizontal stiffness slope.As a result, the displacement range covered 50% of the reference range and had a bounding box of 66 × 50 × 61 mm (Figure 14(c)).Its BB design parameters are shown in Figure 15.
The tailored springs of the reference study were manufactured using the LPBF technique, similar to the randomised study presented in Sec. 3. The material chosen was 17-4 PH, and the annealing and hardening cycle applied was H900.Due to the low repeatability error (< 5%) of the AM springs within the LPBF machine (see Table 1), the springs were fabricated with a single repetition and subsequently validated.Each spring was manufactured as a single batch, as the design space of the springs encompassed the maximum build volume.The stiffness performance of the fabricated springs was tested using a tensile machine in a compressive force setup (Shimadzu AGS-X Series with a load cell of 20 N and a loading rate of 2 mm/min).The setup is identical to that described in Sec. 3. The as-tested stiffness curves are shown in Figure 16(a-c).The stiffness deviation between the target and as-tested curves were compared.
The accuracy error between the as-tested and customised curves, as indicated in Table 3, ranged from SMAPE = 0.5% for the linear curve to SMAPE = 4.23% for the degressive curve.The progressive curve exhibited the highest accuracy error, with a SMAPE of 6.48%.Conversely, the linear curve demonstrated the lowest accuracy error, with a SMAPE of 0.5%.

Design synthesis approach
The design of customised metal AM springs was achieved through the combination of two synthesis approaches.The BB approach specified the overall topology of the springs, while structural optimisation defined the shape and size of the beam-based compliant elements.The single AM spring were aggregated in a parallel arrangement of four BBs and in serial arrangements of four and ten BBs (see Figure 15).Such aggregations allowed for a high degree of tailoring, as different (non-)linear force-displacement curves can be combined in any conceivable configuration, as long as the constraints of the LPBF process are not violated.Furthermore, the integration of  snap-through buckling effects within the aggregated mechanism can also extend the stiffness functionalities.During a randomised case study, such effects were observed among some design iterations of degressive BBs (Figure 7).These effects can also be extended by embedding built-in switching elements within the CMs, enabling the tuning of S-shaped stiffness curves with a constant stiffness slope (Farzaneh et al. 2022;Song et al. 2019b).More design freedom can be explored with further individualised BBs and more design parameters.Furthermore, the BBs and their arrangement can also be used to synthesise rotational (non-)linear springs (Danun et al. 2021), which are in demand for robotic applications (Irmscher et al. 2018;Song, Lan, and Dai 2019a).The current computational framework for generating customised springs relied on manual intervention by the designer for the design synthesis of the BBs and the definition of their parameters.Ongoing research efforts focus on fully automating the design synthesis and stiffness customisation of single BBs and their aggregates.

Computational customisation framework
The results of the FEA optimisation method for creating customised (non-)linear beam-based BBs were reliable, as demonstrated by both the randomised case study and the concrete reference case study.The deviation between the target and customised springs was within the SMAPE range of 0.2% to 4.5%.The linear stiffness curves exhibited the lowest SMAPE, while non-linear curves tended to have higher SMAPE.However, the optimised results were well-approximated in terms of curve shape and stiffness.One drawback of this approach was the high computational power (32x-CPU @ 3.70 GHz, 64 GB of RAM) required, which scaled with the number of design iterations being optimised.Computation times ranged from 5.5 h to 8 h for design iterations in the range of 103-189.To reduce the computational effort, simplifying with 2D shell models or creating new BB models that are analytically describable can be envisioned.An analytical model also simplifies the difficulty of setting parameter boundaries for optimisation.A model whose bounds can be described by a set of solvable equations is key to automating the entire optimisation and data generation process.The model can be checked without user intervention, and it will reduce breakdown during optimisation because it is always constrained to a feasible bound domain.Furthermore, the analytical model can be further extended with a machine learning algorithm to transform the given (non-)linear curve into a spring design (Ma et al. 2020;Hanakata et al. 2018).

Process-induced deviation of LPBF
All of the generated designs were proven to be manufacturable using the LPBF process.The customised BBs and their aggregations conformed to the design rules of the LPBF process and were designed with a reduced amount of support structure, due to their self-supported nature Table 3.The stiffness performance of the (non-)linear springs in the reference study was quantified using the accuracy error SMAPE.The SMAPE had a range of 0.2% to 6.48% across the three combinations: target and customised, customised and as-tested, and target and as-tested.(Danun et al. 2021).Process-related limitations, such as a minimum wall thickness of 0.4 mm, was taken into account during the customisation process, as this limitation specifies the maximum range of the springs' stress peaks and flexibility.The printer build volume constrained the overall size of the spring structure, but this can be overcome by using machines with larger build volumes.The structural flexibility of the AM springs was determined among others by the material properties of 17-4 PH (H900), which demonstrated its suitability as a spring steel due to its high modulus of resilience.
Alternative metallic AM materials with high modulus of resilience (Howell, Magleby, and Olsen 2013) can also be considered to further enhance the structural flexibility of the springs.Nevertheless, the target (non-)linear curves of both studies covered a wide range of possible stiffness curves, from low to high stiffness (Figure 17).
The randomised case study represented arbitrarily (non-)linear stiffness curves that met at a single force-displacement point.The referenced case study consisted of stiffness curves that were based on the work of Jutte and Kota (Jutte and Kota 2008).The potential customisation area (grey marked area) is indicated in Figure 17, and was specific to the printer build volume of 90 × 90 × 80 mm and material 17-4 PH (H900).
Both studies have shown that the accuracy error of the stiffness curves between the customised and astested was a maximum of 4.23%.The maximum precision error for five replicas was 5%.Therefore, the AM springs complied with DIN 2095 to an accuracy of grade 2, with an accuracy error of less than ±10% being achieved as a design performance for the AM springs' stiffness curve.The largest SMAPE between the target and as-tested was 6.48% for the progressive stiffness curve within the reference case study.The linear springs were predicted with the highest accuracy and the least variance, with a SMAPE of 0.2%.The lower stiffness curves in the reference study indicated an accuracy error that was about three times higher than their stiffer counterparts in the randomised study.This variance was likely caused by process-induced manufacturing defects (e.g.surface roughness [Alghamdi et al. 2020], machine tolerance [Moesen et al. 2011], or thermal distortion effects [Hartmann et al. 2019], etc.).In particular, thin-walled filigree structures might be prone to warpage, as already investigated in various studies (Hartmann et al. 2019;Lu et al. 2021).Such distortion effects might cause the shape deviation of progressive and degressive stiffness curves as the transition between stretch-dominated and bending-dominated beam elements change to geometrical distortions.Furthermore, from an analytical perspective, the thickness of the beam exhibits the greatest sensitivity to dimensional manufacturing errors (see Eq. 1).Despite implementing a compensation method that accounted for sensitivity in the beam thickness, the variance between the customised and as-tested stiffness values was not eliminated.To address such variance for AM CMs, further studies could lead to explicit correction strategies that address in-process, pre-process, and postprocess compensation or optimisation techniques.Potential areas of in-process optimisations include three categories of process parameters: laser-related parameters (e.g.laser power, scanning speed), powderbed-related parameters (e.g.layer thickness, powderbed density) and powder-related parameters (e.g.particle size, shape, and distribution) (Gibson, Rosen, and Stucker 2015).Additionally, optimisation techniques applied prior to the process, such as the design aspects of the support structure or the thermomechanical simulation of the part, can enhance accuracy by predicting and reducing distortion and improving critical Figure 17.In this work, both the high and low force-displacement ranges were considered.The randomised study focused on analysing target (non-)linear stiffness curves in the high range, while the concrete reference study examined lower range of (non-)linear stiffness curves.The grey-marked area encompasses any potential achievable (non-)linear stiffness curves.
surface roughness areas (Calignano 2014;Sehhat, Sutton, and Leu 2022).Post-process optimisation techniques, such as visual scan detection with subsequent design-related compensation of defects, can also provide further accuracy (Sehhat, Sutton, and Leu 2022).It is also worth noting that the tests for each spring replica in this study were not repeated to enhance the statistical significance of the results.This should be taken into consideration when interpreting the findings.Despite this, the sample size and variance were deemed adequate to obtain reliable results.However, repeating tests could strengthen the statistical power of the outcomes.Additionally, future research should specifically focus on the mechanical properties of metallic AM springs, such as their fatigue behaviour under bending conditions, as was done for other mechanical components (Bonaiti et al. 2019;Concli et al. 2021).Furthermore, in specific applications, the evaluation of the springs would be reassessed in light of its compatibility with relevant standards for the field of application.

Conclusion
The combination of the BB and structural optimisation approaches enable the customisation of linear, progressive, and degressive AM springs.The BB approach guides the complex design phase of CMs, while the structural optimisation approach handles the numerical complexity.The stiffness curves and ranges of both force and displacement of the synthesised springs are tailorable in terms of linear, progressive, and degressive loading behaviours (e.g.low-and high stiffness ranged case studies).The interaction between the base and top beams of BBs establishes the ratio between softening (bending-dominated) and stiffening (axial stretch-dominated) forces within the beam material.This behaviour is mainly affected by the design parameters α, L, and T4, which are most sensitive to the stiffness and non-linearity of the curves.A manually guided workflow and an automated customisation framework provide the capability to customise the BBs in terms of their stiffness curves.The optimisation software uses a GA to tailor the target curves within the automated FEA customisation framework.The range of force and displacement is scaled analytically through the serial and parallel aggregation of a single customised BB.In two case studies, a randomised and a concretely referenced, the target linear, progressive, and degressive curves are prescribed within SMAPE ranges of 0.2% and 4.5%.The manufacturability of the metallic springs has been proven for the metallic LPBF and the stainless-steel material 17-4 PH (H900).The accuracy error complies with the tolerance requirement of DIN 2095 (Grade 2) for translational springs, with a maximum error of 4.23%.The precision errors are maximum of 5% for five replicas.The SMAPE between the target and final as-tested stiffness curves is between 0.5% and 6.48%.The potential for this approach to customise rotational (non-)linear springs is also envisioned.

Note on contributor
Aschraf N. Danun is a doctoral student and research associate at ETH Zurich, Switzerland.He holds a master's and bachelor's degrees in mechanical engineering from TU Munich, Germany, which he received before joining ETH Zurich in 2018.His doctoral research focuses on the design for additive manufacturing (DfAM) of compliant mechanisms, with a specific emphasis on potential applications in the medical and robotic field.
Oliver Poole holds a master's and bachelor's degrees in mechanical engineering from ETH Zurich with a focus on mechanical design, optimization, and machine learning.His work focuses on interpretable computer vision applications and automated anomaly detection in mechanical systems using machine learning.Currently, he is working as a data scientist in the private sector.
Edouard Tarter is a doctoral student at CMASLab, ETH Zurich.He obtained both his bachelor's and master's degree in mechanical engineering from ETH Zurich.He is now conducting his PhD on the development of a bistable composite system with integrated actuation.
Patrick Beutler is a PhD student at ETH Zurich and inspire AG in the field of metal additive manufacturing.He completed his master's degree in mechanical engineering at ETH Zurich in 2020.His research focuses on the combination of design automation and design for additive manufacturing (DfAM).In his research, he works on applications of AM in superconducting magnets, fluid applications, medical technology, and clamping technology.
Mirko Meboldt is a Full Professor for Product Development and Engineering Design at ETH Zürich since 2012.His research focuses on design for new technologies with a special emphasis on AM and digital value chain.He develops methods and tools for design automation that combine application, design, and process knowledge for the best use of AM in an industrial context.orientations were determined beforehand through a tensile test and defined as transversely orthotropic.The properties in 0°were chosen along the print direction (X-Y axis in the FEA model) and those in 90°were chosen perpendicular to it (Z axis in the FEA model).Furthermore, the as-built mechanical properties of 17-4 PH are consistent with those reported in the literature.Specific data can be found in references (GE: Concept Laser GmbH 2017;Nalli, Cortese, and Concli 2021).
Table A1.The orthotropic material properties of martensitic precipitation-hardening stainless steel 17-4 PH (H900) were selected for use within the numerical simulations.

Figure 1 .
Figure 1.Customising translational BB-based metallic LPBF springs regarding their (non-)linear stiffness curves (A).The metallic translational springs are tailorable to specific force and displacement ranges, as indicated by the bluish solid arrow.

Figure 2 .
Figure 2. Schematic overview of the work.

Figure 3 .
Figure 3. a) Stretch-dominated beam elements induce axial stresses within the material and cause a stiffening behaviour.Transverse bending-dominated beam structures, on the other hand, decrease the stiffness and lead a softening behaviour.b) The BB is a simple bending beam that is fixed at one end and loaded at the other.A dominated bending within a BB produces a linear characteristic.c) By adding a geometrical BC, the beam can not move towards the fixed end, which induces axial forces and results in a stiffening of the beam.The same occurs in d) when both ends are constrained and axial stresses are caused.The dashed black lines indicate the rotational symmetric configuration of the spring.

Figure 4 .
Figure 4.The parametric model of the BBs consists of nine design parameters that specify the stiffness characteristic of the BBs.The BBs' length (L) and height (H) describe the overall design space.The four thicknesses (T1, T2, T3, and T4) regulate the shape and size of the top and base beam.The radii (R2 and R1) set the radial transition at both beam edges.The parameter α defines the angle of the top beam.

Figure 7 .
Figure 7. a) 121 design iterations were required to calculate the customised linear curve.b) The progressive curve took 155 design iterations.c) The degressive curve was calculated within 158 design iterations.

Figure 9 .
Figure 9.The calculated design parameters of the customised BBs and their four-fold parallel and serial aggregation.

Figure 8 .
Figure8.The accuracy error of linear and non-linear target curves crossing at a force-displacement point were evaluated using a SMAPE of 0.77% for linear, 0.75% for progressive, and 1.56% for degressive curves.

Figure 10 .
Figure 10.The serial and parallel aggregation of single BBs led to scaling of the stiffness curve, resulting in a scaling of the force and displacement range.

Figure 12 .
Figure12.The stiffness curves of the translational springs were tested using a tensile machine and a compression force.
. This approach spreads the displacement range over multiple BBs and avoids collisions between adjacent top beams, as well as contact between the top beam and the rigid base.The design parameters and the aggregated spring design are shown in Figure15.

Figure 14 .
Figure 14.Linear and non-linear examples of target curves were used as references to customise AM springs using an automated computational framework.The AM springs were created with a) linear, b) progressive, and c) degressive stiffness curves.

Figure 15 .
Figure 15.The BB design parameters of customised linear, progressive, and degressive AM springs.

Figure 16 .
Figure16.In the reference study, the as-tested (non-)linear stiffness curves (represented by the grey dashed line) follow closely the target curves (represented by the black solid line) of the LPBF springs.The deviation of the non-linear curves was higher than the linear curves.