Transversally reciprocating heat pipes (TRHP) are proposed and analyzed in this paper. The transversally reciprocating heat pipe is a wickless heat pipe that utilizes the transversally reciprocating motion for the liquid return from the condenser section to the evaporator section. The concept of the transversally reciprocating heat pipe is introduced, and its application for the cooling of cutting tools for shaping machines is described. Like many other types of heat pipes, however, the TRHP is subject to certain working limitations. The most important working limitation of a TRHP is the minimum transversally reciprocating frequency (or the critical working frequency) below which the heat pipe ceases to function. Based on the working mechanism of the TRHP, a simplified analytical model is proposed and a closed-form analytical solution for the prediction of the critical working frequency is obtained. It is concluded from the analysis that both the heat pipe inclination angle and the crank radius of the reciprocating mechanism have a significant effect on the critical working frequency. A reasonable selection for the dimensions of the reciprocating mechanism and the length of the heat pipe would also ensure a higher heat-transfer capacity of the heat pipe for effective cooling of the cutting tool.