A machine slide weighing 4 KN is elevated by a two start square threaded 40 mm dia. Screw, 6 mm pitch at the rate of 0.6 m/min. If the coefficient be 0.12 at thread and collar. Calculate the power of motor to drive the slide. The end of the screw is carried on a thrust collar 30 mm inside and 50 mm outside diameter.

 To calculate the power of the motor required to drive the slide, we need to determine the torque required to overcome friction in the screw threads and the collar bearing. Then, we can calculate the power using the rotational speed of the screw.

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Given Data:

1. Slide Details:

   • Weight of slide ($W$) = 4 kN = 4000 N

   • Speed of elevation ($v$) = 0.6 m/min = 0.01 m/s

2. Screw Details:

   • Nominal diameter ($d_o$) = 40 mm

   • Pitch ($p$) = 6 mm

   • Number of starts ($n$) = 2

   • Coefficient of friction for screw and nut ($\mu$) = 0.12

3. Collar Bearing Details:

   • Inside diameter ($D_i$) = 30 mm

   • Outside diameter ($D_o$) = 50 mm

   • Coefficient of friction for collar (${\mu }_c$) = 0.12

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Step 1: Calculate Mean Diameter of Screw ($d_m$)

For a square thread:

• Core diameter ($d_c$) = Nominal diameter ($d_o$) - Pitch ($p$)

  $$d_c=40-6=34\text{mm}$$

• Mean diameter ($d_m$):

  $$d_m=\frac{d_o+d_c}{2}=\frac{40+34}{2}=37\text{mm}$$

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Step 2: Calculate Lead ($l$)

For a multi-start thread, the lead is the distance the nut moves in one full rotation of the screw. For a two-start thread:

$$l=n\times p=2\times 6=12\text{mm}$$

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Step 3: Calculate Helix Angle ($\alpha$)

The helix angle is the angle between the thread helix and a plane perpendicular to the screw axis. It is given by:

$$\tan (\alpha )=\frac{l}{\pi d_m}$$$$\tan (\alpha )=\frac{12}{\pi \times 37}=0.103$$$$\alpha ={\tan }^{-1}(0.103)=5.89\mathrm{°}$$

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Step 4: Calculate Friction Angle ($\varphi$)

The friction angle is related to the coefficient of friction ($\mu$):

$$\varphi ={\tan }^{-1}(\mu )={\tan }^{-1}(0.12)=6.84\mathrm{°}$$

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Step 5: Torque Required to Lift the Load ($T_{lift}$)

The torque required to lift the load is the sum of the torque to overcome screw friction and the torque to overcome collar friction.

1. Torque to Overcome Screw Friction:

   $$T_{screw}=W\cdot \frac{d_m}{2}\cdot \tan (\alpha +\varphi )$$$$T_{screw}=4000\cdot \frac{37}{2}\cdot \tan (5.89\mathrm{°}+6.84\mathrm{°})$$$$T_{screw}=4000\cdot 18.5\cdot \tan (12.73\mathrm{°})$$$$T_{screw}=4000\cdot 18.5\cdot 0.226$$$$T_{screw}=16,724\text{Nmm}=16.724\text{Nm}$$

2. Torque to Overcome Collar Friction:
   The collar friction torque depends on the mean radius of the collar ($r_c$):

   $$r_c=\frac{D_o+D_i}{4}=\frac{50+30}{4}=20\text{mm}$$$$T_{collar}={\mu }_c\cdot W\cdot r_c$$$$T_{collar}=0.12\cdot 4000\cdot 20$$$$T_{collar}=9,600\text{Nmm}=9.6\text{Nm}$$

3. Total Torque to Lift the Load:

   $$T_{lift}=T_{screw}+T_{collar}$$$$T_{lift}=16.724+9.6=26.324\text{Nm}$$

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Step 6: Calculate Rotational Speed of Screw ($N$)

The rotational speed of the screw is related to the linear speed of the slide and the lead of the screw:

$$v=l\cdot N$$$$N=\frac{v}{l}=\frac{0.01}{0.012}=0.833\text{rev/s}$$

Convert to revolutions per minute (rpm):

$$N=0.833\times 60=50\text{rpm}$$

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Step 7: Calculate Power of Motor ($P$)

The power required to drive the screw is given by:

$$P=T_{lift}\cdot \omega$$

where $\omega$ is the angular velocity in radians per second:

$$\omega =2\pi N=2\pi \times 0.833=5.236\text{rad/s}$$$$P=26.324\cdot 5.236=137.8\text{W}$$

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Final Result:

The power of the motor required to drive the slide is approximately 137.8 W.

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Conclusion

To elevate the 4 kN slide at a rate of 0.6 m/min, a motor with a power output of 137.8 W is required. This calculation accounts for the friction in both the screw threads and the collar bearing.