Vibrational Transportation on a Platform Subjected to Sinusoidal Displacement Cycles Employing Dry Friction Control

Abstract

Currently used vibrational transportation methods are usually based on asymmetries of geometric, kinematic, wave, or time types. This paper investigates the vibrational transportation of objects on a platform that is subjected to sinusoidal displacement cycles, employing periodic dynamic dry friction control. This manner of dry friction control creates an asymmetry, which is necessary to move the object. The theoretical investigation on functional capabilities and transportation regimes was carried out using a developed parametric mathematical model, and the control parameters that determine the transportation characteristics such as velocity and direction were defined. To test the functional capabilities of the proposed method, an experimental setup was developed, and experiments were carried out. The results of the presented research indicate that the proposed method ensures smooth control of the transportation velocity in a wide range and allows it to change the direction of motion. Moreover, the proposed method offers other new functional capabilities, such as a capability to move individual objects on the same platform in opposite directions and at different velocities at the same time by imposing different friction control parameters on different regions of the platform or on different objects. In addition, objects can be subjected to translation and rotation at the same time by imposing different friction control parameters on different regions of the platform. The presented research extends the classical theory of vibrational transportation and has a practical value for industries that operate manufacturing systems performing tasks such as handling and transportation, positioning, feeding, sorting, aligning, or assembling.

Keywords: control; dry friction; handling and transportation; motion control; sinusoidal excitation; vibrations.

Figure 1

Lumped-element model of vibrational transportation: (1) object to be transported; (2) platform subjected to sinusoidal displacement cycles.

Figure 2

Duty cycle function ψ(τ) for dry friction control with respect to the period of sinusoidal displacement cycles.

Figure 3

Normalized average transportation velocity depending on: (a) φ when A = 1 mm, ω = 125.66 rad/s, $${\mu }_0$$ = 0.1, $${\mu }_0/{\overline{\mu }}_m$$ = 8; (b) λ when A = 1 mm, ω = 125.66 rad/s, $${\mu }_0$$ = 0.1, $${\mu }_0/{\overline{\mu }}_m$$ = 8; (c) $${\mu }_0/{\overline{\mu }}_m$$ when A = 1 mm, ω = 125.66 rad/s, φ = π/2; (d) $${\mu }_0$$ when A = 1 mm, ω = 125.66 rad/s, λ = π, φ = π/2.

Figure 4

Normalized average transportation velocity depending on: (a) φ and λ when A = 1 mm, ω = 125.66 rad/s, $${\mu }_0$$ = 0.1, $${\mu }_0/{\overline{\mu }}_m$$ = 8; (b) μ_c1 and $${\mu }_0/{\overline{\mu }}_m$$ when A = 1 mm, ω = 125.66 rad/s, λ = π, φ = π/2; (c) φ and $${\mu }_0/{\overline{\mu }}_m$$ when A = 1 mm, ω = 125.66 rad/s, λ = π, $${\mu }_0$$ = 0.1; (d) λ and $${\mu }_0/{\overline{\mu }}_m$$ when A = 1 mm, ω = 125.66 rad/s, $${\mu }_0$$ = 0.1 φ = π/2.

Figure 5

Average transportation velocity depending on: (a) A when $${\mu }_0$$ = 0.1, $${\mu }_0/{\overline{\mu }}_m$$ = 8, λ = π, φ = π/2; (b) ω when $${\mu }_0$$ = 0.1, $${\mu }_0/{\overline{\mu }}_m$$ = 8, λ = π, φ = π/2.

Figure 6

Three-dimensional representation of the average transportation velocity as a function of the amplitude and the angular frequency of the sinusoidal excitation in a range of low A and ω values.

Figure 7

Schematic diagram of the experimental setup used to measure the average transportation velocity: (1) platform; (2) part to be transported; (3) piezoelectric actuator; (4) signal generator for photodiode sensors (PD1 and PD2); (5) digital oscilloscope; (6) arbitrary waveform generator; (7) high-frequency vibration amplifier; (8) power amplifier for the sinusoidal excitation; (9) electrodynamic shaker.

Figure 8

Oscillograms of the sinusoidal signal and the signal for the piezoelectric actuator excitation as ω = 125.66 rad/s, λ = 3π/8, and φ = 5π/8.

Figure 9

Experimental and modeled dependences of the average transportation velocity depending on: (a) φ when A = 3.8 mm, ω = 125.66 rad/s, $${\mu }_0$$ = 0.24, $${\mu }_0/{\overline{\mu }}_m$$ = 4; (b) λ when ω = 125.66 rad/s, $${\mu }_0$$ = 0.24, $${\mu }_0/{\overline{\mu }}_m$$ = 4; (c) A when $${\mu }_0$$ = 0.24, $${\mu }_0/{\overline{\mu }}_m$$ = 4, λ = 8π/9, φ = 7π/15; (d) ω when $${\mu }_0$$ = 0.24, $${\mu }_0/{\overline{\mu }}_m$$ = 4, λ = 8π/9, φ = 7π/15.