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3 2.2 Lead Screw Engineering 9 Fig. 2.3 Dimensions of screw threads Fig. 2.4 Screws with various numbers of starts where d m is the pitch diameter and l is the lead and it is defined as l ¼ n s p; (2.2) where p is the screw pitch (distance between identical points of two consecutive threads) and n s is the number of starts (or starts). Figure 2.4 shows three 1-in. lead screws with one, two, and ten starts. Increasing the number of starts increases the lead thus increasing the translational velocity of the nut for a given fixed angular velocity of the screw. Based on (2.1) and (2.2) the lead angles for these screws are found as follows: l ðaþ ffi 5:20, l ðbþ ffi 10:31, and l ðcþ ffi 18:52. In these examples, the pitch diameter was found according to the following equation: d m ¼ D ðp=2þ ¼d þðp=2þ. 2.2 Lead Screw Engineering For design and selection purposes, the mechanical analysis of lead screws usually is limited to the factors affecting their static or quasi-static performance, such as efficiency, driving torque requirements, and load capacity [33 35]. There are

5 2.2 Lead Screw Engineering 11 Fig. 2.5 Lead screw design and selection factors with preloaded springs that can move with respect to one another to compensate backlash and wear [36, 41 44]. The drawback of using these nuts is in the increased friction force, which lowers the efficiency and increases the required driving torque. 4 4 See Sect. 5.4 for a mathematical model of a lead screw with an anti-backlash nut.

7 2.4 Effect of Thread Angle 13 Fig. 2.7 Sign convention for contact forces between nut and lead screw 2.4 Effect of Thread Angle Before moving on to the dynamic models of lead screw systems, the effect of thread geometry on the contact forces is considered here. The force interaction shown in Fig. 2.7 is essentially correct for the square threads where the normal force is parallel to the lead screw axis. For Acme or other types of threads, a slight modification is needed to take into account the thread angle. Figure 2.8 shows the thread semi-angles as measured on a section through the axis of a screw, c a, and as measured on a section perpendicular to the helix, c n. Using the geometric relationship in Fig. 2.9, one can write [47] tan c n ¼ x n y ; tan c a ¼ x a y ; (2.5) x n ¼ x a cos l: (2.6)

8 14 2 Lead Screws Fig. 2.8 Effect of lead angle on the measurement of thread angle Fig. 2.9 Geometry of the threads on two different section planes Combining (2.5) and (2.6) gives tan c n ¼ tan c a cos l: Figure 2.10 shows a portion of a lead screw with localized contact force ^N (perpendicular to the thread surface) and friction force F f. The X-axis of XYZ coordinate system is parallel to the lead screw axis. The x-z plane is perpendicular to the helix. The projection of contact force on the x-y (or X-Y) plane is calculated as N ¼ ^N cos c n : (2.7) Since ^N is the normal force, using (2.4) the friction force for trapezoid threads is calculated by F f ¼ ^m ^N sgnð yþ, _ where ^m is the true coefficient of friction. One can define the apparent coefficient of friction as

9 2.4 Effect of Thread Angle 15 Fig Forces acting on a thread m m ¼ ^m cos c n ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (2.8) tan 2 c a cos 2 l þ 1 Using (2.7) and (2.8), the friction force is written conveniently as F f ¼ mjnjsgnð yþ, _ which is the same as (2.4) and will be used in the subsequent chapters.

10

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