# Bistable Compliant Mechanism 3D Printing for Tuned Mass Damper Vibration Control

Ali Zolfagharian, Frédéric Demoly, Mohammad Lakhi, Bernard Rolfe, Mahdi Bodaghi

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Ali Zolfagharian, Frédéric Demoly, Mohammad Lakhi, Bernard Rolfe, Mahdi Bodaghi. Bistable Compliant Mechanism 3D Printing for Tuned Mass Damper Vibration Control. Advanced Engineering Materials, In press, 10.1002/adem.202402233. hal-05017116

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Submitted on 2 Apr 2025

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# Bistable Compliant Mechanism 3D Printing for Tuned Mass Damper Vibration Control

Ali Zolfagharian¹, *, Frédéric Demoly²,³, Mohammad Lakhi¹, Bernard Rolfe¹, Mahdi Bodaghi⁴

¹ School of Engineering, Deakin University, Geelong, VIC, 3216, Australia

² ICB UMR 6303 CNRS, Belfort-Montéeliard University of Technology, UTBM, 90010 Belfort Cedex,
France

³ Institut universitaire de France (IUF)

⁴³ Department of Engineering, School of Science and Technology, Nottingham Trent University,
Nottingham NG11 8NS, UKUK

*Corresponding author E-mail address: a.zolfagharian@deakin.edu.au

## Abstract

Bistable mechanisms, characterized by their ability to maintain two distinct stable states under external stimuli, offer a versatile solution to engineering challenges, particularly in vibration suppression systems. While traditional tuned mass dampers (TMDs) effectively mitigate vibrations at specific frequencies, their efficacy falls short across a broader spectrum, necessitating a design evolution to address varying oscillatory conditions. This work bridges this gap by integrating bistable mechanisms into TMD systems, promising enhanced adaptability for passive vibration reduction without the dependency on complex control systems or external power sources. A bistable mechanism is designed based on numerical simulations and is 3D-printed via fused filament fabrication (FFF). The applicability of a bistable mechanism for adaptively tuned mass damping of flexible links is presented. Following rigorous experimental testing, the optimized design for bistable mechanisms has been determined. Then, a chain of bistable mechanisms is introduced to achieve the desired linear displacement for carrying the TMD payload. The stroke length for the TMD vibration test has been customized to reduce system vibrations by carefully selecting material and optimizing geometric configurations of bistable mechanisms.

**Keywords:** Bistable; compliant; 4D printing; TMD; vibration

## 1. Introduction

Mechanical vibration is an recurrent phenomenon in various engineering systems. It is defined as the oscillatory motion of structures out of its equilibrium position, caused by external forces, leading to restrictions on performance, potential structural damage, and even compromising the safety of the operator working with the system. Tuned mass damper (TMD) is a practical passive engineering solution designed to neutralize the vibrations and improve the performance of the operating system subjected to an external force [1]. There have been a few studies exploring the use of TMDs as passive vibration control devices for structural vibrations. TMDs are effective in mitigating oscillations induced by dynamic loads [2, 3], but their fixed
mechanical properties limit their adaptability and suitability to changing dynamic conditions. A traditional TMD uses an additional mass-spring system connected to the main structure to dissipate the unwanted oscillation at a specific frequency. However, a major drawback of this solution is its limited effectiveness, as it operates only at the specific frequency for which it is designed. In cases where a broader range of frequencies is encountered, traditional TMDs become inefficient. Altering parameters such as the weight or spring stiffness aids in adapting the TMD system to various oscillating frequencies.

The exploration of bistable designs unveils their potential for creating more responsive and efficient TMDs capable of dynamically adjusting to different vibrational frequencies [4]. This adaptability stems from the inherent characteristics of bistable mechanisms, where the transition between their two stable states can be harnessed to reposition the mass within the TMD system, thus altering its dynamic response. To address this, the integration of magnetorheological (MR) dampers with TMD systems for active vibration control was studied [5]. MR dampers, known for their controllable damping properties, can dynamically adjust damping forces in response to real-time structural vibrations. This integration enables the development of MR-TMD systems capable of actively modulating damping characteristics based on instantaneous feedback from sensors monitoring structural dynamics. The MR-TMD system has demonstrated superior performance compared to traditional passive TMDs, achieving enhanced vibration reduction and adaptability to changing dynamic conditions. This innovative approach offers a robust solution for mitigating structural vibrations and enhancing system performance in various engineering domains.

In this work, the same approach is employed on a flexible stainless-steel structure, where the placement of the mass from the reference point determines the stiffness of the spring in that region. To change the position of the mass along the structure without adding any complex control system or electronic devices requiring an external power supply, bistable mechanisms are considered. A bistable mechanism is a mechanical system with two stable equilibrium positions that allows it to hold and switch between two distinct states based on an external stimulus. The distance between the two states is manipulated to move the TMD mass accordingly.

According to Damanpack et al. [6], the stability of hyper-elastic curved beams through a combination of experimental testing and numerical simulations was explored. They fabricated curved beam specimens with varying opening angles using 3D printing techniques and subjected them to mechanical testing to characterize their behavior under different loading conditions. Finite element analysis (FEA) was used to predict the nonlinear behavior of structures. Results from experimental techniques were compared to verify the behavior of the structures as per the curvature beam. Further analysis of the performance prediction of the bistable mechanisms provided more information on their design [7]. The curve beam, a crucial component in compliance beam design, plays a significant role in exhibiting bistability performance [8]. The geometry and materials used in the design and fabrication process determine the component's capacity to store mechanical energy and exhibit bistability. It was shown that the choice of appropriate material directly impacts the component's performance [9].
Selecting the appropriate material is crucial as it directly affects the bistability performance of the component. Ideally, a material with a high strength-to-modulus ratio, which is an important characteristic for compliant mechanisms, is desired because it allows a larger deflection before failure. Zirbel et al. have used acrylonitrile butadiene styrene (ABS) to create a prototype of bistable mechanism as it can demonstrate similar effects to other materials such as metallic glass or titanium, as well as the ease of fabrication making it a good option for prototyping purposes [10]. Therefore, ABS is used as the material for the bistable component in this study.

Using 3D printing technology to fabricate bistable components further amplifies the feasibility and innovation of this approach, offering a pathway to tuned vibration control solutions [11]. Reconfigurable structures using bi-stability properties were printed to serve as mechanical switches, simplifying actuation and motion control without the need for electronics-controlled systems [12]. Moreover, other researchers developed bistable components using shape memory polymers (SMP), which allow twisting and rotational motion based on compliant mechanisms with built-in joints [13,14]. Bistable structures have two stable states separated by an energy barrier, known as snap through instability. The transition between these stable states can be done by applying enough energy that passes the hill of the barrier, resulting in a stable state with lower energy without additional energy. The shape of the component remains in a stable position over time without any energy consumption.

The main objective of this research is to design and optimize a bistable component within a TMD system for a flexible stainless-steel structure, therefore enhancing its attenuation performance under various oscillating frequencies without using any active control system. The work will report the study of multiple bistable designs and related prototypes will be 3D printed. The efficient vibration suppression will be achieved through numerical simulations and optimization, then contributing to the advancement of vibration control technologies. This work goes beyond the mere theoretical exploration of bistable mechanisms; it encompasses the practical realization of a bi-stable-enhanced TMD system tailored for a compliant structure. The objective is twofold: to demonstrate the viability of bistable mechanisms in broadening the operational bandwidth of TMD systems and to advance the field of vibration control through the application of cutting-edge manufacturing techniques.

## 2. Methodology

The integration of bistable mechanisms in vibration isolation systems through additive manufacturing offers a significant leap in design flexibility and functionality. By utilizing 3D printing techniques, complex bistable structures, capable of switching between two stable states under specific stimuli, external force here, can be precisely fabricated. This capability enhances vibration isolation by allowing dynamic adjustment of system parameters in response to varying vibrational frequencies, thereby broadening the effective operational bandwidth of such systems. The combination of bistable designs with additive manufacturing techniques paves the way for innovative, adaptive vibration control solutions.

Integrating cells as cylinders and stacking bistable mechanisms in the paper are justified as innovative strategies to enhance structural complexity and functionality. This approach allows
for the creation of more adaptable and efficient systems with tailored mechanical responses, suitable for a wide range of applications that are used in this work.

## 2.1. Design of bistable mechanism (cell)

Designing gradient bistable cells based on prescribed maximum instability forces is a strategic approach to creating programmable multistable mechanical metamaterials [15,16]. Researchers have varied the thickness, geometry, or material properties of curved shells to develop these innovative materials, enabling the design of structures with complex, programmable behaviors suitable for a wide range of applications [17]. This method leverages the intrinsic properties of materials and geometric configurations to achieve the desired mechanical responses. The novelty of this approach lies in its precise control over the mechanical behavior of structures, allowing for the creation of programmable multistability.

Two conditions must be met for the curved beam to exhibit bistability [18], i.e., the ratio of beam apex height to thickness must exceed 2.31 and the buckling mode 2 (Figure 1) must be constrained [19]. Therefore, a double curved beam system with a membrane connecting beams at the mid-span was used in this work to ensure buckling mode 3 (Figure 1) and bistability. The membrane transfers the rotational motion at either beam to the axial motion of the other beam. The two beams have the same pattern of motion, and the rotation will be reduced; therefore, no asymmetric modes can be generated under this circumstance, and the purpose of suppressing buckling mode 2 could be achieved. Since buckling can take either an up or down direction, this creates two stable positions for the bistable mechanism.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Different buckling modes of the compliance beam. (a) deformation of a single curved beam with recoverability. (b) deformation of a double curved beam system with bistability [19]. (c) double curved beam profile.</figcaption></figure>

Qiu et al. [18] proposes the profile of the curved beam in the shape of buckling mode 1 of a fixed-end elastic beam, with the configuration of the beam expressed as:

$$l(x) = \frac{h}{2} \left[ 1 - \cos \left( 2\pi \frac{x}{l} \right) \right] \quad (1)$$

where, h is the apex of the cosine curve, l is the horizontal span of the curved beam. Besides the parameters that affect the profile of the curve beam, other parameters are also introduced
to the formation of the component, including the beam thickness *t* and the distance gap between two beams *g* (Figure 2b). The straight beam in the middle and the side walls were expected to constrain the lateral movement of the curved beams.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Schematic of the bistable component. (a) 2D CAD drawing with annotations. (b) 2D CAD drawing with dimensions.</figcaption></figure>

A 3D model of the beam profile is generated using Autodesk Inventor (San Francisco, USA) with its parameters to comply with the profile configuration and constraints to maximize the ability to achieve bistability (Figure 1a). While there are constraints to follow during designing the compliance beam, there is no specific shape or outline that has been tested or proven to achieve bistability. Therefore, the shape of the compliance beam of the component is designed using simple curves. The beam is constructed with two connecting parts: the hinges that connect the walls to the beam and the curve beam that initiates the snap-through behavior (Figure 2).

The hinges are created using Tangent function with a 15-degree sweep path length; the circle radius starts at 20 mm and increments by 0.6 mm to create the thickness of a single beam; each beam is 1.2 mm apart. The curved beam is designed using a 3-point arc with two of its ends connected to the hinges; the center of the curved beam is placed on the center line of the component with a radius of 23 mm. Once both beams are formed, a middle linking block between the beams is created to link the beam movements together, improving the response of the compliance beam.

## 2.2. *Design of stacking mechanism*

To improve the performance of the bistable component by varying the stroke length (the distance covered between two stable states), a stackable design is implemented that enables each unit to be connected by simply sliding the connecting joint together. The female joint is made with a 3% tolerance, ensuring the bistable cells can be connected firmly and will not loosen during vibration tests. A slot is also designed to fit the component onto a flexible link.

## 2.3. Equivalent spring-mass system of the flexible link TMD

The flexible link is modelled to represent the simple mass-spring mechanical system, and the TMD due to its simplicity and easy to implement [3]. A simulation model of mass-spring
system with a TMD attached on the primary mass is created to observe the behavior of the main system and the performance of the TMD.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Proposed TMD system model.</figcaption></figure>

The primary system profile is set to be weight of the main mass $m_{main} = 0.15$ Kg; spring constant $k_{main} = 226.8$ N/m, therefore the oscillating frequency of the main mass can be computed:

$$ \omega_{main} = \sqrt{\frac{k_{main}}{m_{main}}} \qquad (2) $$

For the TMD system to successfully stabilize the main system, ideally, the natural frequency of the TMD $\omega_{TMD}$ should approximately equals to the natural frequency of the main system. With that in mind, we can choose the TMD system profile such that its natural frequency is equals to $\omega_{main}$. The mass chosen is 0.01875 Kg, using the natural frequency formula, we can get the value for the TMD spring constant using:

$$ k_{TMD} = \omega_{main}^2 \times m_{TMD} \qquad (3) $$

When the TMD system is attached to the main mass, the TMD mass oscillates with the same natural frequency, which reduces the main mass's vibration; hence, the position of the main mass is significantly reduced. To reduce the vibration for a flexible structure, the parameters from the mass-spring system must be related to the flexible structure, as the stiffness of the spring is assumed to be proportional to its length. In case of a flexible structure, the placement of the mass on the structure from the reference point determines the stiffness of the structure in that region, this relationship is described as the following equation:

$$ k = \frac{3EI}{L^3} \qquad (4) $$

where $E$ is the constant modulus of elasticity of the material, $L$ is the placement of the mass from the reference point, and $I$ is the moment of inertia. For the numerical simulation and experiment, the flexible structure is represented by a 600 mm stainless steel ruler, so some parameters of the spring constant is given based on the material and the profile of the ruler, specifically, $E = 210$ GPa, where $b$ is the base width of the shape given as 0.02 mm; and $h$ is the height of the shape given as 0.0015 mm with rectangular cross section.
Given $m_{main}$, $k_{main}$, and $m_{TMD}$ t, the distance between $m_{TMD}$ and $m_{main}$ on the flexible stainless-steel structure can be computed using the natural frequency formula $\omega_{main} = \omega_{TMD}$ and Eq. (2) to come up with the relationship between the mass and spring stiffness of the main system and the TMD system for the best performance of vibration control. Finding the distance between $m_{TMD}$ and the main mass one could have the distance of the TMD mass from the main mass ($L$ is the distance of the TMD mass from the main mass) using Eq. (4).

### 3. Results and Discussions

During the fused filament fabrication (FFF) process, there are several parameters adjusted so the component can fit precisely on the scale of the vibration test and increase the ability to achieve bistability, easing the post-processing steps of removing the support structures and excess materials, separating the model from the printing bed, etc. The main parameter that directly affects the performance of the component is the initial infill percentage, as it is also a factor that affects the rigidity of the walls. By increasing the infill percentage to 50% (with hexagon infill pattern), the walls can be stronger and can withstand the force applied from the compliance beam during compression. The 3D printer used is the Flashforge Guiders IIS (Jinhua, China), using a general 1.75 mm ABS filament, giving the component a suitable strength and flexibility for the fabrication of compliance components. The settings to use for the given printer and filament are shown in Table 1.

<table><caption>**Table 1.** FFF 3D printing processing parameters.</caption><thead><tr><td>Name</td><td>Value</td></tr></thead><tbody><tr><td>Nozzle size (mm)</td><td>0.4</td></tr><tr><td>Filament size (mm)</td><td>1.75</td></tr><tr><td>Extruder temperature (°C)</td><td>240</td></tr><tr><td>Platform temperature (°C)</td><td>105</td></tr><tr><td>Infill percentage (%)</td><td>50</td></tr><tr><td>Infill pattern</td><td>Hexagon</td></tr></tbody></table>

Multiple bistable components have been fabricated to test the functionality of the compliance beam. Each model is adjusted with different factors such as hinge angle, hinge length, curve profile, and printing infill, which influence the performance of the beam, all of which have been shown in Table 2 and Figure 4.

Through the printing tests, each model is analyzed in terms of bistability, scale factor, and reliability of the beam. With each factor adjustment that is tested to have a positive effect on the component, it will be kept for the next re-design stage. Any factor that causes failure will be adjusted or eliminated in the next test to improve the quality of the bistable component.

Multiple experiments demonstrate the instability of the walls' rigidity, as they bend during compression and fail to hold the compliance beam in place, leading to monostable behavior. To solve this problem, a solution proposed during the design process is to create a wall-holder structure, bounding the walls towards the central structure of the component. The wall holder is created using the tangent function, in which one side is connected to the wall and the other
side is connected to the center line of the component, with a radius of 42 mm. Through the experiment, the structure was proven to be effective in holding the walls in place during compression, increasing the rigidity of the walls.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Different 3D-printed bistable designs.</figcaption></figure>

There are three milestones during the design process that contribute to the success of fabricating the bistable component. The first milestone occurred at model 6, where the profile of the curve beam was changed from only using the 3-point arc method for the whole compliance beam to adding a hinge using the 3-point arc method and designing the curve beam using the tangent method to give more control over the angle, curvature, and thickness of the whole compliance beam. The second milestone occurred at model 10, when the side walls of the component were observed to have been deflected outward when under compression, preventing energy from the beam from being kept in a second stable state. Therefore, a change in the printing process was made to give the walls more rigidity, which is to increase the infill percentage from 20% to 50%. The result was observed to have achieved bistability. To improve the capability of reserving the energy of the compliance beam, an additional feature is added to the design at test 8 (the last milestone), which is the wall holder. It is designed as a curve connecting the walls to the base, which helps to reduce load on the walls when under compression and alter the outward force towards the base.

All the bistable systems have been modeled in the FEA ABAQUS software (Dassault Systems, Vélizy-Villacoublay, France) to find out the force displacement results, as demonstrated in Figure 5. From the graphs shown in Figure 6, it could be seen that designs 6, 8, and 9 were reflecting bistable characteristics where the load snapped through the negative values. However, as explained in Table 2, design 6 experiences near bistability (hold at 2^nd^ shortly, then snap back to 1^st^ steady state without external force applied in experimental tests). Both designs 8 and 9 satisfy bistability, but design 8 is shown to be fragile under fatigue in experimental tests, and the bottom beam breaks after several tests. The final design was also tested under
compression loading experimentally to measure the maximum force before snapping through, which was shown to be reasonably close to the FEA results with a 2% error average error of three repeated tests (Figure 7).

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<figure><img src="image_6.png" /><figcaption>**Figure 6.** Force-displacement FEA results of the 3D-printed bistable samples.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 7.** Force-displacement measurements of the 3D-printed bistable samples.</figcaption></figure>
<table><caption>**Table 2.** Bistability test models.</caption><thead><tr><th>Design #</th><th>Curve profile</th><th>Hinge angle</th><th>Hinge length</th><th>Print profile</th><th>Beam thickness</th><th>Results</th><th>Observations</th><th>Comments</th></tr></thead><tbody><tr><td>1</td><td>3-point arc (38 mm radius)</td><td>75 deg from base</td><td>1.5 mm</td><td>120% scale, infill 20%</td><td>1.5 mm</td><td>Monostable, beam is flexible, no fracture was found</td><td>High damping effect (require larger force for linear motion)</td><td>Missing connection block between upper and bottom beam</td></tr><tr><td>2</td><td>3-point arc (38 mm radius)</td><td>85 deg from base</td><td>1.5 mm</td><td>120% scale, infill 20%</td><td>1.5 mm</td><td>Monostable</td><td>Break at peak point of the curve</td><td>Missing connection block between upper and bottom beam, monostable can still be observed for the initial stress tests, however the beam break after several tests</td></tr><tr><td>3</td><td>3-point arc (38 mm radius)</td><td>90 deg from base</td><td>1.5 mm</td><td>120% scale, infill 20%</td><td>1.5 mm</td><td>Monostable, beam is flexible, no fracture was found</td><td>Nil</td><td>Hinge is not curve/long enough to hold the curve beam</td></tr></tbody></table>
<table><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>at the 2nd stable state</td></tr><tr><td>4</td><td>3-point arc (38 mm radius)</td><td>90 deg from base</td><td>1.5 mm</td><td>100% scale</td><td>1 mm</td><td>Broken hinge during compression</td><td>Nil</td><td>&lt;2 mm beam thickness is too thin for the 3D printer to fabricate, causing printing error at curvature point, hinge too weak causing it to break during compression</td></tr><tr><td>5</td><td>Tangent and 3-point arc (11 mm)</td><td>Tangent angle (26 deg)</td><td>Tangent curve length (20 mm)</td><td>120% scale, infill 20%</td><td>2 mm</td><td>Beam is not flexible and does not respond to input force</td><td>Nil</td><td>Beam is too thick, prevent the ability to bend the structure, designed curvature of the beam and hinges is not suitable preventing the</td></tr></table>
<table><tr><td>6</td><td>Tangent and 3-point arc (23 mm)</td><td>Tangent angle (15 deg)</td><td>Tangent curve length (20.8 mm)</td><td>120% scale, infill 20%</td><td>0.8 mm</td><td>Monostable, near bistability (hold at 2nd steady state for 30 seconds then snap back to 1st steady state without external force applied</td><td>High damping effect (require larger force for linear motion)</td><td>2 steady states provide a 5mm stroke</td></tr><tr><td>7</td><td>Tangent and 3-point arc (23 mm)</td><td>Tangent angle (12 deg)</td><td>Tangent curve length (20.8 mm)</td><td>120% scale, infill 20%</td><td>0.8 mm</td><td>Broken beam</td><td>High damping effect (require larger force for linear motion)</td><td>Hinge too short causing more curvature to the beam, leading to more tension applied at peak point when bend, causing the beam to break</td></tr><tr><td>8</td><td>Tangent, 3-point arc (23 mm), and wall holder (3-point arc 42 mm)</td><td>Tangent angle (15 deg)</td><td>Tangent curve length (20.8 mm)</td><td>120% scale, infill 50%</td><td>0.8 mm</td><td>Bistability achieved; Bottom beam break after several tests</td><td>High damping effect (require larger force for linear motion)</td><td>Beam thickness might be too much causing fracture during compression</td></tr></table>
<table><tr><td>9</td><td>Tangent, 3-point arc (23 mm), wall holder (3-point arc 42 mm), and widen connecting beam from 0.5 mm to 2 mm</td><td>Tangent angle (15 deg)</td><td>Tangent curve length (20.8 mm)</td><td>120% scale, infill 50%</td><td>0.6 mm</td><td>Bistability achieved; Bottom beam is flexible, no fracture was found</td><td>High damping effect (require larger force for linear motion)</td><td>Final design for bistable component</td></tr></table>
A vibration test is then conducted to assess the performance of the bistable component in changing the working frequency of the TMD system. The experiment is set up by attaching a flexible stainless-steel ruler to a shaker (Name, City, Country if possible) at one end as a reference point where the external force would be applied to vibrate the system (Figure 8). From the reference point, the main mass is placed 25 cm away to replicate the main structure, which the vibration induced by the shaker should be minimized. Two accelerometer sensors are placed at the reference point and the main mass to observe their acceleration and displacement according to the frequency of the shaker. The TMD mass is attached to the ruler on the other end, and connecting to it is a stack of five bistable components, which gives the total stroke length of about 3.3 cm. Each unit is firmly attached to the ruler by a housing at the back to hold it in place.

<figure><img src="image_8.png" /><figcaption>**Figure 8.** Vibration test set up for assessing the effect of the proposed enhanced TMD with stacked bistable cells.</figcaption></figure>

The initial test is conducted with only the *m*<sub>main</sub> = 150 g to verify the natural frequency of the system by varying the induced frequency of the shaker from 5 Hz to 50 Hz. The peak point of the system in terms of acceleration and displacement in the frequency domain graph generated from the test can be seen in Figures 9 and 10, respectively. The next test is conducted with the TMD mass system attached to the ruler on the other end with the bistable component at its compressed mode and another test in its extended mode. These are to define the working frequency of the TMD system as well as to demonstrate how the TMD system can effectively reduce the amplitude of the acceleration over a range of frequencies.

Once the most optimized bistable component has been designed and fabricated, it is implemented in the TMD mass of the vibration test to help shift the TMD mass position, resulting in the improvement of the vibration mitigation effect as shown in Figures 9 and 10.

During the first test, only the main mass of 150 g is attached to the system. With the induced vibration from the other end of the structure, the main mass suffers from the vibration intensively during the system's natural frequency of about 35 Hz. As shown, the acceleration
force measured from the main mass reaches about 3.5G. The second test is conducted with the TMD mass introduced and connected to the bistable component in its compressed mode, bringing the TMD mass outside of the working frequency of the system. As a result, the natural frequency of the system is reduced to around 25 Hz, with its amplitude slightly dropping to 3G. The last test is conducted with the same TMD mass, but the bistable component is now extended, bringing the TMD mass closer to the main mass and inside the working range of the system's natural frequency. The results show that, while the system's frequency peak point is shifted down to 20 Hz, its acceleration peak point also decreases to only 1.3G, which has 63% lower effects when under the same range of oscillation frequency.

<figure><img src="image_9.png" /><figcaption>**Figure 9.** Effect of TMD bistable mechanism on acceleration of the system.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 10.** Effect of TMD bistable mechanism on displacement of the system.</figcaption></figure>
4. Conclusions

In conclusion, the fabrication of a bistable component using additive manufacturing marks a significant stride towards enhancing vibration control systems. This work has been devoted to optimizing the design of the compliance beam. By carefully selecting the material's stiffness and refining geometric configurations, the stroke length has been effectively tailored to meet the specific requirements of the TMD vibration test, offering precise and reliable performance. Future research could explore the integration of smart materials into the bistable component via four-dimensional (4D) printing, thereby having more control over the stroke lengths between different stable states of the compliance beam and enabling a self-triggering mechanism between stable states. This improvement holds immense potential for further enhancing the efficacy and adaptability of mechanical vibration control systems, paving the way for more robust and responsive structural dynamics management in diverse engineering applications.

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We can expand a bit the reference list with the following items:
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* Wu L., Pasini D. In situ activation of snap-through instability in multi-response metamaterials through multistable topological transformation. Advanced Materials 2023, 35(36), 2301109.

* Seyedkanani A., Akbarzadeh A. Magnetically assisted rotationally multistable metamaterials for tunable energy trapping-dissipation. Advanced Functional Materials 2022, 32(52), 2207581.

* Fancher R., Frankel I., Chin K.C.H., Abi Ghanem M., MacNider B., Shannahan L.S., Berry J.F., Fermen-Coker M., Boydston A.J., Boechler N. Dependence of the kinetic energy absorption capacity of bistable mechanical metamaterials on impactor mass and velocity. Extreme Mechanics Letters 2023, 63, 102044.

* Zhao A., Liu C., Zheng Y., Zhang A., He P., Chen H., Wu G., Zhang M., Wu T., Gu G. Evaluation of shock migration performance for a multi-stable mechanical metamaterial. Composite Structures 2023, 321, 117312.

* Perez-Garcia C., Aranda-Ruiz J., Lopez-Donaire M.L., Zaera R. Garcia-Gonzalez D. Magneto-responsive bistable structures with rate-dependent actuation modes. Advanced Functional Materials, 2024, 2313865.

* Hu B., Fang X., Wen J., Yu D. Effectively reduce transient vibration of 2D wing with bis-stable metamaterial. International Journal of Mechanical Sciences 2024, 272, 109172.

* Jiang L., Li B., Ma W., Wu Y., Bai R., Sun W., Wang Y., Chen G. Electroactive soft bistable actuator with adjustable energy barrier and stiffness. IEEE Transactions on Robotics 2023, 40, 472-482.

* ten Wolde M.A., Farhadi D. A single-input state-switching building block harnessing internal instabilities. Mechanism and Machine Theory 2024, 196, 105626.