Abstract
We consider an oscillation-based actuator in which directed motion is generated by a spherical contact subjected to superimposed oscillations in vertical and horizontal directions. We consider the full dynamical problem with account of contact forces arising due to the superimposed periodic loading. We find that the system can have two regimes: either motion in one direction or stick–slip motion, when velocity periodically changes its sign. The actuator can also produce directed motion against an external horizontal force (modeling dragging a “cargo”).
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Abbreviations
- \( z = f(r) \) :
-
Function of 3D shape of the profile
- \( r \) :
-
Radius-vector
- \( g(x) \) :
-
Function of 1D profile
- \( x,\,x_{i} \) :
-
Coordinates in 1D dimension
- \( R \) :
-
Radius of the indenter
- \( k_{z} ,\,k_{x} \) :
-
Stiffnesses of the springs in normal and tangential directions
- \( E^{*} \) :
-
Effective elastic modulus
- \( G^{*} \) :
-
Effective shear modulus
- \( \Delta x \) :
-
Step of discretization
- \( E,\,E_{1} ,\,E_{2} \) :
-
Elastic moduli
- \( \nu ,\,\nu_{1} ,\,\nu_{2} \) :
-
Poisson ratios
- \( G,\,G_{1} ,\,G_{2} \) :
-
Shear moduli
- \( u_{z} \) :
-
Normal coordinate of the spring
- \( u_{x} \) :
-
Tangential coordinate of the spring
- \( d \) :
-
Indentation depth
- \( a \) :
-
Radius of the contact
- \( F_{z} \) :
-
Normal force
- \( F_{x} \) :
-
Tangential force
- \( \mu ,\,\mu_{1} \) :
-
Friction coefficients
- \( M \) :
-
Mass of the slider
- \( m \) :
-
Mass of the cargo
- \( V \) :
-
Velocity of the slider
- \( X \) :
-
Tangential coordinate of the slider
- \( F_{base} \) :
-
Friction force between the base and the slider
- \( \Delta \tilde{x} \) :
-
Relative displacement of the indenter against the slider
- \( t \) :
-
Time
- \( \Delta t \) :
-
Step of time discretization
- \( t_{i} \) :
-
Moment of time
- \( t_{1} ,\,t_{2} \) :
-
Time intervals
- \( N \) :
-
Integer number
- \( A \) :
-
Coordinate of normal (tangential) deflection of the indenter
- \( A_{\rm{max} } \) :
-
Amplitude
- \( x,\,z \) :
-
Tangential and normal coordinates of the indenter (earlier coordinates in 1D dimension)
- \( m_{c1} ,\,m_{c2} \) :
-
Critical masses of the cargo
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Acknowledgements
V.N. Borysiuk has received research Grants from Ministry of Education and Science of Ukraine (Research Project No. 0117U003923).
Funding
This study was funded in parts by Ministry of Education and Science of Ukraine (Research Project No. 0117U003923).
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Lyashenko, I.A., Borysiuk, V.N. & Popov, V.L. Dynamical model of asymmetric actuator of directional motion. Meccanica 54, 1681–1687 (2019). https://doi.org/10.1007/s11012-019-01045-9
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DOI: https://doi.org/10.1007/s11012-019-01045-9