Week 8 - Analysis on the Bottom Housing Sub-assembly

Note: This section was updated after week 12&13 to provide appropriate sub-assembly drawings for reference.

Motor Choice

For our sub-section, we needed to find a motor that would allow the rack grippers to expert a 300 Newton force on the jar/bottle that needs to be unscrewed; the 300 N force benchmark was chosen due to it being an average benchmark strength found in males aged 20 to 49. When choosing our motor, we had a few limitations to work with, such as the housing size of the rack and pinion sub-assembly, the amount of space needed to create the a gear train that would allow the motor produce a 300N force for the rack, and the amount of money to keep the project within budget. From these limitations, we decided to go with the Uxcell DC 12V 220RPM Worm Gear Motor that is also used for the top gripper sub-assembly due to its user-friendly wiring interface and generally small shape; however, to achieve the 300 N force from this motor using a gear train, the revolutions per minute of the pinion-driving shaft was calculated to be less that 1RPM, meaning that the process of griping the bottle using this motor would be slow and not up to the user’s standards.

Motor Analysis

To determine the power of the of motor, the following equation is used. Since this motor is going to be loaded and only the non-loaded values of the voltage and current are given by the manufacturer, a 70% efficiency is assumed to exist in the the motor.

power= Voltage* Amps* 0.7

Using the equation, the power is found to be 5.04 Watts or 0.00504 kW.

Gear Analysis: Output Force

We need to determine the RPM required to drive the pinion so that the force the rack exerts on the bottle/jar is roughly the 300N benchmark; to calculate the RPM required, the following equation was used.

Where n is the revolutions per minute of the pinion gear, H is the power from the motor in kW, d is the outer diameter of the pinion in mm, Wt is the translated load of the rack in kN. To calculate the value of d for the spur pinion, the smallest number of teeth for said gear needs to first be found using this equation.

Where Np is the number of teeth on the pinion, 𝜙 is the pressure angle of the gear in degrees, and k is a constant dependent on the type of teeth on gears. In this case, the pressure angle is 20 degrees and the value of k is 1. Because the number of teeth can only be a whole number, the amount found is rounded up to its current value of 18 teeth. Using McMaster-Carr as the basis of the gear dimensions, the outer diameter of the of the spur pinion is 21mm since this is the the smallest 18 tooth gear available has an outer diameter of 21mm; because we want to conserve space, we ideally want to use the smallest size for gear components possible.

To get the RPM output from the motor to the desired 15.27RPM, a gear train must be implemented. We used a gear generator application to model our potential gear train; when it came to selecting gears, we put a few restrictions in place for decision making: the gears had to have a pressure angle of 20 degrees, have a diametral pitch of 24 teeth per inch, be made of plastic, and be available from a vendor like McMaster-Carr. With these restrictions in place, we were able to find a suitable gear train.

The gear train used to drive the rack and pinion consists of three 18 tooth gears and three 48 tooth gears; this combination produces a total gear ratio of 18.96:1 and causes the pinion to turn ideally at a speed of 11.6 RPM. To calculate the resulting translated load from the found RPM while taking account of the 70% efficiency of the motor and a 10% deficiency from the gears, the following equation was used:

The resultant translated load that the rack exerts on the bottle/jar using the gearbox setup and taking in account the 70% efficiency cap of the motor and the 10% deficiency rate from the gears is 316N. This is 16 N greater than 300N goal that we had strived for, which means that the system should work in theory. The factor of safety is rather low comparing the actual translated load to the desired translated load with the factor of safety being 1.05.

Gear Analysis : Forces and Stresses

Even though the force the rack exerts on the bottle/jar has been found to be a sufficient amount to meet the 300 N benchmark, it is important to look at the gear to gear forces and stresses present to see if the gears will not fracture. The gear will fracture if the stress found in the gears is greater than the material's yield strength, which is 48.3 MPa for ABS plastic. First, the force between the gears needs to be calculated using the following equation, which is similar to the equation used to calculate the translated load.

Where Wt is the translated load in kN, H is the power output from the motor in kW, d is the outer diameter of the gear in mm, and n is the rotational speed of the gear in RPM. With the translated load found, the Lewis Bending Equation is used to find the stress in the gear.

Where σ is the Lewis bending stress in MPa, P is the pitch of the gear in teeth per mm, F is the face width of the gear in mm, Wt is the translated load in Newtons, and Y is the Lewis Form Factor. The Lewis Form Factor for a 18 tooth and 48 tooth gear is 0.309 and 0.409 respectively.

Gear Analysis: Miscellaneous Analysis

From the info given by the manufacturer, we can also calculate a few other values of interest for analysis , such as a gear pairing's contact ratio, base pitch, path of approach/recess, addendum, and length of path of contact. The equation to to find the addendum is as follows:

Where r is the addendum in inches, dg is the pitch diameter in inches, and N is the number of teeth. with the addendum for both gears in a gear pairing calculated, the path of approach (for the driving gear) and the path of recess (for the driven gear) can be found using the same equation and should use the gear's corresponding r and dg values.

Where PL is the path of recess in inches, KP is the path of approach in inches, and 𝜙 is the pressure angle of the gear. To find the length of path of contact, the following equation is used :

Where KL is the length of path of contact in inches. To find the base pitch, the follow equation is used :

Where pb is the base pitch in inches. To find the contact ratio, the following equation is used:

Gear Table and Key

The following is a table of the gears used in this sub-assembly and their properties that are either provided by their manufacturer, McMaster Carr, or where calculated using the formulas described the the previous sections. Since some of the length dimensions provided by the manufacturer are in English units, both English and Metric units are posted for these length dimensions. the diagram below shows what gear is what in the table with the corresponding shorthand.

References

Budynas, Richard Gordon., and J. Keith. Nisbett. Shigley's Mechanical Engineering Design. McGraw-Hill Education, 2015.

Nilsen, Tove, et al. “Grip Force and Pinch Grip in an Adult Population: Reference Values and Factors Associated with Grip Force.” Scandinavian Journal of Occupational Therapy, vol. 19, no. 3, 2011, pp. 288–296., doi:10.3109/11038128.2011.553687.

“Thermoplastics - Physical Properties.” Engineering ToolBox, 2005, www.engineeringtoolbox.com/physical-properties-thermoplastics-d_808.html.