An ideal mass-spring-damper system with mass m (in kg), spring constant k (in N/m) and viscous damper of damping coeficient c (in N-s/m) can be described by:
- is the acceleration (in m/s2) of the mass
- is the displacement (in m) of the mass relative to a fixed point of reference
1.1 Differential equation
The above equations combine to form the equation of motion, a second-order differential equation for displacement x as a function of time t (in seconds).
Rearranging, we have
Next, to simplify the equation, we define
The first parameter, , is called the (undamped) natural frequency of the system . The second parameter, , is called the damping ratio. The natural frequency represents an angular frequency, expressed in rad/s. The damping ratio is a dimensionless quantity.
After the substitutions the differential equation becomes
2 2nd Order Systems
The mass-spring-dashpot system is the inspiration of the ideal (or standard) 2nd order transfer function. Most closed loop systems and sensors are designed so that an ideal 2nd order transfer function describes them accurately.
Sensors are typically designed to be linear with a known gain (usually K = 1), damping ratio of , and a bandwidth () at least 10x the system closed loop bandwith.
The ideal 2nd order transfer function is
|Generic 2nd Order System|
- is the system gain
- is the system's natural frequency
- is the system's damping ratio.
At low frequencies, relative to ,
and at high frequencies
The double integrator leads to the -40 dB/decade (order of magnitude) at higher frequencies.
2.1 Damping Ratio
The natural frequency in the images is in Hz while the natural frequency in Eqn. O2 1 is in rad/sec. The conversion between the 2 parameters is .
The damping ratio originated from the mass-spring-dashpot system. In an ideal 2nd order transfer function the damping ratio can have a dramatic effect on the system response.
When modeling a system to match measured data the phase is a better way to match than the magnitude. Measurements happen at discrete points and can miss the true peak magnitude. Matching the slope of the phase shift will provide for better matching.
Figure 1 shows that the peak magnitude of the transfer function is dependent upon and . Adjusting the overall system gain is a simple matter so is not particularly important. However, there isn't any way to compensate or adjust in real systems. Figure 2 shows the dependency of the phase shift at on . Clearly the smaller is the steeper, more rapidly, the phase shifts from 0 degrees to -180 degrees.
2.2 Peak Magnitude
The peak magnitude is a function of . That relation is
2.3 Peak Frequency
The frequency of the peak magnitude () is not the same as . Often and are very close. The relationship of to is
3 Example Models
Honeywell GG5300 MEMS Rate Gyro Plant
The Honeywell GG5300 Three Axis MEMS Rate Gyro is advertised as being for "Missiles & Munitions." The datasheet available on Honeywell's website states
|Bandwidth @ 90 deg phase||100 Hz typical|
The typical sensor model would have the following parameters
The sensor has a scale factor (Volts per radian) that we are ignoring for now. (The scale factor here is just a gain. External electronics can make this scale factor whatever is convenient.) Using the parameters in the above table a transfer function can be formed in MATLAB with the following commands
>> wn = 100 * (2*pi); >> z = 1/sqrt(2); >> K = 1; >> GG5300 = tf(K * [wn^2], [1, 2*z*wn, wn^2]);
The resulting frequency response for a single axis is
It is fine to model this sensor as a transfer function. However, when combined into a larger LTI model it is recommended that every individual piece of the larger model be converted to a state-space object. State-space objects are more accurate.
For most models each axis can be modelled with an identical transfer function.
- Wikipedia article on Damping.