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1 Mass-Spring-Damper

A mass attached to a spring and a damper. The F in the diagram denotes an external force, which this example does not include.

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:

$LaTeX: F_\mathrm{s} = - k x$

$LaTeX: F_\mathrm{d} = - c v = - c \dot{x} = - c \frac{dx}{dt}$

Treating the mass as a free body and applying Newton's 2^nd law, we have:

$LaTeX: \sum F = ma = m \ddot{x} = m \frac{d^2x}{dt^2}$

where

$LaTeX: a$ is the acceleration (in m/s²) of the mass

$LaTeX: x$ 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).

$LaTeX: m \ddot{x} + c \dot{x} + k x = 0.\,$

Rearranging, we have

$LaTeX: \ddot{x} + { c \over m} \dot{x} + {k \over m} x = 0.\,$

Next, to simplify the equation, we define

$LaTeX: \omega_n = \sqrt{ k \over m } \mbox{ and } \zeta = { c \over 2 \sqrt{k m} }$

The first parameter, $LaTeX: \omega_n$, is called the (undamped) natural frequency of the system . The second parameter, $LaTeX: \zeta$, 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

$LaTeX: \ddot{x}+2\zeta\omega_n\dot{x}+\omega_n^2x=0$

2 2^nd Order Systems

The mass-spring-dashpot 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 $LaTeX: \zeta=\frac{1}{\sqrt{2}}$, and a bandwidth ($LaTeX: \omega_n$) at least 10x the system closed loop bandwith.

The ideal 2^nd order transfer function is

$LaTeX: H(s)=K\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \mbox{ for } 0 \le \zeta \le 1$

Generic 2nd Order System

where

$LaTeX: K$ is the system gain

$LaTeX: \omega_n$ is the system's natural frequency

$LaTeX: \zeta$ is the system's damping ratio.

At low frequencies, relative to $LaTeX: \omega_n$,

$LaTeX: H(s) \approx K \frac{\omega_n^2}{\omega_n^2} \approx K$

and at high frequencies

$LaTeX: H(s) \approx K \frac{1}{s^2+2\zeta\omega_n s} \approx \frac{K}{s^2}$

The double integrator $LaTeX: \left(\frac{1}{s^2}\right)$ leads to the -40 dB/decade (order of magnitude) at higher frequencies.

2.1 Damping Ratio

Figure 1: Bode magnitude of ideal 2^nd order Transfer Function

Figure 2: Bode phase of ideal 2^nd order Transfer Function

Notation

The natural frequency in the images is $LaTeX: f_n$ in Hz while the natural frequency in Eqn. O2 1 is $LaTeX: \omega_n$ in rad/sec. The conversion between the 2 parameters is $LaTeX: \omega_n=2 \pi f_n$.

The damping ratio originated from the mass-spring-dashpot system. In an ideal 2^nd order transfer function the damping ratio $LaTeX: \zeta$ can have a dramatic effect on the system response.

Modeling Note

When modeling a system to match measured data the phase is a better way to match $LaTeX: \zeta$ 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 $LaTeX: \zeta$ matching.

Figure 1 shows that the peak magnitude of the transfer function is dependent upon $LaTeX: K$ and $LaTeX: \zeta$. Adjusting the overall system gain is a simple matter so $LaTeX: K$ is not particularly important. However, there isn't any way to compensate or adjust $LaTeX: \zeta$ in real systems. Figure 2 shows the dependency of the phase shift at $LaTeX: f_n$ on $LaTeX: \zeta$. Clearly the smaller $LaTeX: \zeta$ is the steeper, more rapidly, the phase shifts from 0 degrees to -180 degrees.

2.2 Peak Magnitude

Figure 3: Bode Plot detailing the height of the peak magnitude

The peak magnitude $LaTeX: M_p$ is a function of $LaTeX: \zeta$. That relation is

$LaTeX: M_p = \left| H \left( j \omega_p \right) \right| = \begin{cases} 1, & \mbox{if }\zeta \ge \frac{1}{\sqrt{2}} \\ \frac{1}{2 \zeta \sqrt{1-\zeta^2}}, & \mbox{if }0 \le \zeta < \frac{1}{\sqrt{2}}\end{cases}$

O2_2

2.3 Peak Frequency

The frequency of the peak magnitude ($LaTeX: \omega_p$) is not the same as $LaTeX: \omega_n$. Often $LaTeX: \omega_p$ and $LaTeX: \omega_n$ are very close. The relationship of $LaTeX: \omega_p$ to $LaTeX: \omega_n$ is

$LaTeX: \omega_p = \begin{cases} 0, & \mbox{ if }\zeta \ge \frac{1}{\sqrt{2}} \\ \omega_n \sqrt{1-2 \zeta^2}, & \mbox{ if }0 \le \zeta < \frac{1}{\sqrt{2}}\end{cases}$

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

$LaTeX: \omega_n=100$ Hz,

$LaTeX: \zeta=\frac{1}{\sqrt{2}}$,

$LaTeX: K=1$

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

Honeywell GG5300 MEMS Gyro Sensor Model

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 state-space object. State-space objects are more accurate.

For most models each axis can be modelled with an identical transfer function.

DataSheet

5 External Resources

• Damping
• Damping Ratio
• Q factor
• Harmonic Oscillator