This paper investigates the vibratory bowl feeder for automatic assembly, presents a geometric model of the feeder, and develops force analysis, leading to dynamical modeling of the vibratory feeder. Based on the leaf-spring modeling of the three legs of the symmetrically arranged bowl of the feeder, and equating the vibratory feeder to a three-legged parallel mechanism, the paper reveals the geometric property of the feeder. The effects of the leaf-spring legs are transformed to forces and moments acting on the base and bowl of the feeder. Resultant forces are obtained based upon the coordinate transformation, and the moment analysis is produced based upon the orthogonality of the orientation matrix. This reveals the characteristics of the feeder, that the resultant force is along the $z$-axis and the resultant moment is about the $z$ direction and further generates the closed-form motion equation. The analysis presents a dynamic model that integrates the angular displacement of the bowl with the displacement of the leaf-spring legs. Both Newtonian and Lagrangian approaches are used to verify the model, and an industrial case-based simulation is used to demonstrate the results.

1.
,
G. M. L.
, and
Masour
,
W. M.
, 1971, “
Simulation of Vibratory Feeders, Proc. of the Symp. on Computer-Aided Engineering
,”
Univ. Waterloo
, pp.
215
249
.
2.
Mansour
,
W. M.
, 1972, “
Analog and Digital Analysis and Synthesis of Oscillatory Tracks
,”
ASME J. Eng. Ind.
0022-0817,
94
(
2
), pp.
488
494
.
3.
Okabe
,
S.
,
Yokoyama
,
Y.
, and
Boothroyd
,
G.
, 1988, “
Analysis of Vibratory Feeding Where the Track has Directional Characteristics
,”
, Z
3
(
4
), pp.
73
86
.
4.
Hongler
,
M.-O.
,
Cartier
,
P.
, and
Flury
,
P.
, 1989, “
Numerical Study of a Model of a Vibro-Transporter
,”
Phys. Lett. A
0375-9601,
135
(
2
), pp.
106
112
.
5.
Maul
,
G. P.
, and
Thomas
,
M. B.
, 1997, “
A Systems Model and Simulation of the Vibratory Bowl Feeder
,”
ASME J. Manuf. Systems
,
16
(
5
), pp.
309
314
.
6.
Doi
,
T.
,
Yoshida
,
K.
,
Tamai
,
Y.
,
Kono
,
K.
,
Naito
,
K.
, and
Ono
,
T.
, 2001, “
Feedback Control for Electromagnetic Vibration Feeder (Applications of Two Degrees-of-Freedom Proportional Plus Integral Plus Derivative Controller With Nonlinear Element)
,”
JSME Int. J., Ser. C
1340-8062,
44
(
1
), pp.
45
52
.
7.
Mahdavian
,
S. A.
,
Mahdavian
,
S. M.
, and
Brooy
,
R. La
, 1992, “
Effects of Driving Signal Forms and Frequency in the Performance of a Vibratory Bowl Feeder
,”
Proc. of International Conference on Manufacturing Automation
,
Univ. Hong Kong.
, pp.
836
841
.
8.
Du
,
W.
, and
Dickerson
,
S. L.
, 1999, “
Modelling and Control of a Novel Vibratory Feeder
,”
1999 IEEE/ASME Int. Conference on Advanced Intelligent Mechatronics
, pp.
496
501
.
9.
Dai
,
J. S.
, and
Kerr
,
D. R.
, 2000, “
A Six-Component Contact Force Measurement Device Based on the Stewart Platform
,”
J. Mech. Eng. Sci.
0022-2542,
214
(
C5
), pp.
687
697
.
10.
Dai
,
J. S.
,
Sodhi
,
C.
, and
Kerr
,
D. R.
, 1994, “
Design and Analysis of a New Six Component Force Transducer for Robotic Grasping
,”
Proc. of Second Biennial European Joint Conference on Engineering Systems Design and Analysis
, London
64
(
8-3
), pp.
809
817
.
11.
Kozak
,
K.
,
Ebert-Uphoff
,
I.
, and
Singhose
,
W.
, 2004, “
Locally Linearized Dynamic Analysis of Parallel Manipulators and Application of Input Shaping to Reduce Vibrations
,”
J. Mech. Des.
1050-0472,
126
, pp.
156
168
.
12.
Sugar
,
T. G.
, and
Kumar
,
V.
, 2002, “
Design and Control of a Compliant Parallel Manipulator
,”
J. Mech. Des.
1050-0472,
124
, pp.
676
683
.
13.
Matsuoka
,
Y.
,
Kawai
,
K.
, and
Sato
,
R.
, 2003, “
Vibration Simulation Model of Passenger-Wheelchair System in Wheelchair-Accessible Vehicle
,”
J. Mech. Des.
1050-0472,
125
, pp.
779
785
.
14.
Fung
,
E. H. K.
,
Zou
,
J. Q.
, and
Lee
,
H. W. J.
, 2004, “
Lagrangian Formulation of Rotating Beam with Active Constrained Layer Damping in Time Domain Analysis
,”
J. Mech. Des.
1050-0472,
126
, pp.
351
358
.
15.
Kang
,
J. S.
,
Bae
,
S.
,
Lee
,
J. M.
, and
Tak
,
T. O.
, 2003, “
Force Equilibrium Approach for Linearization of Constrained Mechanical System Dynamics
,”
J. Mech. Des.
1050-0472,
125
, pp.
143
149
.
16.
Hunt
,
J. B.
, 1979,
Dynamic Vibration Absorbers
,
Mechanical Engineering Publications Ltd
, London.
17.
Rao
,
S. S.
, 1995,
Mechanical Vibrations
,