Hi all,
New to the forum, so apologise if I commit any forum sins (let me know!).
I am trying to solve a practical problem controlling two machines in a factory. One machine supplies empty bottles which are carried to a second machine which fills them. The second machine consumption rate can be set, as can the feed rate of the incoming feed. The belt between the two also runs at a set speed.
I am trying to find the optimum location to place a number of sensors which control the feed of bottles to the second machine such that stop-starting is limited.
At this moment in time, there are sensors positioned at various points, represented in the sketch above.
H_machine,out - the position of the consuming machine
H_stop,out - position of the sensor that starts the consuming machine i.e. when this sensor is covered i.e. h>H_stop, the consuming machine will run
H_stop,in - position of sensor to prevent overfill of the belt i.e. if h>H_stop,in, the feeding machine will stop feeding
H_slow,out - smooths the consumption of the consuming machine: if h<H_slow,out, slow consumption rate to allow bottles coming on conveyor to "catch up"
H_machine,in - position of feed machine
I am interested in positioning the sensors in such a way that dh/dt > 0, that is, bottles never stop feeding the consuming machine.
Thus far, I have modelled it as follows:
dh/dt = F_in - F_out
where F_out =
0 if h<H_stop,out
F_out,lo if H_stop,out<h<H_slow,out
F_out,hi if H_slow,out<h
and where F_in =
0 if H_stop,in<h(t-t_lag)
F_in if h(t-t_lag)<H_stop,in
My first issue is solving this as is. I would ave expected to use Laplace transforms and representing the above with Heaviside functions but the fact that the differential piecewise functions are dependent on the value of the function itself confuses me. Another issue is working out what the t_lag time is, since any infeeding signals or impulses depend on the height of the bottle queue from a few seconds back (there is a lag between when the signal is sent and when the new feed of bottles arrives).
There is also the option to switch between a high and low speed for the feeding machine too, also dependent on a sensor.
I hope you can help!
New to the forum, so apologise if I commit any forum sins (let me know!).
I am trying to solve a practical problem controlling two machines in a factory. One machine supplies empty bottles which are carried to a second machine which fills them. The second machine consumption rate can be set, as can the feed rate of the incoming feed. The belt between the two also runs at a set speed.
I am trying to find the optimum location to place a number of sensors which control the feed of bottles to the second machine such that stop-starting is limited.
At this moment in time, there are sensors positioned at various points, represented in the sketch above.
H_machine,out - the position of the consuming machine
H_stop,out - position of the sensor that starts the consuming machine i.e. when this sensor is covered i.e. h>H_stop, the consuming machine will run
H_stop,in - position of sensor to prevent overfill of the belt i.e. if h>H_stop,in, the feeding machine will stop feeding
H_slow,out - smooths the consumption of the consuming machine: if h<H_slow,out, slow consumption rate to allow bottles coming on conveyor to "catch up"
H_machine,in - position of feed machine
I am interested in positioning the sensors in such a way that dh/dt > 0, that is, bottles never stop feeding the consuming machine.
Thus far, I have modelled it as follows:
dh/dt = F_in - F_out
where F_out =
0 if h<H_stop,out
F_out,lo if H_stop,out<h<H_slow,out
F_out,hi if H_slow,out<h
and where F_in =
0 if H_stop,in<h(t-t_lag)
F_in if h(t-t_lag)<H_stop,in
My first issue is solving this as is. I would ave expected to use Laplace transforms and representing the above with Heaviside functions but the fact that the differential piecewise functions are dependent on the value of the function itself confuses me. Another issue is working out what the t_lag time is, since any infeeding signals or impulses depend on the height of the bottle queue from a few seconds back (there is a lag between when the signal is sent and when the new feed of bottles arrives).
There is also the option to switch between a high and low speed for the feeding machine too, also dependent on a sensor.
I hope you can help!
