The square root rule – a case study of a scaling factor for machines with dynamic similitude

Abstract

This paper aims to present a case study where a scaling factor was used to design a series of toggle style injection molding machines for the plastics industry. The series ranged from 100 to 1000 US tons of clamping capacity. Every machine in the series was scaled down or up from a prototype that had undergone extensive fatigue testing before the decision to design an entire range of machines. In this study, a comparison is made between a prototype machine with 250 US tons of clamping force and a scaled-up machine with 1000 US tons of clamping force. Finding dimensionless scaling factors when designing a series of machines requires machines that have dynamic similitude. To qualify, there must be geometric similarity, dynamic similarity, and kinematic similarity. In the case study presented below, the geometric (static) similarity will be forced to exist, and the dynamic similarity and kinematic similarity will be shown to result from this. A dimensionless scale factor (square root rule) was derived by scaling dimensions based on the square root of the clamping force ratio. As each size machine was built, measurements showed that stresses in all critical points were the same as at corresponding points on the prototype. Also, deflections in each direction followed the scale factor. Static relationships in the toggle systems followed the scale factor, and critical positions of the linkage system all occurred at the same link angles. By using this scale factor, the company saves thousands of man-hours in design effort, completes the design for the entire line and gets to the market in less than half the weeks normally required for such an effort. Additional benefits are: (i) very high confidence level in resistance to failure by metal fatigue, (ii) very accurate advanced information on tooling requirements for manufacturing, and (iii) very accurate advanced marketing and sales information on performance specifications, floor space requirements, and energy requirements for the entire range.

Keywords:

• Design process
• static relationships
• kinematics
• linkages
• product family design
• scale factor
• reliability in design
• fatigue failure
• energy requirements
• injection molding machines

Acknowledgments

The authors would like to extend thanks to the following professionals for their help:

1. Reed Division of Package Machinery Company Limited (thanks to Marden Edwards for permission to use a photo of a Reed 250 ton molding machine)

2. Carl Hanser Verlag Munich for permission to use some figures from the works in Farrell (1997) and Osswald et al. (2008).