In mechanical engineering, a **helix angle** is the angle between any helix and an axial line on its right, circular cylinder or cone.^{ [1] } Common applications are screws, helical gears, and worm gears.

The helix angle references the axis of the cylinder, distinguishing it from the lead angle, which references a line perpendicular to the axis. Naturally, the helix angle is the geometric complement of the lead angle. The helix angle is measured in degrees.

In terms specific to screws, the helix angle can be found by unraveling the helix from the screw, representing the section as a right triangle, and calculating the angle that is formed. Note that while the terminology directly refers to screws, these concepts are analogous to most mechanical applications of the helix angle.

The helix angle can be expressed as:^{ [2] }

where

*l*is lead of the screw or gear*r*is mean radius of the screw thread or gear_{m}

The helix angle is crucial in mechanical engineering applications that involve power transfer and motion conversion. Some examples are outlined below, though its use is much more widely spread.

Cutting a single helical groove into a screw-stock cylinder yields what is referred to as a single-thread screw. Similarly, one may construct a double-thread screw provided that the helix angle is the same, and a second thread is cut in the space between the grooves of the first. For certain applications, triple and quadruple threads are in use.^{ [3] } The helix may be cut either right hand or left hand. In screws especially, the helix angle is essential for calculating torque in power screw applications.

The maximum efficiency for a screw is defined by the following equations:^{ [4] }

Where is the helix angle, is the friction angle, and is the maximum efficiency. The friction value is dependent on the materials of the screw and interacting nut, but ultimately the efficiency is controlled by the helix angle. The efficiency can be plotted versus the helix angle for a constant friction, as shown in the adjacent diagram. The maximum efficiency is a helix angle between 40 and 45 degrees, however a reasonable efficiency is achieved above 15°. Due to difficulties in forming the thread, helix angle greater than 30° are rarely used. Moreover, above 30° the friction angle becomes smaller than the helix angle and the nut is no longer self-locking and the mechanical advantage disappears.^{ [4] }

In helical and worm gears, the helix angle denotes the standard pitch circle unless otherwise specified.^{ [1] } Application of the helix angle typically employs a magnitude ranging from 15° to 30° for helical gears, with 45° capping the safe operation limit. The angle itself may be cut with either a right-hand or left-hand orientation.^{ [5] } In its typical parallel arrangement, meshing helical gears requires that the helix angles are of the same magnitude and cut oppositely .

Worm gears resemble helical gear seats, the difference being that the shafts of a worm train are aligned perpendicularly. In this case, the helix angle of the worm meshes with the lead angle of the worm gear.^{ [6] }

A ** simple machine** is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage to multiply force. Usually the term refers to the six classical simple machines that were defined by Renaissance scientists:

An **inclined plane**, also known as a **ramp**, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defined by Renaissance scientists. Inclined planes are widely used to move heavy loads over vertical obstacles; examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.

A **gear** is a rotating circular machine part having cut teeth or, in the case of a **cogwheel** or **gearwheel**, inserted teeth, which mesh with another toothed part to transmit torque. A gear may also be known informally as a **cog**. Geared devices can change the speed, torque, and direction of a power source. Gears of different sizes produce a change in torque, creating a mechanical advantage, through their *gear ratio*, and thus may be considered a simple machine. The rotational speeds, and the torques, of two meshing gears differ in proportion to their diameters. The teeth on the two meshing gears all have the same shape.

A **helix**, plural **helixes** or **helices**, is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word *helix* comes from the Greek word *ἕλιξ*, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

The **great-circle distance**, **orthodromic distance**, or **spherical distance** is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called *great circles*.

**Blade element theory** (**BET**) is a mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements. These forces are then integrated along the entire blade and over one rotor revolution in order to obtain the forces and moments produced by the entire propeller or rotor. One of the key difficulties lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is often combined with momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk, producing blade element momentum theory. At the most basic level of approximation a uniform induced velocity on the disk is assumed:

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following

In physics, **Washburn's equation** describes capillary flow in a bundle of parallel cylindrical tubes; it is extended with some issues also to imbibition into porous materials. The equation is named after Edward Wight Washburn; also known as **Lucas–Washburn equation**, considering that Richard Lucas wrote a similar paper three years earlier, or the **Bell-Cameron-Lucas-Washburn equation**, considering J.M. Bell and F.K. Cameron's discovery of the form of the equation in 1906.

In experimental particle physics, **pseudorapidity**, , is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as

A **worm drive** is a gear arrangement in which a **worm** meshes with a **worm gear**. The two elements are also called the **worm screw** and **worm wheel**. The terminology is often confused by imprecise use of the term *worm gear* to refer to the worm, the worm gear, or the worm drive as a unit.

A **leadscrew**, also known as a **power screw** or **translation screw**, is a screw used as a linkage in a machine, to translate turning motion into linear motion. Because of the large area of sliding contact between their male and female members, screw threads have larger frictional energy losses compared to other linkages. They are not typically used to carry high power, but more for intermittent use in low power actuator and positioner mechanisms. Leadscrews are commonly used in linear actuators, machine slides, vises, presses, and jacks. Leadscrews are a common component in electric linear actuators.

A **screw** is a mechanism that converts rotational motion to linear motion, and a torque to a linear force. It is one of the six classical simple machines. The most common form consists of a cylindrical shaft with helical grooves or ridges called *threads* around the outside. The screw passes through a hole in another object or medium, with threads on the inside of the hole that mesh with the screw's threads. When the shaft of the screw is rotated relative to the stationary threads, the screw moves along its axis relative to the medium surrounding it; for example rotating a wood screw forces it into wood. In screw mechanisms, either the screw shaft can rotate through a threaded hole in a stationary object, or a threaded collar such as a nut can rotate around a stationary screw shaft. Geometrically, a screw can be viewed as a narrow inclined plane wrapped around a cylinder.

**Lateral earth pressure** is the pressure that soil exerts in the horizontal direction. The lateral earth pressure is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering structures such as retaining walls, basements, tunnels, deep foundations and braced excavations.

**Rotational diffusion** is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. Rotational diffusion is the counterpart of translational diffusion, which maintains or restores the equilibrium statistical distribution of particles' position in space.

**Lead** is the axial advance of a helix or screw during one complete turn (360°) The lead for a screw thread is the axial travel for a single revolution.

In machine learning and mathematical optimization, **loss functions for classification** are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems. Given as the space of all possible inputs, and as the set of labels, a typical goal of classification algorithms is to find a function which best predicts a label for a given input . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same to generate different . As a result, the goal of the learning problem is to minimize expected loss, defined as

A **transmitarray antenna** is a phase-shifting surface (PSS), a structure capable of focusing electromagnetic radiation from a source antenna to produce a high-gain beam. Transmitarrays consist of an array of unit cells placed above a source (feeding) antenna. Phase shifts are applied to the unit cells, between elements on the receive and transmit surfaces, to focus the incident wavefronts from the feeding antenna. These thin surfaces can be used instead of a dielectric lens. Unlike phased arrays, transmitarrays do not require a feed network, so losses can be greatly reduced. Similarly, they have an advantage over reflectarrays in that feed blockage is avoided.

**Propeller theory** is the science governing the design of efficient propellers. A propeller is the most common propulsor on ships, and on small aircraft.

- 1 2
*Gear Nomenclature, Definition of Terms with Symbols*, American Gear Manufacturers Association, p. 72, ISBN 1-55589-846-7, OCLC 65562739, ANSI/AGMA 1012-G05 - ↑ Shigley, p. 401.
- ↑ Norton, Robert L., Machine Design: An Integrated Approach. 3rd ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2006.
- 1 2 Karwa, p. 252.
- ↑ Shigley, Joseph E., and Larry D. Mitchell Mechanical Engineering Design. 4th ed. New York: McGraw-Hill, Inc, 1983.
- ↑ Spotts, M F., and T E. Shoup. Design of Machine Elements. 7th ed. Upper Saddle River, NJ: Prentice Hall, 1998.

- Bhandari, V B (2007),
*Design of Machine Elements*, Tata McGraw-Hill, ISBN 978-0-07-061141-2 . - Karwa, Rajendra (2005),
*A textbook of machine design*, Firewall Media, ISBN 978-81-7008-833-2 .

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