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6.4: Power Screws

  • Page ID
    53830
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    A power screw (also sometimes called a lead screw) is another simple machine that can be used to create very large forces. The screw can be thought of as a wedge or a ramp that has been wound around a shaft. By holding a nut stationary and rotating the shaft, we can have the nut sliding either up or down the wedge in the shaft. In this way, a relatively small moment on the shaft can cause very large forces on the nut.

    A cider press consisting of a wooden bucket and frame connected by a large metal screw, rotated by turning a large wheel-shaped handle.
    Figure \(\PageIndex{1}\): The screw in this cider press is rotated with the handle on the top. The stationary nut on the frame of the press forces the shaft downward as it is turned. Public Domain image by Daderot.
    A scissor jack being used to raise or lower a car.
    Figure \(\PageIndex{2}\): By rotating the screw in this scissor jack, the user can move the nut (on the left end) closer or further from the anchor on the right end, which will raise or lower the car. Public Domain image by nrjfalcon1.

    Static Analysis of Power Screws:

    The easiest way to analyze a power screw system is to turn the problem into a 2D problem by "unwrapping" the ramp from around the shaft. To do this we will need two numbers. First we will need the diameter of the shaft, and second we will need either the threads-per-inch/centimeter or the pitch of the screw. The threads-per-inch tells you how many threads you have per inch/centimeter of screw. With a single thread design (most screws) this will also be the number of times the thread wraps around the screw in one inch/centimeter. The pitch, on the other hand, gives you the distance between two adjacent threads. Either of these numbers can be used to find the other.

    Once we have these numbers, we can imagine unwrapping the ramp from around the screw and ending up with a ramp in one of the two situations below. In either case, we can use the inverse tangent function to find the lead angle, which can be thought of as the angle of the thread that the nut is climbing up. Finding the lead angle is the first step in analyzing a power screw system.

    Two copies of a 2D representation of a power screw, with the thread represented as a straight line sloping upwards from left to right, the nut represented as a right triangle whose hypotenuse is in contact with the upper side of the thread, and the lead angle (theta) as the angle the thread makes above the horizontal. In one version of the diagram, the vertical distance that the thread covers is given as 1 inch, and the horizontal distance it covers (l) is the product of pi, the shaft diameter, and the threads per inch. In the other version, the vertical distance the thread covers is given by the pitch, and l is the product of pi and the shaft diameter.
    Figure \(\PageIndex{3}\): The lead angle of a screw is the angle of the thread that the nut will be climbing up as the shaft rotates.

    Once we find the lead angle, we can draw a free body diagram of the "nut" in our unwrapped system. Here we include the pushing force which is pushing our nut up the incline, the load force which is the force the nut exerts on some external body, the normal force between the nut and screw, and the friction force between the nut and the screw.

    Free body diagram of the nut from the power screw diagram in Figure 3 above. The nut, which points to the right, experiences a pushing force on its base, a downwards load force on its flat side (which faces upwards and is not in contact with the thread), a friction force down and to the left along its side in contact with the thread, and a normal force pointing up and to the left that makes an angle to the vertical that is equal to the lead angle (theta).
    Figure \(\PageIndex{4}\): The free body diagram of the "nut" in our power screw system, with a standard-orientation \(xy\)-coordinate system.

    If our screw is pushing a load at some constant rate, then we can assume two things: First, the nut is in equilibrium, so we can write out the equilibrium equations for the nut. Second, the nut is sliding, indicating that the friction force will be equal to the normal force times the kinetic coefficient of friction.

    \[ F_f = \mu_k * F_N \]

    \[ \sum F_x = F_{push} - F_N * \sin (\theta) - \mu_k * F_N * \cos (\theta) = 0 \]\[ \sum F_y = F_{push} - F_N * \cos (\theta) - \mu_k * F_N * \sin (\theta) = 0 \]

    We can then simplify the equations above into a single equation relating the load force and the pushing force.

    \[ F_{push} = \frac{\sin (\theta) + \mu_k * \cos (\theta)}{\cos (\theta) - \mu_k * \sin (\theta)} * F_{load} \]

    In reality, the pushing force is not a single force at all. It is the forces preventing the nut from rotating with the screw. The cumulative pushing force will really cause an equal and opposite moment to the input moment that is spinning the shaft.

    Diagram showing the nut as a smaller circle concentric with the shaft. The push force is applied upwards at the rightmost edge of the nut, creating a clockwise moment about the circle center whose magnitude equals the product of the nut radius and the push force magnitude.
    Figure \(\PageIndex{5}\): The pushing force is really just a representation of the forces keeping the nut from rotating. If the nut is not rotating, then these forces must cause an equal and opposite moment to the torque that is driving the screw.

    Finally, if we replace the pushing force with the moment that is driving the screw in our system (in this case the torque \(T\)), we can relate the input torque that is driving our screw to the force that the nut on the screw is pressing forward with. Screw systems are usually designed to allow fairly small input moments to push very large load forces.

    \[ T = \frac{\sin (\theta) + \mu_k \cos (\theta)}{\cos (\theta) - \mu_k \sin (\theta)} * F_{load} * r_{shaft} \]

    Self-Locking Screws

    Imagine that we apply a torque to a power screw to lift a body; then when we get the load to the desired height we stop applying that torque to let the body sit where it is. If we were to redraw our free body diagram from earlier for the new situation, we would find two things.

    1. The pushing force is missing (since a torque is no longer applied to the shaft).
    2. The friction is now fighting against the nut sliding back down the ramp.
    The free body diagram from Figure 4 above, but with the pushing force eliminated and with the frictional force now pointing up along the slope of the thread.
    Figure \(\PageIndex{6}\): Without the pushing force from an applied torque, the friction force acts to prevent the nut from sliding down the ramp.

    With this new free body diagram, there are two possible scenarios that could occur:

    1. The friction force is large enough to keep the nut from sliding down the ramp, meaning everything will remain in static equilibrium if released.
    2. The friction force will not be sufficient to keep the nut from sliding down the ramp, meaning that the load would begin to fall as soon as the torque is removed from the shaft.

    With power screw applications such as a car jack, the second option could be very dangerous. It is therefore important to know if a power screw system is self-locking (scenario 1 above) or not self-locking (scenario 2 above).

    To define the boundary between self-locking systems and non-self-locking systems, we use something called the self-locking angle. As intuition would tell us, slipping does not occur on very gentle slopes (small lead angles) while it does occur on very steep slopes (large lead angles). The angle at which the nut would begin to slip is known as the self-locking angle.

    To find the self-locking angle, we will assume impending motion (relating the friction force to the normal force) and leave the lead angle as an unknown. This lets us create the free body diagram as shown below and gives us the equilibrium equations below.

    The free body diagram from Figure 6 above, with the friction force expressed as the coefficient of static friction times the normal force and the lead angle, as well as the angle the normal force vector makes with the vertical, being labeled as the locking angle (theta_locking).
    Figure \(\PageIndex{7}\): To find the self-locking angle we will assume impending motion. We then draw our free body diagram (above) and write out our equilibrium equations (below) accordingly.

    \[ \sum F_x = - F_N * \sin (\theta) + \mu_s * F_N * \cos (\theta) = 0 \] \[ \sum F_y = -F_{load} + F_N * \cos (\theta) + \mu_s * F_N * \sin (\theta) = 0 \]

    Using the \(x\) equilibrium equation as a starting point, we can solve for the angle \(\theta\) (eliminating the normal force all together in the process). This new equation shown below gives us the self-locking angle. \[ \theta_{locking} = \tan ^{-1} (\mu_s) \]

    Systems with lead angles smaller than this will be self-locking, while systems with lead angles larger than this will not be self-locking.

    Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/mci-kH14JGw.

    Example \(\PageIndex{1}\)

    The power screw below is being used to lift a platform with a weight of 12 pounds. Based on the information below...

    • What is the required torque on the shaft to lift the load?
    • Would the load fall if the toque was removed from the shaft?
    A power screw being used to lift a platform. The screw diameter is 0.375 inches, the threads-per-inch is 12, and the coefficient of static and kinetic friction is 0.16.
    Figure \(\PageIndex{8}\): problem diagram for Example \(\PageIndex{1}\). A power screw with a 0.375-inch screw diameter, 12 threads per inch, is used to lift a platform, with a coefficient of static and kinetic friction of 0.16.
    Solution
    Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/uB2r3AtxCRs.

    This page titled 6.4: Power Screws is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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