Bidirectional actuation of buckled bistable beam using twisted string actuator

Abstract

A new actuation mechanism using the twisted string concept to trigger the snap-through of a bistable buckled beam to produce an effective on/off bistable actuator is proposed. The twisted string concept combined with a pin utilizes actuation moment to actuate a bistable beam. The required actuation loads are analytically formulated using the Euler–Bernoulli beam model and solved with the proposed solution algorithm. The actuation mechanism is fabricated to meet the 24.3 N mm actuation requirements. A prototype of the actuator was built, and its performance was evaluated. In a unidirectional actuation, an actuator response time of 104 ms was achieved. The overall response time of the actuator is affected by the length of the string. The twisted string mechanism was also placed in an antagonistic configuration to enable bidirectional actuation. The shape of the input voltage signal also affected the bidirectional performance of the actuator. The actuator produced an actuation bandwidth of 2 and 5 Hz with sine and square input voltages, respectively, while generating 10 mm output displacement.

Keywords

Twisted string concept bistable structure buckled beam actuation moment Euler–Bernoulli model snap-through

Introduction

Bistable structures have been applied in various engineering fields, such as control surface actuation (Barrett et al., 2005; Kang et al., 2014; Mallick et al., 2014), deployment mechanisms (Suh et al., 2018), energy harvesting (Lee and Inman, 2018; Li and Qin, 2015), morphing wings (Ai et al., 2017), and relays in microelectromechanical (MEMS) systems (Qiu, 2003). A bistable structure has two distinct stable configurations. These stable equilibrium states show robustness against disturbances under a certain critical energy level, and they are sustained with no additional power consumption. Therefore, bistable structures are considered effective components for systems that only need two simple operating states, that is, on-and-off, closed-and-open, or up-and-down.

Bistable structures exhibit a nonlinear snap-through phenomenon; it enables a large geometric deformation only with a small amount of trigger energy input to the system. Taking advantage of this geometric nonlinearity of the snap-through effect, many studies have been conducted to amplify actuation displacement. To actively induce snap-through, various smart materials have also been employed, such as piezo-materials (Aimmanee and Tichakorn, 2018; Kang et al., 2014), shape memory alloy (SMA; Ryu et al., 2011), and micro-fiber composites (MFC; Cazottes et al., 2008). Barrett et al. (2005) reported the displacement amplification of a piezo bimorph beam actuator operating in a post-buckling regime. Cazottes et al. (2008) demonstrated the feasibility of a buckled bistable beam with initial compression as a snap-through actuator, but only a unidirectional snap-through was achieved.

Han et al. (2013) reported that reciprocating control fins with the stroke amplitude of 8 mm could significantly improve the circular error probability (CEP) of a projectile even with a low control bandwidth. Kang et al. (2014) proposed a reciprocating control fin comprising a piezo-stack actuator, a lever-arm amplification mechanism, and a bistable beam. This new actuator concept was proposed to produce additional control forces and moments during projectile flight. A prototype actuator was developed to achieve a stroke amplitude of 8 mm and operating frequency of 2 Hz. Although the design requirements were met, we realized that the actuation mechanism deployed in that actuator was rather bulky and expensive for its application in projectile control. Furthermore, the piezo-stack actuator may not work properly due to shock during the launch phase. Therefore, there was the need to develop a simpler and more cost-effective actuation mechanism that can be used in a wider range of applications than the previous one.

In this article, a new actuation mechanism using a twisted string actuator (TSA) is proposed to overcome the limitations in the previous design. This new mechanism consists of a pin, strings, and a motor. The new concept induces bending moment, while the previous concept induces force to trigger the snap-through of the bistable beam. The trigger moment is mathematically formulated and solved using the proposed solution algorithm. Experiments were performed to validate the results and to evaluate the performance of the actuator. Although direct current (DC) motor was used in the prototype, it could be replaced by a piezo motor depending on the purpose for which the bistable actuator must serve and the actuator requirements in terms of operating frequency.

A TSA is a linear actuator. The string used in this type of actuator acts as a transmission gear. The transmission ratio of a TSA, which is the ratio of the string contraction (output) to the angle of twist of the motor (input), shows nonlinear behavior (Mehmood et al., 2015). TSA is inexpensive and relatively has high-speed, lightweight, compact, and simple to operate. It consists of a high-speed low-torque DC motor and strings. Its characteristics have made it a good candidate for robotic applications (Palli et al., 2015; Shin et al., 2012; Suzuki et al., 2007). However, TSAs have nonlinear behavior with a serious limitation; they transmit only pulling force (unidirectional). Palli et al. (2015) placed two twisted strings in an antagonistic configuration to achieve bi-direction movement in a robotic joint.

Description of the new actuation concept

The TSA concept is proposed to actuate a bistable beam. This combination is called a twisted string bistable actuator (TSBA). The TSBA actuation mechanism comprises DC motors, strings, and a pin as shown in Figure 1. The pin serves as the moment arm, while the twisting of the strings produces the linear force. In the actuation process, a pair of strings is attached to the rotating shaft of the motor, and the other end is attached to the pin. The rotation of the motor’s shaft causes the strings to twist, and their length is reduced. The reduction in the length of the string produces a pulling force on the pin. Since the pin is slotted into the buckled beam, the point of attachment on the beam serves as the bending moment location. This pulling on the lower end of the pin by the twisted strings produces a rotation of the pin about the point of attachment on the beam. A bending moment is induced on the bistable beam for snap-through to occur.

Modeling of actuation moment forsnap-through of bistable beam

The bistable beam is analytically modeled to predict the necessary trigger moment required for snap-through. The formulation considers the buckling of a beam with small deflection; the buckling caused by up to 2% length shortening can be analyzed using the present theory (Cazottes et al., 2009; Cleary and Su, 2015). For large deformations, the elastica beam theory can be applied (Camescasse et al., 2013, 2014). In solving for the equations obtained from the modeling, a new solution algorithm is proposed. The trigger moment obtained in this section served as a guide in designing the actuation mechanism, as will be presented in subsequent sections.

Mathematical modeling

A Euler–Bernoulli beam clamped at both ends with initial length $L_{0}$ , thickness $t$ , and width $b$ is considered. The beam is initially straight and buckling is achieved by applying an external compressive force $P$ , as shown in Figure 2. An external trigger moment is applied at a location $L_{M}$ to actuate the beam. The first two buckling modes are enough to estimate the bistable beam shape (Cleary and Su, 2015; Vangbo, 1998) and are defined as

y_{1} (x) = a_{1} (1 - \cos (2 π \frac{x}{L}))

(1)

y_{2} (x) = a_{2} (1 - \frac{2 x}{L} - \cos (2.86 π \frac{x}{L}) + \frac{2}{2.86 π} \sin (2.86 π \frac{x}{L}))

(2)

where $a_{1}$ and $a_{2}$ are the first and second buckling mode amplitudes, respectively. The final shape of the pre-compressed beam during snap-through is defined as a combination of the first two buckling modes (general solution)

y (x) = y_{1} (x) + y_{2} (x)

(3)

The threshold moment necessary to actuate the beam, as shown in Figure 2(b), is modeled by considering the energy of the system related to equation (3). The total energy of the bistable buckled beam is given as the sum of the internal (compressive and bending) and external (actuating force or moment) energies. Cazottes et al. (2009) presented the energy expression as follows

U_{Total} = U_{bending} + U_{Compression} + U_{Ext . Moment}

(4)

The highest order of polynomial in equation (4) is four. The maximum actuation moment is obtained by considering the extrema of the total energy. This is driven by finding the first derivative of the total energy with respect to $a_{1}$ and $a_{2}$ in the following form

{\frac{\partial U_{Total}}{\partial a_{i}}}_{i = 1, 2} = 0

(5)

Since bistability in this study is achieved by the pre-compression of the beam, the pre-compression of a beam of initial length $L_{0}$ and final length $L$ can be defined as

η (%) = \frac{L_{0} - L}{L_{0}} = \frac{Δ L}{L_{0}} %

(6)

Proposed mathematical solution algorithm

In previous studies, equation (5) was solved using the guessed method (Cazottes et al., 2009). A third equation is required if a continuous set of solutions are needed (Cleary and Su, 2015). This section proposes a simple method to solve for the same equation without the third equation.

Equation (5) is a third-order polynomial with two simultaneous equations and three unknowns $a_{1}$ , $a_{2}$ , and $M$ . To solve for the unknowns, it is assumed that $a_{2}$ does not contribute to the initial and final states of the beam and is therefore set to zero. This reduces the unknowns at two equilibrium configurations to two. Since the highest order of the polynomial in equation (5) is three, three solutions will be obtained for both $a_{1}$ and $a_{2}$ with two solutions having the same value, but with opposite signs. The third solution gives zero for both $a_{1}$ and $M$ , which is meaningless. Here, $a_{1}$ and $a_{2}$ are inserted in equation (3) to define the shape of the beam with its corresponding moment. The change in sign of $a_{1}$ values represents the beam in its two stable configurations. Since the snap-through of the beam occurs between these two stable states, an iteration of $a_{1}$ between the stable state values will produce a corresponding $a_{2}$ and $M$ . An in-house MATLAB code is developed to perform the iteration process as summarized in Figure 3. The maximum moment obtained from the iteration process is the minimum external moment required to actuate the beam. This algorithm also helps to obtain the various shape deformations of the beam during the snapping process.

Experimental validation of analytical modeling

The bistable beam used in this study was fabricated from SK5 (high-carbon tool steel) with an initial length of 140 mm, a width of 25 mm, a thickness of 0.3 mm, Poisson’s ratio of 0.29, and Young’s modulus of 206.8 GPa. These beam parameters selected were the same as those in our previous study (Kang et al., 2014), which included post-buckling analysis. Figure 4 shows the experimental setup used to measure the required trigger moment for snap-through to occur. A linear spring and a force sensor were attached to a 30 mm pin. A laser displacement sensor (LDS) was used to measure the displacement at the center of the beam during snap-through. The motor was rotated slowly, and the measured force was retrieved using an NI DAQ device and LabVIEW.

Results and discussion

The straight beam was initially pre-compressed by 0.29, 0.45, and 0.65 mm to achieve output displacements of 8, 10, and 12 mm, respectively. These pre-compressions represent 0.21%, 0.32%, and 0.46% of the initial beam length according to equation (6). These values were chosen because they are small enough that they would not violate the small-deflection theory and to avoid plasticity setting in the beam during the experiment. Figure 5 shows a comparison of the analytical and experimental results. An increase in pre-compression resulted in nonlinear behavior of the output displacement. The analytical prediction matches well with the experimental results.

The bistable beam is triggered at 15 and 20 mm from one end of the beam. The trigger moment results obtained from the experiment are compared to the analytical results and summarized in Table 1. The percentage difference between the experimental and analytical results shows a maximum percentage difference of 6.82%. The percentage difference increased as the output displacement increased because the Euler–Bernoulli equation for elastic beam considers only small deflections. Therefore, the large deflection theory should rather be considered when modeling large output displacement, $d$ . At each trigger location, the required trigger moment is observed to increase as the output displacement increases. In addition, the trigger moment decreases as the actuation location moves farther away from one fixed end of the beam. These trends are seen in both the analytical and experimental results.

Table 1.

Necessary trigger moment requirement at an actuation location of 15 and 20 mm.

Output displacement, d (mm)	Trigger location (15 mm)			Trigger location (20 mm)
	Experiment	Analytical	Diff. (%)	Experiment	Analytical	Diff. (%)
8	24.43	24.92	2.00	21.58	20.79	3.66
10	30.59	32.09	4.90	27.18	26.01	4.30
12	36.72	38.63	5.20	29.16	31.15	6.82

Diff.: percentage difference; unit: N mm.

Based on the observations summarized in Table 1, further analytical study was conducted to investigate the effect of the actuation location on the trigger moment and results are presented in Figure 6. As seen in Figure 6, the necessary trigger moment along the length of the pre-compressed beam tends to decrease to a minimum and then increase again. This trend was observed to be the same regardless of the pre-compression magnitude. The minimum actuation moment was found to be at 18% of the pre-compressed length as already reported by Cleary and Su (2015). This represents an actuation location of approximately 25 mm in this study.

A bistable actuator of output displacement, 10 mm, was selected for further analysis; this is a little larger than the stroke requirement analyzed by Kang et al. (2014). From the analysis, we determined that a trigger moment of 24.3 Nmm is required. The actuation mechanism presented in the later sections of this article was designed to meet that requirement.

Design of actuation mechanism

To produce a compact actuation mechanism, the length of the pin, length of the string, and size of the motor are very important in the final design. As seen in Figure 7(a), the movement of the tip of the pin from point 1 to point 2 by an actuation displacement $Δ X$ and angle $α$ will cause snap-through. The relationship between the length of the pin $P_{M}$ , the actuation angle of the pin $α$ , and the actuation displacement $Δ X$ is given by

P_{M}^{2} = \frac{{(Δ X)}^{2}}{2 (1 - \cos α)}

(7)

From equation (7), the length of the pin is proportional to the actuation displacement. In the actuation process, the actuation displacement $Δ X$ is assumed to be the same as the amount of reduction in the length of the string after twisting. The reduction in the length of the string is a function of the properties of the string, the rotational angle of the motor, and the amount of force to be overcome. For a string of initial length $L_{x}$ , radius $r$ , and stiffness $K_{x}$ , attached to a mass $F_{x}$ , as shown in Figure 7(b), the reduction in the length of the string due to the rotation of the motor through an angle $θ$ is given by

Δ X = (\frac{F_{x}}{K_{X}} + L_{x}) - \sqrt{{(\frac{F_{x}}{K_{X}} + L_{x})}^{2} - θ^{2} r^{2}}

(8)

Equation (8) is a well-established formulation in which the twisted string is treated as a homogeneous cylinder; the radius of the string remains constant during the twisting process (Palli et al., 2015; Suzuki et al., 2007). The trigger moment is related to the actuation displacement, TSA force, and length of pin by

M_{act} = F_{TSA} h = \frac{F_{TSA}}{2} \sqrt{4 {P_{M}}^{2} - Δ X^{2}}

(9)

TSA design

In this study, the measured actuation angle $α$ , when the pin is placed at the optimized location 25 mm from one fixed end of the beam, is 9.5°. The actuation angle depends on the location of the pin along the length of the pre-compressed beam. For a pin of length 30 mm, the actuation displacement $Δ X$ is 4.96 mm according to equation (7). Based on the analysis from the previous section, the critical trigger moment for the selected beam configuration is 24.3 Nmm. This implies that the TSA needs to produce a minimum force of 0.81 N according to equation (9).

The design of the TSA required a relatively small and lightweight high-speed low-torque motor. A Faulhaber 1024K003SR DC motor was selected, and its properties are presented in Table 2. A fishing line with a stiffness of 5000 Nm and a radius of 0.25 mm is attached to the DC motor through a connector. Since the minimum required TSA force is 0.81 N, a mass of 100 g (1 N) is hung from the other end of the string. The LDS (LK2101, Keyence) is used to measure the displacement of the mass as the motor rotates. The experimental setup is shown in Figure 8(a). A relatively low rotational angle of the motor is desirable during the actuation process; snap-through of the beam would be achieved within a very short time. An experiment was conducted with strings with lengths of 35, 45, and 55 mm to investigate the effect of the string length on the string contraction and the rotational angle of the motor. The motor was made to rotate slowly for 32 complete turns in each case. The displacement of the mass due to the reduction in the length of the string was recorded. As seen in Figure 8(b), the string length significantly affects the magnitude of the string contraction. For the same number of motor rotations, the 35 mm string was reduced in length by 13.2 mm, while the 55 mm string was reduced by 10.48 mm. An increase in the initial length of the string necessitated a larger rotational angle of the motor to obtain the same string contraction.

Table 2.

Properties of Faulhaber 1024K003SR DC motor.

Parameter	Value
No-load speed	12,200 min⁻¹
Nominal voltage	3 V
Stall torque	5.1 mN m
Rated torque	1.6 mN m
Mass	10.8 g

Performance evaluation of bistable actuator

A prototype of the TSBA was built to serve in systems where a unidirectional (static) or bidirectional (dynamic) application is required. The key actuator parameters considered in this study were the control bandwidth and stroke. A fast response time is also desirable for unidirectional application. Based on the analysis presented in the previous sections, a summary of the final TSBA design is shown in Figure 9.

Integration of actuation mechanism with bistable beam

The pre-curved thin plate for the snap-through with a fixed–fixed boundary condition was set up for the performance evaluation. Figure 10 shows the procedure of integrating the pre-curved thin plate with the actuation mechanism described in the previous section. The bistable thin plate structure receives the trigger moment at the trigger interface. The trigger interface consists of a pin attached to strings. The strings are attached to the shaft of the motor through a connector. The transition from the second equilibrium shape (curved down) to the first equilibrium shape (curved up) is achieved by moving the pin to the left side and vice versa. The point at which the pin is attached to the beam is tightened sufficiently not to disturb the snap-through behavior.

Static performance

The string length was varied to evaluate its effect on the static performance of the actuator for unidirectional application in systems. A unit step input voltage was supplied to the Faulhaber DC motor. The voltage signal was sent from a DS1103 DSP board dSPACE through an amplifier. For static or unidirectional actuation, a single motor with strings is enough to actuate the bistable beam. For each of the string lengths, the duration of the unit step voltage was 0.3 s. The experimental setup for evaluating the performance of the actuator is shown in Figure 11. The length of the pin was 30 mm.

The response time of the actuator with its corresponding output motor speed and torque was evaluated based on three string lengths: 40, 50, and 60 mm. As seen in Figure 12, an increase in the length of the string increased the response time of the actuator. The output motor speed and torque were also affected as seen in Table 3. The experimental results show that a faster response of the actuator was achieved by reducing the string length. The beam was also observed to vibrate after the snap-through occurred. This makes the proposed design a good candidate for energy harvesting (Cottone et al., 2013; Zhu and Zu, 2014). The behavior of the beam during the snap-through process was captured by a high-speed camera as shown in Figure 13.

Table 3.

Summary of the actuator response for various string lengths.

Stringlength (mm)	Responsetime (s)	Motor speedmaximumoutput (r/min)	Motor torquemaximumoutput(mN m)
40	0.1039	14,710	3.202
50	0.1099	15,380	3.179
60	0.1717	15,620	3.171

Dynamic performance

Two motors were placed in an antagonist configuration to achieve bidirectional actuation. In this configuration, the motors were made to twist and untwist the string by changing directions simultaneously; one motor twists the string while the other untwists to allow the pulling of the pin in one direction. Two cases were studied in which sinusoidal and square voltage was input to the motors. The input voltage in both cases was 4.5 V, and the lengths of the strings and pin were 40 and 30 mm, respectively.

The actuation mechanism generated slightly unequal endurance times $(Δ t_{2} > Δ t_{1})$ regardless of the input voltage shape as shown in Figure 14. This was attributed to the separate power supply source used for the two motors. In addition, one significant observation was that the output displacement of the actuator was 12.2 mm instead of 10 mm at a 1 Hz operating frequency because precision control of the motors to produce an exact pulling force was not considered in this study. Thus, the motor generated more pulling force on the pin than expected. The input voltage amplitude in this frequency range can be reduced to obtain a specific desired pulling force.

Figure 15 shows the performance of the TSBA for both signals at various operating frequencies. An increase in the operating frequency causes a decrease in output displacement d from both input signals. This may be attributed to the fact that the increase in motor speed at a constant voltage amplitude causes insufficient twisting of the strings. This reduction in the length of the string, which is proportional to the pulling force, is insufficient to cause snap-through. Therefore, the speed and torque of the motor must be considered when this method of actuation is used at a high operating frequency. The required output displacement is 10 mm, and the actuator meets that requirement when operated to maximum frequencies of 5 and 2 Hz with sine and square input signals, respectively. This also meets the actuator requirements of the reciprocating control fin described in our previous paper.

Conclusion

A new actuation mechanism has been proposed and developed to trigger the snap-through of a bistable beam. The TSA concept is employed, and the combination with the bistable beam is called a TSBA. The TSA with a pin induced bending moment on the bistable beam for snap-through to occur. The trigger moment is analytically formulated using the Euler–Bernoulli model and the energy of the system. The location of the minimum trigger moment was found at 18% of the pre-compressed length. A prototype of the actuator was built to verify the feasibility of using the twisted string concept to produce bidirectional actuation. At high operating frequency, the actuator displacement was observed to decrease due to the low motor torque at high speed. The operating frequency of the actuator was 5 Hz with a stroke of 10 mm when a sine input voltage was provided. The TSBA developed in this study is very simple to build and cost-effective so that it can be used in a wider range of applications than previous designs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the New & Renewable Energy Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), which was granted financial resources by the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20153030023880).

ORCID iD

Jae-Hung Han

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