A buckling sheet ring oscillator for multimodal locomotion without electronics
or VSBSA 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). VSBSA could be tuned by varying design parameters depending on scaling VSBSA = 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, Eand the moment of inertia of the area, I (which, in turn, is proportional to the cube of the thickness of the sheet, you3); VSBSA increases with larger bladder diameters but decreases with stiffer sheets that resist bladder deformation and expansion. The effective pressure of the system, PEFF = (PSUPP RSHOOT + PAT M RBSA)/(RSHOOT + RBSA), senses the contributions of atmospheric and supply pressures, and the effective pneumatic resistance of the circuit, REFF = (RTUBE RBSA + RSHOOT RBSA + RSHOOT RTUBE)/(RSHOOT + RBSA), takes into account the contributions of the three relevant pneumatic resistances: (i) the pull-up resistance (RSHOOT), (ii) the inter-device pneumatic resistance (RTUBE), and (iii) the pneumatic resistance of the flow control tubing on the BSA (RBSA). 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 (youyou) of a buckling sheet inverter can be expressed as Eq. 2 (derivation in additional materials and fig. S7)
= RTUBE + RSHOOT is a simplified form of REFF corresponding to the unfolded RC circuit, in which RBSA approaches infinity due to kinking of the flow control tube. The resulting equation for the period of total oscillation (youPERIOD) of a ring oscillator containing not the buckling sheet undulators is therefore given by Eq. 3, where youB is the buckling time and youyou is the unwinding time
Additionally, we derived an analytical expression for the pressure amplitude (A) during the oscillation as A=PEFF − PHAPPEN (derivation in supplementary materials).