In a system, damping explains how quickly the system will return to a state of rest after some external force has acted upon it. For example, in control systems, the goal is often to stop a system from spinning or vibrating. The damping ratio expresses that response as a ratio between the actual damping of the system and the critical damping coefficient. It is expressed as follows:
where:
- ζ is the damping ratio
- c is the actual damping coefficient
- cÂc is the critical damping coefficient
The value of ζ will determine the kind of damping that is seen by the system, which will enable an engineer to make necessary changes to the system to achieve the desired end state.
The ideal damping state of the system is referred to as critical damping. When a system is critically damped, the damping coefficient is equal to the critical damping coefficient and the damping ratio is equal to 1.
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Critical Damping Explained
The critical damping coefficient is the solution to a second-order differential equation that is used to evaluate how quickly the system will return to its original (unperturbed) state. For a single degree of freedom system, this equation is expressed as:
where:
- m is the mass of the system
- c is the damping coefficient
- k is the spring rate
In the case of critical damping, c will be cc and can be determined from the following:
Because damping is only considered for harmonic (i.e., oscillating) systems, the equation for critical damping can be written as follows:
where ωn is the natural frequency of the system as it oscillates. The natural frequency of the system can be determined in several ways, depending on the actual parameters of the system.
The critical damping coefficient is the idealized coefficient for damping the system. The actual damping of a system will be some value other than the critical damping coefficient, which is where the critical damping ratio comes into play.
Using the Critical Damping Coefficient
Using the critical damping coefficient, it is possible to determine the critical damping ratio with the equation introduced above. The value of the critical damping ratio may be a desired value, or it may be a calculated value.
If the value is established as a desired value for an engineering application, the actual damping coefficient will be calculated. If the actual damping coefficient is known, then the critical damping ratio can be calculated to observe how the system is operating.
Finding the actual damping coefficient
If the system has been modeled or can be tested in some way, it is possible to determine what the actual damping coefficient is. That value, c in the equation, is divided by the critical damping coefficient, cc in the equation, to find the ratio.
There are a variety of ways to calculate the damping coefficient of a system, most of which are specific to certain applications. However, they all come down to solving the second-order differential equation. This can be accomplished with numerical methods, analytical methods, or test methods.
From damping coefficient to damping ratio
As can be seen in the equation for the damping ratio, the actual damping coefficient is divided by the critical damping coefficient. This means that the ratio will either be greater than or less than one. If the value of ζ is greater than one, the system is said to be overdamped. If the value is less than one, the system is considered underdamped.
Critically damped system
If the value of the damping ratio works out to be exactly one (or as close to one as possible), then the system is said to be critically damped. This means that the actual damping coefficient is equal to (or very nearly equal to) the critical damping coefficient. In this case, the system will return to the ideal state more quickly than the overdamped system but will not overshoot that ideal state as the underdamped system will do.
Overdamped system
An overdamped system will return to its ideal state more slowly than the idealized (i.e., critical) system. This may be desired, or it may prove to be problematic. Overdamping a system means that there is more time to correct any problems, which can be beneficial for systems where overshooting is not a good outcome.
Underdamped system
An underdamped system will return to its ideal state more quickly than under a critical system. The system will oscillate back and forth, overshooting the ideal state in both the positive and negative direction, and the magnitude of that oscillation will diminish over time until it is no more. This can be a good thing if the goal is achieving a stable state quickly, but if the overshooting will be a problem, then underdamping the system is not the way to go.
Undamped system
It is possible for a system to have no damping at all. This would be a case where the damping ratio is equal to zero and means that the system will continue to oscillate forever, never reaching that ideal state. Undamped systems are generally not something that engineers aim for, although some applications do exist. However any system will have some damping due to the action of external forces.
Applications of the Critical Damping Ratio
When solving engineering problems, understanding and applying damping to a system can be very important. Because the system in question can be described using a second-order differential equation, it is possible to calculate the critical damping coefficient relatively easily. This value can then be used to calculate other parameters for the system.
Depending on the desired system outcome, underdamped, critically damped, or overdamped, the damping ratio can be used to determine the ideal actual damping coefficient. With the calculated actual damping coefficient, the system can be designed in such a way as to express that coefficient.
For a system that is to be critically damped, the actual damping coefficient will be set to the value of the critical damping coefficient. From that point, the value of k in the system equation can be determined to complete the system design.
By taking into consideration that it would be nearly impossible to have an actual damping coefficient that was exactly equal to the critical damping coefficient, some sort of tolerance can be applied such that the critical damping ratio was equal to some predetermined value very close to one. That ratio would then be applied to the critical damping coefficient, and that value c would be used in the second-order differential equation to solve for k and the motion of the system.
Expanding Beyond One Dimension
Most real systems are going to have more than just a single dimension. In fact, most systems are going to have as many as six degrees of freedom (DOF). This means that the equations of motion will include x-, y-, and z-directions as well as rotations about each of those axes.
However, expanding the critical damping ratio beyond a one-dimensional system only involves including the additional dimensions in the differential equation. Moving from one dimension to multiple dimensions will include some additional calculations, but the key concepts of solving for and using the damping coefficients will remain the same.
Key Considerations
When using the critical damping ratio to evaluate the damping of a system, there are some things that need to be taken into consideration.
The system that is being modeled needs to have some sort of harmonic motion. For a control system, this could be simple harmonics (oscillations) about a single axis, or up to a full 6 DOF system with oscillations and rotations about all three axes like a flight simulator.
The critical damping coefficient is an expression of the idealized and describes the system as it returns to the ideal state after a force has acted upon it. The ratio of the actual damping to this critical damping is the critical damping ratio and can be used to express whether the system is underdamped, critically damped, or overdamped (or, undamped).
Using the critical damping ratio, an engineer can determine other parameters of the system to model either the desired behavior or the observed behavior. Modeling the desired behavior allows an engineer to change parameters to achieve a desired outcome. Modeling the observed behavior of a system allows an engineer to determine a way to control the behavior of the system to achieve some desired end state.
Because the critical damping ratio is one coefficient divided by another coefficient, there are no units associated with it.
