Based on the design requirements of the motion
curve of blow-station, a Bezier curve is used as the motion curve
of open/close blow-station cam. A multi-objective function for
minimize angular acceleration and pressure angle with
continuous angular jerk constraint is defined. By using
Augmented Lagrange Multiplier (ALM) method and the theory
of Envelope, the profile of the open/close blow-station cam with
optimal motion curve is obtained. The result of this work
shows the productivity of this machine is increased from
8,000bottles/hrs to 11,000bottles/hrs after using this optimum
designed blow-station cam. It is 37.5% increscent to the
available machine provided by a local company.
Keywords: Rotary type blow-molding machine,
blow-station cam, optimal design
The advantages of PET bottles are low cost, light
weight, high toughness, stable structure, easy to
transparent and burnish, and high security [1]. Figure 1
shows the skeleton of a rotary type PET bottle
blow-molding machine. The preform is heated up in
the heating zone (VIII), in-feed preform wheel (I)
transmits the perform, and the catch arm of in-feed
preform cam-linkage mechanism (II) picks the preform
from the in-feed preform wheel then send it into the
blow-station (III) to form a PET bottle. The catch arm
of outlet bottle cam-linkage mechanism (IV) picks the
PET bottle take it out of the blow-station and transmits to
the outlet bottle wheel (V) for packing.
In the blow-station device, the blow-station linkage
mechanism is driven by the open/close blow-station cam
mechanism as shown in Figure 2. SI(OIXIYI) is the
fixed coordinate system. Pivots A0 and B0 are incident
to the blow wheel and the blow-station linkage
mechanism, respectively. Coordinate system OMXMYM
is a moving coordinate system attached to the
blow-station linkage mechanism. Angle b is between
start line OIC1 and end line OIC2 of the open/close
blow-station cam. b1 is the open phase angle and b2 is
the close phase angle of the blow-station. RP is a
position vector from OI to A0, RB is a position vector
from OI to B0, qp and qB are the vector angles of the
vectors RP and RB, respectively. qopen is the angle that
blow-station can open. The blow-station is attached to
the blow-station linkage mechanism and the blow-station
linkage mechanism is attached to the blow wheel. The
blow wheel rotates with constant angular velocity
wB counterclockwise. When the blow wheel rotates, the
roller follower contacts the cam and the crank of the
linkage mechanism will be pushed to rotate. The crank
drives the blow-station linkage mechanism to generate
open and close actions. When angle qB is at -d, the
blow-station is about to open. The blow-station is
closed when qB is at (-d+b1+b2). The kinematic
characteristics of the blow-station of a rotary type
blow-molding machine are dominated by its open/close
blow-station cam mechanism.
In the target commercial available machine, the
motion curves, are discontinuous at the interface
position. In order to improve this disadvantage, this
paper proposes an optimal design approach for designing
the profile of the open/close blow-station cam with
continuous angular jerk at the interface position. In
1988, Tsay and Huey proposed an application of
spline functions to the general synthesis of cam motion
programs. In the same year, MacCarthy proposed a
very general spline construction in kinematic design. In
1994, Ting et al. applied the Bezier technique for
motion program synthesis of cam-follower mechanisms.
It shows that any polynomial curve can be obtained
through the Bezier technique by selecting control points.
In 2005, Qiu et al. proposed a new method for
synthesizing the motion curves of cams based on
kinematic or dynamic optimization. In this paper, an
optimal design of the Bezier motion curve with
continuous angular jerk constraint of open/close
blow-station cam is proposed by using ALM optimal
approach. Based on Envelope theory, the profile of the
open/close blow-station cam is obtained. According to
the vector loop method and the definition of pressure
angle, the motion of the follower and the pressure angle
of the cam are analyzed.
II. Design requirements of the blow-station
In Figure 3, PL is the most right side point of the left
mold, PR is the most left side point of the right mold, n1
is a vector perpendicular to the line bf , n2 is a vector
characteristics of the blow-station with different motion
curve. the angular velocity curve is the smoothest curve at the interface position
between the open phase and the close phase. And, its
peak value is the lowest among motion curves. A
discontinuous angular jerk Jg in MS, MT, and MCV
motion curves will be generated at the interface position, respectively. And, this will
cause the blow-station to vibrate and noise. However,
the variation of angular acceleration ag at the interface
position is eliminated and its peak value is decreased in
the optimum Bezier motion curve. Therefore, the
angular jerk Jg in the optimum Bezier motion curve is
continuous. Its peak value is also decreased.
According to the results shown in Tables 1 to 3, the r.m.s.
value of angular acceleration of the blow-station in
Bezier curve is decreased by 6.82 %, 13.67 %, and 8.87
% relative to MS, MT, and MCV, respectively. And, the
fluctuation value of angular acceleration is decreased by
11.40 % and 38.87 % relative to MS and MCV,
respectively; but it is increased by 0.19 % relative to MT
curve. The r.m.s. value of angular jerk is decreased by
25.50 %, 31.21 %, and 63.70 % relative to MS, MT, and
MCV curves, respectively. The fluctuation value of
angular jerk is decreased by 43.06 % and 61.71 %
relative to MS and MCV curves, respectively; but it is
increased by 13.2 % relative to MT curve.
Figure 7 is the profiles of open/close blow-station
cam with different motion curves. Figure 8 is the
pressure angles in different motion curves. The peak
value of pressure angle in Bezier curve is obviously
smaller than the other three curves. And the r.m.s.
value of pressure angle with Bezier curve is decreased by
10.03 %, 11.72 %, and 3.82 % relative to MS, MT, and
MCV curves, respectively. And the fluctuation value is
improved by 29.57 %, 30.58 %, and 30.36 % relative to
MS, MT, and MCV curves, respectively.
Angular acceleration characteristics of the blow-station and the pressure angle
of the cam are improved. And, by using the designed
open/close blow-station cam mechanism, the vibration
curve of blow-station, a Bezier curve is used as the motion curve
of open/close blow-station cam. A multi-objective function for
minimize angular acceleration and pressure angle with
continuous angular jerk constraint is defined. By using
Augmented Lagrange Multiplier (ALM) method and the theory
of Envelope, the profile of the open/close blow-station cam with
optimal motion curve is obtained. The result of this work
shows the productivity of this machine is increased from
8,000bottles/hrs to 11,000bottles/hrs after using this optimum
designed blow-station cam. It is 37.5% increscent to the
available machine provided by a local company.
Keywords: Rotary type blow-molding machine,
blow-station cam, optimal design
The advantages of PET bottles are low cost, light
weight, high toughness, stable structure, easy to
transparent and burnish, and high security [1]. Figure 1
shows the skeleton of a rotary type PET bottle
blow-molding machine. The preform is heated up in
the heating zone (VIII), in-feed preform wheel (I)
transmits the perform, and the catch arm of in-feed
preform cam-linkage mechanism (II) picks the preform
from the in-feed preform wheel then send it into the
blow-station (III) to form a PET bottle. The catch arm
of outlet bottle cam-linkage mechanism (IV) picks the
PET bottle take it out of the blow-station and transmits to
the outlet bottle wheel (V) for packing.
In the blow-station device, the blow-station linkage
mechanism is driven by the open/close blow-station cam
mechanism as shown in Figure 2. SI(OIXIYI) is the
fixed coordinate system. Pivots A0 and B0 are incident
to the blow wheel and the blow-station linkage
mechanism, respectively. Coordinate system OMXMYM
is a moving coordinate system attached to the
blow-station linkage mechanism. Angle b is between
start line OIC1 and end line OIC2 of the open/close
blow-station cam. b1 is the open phase angle and b2 is
the close phase angle of the blow-station. RP is a
position vector from OI to A0, RB is a position vector
from OI to B0, qp and qB are the vector angles of the
vectors RP and RB, respectively. qopen is the angle that
blow-station can open. The blow-station is attached to
the blow-station linkage mechanism and the blow-station
linkage mechanism is attached to the blow wheel. The
blow wheel rotates with constant angular velocity
wB counterclockwise. When the blow wheel rotates, the
roller follower contacts the cam and the crank of the
linkage mechanism will be pushed to rotate. The crank
drives the blow-station linkage mechanism to generate
open and close actions. When angle qB is at -d, the
blow-station is about to open. The blow-station is
closed when qB is at (-d+b1+b2). The kinematic
characteristics of the blow-station of a rotary type
blow-molding machine are dominated by its open/close
blow-station cam mechanism.
In the target commercial available machine, the
motion curves, are discontinuous at the interface
position. In order to improve this disadvantage, this
paper proposes an optimal design approach for designing
the profile of the open/close blow-station cam with
continuous angular jerk at the interface position. In
1988, Tsay and Huey proposed an application of
spline functions to the general synthesis of cam motion
programs. In the same year, MacCarthy proposed a
very general spline construction in kinematic design. In
1994, Ting et al. applied the Bezier technique for
motion program synthesis of cam-follower mechanisms.
It shows that any polynomial curve can be obtained
through the Bezier technique by selecting control points.
In 2005, Qiu et al. proposed a new method for
synthesizing the motion curves of cams based on
kinematic or dynamic optimization. In this paper, an
optimal design of the Bezier motion curve with
continuous angular jerk constraint of open/close
blow-station cam is proposed by using ALM optimal
approach. Based on Envelope theory, the profile of the
open/close blow-station cam is obtained. According to
the vector loop method and the definition of pressure
angle, the motion of the follower and the pressure angle
of the cam are analyzed.
II. Design requirements of the blow-station
In Figure 3, PL is the most right side point of the left
mold, PR is the most left side point of the right mold, n1
is a vector perpendicular to the line bf , n2 is a vector
characteristics of the blow-station with different motion
curve. the angular velocity curve is the smoothest curve at the interface position
between the open phase and the close phase. And, its
peak value is the lowest among motion curves. A
discontinuous angular jerk Jg in MS, MT, and MCV
motion curves will be generated at the interface position, respectively. And, this will
cause the blow-station to vibrate and noise. However,
the variation of angular acceleration ag at the interface
position is eliminated and its peak value is decreased in
the optimum Bezier motion curve. Therefore, the
angular jerk Jg in the optimum Bezier motion curve is
continuous. Its peak value is also decreased.
According to the results shown in Tables 1 to 3, the r.m.s.
value of angular acceleration of the blow-station in
Bezier curve is decreased by 6.82 %, 13.67 %, and 8.87
% relative to MS, MT, and MCV, respectively. And, the
fluctuation value of angular acceleration is decreased by
11.40 % and 38.87 % relative to MS and MCV,
respectively; but it is increased by 0.19 % relative to MT
curve. The r.m.s. value of angular jerk is decreased by
25.50 %, 31.21 %, and 63.70 % relative to MS, MT, and
MCV curves, respectively. The fluctuation value of
angular jerk is decreased by 43.06 % and 61.71 %
relative to MS and MCV curves, respectively; but it is
increased by 13.2 % relative to MT curve.
Figure 7 is the profiles of open/close blow-station
cam with different motion curves. Figure 8 is the
pressure angles in different motion curves. The peak
value of pressure angle in Bezier curve is obviously
smaller than the other three curves. And the r.m.s.
value of pressure angle with Bezier curve is decreased by
10.03 %, 11.72 %, and 3.82 % relative to MS, MT, and
MCV curves, respectively. And the fluctuation value is
improved by 29.57 %, 30.58 %, and 30.36 % relative to
MS, MT, and MCV curves, respectively.
Angular acceleration characteristics of the blow-station and the pressure angle
of the cam are improved. And, by using the designed
open/close blow-station cam mechanism, the vibration
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