Figure 11.1 Commonly seen Bode plot

The mass-spring-damper transfer function from the previous chapter is expanded in Figure 11.2 A phasor transform example. In this example the transfer function is multiplied by the complex conjugate to eliminate the complex number in the denominator. The magnitude of the resulting transfer function is the gain, and the phase shift is the angle. Note that to correct for the quadrant of the phase shift angles pi radians is subtracted for certain frequency values.

Figure 11.2 A phasor transform example

The results in Figure 11.2 A phasor transform example are normally left in variable form so that they may be analyzed for a range of frequencies. An example of this type of analysis is done in Figure 11.3 A phasor transform example (continued). A set of frequencies is used for calculations. These need to be converted from Hz to rad/s before use. For each one of these the gain and phase angle is calculated. The gain gives a ratio between the input sine wave and output sine wave of the system. The magnitude of the output wave can be calculated by multiplying the input wave magnitude by the gain. (Note: recall this example was used in the previous chapter) The phase angle can be added to the input wave to get the phase of the output wave. Gain is normally converted to 'dB' so that it may cover a larger range of values while still remaining similar numerically. Also note that the frequencies are changed in multiples of tens, or magnitudes.

Figure 11.3 A phasor transform example (continued)

In this example gain is defined as x/F. Therefore F is the input to the system, and x is the resulting output. The gain means that for each unit of F input to the system, there will be gain*F=x output. If the input and output are sinusoidal, there is a difference in phase between the input and output wave of θ (the phase angle). This is shown in Figure 11.4 A phasor transform example (continued), where an input waveform is supplied with three sinusoidal components. For each of the frequencies a gain and phase shift are calculated. These are then used to calculate the resulting output wave, relative to the input wave. The resulting output represents the steady-state response to the sinusoidal output.

Figure 11.4 A phasor transform example (continued)

11.2 BODE PLOTS

In the previous section we calculated a table of gains and phase angles over a range of frequencies. Graphs of these values are called Bode plots. These plots are normally done on semi-log graph paper, such as that seen in Figure 11.5 4 cycle semi-log graph paper. Along the longer axis of this paper the scale is logarithmic (base 10). This means that if the paper started at 0.1 on one side, the next major division would be 1, then 10, then 100, and finally 1000 on the other side of the paper. The basic nature of logarithmic scales prevents the frequency from being zero. Along the linear axis (the short one) the gains and phase angles are plotted, normally with two graphs side-by-side on a single sheet of paper.

Figure 11.5 4 cycle semi-log graph paper

Figure 11.6 Drill problem: Plot the points from Figure 11.3 A phasor transform example (continued) on graph paper

Figure 11.7 Aside: Bode Plot Example with Scilab

Figure 11.8 Drill problem: Plot the points from Figure 11.2 A phasor transform example with a computer

Figure 11.9 Drill problem: Draw the Bode plot for gain and phase

11.3 STRAIGHT LINE APPROXIMATIONS

An approximate technique for constructing a gain Bode plot is shown in Figure 11.10 The method for Bode graph straight line gain approximation. This method involves looking at the transfer function and reducing it to roots in the numerator and denominator. Once in that form, a straight line approximation for each term can be drawn on the graph. An initial gain is also calculated to shift the results up or down. When done, the straight line segments are added to produce a more complex straight line curve. A smooth curve is then drawn over top of this curve.

Figure 11.10 The method for Bode graph straight line gain approximation

Figure 11.11 Why the straight line method works

Figure 11.12 Why the straight line method works (cont'd)

Figure 11.13 Drill problem: Slope of second order transfer functions.

An example of the straight line plotting technique is shown in Figure 11.14 An approximate gain plot example. In this example the transfer function is first put into a root form. In total there are three roots, 1, 10 and 100 rad/sec. The single root in the numerator will cause the curve to start upward with a slope of 20dB/dec after 1rad/sec. The two roots will cause two curves downwards at -20dB/dec starting at 10 and 100 rad/sec. The initial gain of the transfer function is also calculated, and converted to decibels. The frequency axis is rad/sec by default, but if Hz are used then it is necessary to convert the values.

Figure 11.14 An approximate gain plot example

The example is continued in Figure 11.15 An approximate gain plot example (continued) where the straight line segments are added to produce a combined straight line curve.

Figure 11.15 An approximate gain plot example (continued)

Finally a smooth curve is fitted to the straight line approximation. When drawing the curve imagine that there are rubber bands at the corners that pull slightly and smooth out. For a simple first-order term there is a 3dB gap between the sharp corner and the function. Higher order functions will be discussed later.

Figure 11.16 An approximate gain plot example (continued)

The process for constructing phase plots is similar to that of gain plots, as seen in Figure 11.18 An approximate phase plot example. The transfer function is put into root form, and then straight line phase shifts are drawn for each of the terms. Each term in the numerator will cause a positive shift of 90 degrees, while terms in the denominator cause negative shifts of 90 degrees. The phase shift occurs over two decades, meaning that for a center frequency of 100, the shift would start at 10 and end at 1000. If there are any lone 'D' terms on the top or bottom, they will each shift the initial value by 90 degrees, otherwise the phase should start at 0degrees.

Figure 11.17 The method for Bode graph straight line gain approximation

The previous example started in Figure 11.14 An approximate gain plot example is continued in Figure 11.18 An approximate phase plot example to develop a phase plot using the approximate technique. There are three roots for the transfer function. None of these are zero, so the phase plot starts at zero degrees. The root in the numerator causes a shift of positive 90 deg, starting one decade before 1rad/sec and ending one decade later. The two roots in the denominator cause a shift downward.

Figure 11.18 An approximate phase plot example

The straight line segments for the phase plot are added in Figure 11.19 An approximate phase plot example (continued) to produce a straight line approximation of the final plot. A smooth line approximation is drawn using the straight line as a guide. Again, the concept of a rubber band will smooth the curve.

Figure 11.19 An approximate phase plot example (continued)

11.3.1 Second Order Underdamped Terms

The previous example used a transfer function with real roots. In a second-order system with double real roots (overdamped) the curve can be drawn with two overlapping straight line approximations. If the roots for the transfer function are complex (underdamped the corner frequencies will become peaked. This can be handled by determining the damping factor and natural frequency as shown in Figure 11.20 Resonant peaks. The peak will occur at the damped frequency. The peaking effect will become more pronounced as the damping factor goes from 0.707 to 0 where the peak will be infinite.

Figure 11.20 Resonant peaks

The approximate techniques do decrease the accuracy of the final solution, but they can be calculated quickly. In addition these curves provide an understanding of the system that makes design easier. For example, a designer will often describe a system with a Bode plot, and then convert this to a desired transfer function.

Figure 11.21 Drill problem: Draw the straight line approximation

11.3.2 Lone Ds on the Top or Bottom

- cancel the Ds on the top and bottom if possible

- consider the Ds to have corner frequencies at zero. This means that Ds on the top start at infinity and come down. Ds on the bottom start at negative infinity and come up.

- The Ds are on the top, and they are second order so the phase angle starts at 180 degrees, and the gain starts going upwards at 40dB/dec.

- eventually the squared term on the bottom begins to work and the top and bottom cancel out to give a gain of 0dB.

- with these types of problems the top and bottom of the transfer function might go to zero or infinity. Normally we would say this is underdefined. In these cases, L'Hospital's rule may be used.

- in this case the graph is drawn from the right hand side, working towards the left.

- The example below shows

Figure 11.22 A System Gain Starting at Negative Infinity

- The problem solution is similar when the lone Ds are on the bottom.

- All of the examples presented up to now have started or stopped and could be drawn from the left or right. This will not always be the case as shown in the example below.

Figure 11.23 Calculating a point to set a position

11.4 FREQUENCY RESPONSE FUNCTIONS

- In many cases the behavior of a system must be determined experiementally.

- One common method is to excite the system with a range of input frequencies, measure the output response, and plot the gain and phase. This is the experimental equivalent of the Bode plot, called the Frequency Response Functions (FRF).

- Given a FRF, a transfer function can be developed by reversing the straight line method for drawing Bode plots.

- Consider the example below

Figure 11.24 Example FRF

- This can be converted by doing a straight line fit to the function. Because the graph is drawn using experimental values there is some need to allow for points that are off the straight line. In other words, fit the curve using a good educated guess.

Figure 11.25 Example FRF

11.5 SIGNAL SPECTRUMS

If a vibration signal is measured and displayed it might look like Figure 11.26 A vibration signal as a function of time. The overall sinusoidal shape is visible, along with a significant amount of 'noise'. When this is considered in greater detail it can be described with the given function. To determine the function other tools are needed to determine the frequencies, and magnitudes of the frequency components.

Figure 11.26 A vibration signal as a function of time

A signal spectrum displays signal magnitude as a function of frequency, instead of time. The time based signal in Figure 11.26 A vibration signal as a function of time is shown in the spectrum in Figure 11.27 The spectrum for the signal in Figure 11.26. The three frequency components are clearly identifiable spikes. The height of the peaks indicates the relative signal magnitude.