Question 15.1: Torque and Efficiency of a Power Screw Problem Determine the...

1    There are several aspects to this problem. We need to compute the lifting and lowering torques for two cases, one with a sliding-friction collar and one with a ballbearing collar. In both cases we will calculate the screw and collar contributions to torque and efficiency separately for comparison and also combine them. First do the case of the sliding collar.

2    Since it is a single-start thread, the lead L equals the pitch p, which is 1 / N = 0.2. The pitch diameter of the thread d_{p} is found in Table 15-3 (p. 867). The torque to lift the load is found from equation 15.5a (p. 869):

T_u=T_{s_u}+T_c=\frac{P d_p}{2} \frac{\left(\mu \pi d_p+L \cos \alpha\right)}{\left(\pi d_p \cos \alpha-\mu L\right)}+\mu_c P \frac{d_c}{2}        (15.5a)

Note that the collar friction exceeds the screw friction.

3    The torque to lower the load is found from equation 15.5b:

T_d=T_{s_d}+T_c=\frac{P d_p}{2} \frac{\left(\mu \pi d_p-L \cos \alpha\right)}{\left(\pi d_p \cos \alpha+\mu L\right)}+\mu_c P \frac{d_c}{2}        (15.5b)

4    The efficiency in the lifting mode will be less than for lowering and is found from equations 15.7 (p. 871). We choose the version shown as equation 15.7c in order to account for both screw and collar components.

W_{i n}=2 \pi T       (15.7a)

W_{\text {out }}=P L        (15.7b)

e=\frac{W_{\text {out }}}{W_{i n}}=\frac{P L}{2 \pi T}      (15.7c)

e=\frac{L}{\pi d_p} \frac{\pi d_p \cos \alpha-\mu L}{\pi \mu d_p+L \cos \alpha}          (15.7d)

e=\frac{\cos \alpha-\mu \tan \lambda}{\cos \alpha+\mu \cot \lambda}          (15.7e)

5    Now recalculate the collar torque and total torque to lift the load with a ball-bearing thrust washer, using equation 15.4f.

T_c=\mu_c P \frac{d_c}{2}      (15.4f)

T_c=\mu_c P \frac{d_c}{2}=0.02(1000) \frac{1.75}{2}=17.5  lb – in          (d)

T_u=T_{s_u}+T_c=122.0+17.5=139.5  lb -\text { in }          (e)

6    The efficiency with the ball-bearing thrust washer is now:

e=\frac{P L}{2 \pi T}=\frac{1000(0.2)}{2 \pi(139.5)}=0.23        (f)

The improvement in efficiency is significant and shows why it is poor practice to use anything but a rolling-element bearing as a thrust washer on a power screw.

7    The self-locking aspects of the screw are independent of the thrust collar friction and can be found from equation 15.6a (p. 870).

\mu \geq \frac{L}{\pi d_p} \cos \alpha \quad \text { or } \quad \mu \geq \tan \lambda \cos \alpha          (15.6a)

\begin{aligned} \mu & \geq \frac{L}{\pi d_p} \cos \alpha \\ 0.15 & \geq \frac{0.2}{\pi(1.15)} \cos \left(14.5^{\circ}\right) \\ 0.15 & \geq 0.06 \end{aligned}    \quad     so it self – locks  (g)

Note that the positive lowering torque in step 3 also indicates that the screw is selflocking. A negative lowering torque means that a braking torque of opposite sense to the lift torque must be applied to hold the load. The files EX15-01A and EX15-01B are on the CD-ROM.