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| United States Patent | 5489250 |
| Link to this page | http://www.wikipatents.com/5489250.html |
| Inventor(s) | Densmore; Bruce D. (Seattle, WA);
Hass; Lester A. (Duvall, WA);
Rotunda; John T. (Renton, WA);
Fissel; Gregory W. (Seattle, WA) |
| Abstract | A high speed deceleration device and method is disclosed that operates on a
D.C. speed-change motor used to change the transmission configurations on
the transmission of a treadmill powered by a constant speed drive motor.
In ordinary use, normal transmission acceleration and deceleration rates
are used to speed up or slow down the treadmill belt. A deceleration
control signal causes a high voltage to be provided to the D.C.
speed-change motor. The high voltage causes the D.C. speed-change motor to
rapidly change the transmission configurations on the treadmill which
causes a rapid deceleration of the treadmill belt. The deceleration
control signal is produced in response to a user direction or in response
to the condition of the treadmill. Once the belt has been slowed below a
threshold speed, the machine shuts itself off. During the deceleration
process, the drive motor of the treadmill continues to provide power to
the treadmill belt through the transmission so that the transmission
continues to move during the deceleration process. |
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Title Information  |
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Drawing from US Patent 5489250 |
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Treadmill deceleration system and method |
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| Publication Date |
February 6, 1996 |
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| Filing Date |
February 25, 1994 |
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| Parent Case |
This application is a division of application Ser. No. 790,750, filed Nov.
8, 1991, now U.S. Pat. No. 5,290,205, issued Mar. 1, 1994. |
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Title Information  |
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References  |
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U.S. References |
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| | Reference | Relevancy | Comments | Reference | Relevancy | Comments | 3737163
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Nov,1991 |      Your vote accepted [0 after 0 votes] | | 4964841 Takayama 474/11 Oct,1990 |      Your vote accepted [0 after 0 votes] | | 4944713 Salerno 474/11 Jul,1990 |      Your vote accepted [0 after 0 votes] | | 4928958 Sabin 482/80 May,1990 |      Your vote accepted [0 after 0 votes] | | 4842266 Sweeney, Sr. 482/54 Jun,1989 |      Your vote accepted [0 after 0 votes] | | 4792134 Chen 482/54 Dec,1988 |      Your vote accepted [0 after 0 votes] | | 4749181 Pittaway 482/7 Jun,1988 |      Your vote accepted [0 after 0 votes] | | 4635927 Shu 482/7 Jan,1987 |      Your vote accepted [0 after 0 votes] | | 4628910 Krukowski 601/26 Dec,1986 |      Your vote accepted [0 after 0 votes] | | 4525658 Yanagida 388/815 Jun,1985 |      Your vote accepted [0 after 0 votes] | | 4364556 Otte 482/4 Dec,1982 |      Your vote accepted [0 after 0 votes] | | 4019106 Van Doren 318/252 Apr,1977 |      Your vote accepted [0 after 0 votes] | | 3984744 Moody 318/442 Oct,1976 |      Your vote accepted [0 after 0 votes] | | 3954018 O'Berto 474/11 May,1976 |      Your vote accepted [0 after 0 votes] | | 3689066 Oscar M Hagen, 116 S. 23rd St. (La Crosse, WI 54601) 482/54 Sep,1972 |      Your vote accepted [0 after 0 votes] | | |
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| Market Size |
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Market Review  |
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Technical Review  |
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Claims  |
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We claim:
1. A method for rapidly decelerating a treadmill belt in a treadmill having
a constant speed drive motor with a rotating member, a treadmill belt, a
transmission for connecting the rotating member to the treadmill belt
causing the treadmill belt to move in response to rotation of the rotating
member, a D.C. speed change motor for varying the internal configurations
of the transmission to vary the speed of the treadmill belt when the
rotating member is rotating at a generally constant speed, the D.C. speed
change motor, in normal operation, responding to voltage applied within a
specified range of voltages, the method comprising the steps of:
producing a rapid deceleration control signal indicating that the treadmill
belt should be decelerated;
producing a D.C. deceleration voltage of a voltage larger than the
specified range of voltages normally applied to the D.C. speed change
motor;
applying the D.C. deceleration voltage to the D.C. speed change motor in
response to said rapid deceleration control signal, thereby causing the
D.C. speed change motor to rapidly vary the internal configurations of the
transmission;
whereby, the speed of the treadmill belt is rapidly decelerated.
2. The method of claim 1 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining that the power to the
treadmill has been interrupted and then restored.
3. The method of claim 1 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining whether the treadmill user
has indicated that the user would like to immediately stop the treadmill
belt.
4. The method of claim 1 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of sensing the occurrence of a drive motor
overload condition.
5. The method of claim 1 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the steps of:
timing the time that a voltage larger than the specified range of voltages
normally applied to the D,C, speed change motor has been applied to the
D.C. speed change motor;
determining that the time that a voltage larger than the specified range of
voltages normally applied to the D,C, speed change motor has been applied
to the D,C, speed change motor exceeds a preset time period.
6. The method of claim 1 further comprising the step of:
sensing the speed of the treadmill belt;
turning off the drive motor once the sensed speed of the treadmill belt
drops below a predetermined speed.
7. The method of claim 6 wherein said step of producing a rapid
deceleration control signal includes the step of determining that the
speed of the treadmill belt exceeds a predetermined speed.
8. The method of claim 6 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining that the power to the
treadmill has been interrupted and then restored.
9. The method of claim 6 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining whether the treadmill user
has indicated that the user would like to immediately stop the treadmill
belt.
10. The method of claim 6 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of sensing the occurrence of a drive motor
overload condition.
11. The method of claim 6 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the steps of:
timing the time that a voltage larger than the specified range of voltages
normally applied to the D.C. speed change motor has been applied to the
D.C. speed change motor;
determining that the time that a voltage larger than the specified range of
voltages normally applied to the D.C. speed change motor has been applied
to the D.C. speed change motor exceeds a preset time period.
12. A method for causing the rapid deceleration of a treadmill belt in a
treadmill having an A.C. drive motor with a rotating member, a treadmill
belt, a transmission for connecting the rotating member to the treadmill
belt so that rotation of the rotating member causes the treadmill belt to
move and a D.C. speed change motor for varying the internal configurations
of the transmission to vary the speed of the treadmill belt when the
rotating member is rotating at a generally constant speed, the D.C. speed
change motor, in normal operation, responding to voltage applied within a
specified range of voltages, comprising the steps of:
sensing the speed of the treadmill belt;
producing a rapid deceleration control signal indicating that the treadmill
belt should be decelerated;
producing a D.C. deceleration voltage of a voltage larger than the
specified range of voltages normally applied to the D.C. speed change
motor;
applying the D.C. deceleration voltage to the D.C. speed change motor in
response to a determination that the speed of the treadmill belt is above
a threshold speed and in response to said rapid deceleration control
signal, thereby causing the D.C. speed change motor to rapidly vary the
internal configurations of the transmission;
whereby, the speed of the treadmill belt is rapidly decelerated.
13. The method of claim 12 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining that the power to the
treadmill has been interrupted and then restored.
14. The method of claim 12 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining whether the treadmill user
has indicated that the user would like to immediately stop the treadmill
belt.
15. The method of claim 12 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of sensing the occurrence of a drive motor
overload condition.
16. The method of claim 12 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the steps of:
timing the time that a voltage larger than the specified range of voltages
normally applied to the D.C. speed change motor has been applied to the
D.C. speed change motor;
determining that the time that a voltage larger than the specified range of
voltages normally applied to the D.C. speed change motor has been applied
to the D.C. speed change motor exceeds a preset time period.
17. The method of claim 12 further comprising the step of turning off the
drive motor once the sensed speed of the treadmill belt drops below a
predetermined speed.
18. A method for causing the rapid deceleration of a treadmill belt in a
treadmill having an A.C. drive motor with a rotating member, a treadmill
belt, a transmission for connecting the rotating member to the treadmill
belt so that rotation of the rotating member causes the treadmill belt to
move and a D.C. speed change motor for varying the internal configurations
of the transmission to vary the speed of the treadmill belt when the
rotating member is rotating at a generally constant speed, the D.C. speed
change motor, in normal operation, responding to voltage applied within a
specified range of voltages, comprising the steps of:
producing a rapid deceleration control signal indicating that the treadmill
belt should be decelerated;
sensing the speed of the treadmill belt;
producing a voltage proportional to the sensed speed of the treadmill belt;
producing a threshold voltage level representative of a predetermined
threshold speed of the treadmill belt;
comparing the voltage representative of the speed of the treadmill belt to
said threshold voltage level;
producing a speed determination signal if the voltage representative of the
sensed speed of the treadmill belt is above said threshold voltage, said
speed determination signal indicating that the speed of the treadmill belt
is above said threshold speed;
producing a D.C. deceleration voltage of a voltage larger than the
specified range of voltages normally applied to the D.C. speed change
motor;
applying said D.C. deceleration voltage to the D.C. speed change motor in
response to the presence of both said speed determination signal and said
rapid deceleration control signal, thereby causing the D.C. speed charge
motor to rapidly vary the internal configurations of the transmission;
whereby, the speed of the treadmill belt is rapidly decelerated.
19. The method of claim 18 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining that the power to the
treadmill has been interrupted and then restored.
20. The method of claim 18 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of determining whether the treadmill user
has indicated that the user would like to immediately stop the treadmill
belt.
21. The method of claim 18 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the step of sensing the occurrence of a drive motor
overload condition.
22. The method of claim 18 wherein said step of producing a rapid
deceleration control signal indicating that the treadmill belt should be
decelerated includes the steps of:
timing the time that a voltage larger than the specified range of voltages
normally applied to the D.C. speed change motor has been applied to the
D.C. speed change motor;
determining that the time that a voltage larger than the specified range of
voltages normally applied to the D.C. speed change motor has been applied
to the D.C. speed change motor exceeds a preset time period.
23. A method for rapidly decelerating and stopping a treadmill belt in a
treadmill having an A.C. drive motor with a rotating member, a treadmill
belt, a transmission for connecting the rotating member to the treadmill
belt causing the treadmill belt to move in response to rotation of the
rotating member, a D.C. speed change motor for varying the internal
configurations of the transmission to vary the speed of the treadmill belt
when the rotating member is rotating at a generally constant speed, the
D.C. speed change motor, in normal operation, responding to voltage
applied within a specified range of voltages, and a rapid deceleration
control signal indicating that the treadmill belt should be decelerated,
comprising the steps of:
producing a D.C. deceleration voltage of a voltage larger than the
specified range of voltages normally applied to the D.C. speed change
motor;
applying the D.C. deceleration voltage to the D.C. speed change motor in
response to the rapid deceleration control signal, thereby causing the
D.C. speed change motor to rapidly vary the internal configurations of the
transmission;
applying electrical power to the A.C. drive motor until the speed of the
treadmill belt is below a threshold speed;
terminating the application of electrical power to the A.C. drive motor
after the speed of the treadmill belt has been slowed below the threshold
speed;
whereby, the speed of the treadmill belt is rapidly decelerated to the
threshold speed whereafter the movement of the treadmill belt is
terminated.
24. The combination of a device for rapidly decelerating a treadmill belt
and a treadmill having a constant speed drive motor with a rotating
member, a treadmill belt, a transmission for connecting the rotating
member to the treadmill belt causing the treadmill belt to move in
response to rotation of the rotating member, a D.C. speed change motor for
varying the internal configurations of the transmission to vary the speed
of the treadmill belt when the rotating member is rotating at a generally
constant speed, the D.C. speed change motor, in normal operation,
responding to voltage applied within a predetermined range of voltages,
the device comprising:
a rapid deceleration control signal producing member to indicate that said
treadmill belt should be decelerated;
a D.C. deceleration voltage producing member for producing a voltage larger
than said predetermined range of voltages normally applied to said D.C.
speed change motor;
a D.C. deceleration voltage application member for applying said D.C.
deceleration voltage to said D.C. speed change motor in response to said
rapid deceleration control signal, thereby causing said D.C. speed change
motor to rapidly vary said internal configurations of the transmission;
whereby, the speed of said treadmill belt is rapidly decelerated.
25. The combination of a device for rapidly deceleratinq a treadmill belt
and a treadmill having a constant speed drive motor with a rotating
member, a treadmill belt, a transmission for connecting the rotating
member to the treadmill belt causing the treadmill belt to move in
response to rotation of the rotating member, a D.C. speed change motor for
varying the internal configurations of the transmission to vary the speed
of the treadmill belt when the rotating member is rotating at a generally
constant speed, the D.C. speed change motor, in normal operation,
responding to voltage applied within a predetermined range of voltages,
the device comprising:
a micro-controller for producing a rapid deceleration control signal
indicating that said treadmill belt should be decelerated;
a high speed deceleration voltage supply member for producing a D.C.
deceleration voltage which is a voltage larger than said predetermined
range of voltages normally applied to said D.C. speed change motor;
a relay member for applying said D.C. deceleration voltage to said D.C.
speed change motor in response to said rapid deceleration control signal,
thereby causing said D.C. speed change motor to rapidly vary said internal
configurations of the transmission;
whereby, the speed of said treadmill belt is rapidly decelerated.
26. The device of claim 25 further comprising a determining member for
determining that the power to said treadmill has been interrupted and then
restored.
27. The device of claim 25 further comprising a determining member for
determining whether the treadmill user has indicated that the user would
like to immediately stop said treadmill belt.
28. The device of claim 25 further comprising a sensing member for sensing
the occurrence of a drive motor overload condition.
29. The device of claim 25 further comprising: a timer for timing the time
that said voltage larger than predetermined range of voltages normally
applied to said D.C. speed change motor has been applied to said D.C.
speed change motor by said high speed deceleration voltage supply means;
and,
a determining member for determining that said timer has timed a time that
exceeds a preset time period.
30. The device of claim 25 further comprising:
a sensing member for sensing the speed of said treadmill belt;
a shut off member for turning off said drive motor once the sensed speed of
said treadmill belt drops below a predetermined speed.
31. The device of claim 30 wherein said sensing member includes an optical
tachometer attached to said transmission.
32. The device of claim 31 wherein said sensing member further includes a
tachometer frequency to voltage converter having an input for receiving
signals from said optical tachometer, said tachometer frequency to voltage
converter receiving signals from said optical tachometer.
33. The device of claim 32 wherein said tachometer frequency to voltage
converter includes:
a frequency doubler;
a filter connected to said frequency doubler for filtering the output of
said frequency doubler to produce a D.C. filtered voltage which is
inversely proportional to the frequency of the signal output from said
frequency doubler;
whereby the filtered voltage produced by said filter is inversely
proportional to the speed of said treadmill belt;
a threshold voltage level producing member for producing a threshold
voltage level representative of a predetermined threshold speed of said
treadmill belt;
a comparison member for comparing the filtered voltage produced by said
filter to the threshold voltage level and for producing a speed
determination signal if the filtered voltage is below the threshold
voltage, said speed determination signal connected to said relay member,
said speed determination signal indicating that the speed of said
treadmill belt is above a threshold speed.
34. The device of claim 30 wherein said device further comprises a member
for applying said D.C. deceleration voltage to said D.C. speed change
motor including a software high speed deceleration command module
connected to said sensing member and to said rapid deceleration control
signal, said command module producing a control signal if said treadmill
belt is running at a speed above the desired threshold and said rapid
deceleration control signal indicates that a rapid deceleration is
desired;
wherein said relay member is connected to said command module and is
enabled in response to said control signal and connects said high speed
deceleration voltage supply member to said D.C. speed change motor.
35. The device of claim 30 wherein said device includes a determining
member for determining that the sensed speed of said treadmill belt
exceeds a predetermined speed.
36. The device of claim 30 wherein said device further comprises a
determining member for determining that the power to said treadmill has
been interrupted and then restored.
37. The device of claim 30 wherein said device further comprises a
determining member for determining whether the treadmill user has
indicated that the user would like to immediately stop said treadmill
belt.
38. The device of claim 30 wherein said device further comprises a sensing
member for sensing the occurrence of an overload condition of said drive
motor.
39. The device of claim 30 wherein said device further comprises:
a timer for timing the time that said voltage larger than said
predetermined range of voltages which is normally applied to said D.C.
speed change motor has been applied to said D.C. speed change motor by
said high speed deceleration voltage supply member; and
determining member for determining that said timer has timed a time that
exceeds a preset time period.
40. The combination of a device for rapidly decelerating a treadmill belt
and a treadmill having a constant speed drive motor with a rotating
member, a treadmill belt, a transmission for connecting the rotating
member to the treadmill belt causing the treadmill belt to move in
response to rotation of the rotating member, a D.C. speed change motor for
varying the internal configurations of the transmission to vary the speed
of the treadmill belt when the rotating member is rotating at a generally
constant speed, the D.C. speed change motor, in normal operation,
responding to voltage applied within a predetermined range of voltages,
the device comprising:
a) high speed deceleration voltage supply member for producing a D.C.
deceleration voltage of a voltage larger than said predetermined range of
voltages normally applied to said D.C. speed change motor;
b) a first determining member for determining that the power to said
treadmill has been interrupted and then restored;
c) a second determining member for determining whether the treadmill user
has indicated that the user would like to immediately stop said treadmill
belt;
d) a first sensing member for determining that the occurrence of an
overload condition of said drive motor;
e) a timer for timing the time that a voltage larger than the specified
range of voltages normally applied to the D.C. speed change motor has been
applied to the D.C. speed change motor by said high speed deceleration
voltage supply means;
f) a third determining member for determining that said timer has timed a
time that exceeds a preset time period;
g) a micro-controller for producing a rapid deceleration control signal
indicating that the treadmill belt should be decelerated in response to:
1) said first determining member;
2) said second determining member;
3) said first member; or
4) said timer;
h) relay member for applying said D.C. deceleration voltage to said D.C.
speed change motor in response to said rapid deceleration control signal,
thereby causing said D.C. speed change motor to rapidly vary said internal
configurations of the transmission;
i) a second sensing member for sensing the speed of said treadmill belt;
j) a shut off member for turning off said drive motor once the sensed speed
of said treadmill belt drops below a predetermined speed;
whereby, the speed of said treadmill belt is rapidly decelerated. |
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Claims  |
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Description  |
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A portion of the disclosure of this patent document contains material which
is subject to copyright protection. The copyright owner has no objection
to the facsimile reproduction by anyone of the patent document or the
patent disclosure, as it appears in the Patent and Trademark Office patent
file or records, but otherwise reserves all copyright rights whatsoever.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to a method and system for rapidly decelerating a
treadmill belt and more specifically relates to a method and system for
rapidly decelerating a treadmill belt on a treadmill having a D.C. motor
that changes the internal configurations of a transmission on a treadmill
powered by a constant speed drive motor, which transmission connects the
constant speed drive motor to the belt of the treadmill.
2. Description of Related Art
Treadmills require a drive motor to provide power to the treadmill's belt.
Most often, these drive motors are AC motors which turn at only one speed
which speed depends on the frequency of the AC power supply. In order to
provide various speeds for the treadmill belt, these AC drive motors are
connected to the treadmill belt through a mechanical transmission which
changes its internal configuration to provide the various speeds for the
treadmill belt. The internal configurations of the mechanical transmission
is typically set by a relatively small speed-change motor. As the
speed-change motor turns, the transmission's internal configuration
changes so that the treadmill belt speed either increases or decreases
depending on the direction that the speed-change motor is turning.
Referring to FIG. 1, a typical treadmill transmission, generally labelled
1, is shown. Typically, these transmissions have a pair of adjustable
pulleys 4,7 connected by a transmission belt 6. A first pulley 4 is
connected to a drive motor 2 through an input shaft 3 so that rotation of
the drive motor 2 rotates this first pulley 4. This first pulley 4 has two
opposing conical plates 5a,b which form a V-shape for frictionally
receiving a correspondingly shaped transmission belt 6.
The transmission 1 also has a second pulley 7. The second pulley 7 has a
set of opposed conical plates attached to an output shaft 9. The conical
plates 8a,b of the second pulley 7 also form a V-shaped recess for
frictionally receiving the transmission belt 6 therebetween. The output
shaft 9 is in turn attached to a drive roller 11 through a drive belt 10.
Rotation of the drive roller 11 causes the treadmill belt (not shown) to
move.
Typically in such a transmission 1, a speed-change motor 12 moves the two
plates 5a,b of the first pulley 4 either closer together or further apart.
Because of the conical shape of the opposed plates 5a,b, as the plates are
moved further apart and tension is placed on the transmission belt 6, the
transmission belt 6 situated around the first pulley 4 will move closer to
the center of the input shaft 3. As the two plates 5a,b are moved closer
together by the speed-change motor 12, the transmission belt 6 is moved
more towards the outer edge of the first pulley 4.
The plates 8a,b of the second pulley 7 are biased in a position toward each
other by a tension spring 13. However, sufficient pressure by the
transmission belt 6 on the conical plates 8a,b of the second pulley 7 will
overcome the inherent bias of the plates to be together and force them
apart thereby allowing the transmission belt 6, which is under tension, to
move from a position more towards the outer edge of the plates 8a,b to a
position closer towards the output shaft 9. In this way the mechanical
advantage imparted to the drive roller 11, and subsequently to the
treadmill belt from the drive motor 2, changes depending on the position
of the transmission belt 6 on the first and second pulleys 4,7.
The rate of change of the speed of the treadmill belt is dependent on both
the current speed of the treadmill belt, that is, on the current position
of the transmission belt 6 on pulleys 4,7, and on how fast the
speed-change motor 12 is turning.
Previous transmissions have used AC speed-change motors. However, under
normal operating conditions, AC motors turn at only one speed regardless
of the current or voltage applied to the motor. Consequently, the speed at
which the opposing conical plates 5a,b are moved together or apart as
driven by the AC speed change motor is a constant. In a transmission using
an AC speed-change motor, the entire change in the speed of the treadmill
belt results from the change in position of the transmission belt 6 on
pulleys 4,7 and not from any change in the rate that the conical plates
5a,b are moved together or apart. A constant rate of moving the conical
plates 5a,b together or apart as a result of the constant speed of the AC
speed change motor 12 results in a non-constant rate of change for the
speed of the treadmill belt as illustrated in FIG. 2.
This non-constant rate of speed change has two primary sources. First, the
belt connecting the pulleys has a constant length. As shown in FIG. 1, as
the plates 5a,b of the first pulley 4 move together or apart in response
to the rotation of the speed-change motor 12, the location of the
transmission belt 6 on plates 5a,b changes. As the conical plates 5a,b
move apart, the transmission belt 6 will have less tension on it. As a
result of the reduction of tension on the transmission belt 6, the tension
spring 13 will push plates 8a,b together thereby moving transmission belt
6 farther away from the output shaft 9. As transmission belt 6 moves away
from output shaft 9, tension is placed on transmission belt 6 by tension
spring 13 pulling it into snug frictional contact with conical plates 5a,b
at a position closer to input shaft 3 than it had been prior to conical
plates 5 a,b moving apart in response to the rotation of speed change
motor 12. Transmission belt 6 will move out from output shaft 9 until the
tension in transmission belt 6 equals the tension applied by tension
spring 13. This causes the mechanical ratio of pulley 4 and pulley 7 to
change. This relationship is illustrated in FIG. 3.
The exact equation for the transmission belt circumference is found by the
sum of the straight portions (l+l) and the two partial circumferences
(c.sub.1 +c.sub.2). This leads to the following equation:
Circumference=2d.multidot.sin .theta.+2.theta..multidot.r.sub.1
+2(.pi.-.theta..multidot.r.sub.2) 1)
where:
.theta.=cos.sup.-1 ((r.sub.2 -r.sub.1)/d), and .theta. is in radians;
r.sub.1 =the radius of the transmission belt 6 around first pulley 4;
r.sub.2 =the radius of the transmission-belt 6 around second pulley 7;
d=the distance between first pulley 4 and second pulley 7; and,
1=d.multidot.sin .theta.=((d.sup.2 -(r.sub.2 -r.sub.1).sup.2).sup.1/2.
This equation is extremely difficult to solve for r.sub.2 in terms of
r.sub.1. Equation 1 can be simplified by making the following
approximations:
c.sub.1 =.pi.r.sub.1
c.sub.2 =.pi.r.sub.2, and
l=((d.sup.2 -(r.sub.2 -r.sub.1).sup.2).sup.1/2 +.pi.(r.sub.2 +r.sub.1)
so that:
Circumference=2((d.sup.2 -(r.sub.2 -r.sub.1).sup.2).sup.1/2 +.pi.(r.sub.2
-r.sub.1) 2)
The error created in this approximation is very small and creates an
equation which is more easily solved. Since Circumference is the
circumference of transmission belt 6 which is a constant "C", and "d", the
distance between pulleys 4 and 7, is a constant, the following quadratic
equation is derived:
(.pi.-4)r.sub.2.sup.2 +((2.pi..sup.2 +8)r.sub.1 -2.pi.C)r.sub.2
+((.pi..sup.2 -4)r.sub.1.sup.2 -2.pi.Cr.sub.1 +C.sup.2 -4d.sup.2)=0.3)
Solving for r.sub.2 gives:
r.sub.2 =(.pi.C-(.pi..sup.2 +4)r.sub.1 .+-.2(4.pi..sup.2 r.sub.1.sup.2
-4.pi.Cr.sub.1 +C.sup.2 +d.sup.2 (.pi..sup.2 -4)).sup.1/2)/(.pi..sup.2
-4)4)
Realistic values show the ".+-." factor is subtracted, only, in this
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