Frequency domain work analysis of machinery including turbomachinery
View Patent ↗Machine performance, including efficiency, is analyzed in direct frequency domain by measuring the input power to the machine over a range of machine speeds (frequencies, ω). A differentiable mathematical function or model is identified that characterizes the input power vs. speed functional relationship. Successive derivatives of experimental power vs. speed data enable the calculation of coefficients of the mathematical function at a plurality of points (frequencies) in the frequency domain. Performance parameters including machine efficiency, pressure, pressure drag, flow, viscous drag, frictional drag, viscosity and work may be calculated from the coefficients of the mathematical function. One or more desired, optimum or best operating point(s) for machine speed may be defined in terms of the coefficients of the mathematical function or the derived performance parameters.
1 . A method of operating a machine comprising a motor rotationally coupled to a load, the method comprising:
operating the motor at a plurality of speeds within a range bounded by a lower speed and an upper speed;
measuring, with a current sensor, electrical current supplied to the motor at each of the plurality of speeds;
correlating each current measurement to its corresponding speed to generate correlated speed-current data;
determining, from the correlated speed-current data, a plurality of numerically calculated derivatives of current with respect to speed;
determining,
from the plurality of numerically calculated derivatives of current, a plurality of derived values of coefficients A and B that fit the correlated speed-current data to a model, having at least two terms, of the form A·ω n1 +B·ω n2 , the model representing a load-dependent relationship between current and speed, where ω is the speed and n 1 and n 2 are real numbers individually selected from the group consisting of 0, 1, 2, 3, and k, wherein k is a non-integer real number;
identifying a speed at which at least one of the determined coefficients A or B satisfies a predetermined criterion; and
operating the motor at a speed determined based on the identified speed.
2 . The method of claim 1 , wherein the load-dependent relationship is expressed as A·ω n1 +B·ω n2 +C·ω n3 , where n1, n2, and n3 are real numbers selected from the group consisting of 0, 1, 2, 3, and k, wherein k is a non-integer real number.
3 . The method of claim 1 , wherein the load-dependent relationship is expressed as A·ω 3 +B·ω 2 +C·ω+D.
4 . The method of claim 1 , wherein the values of coefficients A and B are determined using back-substitution from one or more numerically calculated derivatives of current with respect to speed.
5 . The method of claim 1 , wherein the load is a pump.
6 . The method of claim 1 , wherein the machine/load is selected from the group consisting of: a watercraft/a propeller shaft coupled to a marine propeller, a ground vehicle/a drivetrain shaft coupled to a road wheel, an aircraft/a propeller shaft, and a milling machine/a spindle coupled to a cutting tool.
7 . The method of claim 3 , comprising:
defining normalized coefficients as:
A
*
=
A
/
(
ω
3
)
,
B
*
=
B
/
(
ω
2
)
,
C
*
=
C
/
(
ω
)
,
and
D
*
=
D
;
wherein efficiency is given by n=A*/(A*+B*+C*+D*).
8 . The method of claim 7 , wherein a maximum efficiency point is indicated by a local maximum of normalized coefficient A* with respect to speed.
9 . The method of claim 3 , wherein cavitation onset is indicated by a decrease in coefficient A and a corresponding increase in at least one of coefficients B, C, or D.
10 . A machine comprising:
a motor rotationally coupled to a load;
a sensor configured to measure input power electrical current supplied to the motor; and
a controller in electrical communication with the motor and the sensor, the controller configured to:
(i) operate the motor at a plurality of speeds within a range bounded by a lower speed and an upper speed;
(ii) receive, from the sensor, a current measurement at each of the plurality of speeds;
(iii) correlate each current measurement to its corresponding speed to generate correlated speed-current data;
(iv) determine, from the correlated speed-current data, a plurality of numerically calculated derivatives of current with respect to speed;
(v) determine, from the plurality of numerically calculated derivatives of current, a plurality of derived values of coefficients A and B that fit the correlated speed-current data to a model, having at least two terms, of the form A·ω n1 +B·ω n2 , the model representing a load-dependent relationship between current and speed, where ω is the speed and n1 and n2 are real numbers individually selected from the group consisting of 0, 1, 2, 3, and k (k being a non-integer real number);
(vi) identify a speed at which at least one of the determined coefficients A or B satisfies a predetermined criterion; and
(vii) operate the motor at a speed determined based on the identified speed.
11 . The machine of claim 10 ,
wherein the controller is further configured to determine a coefficient C such that the load-dependent relationship is expressed as A·ω n1 +B· n2 +C·ω n3 , where n1, n2, and n3 are real numbers individually selected from the group consisting of 0, 1, 2, 3, and k (k being a non-integer real number).
12 . The machine of claim 10 , wherein
the controller is further configured to determine coefficients A, B, C, and D such that the load-dependent relationship is expressed as A·ω 3 +B·ω 2 +C·ω+D.
13 . The machine of claim 12 , wherein the controller is further configured to: define normalized coefficients as:
A
*
=
A
/
(
ω
3
)
,
B
*
=
B
/
(
ω
2
)
,
C
*
=
C
/
(
ω
)
,
and
D
*
=
D
;
wherein efficiency is given by n=A*/(A*+B*+C*+D*).
14 . The machine of claim 13 , wherein the controller is further configured to identify a maximum efficiency point by a local maximum of normalized coefficient A* with respect to speed.
15 . The machine of claim 12 , wherein the controller is further configured to detect cavitation onset by a decrease in coefficient A and a corresponding increase in at least one of coefficients B, C, or D.
16 . The machine of claim 10 , wherein the controller is further configured to determine the values of coefficients A and B using back-substitution from one or more numerically calculated derivatives of current with respect to speed.
17 . The machine of claim 10 , wherein the load is a pump.
18 . The machine of claim 10 , wherein the machine/load is selected from the group consisting of: a watercraft/a propeller shaft coupled to a marine propeller, a ground vehicle/a drivetrain shaft coupled to a road wheel, an aircraft/a propeller shaft, and a milling machine/a spindle coupled to a cutting tool.