## Interpreting the Input Shaper Graphs

Content

- Theory - Resonances
- Theory - Power Spectral Density (PSD)
- Understanding the Graph
- How the shaper is recommended
- General Guidelines
- Identifying Mechanical Problems

If you are only interested in the graphs, skip the first two chapters. I recommend reading them, as it will improve your understanding of the graphs afterwards.

### Theory - Resonances

The following explanation should make the idea of resonance tests and their influencing factors a bit more transparent. It does not claim to be scientifically complete.

- A periodic force with a certain frequency is applied to a system (the 3D printer). This is called the excitation frequency.
- The excitation frequency starts at a very low frequency (â€śslowâ€ť vibrations) and steadily increases over the measurement cycle to fast vibrations.
- The â€śresponseâ€ť of the system is measured with an
**I**nertial**M**easurement**U**nit (IMU), also known as an accelerometer. The IMU measures the resulting g-forces (unit: G [m/sÂ˛]) at the location where the IMU is mounted. - Depending on the excitation frequency, the system will respond weakly (low amplitude / low Power Spectral Density) or strongly (high amplitude).
- Local maxima may occur at certain frequencies. These are called resonant or natural frequencies (the peaks in the graph).
- Each part of the system may respond differently to the given excitation frequency. This means that each part has its own natural frequency at which it responds with a high amplitude (peak).
- The result is the complex response
- of the entire system
- at the IMU location
- as a function of the frequency at which the system was excited.

### Theory - Power Spectral Density (PSD)

The Power Spectral Density is a clever (though mathematically quite complex) way to make such time / frequency related measurements comparable. The following explanation tries to keep it in very simple terms and may be oversimplified.

As described in the first chapter, the system is excited at a known frequency in the range of 0 Hz to 200 Hz. The IMU measures this excitation acceleration as well as the resulting / superimposed vibrations.

The goal of the PSD method is to break this response down into individual frequencies and calculate how much power or energy a particular frequency contributes to the overall signal. Finally, the results are normalized to ensure that different measurements can be compared and then plotted graphically.

### Understanding the Graph

#### 1 - Y-Axis (left) - Power Spectral Density

See 2. Theory - Power Spectral Density (PSD)

The magnitude of this axis is given in scientific notation, e.g. â€ś1e4â€ť (see the top of the axis). This means that the value of the axis is multiplied by 10â€‰000. So the peak in the example above has a PSD of

7.5â€‰xâ€‰10â€‰000â€‰=â€‰75â€‰000

Any modification that would make the PSD value â€ś1e5â€ť would add another factor of 10, since â€ś1e5â€ť is a multiplication by 100â€‰000.

This can easily lead to misinterpretation, especially when comparing two graphs: A peak with a PSD value of 2 in a graph scaled â€ś1e4â€ť seems to disappear in a graph scaled â€ś1e5â€ť. In fact, the peak is still there, just no longer graphically visible.

#### 2 - X-Axis - Frequency

As explained above, the measurement is made along a broadband input of frequencies and the resulting frequencies are plotted along the X-axis.

#### 3 - Y-Axis (right) - Shaper Vibration Reduction (Ratio)

This axis belongs to the dotted lines of each shaper (see No 5). It represents the â€śeffectivenessâ€ť of a given shaper at a given frequency in reducing vibration.

For example, the orange line of the MVZ shaper has a ratio of 0.4 at 175 Hz. This means that each vibration at 175 Hz is multiplied by 0.4. The result is the residual vibration. Or to put it another way: Vibrations at 175 Hz are reduced by a factor of x2.5 when the MVZ shaper is used.

Again, this is a simplification: Internally, Klipper uses a mathematical scoring system that evaluates each shaper individually at the given frequencies and the corresponding PSD value. See also the chapter How the shaper is recommended.

#### 4 - The actual graph

- Measured vibrations are displayed in each X, Y, Z direction and their corresponding sum.
- The height of the peak is the power of the vibration (PSD).
- The dotted lines are the effectiveness of each shaper, as discussed above.
- The cyan graph is the resulting vibration after applying the recommended shaper (see No 5).

#### 5 - The Shaper Details

- The available shapers are listed in this box.
- They are sorted from least aggressive (least vibration reduction, but also the least smoothing) to most aggressive with highest smoothing.
- The values in brackets (â€¦) are
- Frequency of the shaper for the
`[input_shaper]`

configuration setting. `vibr`

is the remaining total vibration after this shaper has been applied.`sm`

stands for smoothing and represents the effect of the shaper on the motion. Each shaper causes a certain amount of deviation from Klipperâ€™s calculated target motion. This deviation is necessary for the avoidance of vibration, but can also have the effect of smoothing, i.e. the loss of small / fine details. It is a qualitative statement to compare the effect of the different shapers.`accel`

is the maximum recommended acceleration for a given shaper to avoid additional smoothing. For the example above, this means a printer that can run an acceleration of 20â€‰000 should be limited to 15â€‰600 when using the 3HUMP_EI shaper. It**does not**mean that this value should be used as the new acceleration of your printer, which is rated at, say, 3â€‰000.

- Frequency of the shaper for the
- The last piece of information is the recommended shaper. This tries to find the best balance between smoothing and residual vibration.

### How the shaper is recommended

- The calculation is based on the sum of all axes (shown as a purple line on the graph).
- The script calculates an abstract â€śscoreâ€ť for a given shaper and its frequency in two steps:
- A per-frequency score (shown as a cyan line on the graph) based on the sum of all axes.
- A combined score across the spectrum (shown as
`vibr`

in the graphâ€™s box (No. 5)).

- A separate scoring process is used for different input shaper types, taking into account smoothing and the remaining vibration score, to select the final recommended shaper.
- The goal of this score is to correspond to a good shaper choice that reduces ringing for the printer, often resulting in an optimal or near-optimal shaper configuration.

### General Guidelines

- The resulting vibrations are not bound to any direction. This means that an excitation in Y can result in measured vibrations in X and/or Z.
- Low frequency peaks are worse than high frequency peaks because they require lower frequency shapers to compensate, resulting in more smoothing and thus permitting lower maximum accelerations.
- Mechanical tuning should be aimed at moving the lowest peaks to higher frequencies or eliminating some peaks altogether.
- An ideal graph would be a single sharp peak at a very high frequency (greater than 50 - 70 Hz).
- The tuning process considers the X and Y axes separately / independently. On a system where one axis has limited acceleration (either due to design or due to the required shaper), the other axis could benefit from a more aggressive shaper without (significantly) affecting the overall performance.
- High PSD values are not a bad thing. On the contrary, they can indicate a very stiff system.
- The reason for the resonances cannot be deduced from the graph.

### Identifying Mechanical Problems

In fact, the shaper results can be used to identify mechanical problems with the printer:

The two pairs of graphs are from the same device:

- The first is a Cartesian printer (similar to the Ender 5 layout).
- The second is a high-end CoreXY printer.
- The left graphs have a mechanical defect (binding) on one axis.
- The right graphs have the defect fixed.

The following characteristics **may** indicate a mechanical problem with the motion system

- Low power spectral density (< 1e3).
- Very broad spectrum, almost as prominent as the main peak.
- High proportions of auxiliary directions, e.g. high Z values.

The potential reasons for these effects are:

- A binding axis cannot vibrate freely â†’ low PSD.
- A binding axis runs bumpy â†’ Broad spectrum and additional directions.

This is not hard science, as there are certainly printers that show such a spectrum by default, but it is something to consider and check.