Second Order Systems


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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}

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

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 phase100 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]

5 External Resources