Extension of the definition of tolerance and an application thereof in the calculation of dimension chains

When calculating the tolerance of an assembly dimension chain with compensation loop, the value calculated by the basic formula for dimension chains must be a negative or imaginary number; this value is referred to as virtual tolerance in this paper. Virtual tolerance is not covered by tolerance theory. However, after reviewing such relevant concepts such as negative and positive numbers, the essence of virtual tolerance can be defined as follows: the absolute value of virtual tolerance (or the imaginary part) is the error compensation amount and the size between the upper and lower deviations is the compensation range. Therefore, the amount and range of error compensation are both added to the scope of tolerance so that they can be described in a digital way; in the meantime, the range of tolerance is extended to any number. Based on the concept of virtual tolerance, the tolerance and the error compensation amount can be calculated synchronously, leading to a simplified analysis and calculation process; computer-aided design is also facilitated. According to probability theory and the relevant concepts of dimension chains, the only correct result is that the error compensation amount is the imaginary part of virtual tolerance. But since the error compensation amount calculated by the probabilistic method does not ensure assembly accuracy, a theoretical basis is provided to calculate the assembly dimension chain with compensation loop by the probabilistic method.