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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
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How Natural Frequency Limits
Frequency of Acceleration
Sept. 24, 2018
The natural frequency of a hydraulic system refers to the system’s
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i h hi h
lf
i
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
stiffness. Systems with higher natural frequency are easier to
control when accelerating quicker than systems with low natural
frequency.
Peter Nachtwey
Encoder
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Controlling
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A hydraulic cylinder can be simply modeled as a mass between two springs.
Systems with a low natural frequency (the frequency at which the system
oscillates after a sudden start or stop) have a low spring constant relative to
the mass of the load. Conversely, systems exhibiting a high natural frequency
have a high spring constant relative to the load mass. So how high should the
Feb. 16, 2021
natural frequency of the cylinder be relative to the frequency of acceleration?
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Ideally, the natural frequency should be at least four times the required
March 18, 2020
load accelerate. This is true when using a standard hydraulic motion
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frequency of acceleration —the frequency at which the cylinder’s piston and
1. In this plot, the frequency of
acceleration is 5 Hz and the natural
frequency of the system is 10 Hz. The
actual position and velocity values do
Dec. 30, 2019
not accurately follow the target
Multi-Axis
Motion
values; note the Root Mean Squared
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Motion
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
Error (RMSE). The motion will be “jerky.”
Natural Frequency and Frequency of Acceleration
In a hydraulic system, the natural frequency is calculated from the mass of
the load, the areas of both the rod- and cap-end of the piston, and the bulk
modulus of the hydraulic fluid.
ωn = [4´ βA2/(V´M)]1/2
Where:
ωn is the natural frequency of the cylinder and load;
β is the bulk modulus of oil;
A is the average area of the piston;
V is the total volume of oil trapped between the valve and the piston; and
M is the mass of the load
A simple example illustrating frequency of acceleration is a cylinder and load
following a sinusoidal motion at 5 Hz. In this case, the frequency of
acceleration is 5 Hz. If the cylinder is just making a simple move from one
position to another, the frequency acceleration is 0.5 seconds divided by the
minimum of the acceleration or deceleration time This assumes the
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
minimum of the acceleration or deceleration time. This assumes the
acceleration and deceleration ramps are sinusoidal S-curves having one
fundamental frequency with no harmonics. (Linear ramps have the
fundamental frequency and many harmonics that may cause problems.)
2. The same system as shown in Fig.
1, but with the natural frequency
increased from 10 to 20 so that it is 4
times the frequency of acceleration.
Now the actual motion more
accurately follows the target motion
profile. Note the RMSE value in this plot.
Most motion controllers express the acceleration as a rate. For instance,
2,500 mm/sec2 is about one fourth the acceleration due to gravity. The
acceleration time to 250 mm/sec takes 0.1 second at 2,500 mm/sec2, so the
frequency of acceleration is 5 Hz (0.5/0.1 sec. = 5 Hz). If the acceleration is to
100 mm/sec. the acceleration time is 0.04 sec., so the frequency of
acceleration is 12.5 Hz.
I have been asked many times why the tuning for short moves needs to be
different from that for long moves. The reason is that short moves have very
short acceleration times; consequently, the frequency of acceleration is
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
;
q
y,
q
y
higher. If the natural frequency is not at least four times that of the maximum
frequency of acceleration, the system will not control well.
In the example above, the 12.5 Hz frequency of acceleration may be greater
than 1/4 the natural frequency. Therefore, for good control with a typical
motion controller, it may be better to slow the acceleration ramp down to
1,000 mm/se.c2 so the acceleration time is still 0.1 sec.
3. In this system, the frequency of
acceleration is 8, but the natural
frequency is only 20. Without
advanced control, the following error
is large.
Why Does Natural Frequency
Need to be Four Times the Frequency of Acceleration?
The natural frequency does not actually “need” to be four times the frequency
of acceleration. If it isn’t, you’ll find it difficult or even impossible to optimize
the controller gains so the actual position follows the target position with
little error. If the natural frequency is much greater than the frequency of
acceleration, the controller gains will be much easier to optimize.
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
Here’s a graphical example: In Fig. 1, the frequency of acceleration is 5 Hz,
and the natural frequency is 10 Hz. The natural frequency is only two times
the frequency of acceleration, so the following error is large. (Note the root
mean squared error value, RMSE.) Figure 2 shows that increasing the
natural frequency to 20 Hz reduces the error between the target and actual
position.
This analysis assumes the controller is capable of proportional and integral
(PI) control with both velocity and acceleration feedforwards (feedforwards
predict the needed control output rather than waiting for the control output
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to be generated
by errors).
If the controller
isDIRECTORY
not capable
of PI control, or
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does not support velocity and acceleration feedforwards, the ratio of natural
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frequency to frequency of acceleration needs to be much higher to keep the
following error small.
For instance, a controller with proportional only control will still have 30 deg.
of following error even if the natural frequency is 10 times greater than the
frequency of acceleration. Do not use proportional-only control if the
following error must be close to zero.
The most advanced motion controllers can keep the error between the target
position and actual position small even if the natural frequency is only two
times the frequency of acceleration.
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
4. With an advanced motion
controller with second derivative gain
and jerk feed forward, it’s possible to
achieve precise control even when the
natural frequency is not 4 times the
frequency of acceleration. Note the
reduction in the RMSE value from
Fig. 3.
What Does This Mean?
Saving money! As a general rule, systems with lower natural frequency can
use smaller components (cylinders, pumps, etc.), which cost less. The key is
selecting the right motion controller.
Advanced electrohydraulic motion controllers can perform well when the
ratio of the natural frequency to frequency of acceleration is less than four.
Advanced motion controllers, such as the RMC motion controller family from
Delta Computer Systems, have control algorithms beyond basic PID gain and
velocity and acceleration feed forwards.
The diameter of the cylinder is roughly proportional to the natural frequency.
If the motion controller allows the natural frequency to be only two times the
frequency of acceleration the cylinder diameter may be reduced up to half
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How Natural Frequency Limits Frequency of Acceleration | Power & Motion
frequency of acceleration, the cylinder diameter may be reduced up to half.
How much less depends on the VCCM equation. Reducing the hydraulic
cylinder diameter by half means the pump, valve, piping, and accumulator
sizes can all be reduced. The cost savings can be much more than the total
cost of the hydraulic motion controller. Plus, the machine will be much more
economical to operate.
Figures 3 and 4 demonstrate the benefits of an advanced controller. The first
plot (Fig. 3), shows a short-distance move of a system where the natural
frequency is 20 and the acceleration and deceleration take only 0.125 sec.
Therefore, the frequency of acceleration is 8 Hz. The actual motion doesn’t
follow the target because the natural frequency is only 2.5 times the
frequency of acceleration. The last plot (Fig. 4) assumes the same conditions,
but with control by an advanced motion controller. The following error is
very small.
Peter Nachtwey is president of Delta Computer Systems Inc., Battle Ground,
Wash.
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