# A buckling sheet ring oscillator for multimodal locomotion without electronics

*P*

_{SUPP}), the pneumatic resistance of the pulldown resistor (

*R*

_{SHOOT}), and the inter-device pneumatic resistance (

*R*

_{TUBE}, associated with tubes connecting the buckling sheet undulators). The two alternating states of the ring oscillator, i.e. buckling and unfolding of the connected buckling foil inverters, can be understood through an analog electrical circuit (Fig. 2A), where each event of buckling or unfolding can be modeled as a resistor-capacitor (RC) with each capacitor-like bladder charging or discharging, respectively. To derive expressions for oscillation period and amplitude as a function of

*P*

_{SUPP},

*R*

_{SHOOT}and

*R*

_{TUBE}, we modeled the airflow between two adjacent inverters in the same actuation state. The buckling time (

*you*

_{B}) for an individual buckling sheet undulator can be expressed as Eq. 1 (derivation in additional materials and fig. S7)

$${\mathrm{you}}_{\mathrm{B}}={R}_{\text{EFF}}\times {\mathrm{VS}}_{\text{BSA}}\times \text{in}\left[\frac{({P}_{\text{EFF}}-{P}_{\text{UNFOLD}})}{({P}_{\text{BUCK}}-{P}_{\text{EFF}})}\right]$$

(1)

or *VS*_{BSA} represents the intrinsic pneumatic capacity of an individual BSA, defined as the rate of change of the mass of fluid inside the BSA relative to its internal pressure (*1*). *VS*_{BSA} could be tuned by varying design parameters depending on scaling *VS*_{BSA} = *F*(*D*, *IE*), or *D* is the diameter of the pneumatic bladder and the product *IE* is the bending stiffness of the sheet itself, consisting of the modulus of elasticity of the sheet, *E*and the moment of inertia of the area, *I* (which, in turn, is proportional to the cube of the thickness of the sheet, *you*^{3}); *VS*_{BSA} increases with larger bladder diameters but decreases with stiffer sheets that resist bladder deformation and expansion. The effective pressure of the system, *P*_{EFF} = (*P*_{SUPP} *R*_{SHOOT} + *P*_{AT M} *R*_{BSA})/(*R*_{SHOOT} + *R*_{BSA}), senses the contributions of atmospheric and supply pressures, and the effective pneumatic resistance of the circuit, *R*_{EFF} = (*R*_{TUBE} *R*_{BSA} + *R*_{SHOOT} *R*_{BSA} + *R*_{SHOOT} *R*_{TUBE})/(*R*_{SHOOT} + *R*_{BSA}), takes into account the contributions of the three relevant pneumatic resistances: (i) the pull-up resistance (*R*_{SHOOT}), (ii) the inter-device pneumatic resistance (*R*_{TUBE}), and (iii) the pneumatic resistance of the flow control tubing on the BSA (*R*_{BSA}). Pneumatic resistances can be tuned by changing the length and inner diameter of the tube to suit the fluid mechanics of an internal flow (Additional Materials**)**. Similarly, the running time (*you*_{you}) of a buckling sheet inverter can be expressed as Eq. 2 (derivation in additional materials and fig. S7)

$${\mathrm{you}}_{\mathrm{you}}={R}_{\text{EFF}}^{*}\times {\mathrm{VS}}_{\text{BSA}}\times \text{in}\left[\frac{({P}_{\text{EFF}}-{P}_{\text{ATM}})}{({P}_{\text{UNFOLD}}-{P}_{\text{ATM}})}\right]$$

(2)

or

${R}_{\text{EFF}}^{*}$ = *R*_{TUBE} + *R*_{SHOOT} is a simplified form of *R*_{EFF} corresponding to the unfolded RC circuit, in which *R*_{BSA} approaches infinity due to kinking of the flow control tube. The resulting equation for the period of total oscillation (*you*_{PERIOD}) of a ring oscillator containing *not* the buckling sheet undulators is therefore given by Eq. 3, where *you*_{B} is the buckling time and *you*_{you} is the unwinding time

Additionally, we derived an analytical expression for the pressure amplitude (*A*) during the oscillation as *A=P*_{EFF} − *P*_{HAPPEN} (derivation in supplementary materials).

*P*

_{SUPP}went from 10 kPa (

*P*

_{MALE}) at 22 kPa, the period of oscillation (

*you*

_{PERIOD}) increased from 0.5 to 0.8 s in case

*R*

_{TUBE}is negligible (Fig. 2B). This behavior is explained by Eq. 2 and eq. S5, where

*you*

_{you}increased logarithmically with

*P*

_{SUPP}manufacturing

*P*

_{SUPP}the main factor of increase

*you*

_{PERIOD}. Conversely, when

*R*

_{TUBE}is large (comparable to

*R*

_{SHOOT}), the period of oscillation,

*you*

_{PERIOD}decreases as the supply pressure,

*P*

_{SUPP}increases (when

*P*

_{SUPP}is small, that is

*32*), have not been observed or described in previous work on pneumatic devices. The oscillating pressure amplitude at each exit has been increased from 3 to 15 kPa

*P*

_{SUPP}increased in case

*R*

_{TUBE}is negligible. Given that

*P*

_{HAPPEN}is the same regardless

*P*

_{SUPP}the amplitude

*A*is linearly proportional to

*P*

_{SUPP}(as dictated by Eq. S5 and Eq. S12).

*R*

_{SHOOT}) are also critical components in controlling oscillation behavior (Fig. 2C). Pneumatic resistance can be represented by a length of tubing (

*L*) with a known inside diameter. The period of total oscillation

*you*

_{PERIOD}was linearly proportional to

*R*

_{SHOOT}because the airflow through the pulldown resistor to the atmosphere decreases as

*L*increases. This enhances the “efficient” effect

*P*

_{SUPP}from the previous inverter to the next and results in greater

*you*

_{you}. Like

*R*

_{SHOOT}increases above a threshold value of 0.75 m, the amplitude

*A*begins to saturate. Equations S5 and S12 show that

*A*will reach the value

*P*

_{SUPP}−

*P*

_{HAPPEN}if the length of the tube is infinite. The slight discrepancies between the model predictions and the experimental results for

*R*

_{SHOOT}vs

*you*

_{PERIOD}and

*R*

_{SHOOT}vs

*A*are reasonable for modeling gas flows in small diameter tubes, which often deviate from predictions by more than 10% (

*33*). Error can also result from comparing our analytical model from first principles with a non-ideal real-world system. We discovered that we could gain independent control over

*you*

_{PERIOD}with the same

*A*changing

*R*

_{TUBE}as another design factor for oscillation (Fig. 2D). The model and experimental data confirm that a BRO can access a wide range of oscillation periods, from less than 1 s to several seconds (theoretically from 0.1 s to an infinite period).