Introduction to Threaded Components

Threaded components are any mechanical device that has spiral ridges (threads) that form a helical shape around a central cylinder. One of the most common examples of a threaded components is a screw, which uses the angled threads to create a large force along the length of the screw when a modest input torque is used to tighten the screw. However, screws are not the only mechanical components that use threads. Threads are also present in the feet used to level household appliances, in nuts, as well as bolts. In this page, we will highlight a few tools that can be used to analyze the behavior of any threaded component.

While screws and bolts are often used to create a mechanical advantage, threaded components can also be used in a wide variety of other applications. Some of these other applications include:

• Converting Motion – turning rotation into straight-line movement
• Distributing Force – sharing a load across multiple threads
• Acting as a Spring – handling force and stretching slightly under pressure

Power Screws

Power Screws turn rotation into straight-line motion. This happens because threads form an inclined plane around the cylinder. An input torque can cause the screw to revolve around its axis. As the screw revolves, the threads can move an object along the length of the screw.

Lead Angle

One of the most important parameters governing the behavior of a power screw is its "Lead Angle" \( \lambda \). This angle represents the angle between the thread and a cross sectional cut of the threaded component. The lead angle can be found with the following equation:

• Pitch (p): The distance between two threads
• Lead (L): The vertical distance a screw travels in 1 full rotation (an integer multiple of the pitch if there is more than 1 thread)
• Mean diameter (\( d_m \)): The average of the diameters of the threads and the diameter of the shaft of the screw

Thread Angle

Another key angle to consider when working with threaded components is the "Thread Angle" \( \alpha_n \). This represents the angle between two adjacent threads. However, when analyzing the forces on a threaded component, it is often most useful to find the

1. Begin by listing known information. In this case, we know the link lengths of a mechanism, as well as the angular velocity of the crank.
   
2. Next, compute all of the instant centers on the mechanism.
3. We then can consider the crank of the mechanism. We know that the pin between the crank and ground is an instant center. Since the ground link has a velocity of 0 everywhere, the crank will have a velocity of 0 at that location. Let's call the crank body 2, and the ground link body 1.
4. Then, select another grounded linkage. This is the link that we will analyze. Let's call that body 3. Body 1 and body 3 will share an instant center, which we will call \( A \).
5. Using \( v_a=v_{IC_{1,2}}+r_{IC_{1,2} \rightarrow a} \times \omega_2 \) we can substitute in for \( v_{IC_{1,2}}, r_{IC_{1,2} \rightarrow a} \) and \( \omega_2 \) to find \( v_a \).
   
6. We can reuse the above equation, but instead consider that point \( A \) is also attached to body 3. Thus, we could consider the velocity of point \( A \) as coming from the rotation of body 3 rather than the rotation of body 2. We would write this equation as \( v_a=v_c+r_{c \rightarrow a} \times \omega_3 \).
7. The biggest issue here is that we do not know where to place point \( C \) so that we know its velocity. If we place it at point \( A \), the radius will be 0, and we will be unable to compute \( \omega_3 \). However, there is another location where we know the velocity of a point on body 3. If we consider \( IC_{1,3} \) we know that the ground has a velocity of 0 everywhere. Thus, at \( IC_{1,3} \), the ground and body 3 have the same speed, 0.
8. This allows us to write \( v_a=0+r_{IC_{1,3} \rightarrow a} \times \omega_3 \). Since we know all of these values except \( \omega_3 \), we can then solve for \( \omega_3 \).
9. Now that \( \omega_3 \) is known, we can compute \( V_b \), and \( \omega_4 \). Additionally, this procedure can be repeated to find the velocities of all links in a mechanism.
   

Efficiency of Power Screws

A key performance metric for power screws is their efficiency, which is the output work divided by the input work.

Where

• W: Is the weight being lifted
• L: Is the lead of the screw, which is how far the weight is lifted in one revolution of the screw
• T: Is the torque applied to the screw

Power Screw Forces

When a power a screw lifts a weight, different forces act on the weight itself

• Normal force (N): is the force from the inclined surface of the thread.
• Friction force \( f \cdot n \) : is the force resisting motion.
• Input force (Q): Is the force applied to lift the weight, usually from an input torque.

Balancing these forces gives the following equation:

Power Screw Self Locking and Overhauling

When the external force or torque is removed from a power screw, the weight may fall down if the force from gravity exceeds the friction force. However, if friction is sufficiently high, no motion will occur, even in the absence of an external force.

• Self-locking: A mass lifted by a power screw stays in place when no force is applied.
• Overhauling: A mass lifted by a power screw moves on its own when the force is removed.

Whether a power screw is self locking or overhauling is dependent of the geometry of the screw as well as the friction coefficient. For a screw to be self locking the friciton coefficient must satisfy the equation shown below.

Bolts and Bolted Connections

Bolted connections clamp things together using a bolt and nut. The bolt is stretched (tension), while the material being clamped is squeezed (compression).

• Pre-load force \( F_i \): The force applied when tightening a bolt.
• External force \( F_e \): The extra force trying to separate the joint.

This gives us two equations for the force on the bolt \( F_b \) and the clamping force \( F_c \)

Where

• \( k_b \): stiffness of bolt
• \( k_c \): stiffness of clamped material

Thread Stripping

One of the most common methods for threaded components to fail is due the the threads shearing off. In order to strip a thread, the applied force must cause the stress to exceed the Von Mises yield strength of the material, \( S_{sy}=0.58 S_y \) . For a component with standard threads of width \( .75 p \) the area that must be sheared, \( A_s \) is:

Where

• \( d \): outer diameter of the bolt
• \( t \): height of the nut

This means that the force required to shear the threads is:

Additional Considerations

• Bolt sizes and grades are highly standardized, meaning that it is often far cheaper to use an existing threaded component rather than creating a customized one
• It is often advantageous for a nut to fail in a bolted connection before the bolt, as the bolt can still restrict some directions of motion, even if the nut has failed
• Screws convert rotation into straight-line motion.
• Power screws provide mechanical advantage to lift heavy loads.
• Bolted connections use pre-load force to hold parts together.
• Threads can fail by shearing if overloaded.