Second Order Systems

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## 1 Mass-Spring-Damper[1]

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 2nd 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 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.

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

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

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

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

## 4 References

1. Wikipedia article on Damping. [1]