When a solid body begins to slide on a vibrating surface, the static friction appears to be reduced. In the present report, the effect of vibration in an arbitrary direction is discussed generally, though in the previous report the special case of the vibration in each direction of three axes of rectangular coordinate has been dealt with.
The analytical results show that the vibration in any direction has the effect to reduce the static friction in appearance and the relation between the apparent coefficient of the static friction effected by vibration in the direction of (ψ, γ, ) that is μ
s', and the ratio of acceleration of vibration to that gravity Λ, is expressed as follows:
Λ=√1+tan
2γ/1+μ
s'
2·1/μ
s2-tan
2γ{√
D2-√
D2-(μ
s2-tan
2γ)(μ
s2-μ'
s2)},
where
D= μ'
s.tanγ. cosψ + μ
s2, μ
s: the true coefficient of static friction, γ: the angle of vibratory direction made with z-axis, ψ: the angle with x-axis projected on the xy-plane.
When
D> 0, the phase of vibration is π/2, at beginning to slide, while when
D<0, it is 3 π/2. The most effective direction of vibration to reduce the apparent static friction is as follows: γ= cot
-1μ
s= π/ 2-θ
0 and ψ =0, where θ
0 is the angle of friction, then Λ =sineθ
0.
These theoretical results have been also confirmed in good agreement with that of the experimental studies.
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