## 4 Results and Discussion
### Positional Accuracy with and without Vibrations  
The next task is to identify the amount of positional error caused by the induced vibrations while the material is deposited at a specific point. To determine this, the nozzle is replaced with the pen attachment. The G-code is modified to put single dots at prescribed points instead of continuous material deposition. After the printer's calibration, graph paper is attached to the printer bed and is aligned with the printer's axes. Without vibrations, the printer is expected to make marks on the graph paper at accurate locations where the nozzle will typically deposit material. Hence, these points were used as a reference for marks made with induced vibrations. The error is measured as the distance between points made by the pen attachment when the vibration is on and off.  
Figure 5 shows a set of marks made by the pen attachment. The left dot represents the reference made without the vibration, and the right represents the mark made with the vibration. The dot printed with vibration is given a deliberate offset of 0.5 mm so that the two marks can be distinguished under the microscope. The pen creates a clear circular indentation upon touching the graph paper, and all measurements are performed at the centers of these circles.  
Figure 4: Material distribution within the 3D-printed samples with and without induced vibrations—both images are taken at the same magnification and the scale is as shown. For the samples shown, the pore sizes are measured at 0.6 mm for the sample without vibration and 0.25 mm for the one with vibration.  
Eighty-one dots are printed on each graph paper without turning the vibration motor on. These dots span the entire printer bed (180 \(\times\) 180 mm). The nozzle head travels from one dot to the next at a speed of 2400 mm/min, and each dot is printed 30 mm away from the previous one. Then 0.5 mm is added, and another set of 81 dots is printed with induced vibrations at S225 (maximum) speed. This process is repeated for two more sheets of paper. The pairs of resulting dots are similar to the ones shown in Figure 5. Using the micro-metrology station, the error in the dot's position printed with vibration is measured for each pair. The error in x- and y-directions are counted separately to ensure a consistent measuring technique. This resulted in 243 error measurements, and Figure 6 shows the average error in each direction as measured. The results show that the average error in position is 0.02 mm or less, which is less than the printer's resolution (0.05 mm as advertised). It is also observed that the error in the y-direction is more significant than that in the x-direction, likely due to the lower stiffness of the mounting bracket in the y-direction.  
Figure 5: Sample vibration and offset non-vibration dots printed as observed through the microscope.  
Figure 6: Error in position induced by vibration. The standard error is too small to show in the figure.  
To understand if the induced vibrations lead to an error significantly different from the standard printing conditions of the printer, a Weibull analysis is performed on the data. Figure 7 shows the studies performed on the x- and y-directions separately. These analyses show that the distribution of errors does not vary significantly between the two experimental conditions--with and without induced vibrations. Any significant differences in the Weibull moduli in a reliability analysis typically indicate a difference in the behavior between the conditions being compared. In this study, the Weibull moduli for these conditions are very similar. This suggests that the induced vibrations do not cause significantly more positional errors than normal printing conditions. In other words, the induced vibrations do not cause a significant deviation from the positional accuracy of the prints.