Why Resonance is Important in Control Systems?
Resonance is an interesting phenomenon that occurs when a system is driven at one of its natural frequencies. During resonance, the system oscillates at a peak amplitude, sometimes with just a little amount of input. It can be useful in some system (such as ultrasonic welding machines or sonar heads) but that could cause instability in your system. To understand resonance, you can think of a mass-spring-damper system like below:
Block diagram of a mass-spring-damper system.
The mass-spring-damper system is a classic example in mechanical engineering and physics. The equation of motion for this system is derived from Newton's second law of motion. Here's a brief overview:
Let's denote:
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𝑚 as the mass
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𝑏 as the damping coefficient
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𝑘 as the spring constant
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𝑥(𝑡) as the displacement of the mass as a function of time
The equation of motion is given by:
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Where:
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𝑥¨(𝑡) is the acceleration (second derivative of displacement with respect to time)
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𝑥˙(𝑡) is the velocity (first derivative of displacement with respect to time)
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𝑥(𝑡) is the displacement
If there's an external force 𝐹(𝑡) acting on the system, the equation of motion becomes:
We can derive the transfer function doing Laplace transform, assuming the stationary initial state:
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We can then derive its frequency response by substituting 𝑠=𝑗𝜔:
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Lets draw the bode plot for m = 1, k = 1, b = 0.1:
When you look at the image, you can see the amplitude has a peak point, that is the resonance. At resonance frequency;
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The system can have large amplitude oscillations, that may lead to structural damage
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Even a small amount of noise can reduce the precision due to amplification
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Phase margin between the input and output is large, that reduces the system stability and performance
A system usually has more than one resonance modes, some of them are coupled with the input force, some of them are not. As a rule of thumb, the minimum coupled resonance frequency of a system should be at least 3 times the control bandwidth. Once you know your system's resonance behavior, you can take some actions to improve your performance:
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Filtering your drive signal so that you won't get excessive responses
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Modifying your control algorithm
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Using mechanical means of damping
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Modifying your mechanical design to eliminate low frequency resonance modes
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Tuning your PID parameters to increase the stability
All in all, handling the resonances in a system is crucial for safety and performance especially for precision devices operating at high speeds such as Fast Steering Mirrors. If you need devices that work at higher frequencies safely, you can contact us to select a product for your application or work on a customized solution.