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Contents
1 MassSpringDamper^{[1]}
An ideal massspringdamper system with mass m (in kg), spring constant k (in N/m) and viscous damper of damping coeficient c (in Ns/m) can be described by:
Treating the mass as a free body and applying Newton's 2^{nd} law, we have:
 where
 is the acceleration (in m/s^{2}) 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 secondorder 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 2^{nd} Order Systems
The massspringdashpot system is the inspiration of the ideal (or standard) 2^{nd} order transfer function. Most closed loop systems and sensors are designed so that an ideal 2^{nd} order transfer function describes them accurately.
Design Note 
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 2^{nd} order transfer function is
Generic 2^{nd} Order System 
 where
 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
Notation 
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 massspringdashpot system. In an ideal 2^{nd} order transfer function the damping ratio can have a dramatic effect on the system response.
Modeling Note 
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
O2_2 
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
O2_3 
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
Hz,  , 
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
MATLAB Note 
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 statespace object. Statespace objects are more accurate. 
For most models each axis can be modelled with an identical transfer function.