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# Harmonic Response Analysis Phase Angle Relationships

January 23, 2017 By: Eric Stamper

I thought I’d expand a little more upon an older blog: “How Do You Find the Peak Stress in a Harmonic Response Analysis”. The influence of the phase angle was only briefly mentioned, so this post will provide a little more detail on this topic.

This blog is based on the system shown in Figure 1, above. An overhung shaft with a rotating imbalance. The first two mode shapes are also shown for the first two natural frequencies.

If the amplitude of displacement at the overhung end of the spinning shaft is measured as the speed is increased from zero to 300Hz, the plot would look like what’s shown below in Figure 2. Over this range of speeds, the shaft passes through 2 natural frequencies as indicated by the two amplitude peaks (marked with gray lines).

Figure 2

## phase angle 2.png

The phase angle in this plot refers to the phase of the response relative to the input loading, Zooming in on the first peak, (Figure 3, below) for speeds much lower than the natural frequency, the phase angle is nearly zero. For example, at a near zero shaft speed (Hz<<1), the input loading direction will match the direction of the output displacement and corresponds to the static response. As the shaft rotates faster (increasing in frequency), it will eventually pass through a natural frequency. As this happens, the phase of the input loading and output response separate, as shown by the phase angle curve changing near the natural frequency. In a classic single DOF example, the phase shift at resonance is 90 degrees.

Figure 3: Frequency Response Plot – Zoomed In

## phase angle 3.png

To help illustrate the shift in phase, refer to the Figure 4 below. First, looking down the axis of the shaft, the reference x & y directions and phase angles are shown from the initial position of the eccentric load at 0°. The “Input Load Position Angle 0°” diagram illustrates a snapshot in time when the position of the input load on the shaft, marked with a gray circle, is located at a position of 0°. Each subsequent diagram shows a different snapshot in time as the input load travels around the shaft (load position angles 90°, 180°, and -90° are shown).

All the other colored circles represent the location of the peak displacement output at the varying shaft speeds. The blue circle represents the static solution “near zero speed” and its position tracks with the input load as it travels around the shaft. The response is fully in phase with the load at low speed. As the shaft speed increases and approaches the natural frequency (red circle), it starts to lag the input loading. As the speed continues to increase to the natural frequency and slightly beyond, the lag in phase continues to increase. Also, note that at the natural frequency, the shaft responds with its peak displacement lagging the load by 90°.

Figure 4: Behavior of the Shaft System

## phase angle 4.png

This continual shifting of the phase is further illustrated by looking at a particular component direction of displacement, say X, as the shaft spins up. The “static” response (blue line) at a very low speed in Figure 5 shows that the input loading and output response track the same. If a speed just below the natural frequency is queried, the peak output deflection in the shaft lags the input loading by 45° (red line). At the natural frequency, the peak output deflection in the shaft lags the input loading by 90° (green line), and by 130° at a frequency slightly above the natural frequency (purple line). If this were a single DOF system, the phase shift would asymptotically approach 180°as the speed continues to increase.

Figure 5. Component Direction Responses

## phase angle 5.png

Now if we put everything together in Figure 6, it’s easy to see how the frequency response plot is related to the response of the structure.

Figure 6. Composite Illustration of the Shaft System

## phase angle 6.png

Hopefully, this adds a little more clarity to the phase angle portion of a harmonic response analysis and explains why we see directions of peak response at resonance which appear non-physical relative to the direction of peak load.

I’d be curious to hear about how your structures respond when they’re exposed to loads acting near their natural frequencies.