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69-405: An axle or axletree is a central shaft for a rotating wheel or gear . On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, bearings or bushings are provided at the mounting points where the axle is supported. In the latter case, a bearing or bushing sits inside a central hole in

138-401: A propeller shaft or tailshaft ) which transmits the rotational force to a drive axle at the rear of the vehicle. The drive axle may be a live axle , but modern rear-wheel drive automobiles generally use a split axle with a differential. In this case, one half-axle or half-shaft connects the differential with the left rear wheel, a second half-shaft does the same with the right rear wheel; thus

207-399: A screw . It is possible for objects to have periodic circular trajectories without changing their orientation . These types of motion are treated under circular motion instead of rotation, more specifically as a curvilinear translation. Since translation involves displacement of rigid bodies while preserving the orientation of the body, in the case of curvilinear translation, all

276-463: A chosen reference point. Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through instantaneous center of circle and perpendicular to the plane of motion . In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast,

345-429: A differential (kart racing). However, other go-karts have two rear drive wheels too. Some dump trucks and trailers may be configured with a lift axle (also known as an airlift axle or drop axle ), which may be mechanically raised or lowered. The axle is lowered to increase the weight capacity, or to distribute the weight of the cargo over more wheels, for example, to cross a weight-restricted bridge. When not needed,

414-601: A direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis m {\displaystyle m} can always be written as a rotation having 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} if the axis is replaced with n = − m {\displaystyle n=-m} .) Every proper rotation A {\displaystyle A} in 3D space has an axis of rotation, which

483-448: A full-floating axle is that even if an axle shaft (used to transmit torque or power) breaks, the wheel will not come off, preventing serious accidents. Rotation The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or autorotation ). In that case, the surface intersection of the internal spin axis can be called a pole ; for example, Earth's rotation defines

552-464: A nonzero magnitude. This discussion applies to a proper rotation, and hence det A = 1 {\displaystyle \det A=1} . Any improper orthogonal 3x3 matrix B {\displaystyle B} may be written as B = − A {\displaystyle B=-A} , in which A {\displaystyle A} is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as

621-430: A plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation. The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which

690-406: A point/axis form a group . However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. Rotations around the x , y and z axes are called principal rotations . Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by

759-428: A proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of A {\displaystyle A} is also the eigenvector of B {\displaystyle B} corresponding to an eigenvalue of −1. As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has

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828-413: A protruding hub to which the axle shaft flange is bolted. The semi-floating axle setup is commonly used on half-ton and lighter 4×4 trucks in the rear. This setup allows the axle shaft to be the means of propulsion, and also support the weight of the vehicle. The main difference between the full- and semi-floating axle setups is the number of bearings. The semi-floating axle features only one bearing, while

897-488: A relation of rate of change of unit vector by taking A → {\displaystyle {\vec {A}}} , to be such a vector: d A ^ d t ⋅ A ^ = 0 {\displaystyle {d{\hat {A}} \over dt}\cdot {\hat {A}}=0} showing that d A ^ d t {\textstyle {d{\hat {A}} \over dt}} vector

966-507: A rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion. More generally, due to Chasles' theorem , any motion of rigid bodies can be treated as a composition of rotation and translation , called general plane motion. A simple example of pure rotation is considered in rotation around a fixed axis . The laws of physics are currently believed to be invariant under any fixed rotation . (Although they do appear to change when viewed from

1035-417: A rotating viewpoint: see rotating frame of reference .) In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in

1104-447: A rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant

1173-428: A rotation is a rigid body movement which, unlike a translation , keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line). All rigid body movements are rotations, translations, or combinations of

1242-439: A single plane. 2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity. The question of the existence of such a direction is the question of existence of an eigenvector for

1311-400: A single unit called a transaxle . The drive axle is a split axle with a differential and universal joints between the two half axles. Each half axle connects to the wheel by use of a constant velocity (CV) joint which allows the wheel assembly to move freely vertically as well as to pivot when making turns. In rear-wheel drive cars and trucks, the engine turns a driveshaft (also called

1380-427: A suspension and steering component. Conversely, many front-wheel drive cars have a one-piece rear beam axle. In other types of suspension systems, the axles serve only to transmit driving torque to the wheels: the position and angle of the wheel hubs is made independent from the axles by the function of the suspension system. This is typical of the independent suspensions found on most newer cars, and even SUVs, and on

1449-426: A turn at all. Several manufacturers offer computer-controlled airlifts so that the dead axles are automatically lowered when the main axle reaches its weight limit. The dead axles can still be lifted by the press of a button if needed, for better maneuverability. Lift axles were in use in the early 1940s. Initially, the axle was lifted by a mechanical device. Soon hydraulics replaced the mechanical lift system. One of

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1518-403: Is a group of two or more axles situated close together. Truck designs use such a configuration to provide a greater weight capacity than a single axle. Semi-trailers usually have a tandem axle at the rear. Axles are typically made from SAE grade 41xx steel or SAE grade 10xx steel. SAE grade 41xx steel is commonly known as " chrome-molybdenum steel " (or "chrome-moly") while SAE grade 10xx steel

1587-500: Is a single bearing at the end of the axle housing that carries the load from the axle and that the axle rotates through. To be "semi-floating" the axle shafts must be able to "float" in the housing, bearings and seals, and not subject to axial "thrust" and/or bearing preload. Needle bearings and separate lip seals are used in semi-floating axles with axle retained in the housing at their inner ends typically with circlips which are 3¾-round hardened washers that slide into grooves machined at

1656-407: Is about some axis, although this axis may be changing over time. In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about a plane is not in general a rotation in

1725-441: Is attached to a separate shaft. Modern passenger cars have split-drive axles. In some designs, this allows independent suspension of the left and right wheels, and therefore a smoother ride. Even when the suspension is not independent, split axles permit the use of a differential, allowing the left and right drive wheels to be driven at different speeds as the automobile turns, improving traction and extending tire life. A tandem axle

1794-496: Is called tidal locking ; the Moon is tidal-locked to the Earth. This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator . Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth

1863-535: Is called an axle or axle shaft . However, in looser usage, an entire assembly including the surrounding axle housing (typically a casting ) is also called an axle . An even broader (somewhat figurative) sense of the word refers to every pair of parallel wheels on opposite sides of a vehicle, regardless of their mechanical connection to each other and to the vehicle frame or body. Thus, transverse pairs of wheels in an independent suspension may be called an axle in some contexts. This very loose definition of "axle"

1932-467: Is conserved . Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where

2001-431: Is defined such that any vector v {\displaystyle v} that is aligned with the rotation axis will not be affected by rotation. Accordingly, A v = v {\displaystyle Av=v} , and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle α {\displaystyle \alpha }

2070-444: Is known as " carbon steel ". The primary differences between the two are that chrome-moly steel is significantly more resistant to bending or breaking, and is very difficult to weld with tools normally found outside a professional welding shop. An axle that is driven by the engine or prime mover is called a drive axle . Modern front-wheel drive cars typically combine the transmission (gearbox and differential) and front axle into

2139-546: Is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see Eigenvalues and eigenvectors#Eigenvalues and the characteristic polynomial ). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In

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2208-467: Is not in great enough quantities to provide the corrosion resistance found in stainless steel . Examples of applications for 4130, 4140, and 4145 include structural tubing , bicycle frames , gas bottles for transportation of pressurized gases, firearm parts, clutch and flywheel components, and roll cages . 4150 stands out as being one of the steels accepted for use in M16 rifle and M4 carbine barrels by

2277-620: Is not required to find the rotation axis. If n {\displaystyle n} denotes the unit eigenvector aligned with the rotation axis, and if α {\displaystyle \alpha } denotes the rotation angle, then it can be shown that 2 sin ⁡ ( α ) n = { A 32 − A 23 , A 13 − A 31 , A 21 − A 12 } {\displaystyle 2\sin(\alpha )n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}} . Consequently,

2346-409: Is often used in assessing toll roads or vehicle taxes , and is taken as a rough proxy for the overall weight-bearing capacity of a vehicle, and its potential for causing wear or damage to roadway surfaces. Axles are an integral component of most practical wheeled vehicles. In a solid, "live-axle" suspension system, the rotating inner axle cores (or half-shafts) serve to transmit driving torque to

2415-741: Is perpendicular to the vector, A → {\displaystyle {\vec {A}}} . From: d A → d t = d ( | A → | A ^ ) d t = d | A → | d t A ^ + | A → | ( d A ^ d t ) {\displaystyle {d{\vec {A}} \over dt}={d(|{\vec {A}}|{\hat {A}}) \over dt}={d|{\vec {A}}| \over dt}{\hat {A}}+|{\vec {A}}|\left({d{\hat {A}} \over dt}\right)} , since

2484-856: Is real, it equals its complex conjugate v T v {\displaystyle v^{\text{T}}v} , and v ¯ T v {\displaystyle {\bar {v}}^{\text{T}}v} and v T v ¯ {\displaystyle v^{\text{T}}{\bar {v}}} are both representations of the same scalar product between v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} . This means v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are orthogonal vectors. Also, they are both real vectors by construction. These vectors span

2553-544: Is slightly deformed into an oblate spheroid ; a similar equatorial bulge develops for other planets. Another consequence of the rotation of a planet are the phenomena of precession and nutation . Like a gyroscope , the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth 's axis to its orbital plane ( obliquity of the ecliptic ) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of

2622-534: The geographical poles . A rotation around an axis completely external to the moving body is called a revolution (or orbit ), e.g. Earth's orbit around the Sun . The ends of the external axis of revolution can be called the orbital poles . Either type of rotation is involved in a corresponding type of angular velocity (spin angular velocity and orbital angular velocity) and angular momentum (spin angular momentum and orbital angular momentum). Mathematically ,

2691-943: The Earth around the Sun); and stars slowly revolve about their galaxial centers . The motion of the components of galaxies is complex, but it usually includes a rotation component. 41xx steel 41xx steel is a family of SAE steel grades , as specified by the Society of Automotive Engineers (SAE). Alloying elements include chromium and molybdenum , and as a result these materials are often informally referred to as chromoly steel (common variant stylings include chrome-moly , cro-moly , CrMo , CRMO , CR-MOLY , and similar). They have an excellent strength to weight ratio and are considerably stronger and harder than standard 1020 steel, but are not easily welded , requiring thermal treatment both before and after welding to avoid cold cracking . While these grades of steel do contain chromium, it

2760-400: The axle is lifted off the ground to save wear on the tires and axle, and to increase traction in the remaining wheels, and to decrease fuel consumption. Lifting an axle also alleviates lateral scrubbing of the additional axle in very tight turns, allowing the vehicle to turn more readily. In some situations, the removal of pressure from the additional axle is necessary for the vehicle to complete

2829-426: The bearings, housings, and only a short section of the shaft itself, that also carries all radial loads. The full-floating design is typically used in most ¾- and 1-ton light trucks, medium-duty trucks, and heavy-duty trucks. The overall assembly can carry more weight than a semi-floating or non-floating axle assembly because the hubs have two bearings riding on a fixed spindle. A full-floating axle can be identified by

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2898-405: The degenerate case of a rotation angle α = 180 ∘ {\displaystyle \alpha =180^{\circ }} , the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary). A spectral analysis

2967-399: The direction of the axis of rotation. Similarly, the torque is an axial vector. The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule , the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like

3036-456: The early manufacturers was Zetterbergs, located in Östervåla, Sweden. Their brand was Zeta-lyften. The liftable tandem drive axle was invented in 1957 by the Finnish truck manufacturer Vanajan Autotehdas , a company sharing history with Sisu Auto . A full-floating axle carries the vehicle's weight on the axle casing, not the half-shafts; they serve only to transmit torque from the differential to

3105-438: The effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we do not have a proper rotation, but either the identity or the result of a sequence of reflections. It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in

3174-1071: The eigenvectors of A . A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector A → {\displaystyle {\vec {A}}} which is parameterized by some variable t {\textstyle t} for which: d | A → | 2 d t = d ( A → ⋅ A → ) d t ⇒ d | A → | d t = d A → d t ⋅ A ^ {\displaystyle {d|{\vec {A}}|^{2} \over dt}={d({\vec {A}}\cdot {\vec {A}}) \over dt}\Rightarrow {d|{\vec {A}}| \over dt}={d{\vec {A}} \over dt}\cdot {\hat {A}}} Which also gives

3243-408: The equinoxes and Pole Star .) While revolution is often used as a synonym for rotation , in many fields, particularly astronomy and related fields, revolution , often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planets, planets revolve about their stars (such as

3312-541: The expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that sin ⁡ ( α ) = 0 {\displaystyle \sin(\alpha )=0} . In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of A + I {\displaystyle A+I} that has

3381-513: The first angle moves the line of nodes around the external axis z , the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves. These rotations are called precession , nutation , and intrinsic rotation . In astronomy , rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution. Stars , planets and similar bodies may spin around on their axes. The rotation rate of planets in

3450-443: The first term is parallel to A → {\displaystyle {\vec {A}}} and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against

3519-481: The front of many light trucks. An exception to this rule is the independent (rear) swing axle suspension, wherein the half-axles are also load-bearing suspension arms. Independent drive-trains still need differentials (or diffs), but without fixed axle-housing tubes attached. The diff may be attached to the vehicle frame or body, and/or be integrated with the transmission (or gearbox) in a combined transaxle unit. The axle (half-)shafts then transmit driving torque to

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3588-504: The full-floating assembly has bearings on both the inside and outside of the wheel hub. The other difference is axle removal. To remove the semi-floating axle, the wheel must be removed first; if such an axle breaks, the wheel is most likely to come off the vehicle. The semi-floating design is found under most ½-ton and lighter trucks, as well as in SUVs and rear-wheel-drive passenger cars, usually being smaller or less expensive models. A benefit of

3657-461: The inner end of the shafts and retained in/by recesses in the differential carrier side gears which are themselves retained by the differential pinion gear (or "spider gear") shaft. A true semi-floating axle assembly places no side loads on the axle housing tubes or axle shafts. Axles that are pressed into ball or tapered roller bearings, which are in turn retained in the axle housings with flanges, bolts, and nuts do not "float" and place axial loads on

3726-483: The large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang. In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant . According to Noether's theorem , if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum

3795-562: The matrix A representing the rotation. Every 2D rotation around the origin through an angle θ {\displaystyle \theta } in counterclockwise direction can be quite simply represented by the following matrix : A standard eigenvalue determination leads to the characteristic equation which has as its eigenvalues. Therefore, there is no real eigenvalue whenever cos ⁡ θ ≠ ± 1 {\displaystyle \cos \theta \neq \pm 1} , meaning that no real vector in

3864-542: The plane is kept unchanged by A . Knowing that the trace is an invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by Using the principal arc-cosine, this formula gives a rotation angle satisfying 0 ≤ α ≤ 180 ∘ {\displaystyle 0\leq \alpha \leq 180^{\circ }} . The corresponding rotation axis must be defined to point in

3933-416: The points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation. In rotation, the orientation of the object changes and the change in orientation is independent of the observers whose frames of reference have constant relative orientation over time. By Euler's theorem , any change in orientation can be described by rotation about an axis through

4002-539: The previous topic, v ¯ {\displaystyle {\bar {v}}} is also an eigenvector, and v + v ¯ {\displaystyle v+{\bar {v}}} and i ( v − v ¯ ) {\displaystyle i(v-{\bar {v}})} are such that their scalar product vanishes: because, since v ¯ T v ¯ {\displaystyle {\bar {v}}^{\text{T}}{\bar {v}}}

4071-409: The same subspace as v {\displaystyle v} and v ¯ {\displaystyle {\bar {v}}} , which is an invariant subspace under the application of A . Therefore, they span an invariant plane. This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A , with eigenvalue 1, because of the orthogonality of

4140-455: The solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features. An example is sunspots , which rotate around the Sun at the same velocity as the outer gases that make up the Sun. Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect

4209-457: The two half-axles and the differential constitute the rear axle. The front drive axle is providing the force to drive the truck. In fact, only one wheel of that axle is actually moving the truck and trailer down the road. Some simple vehicle designs, such as leisure go-karts, may have a single driven wheel where the drive axle is a split axle with only one of the two shafts driven by the engine, or else have both wheels connected to one shaft without

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4278-407: The two. A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse ( inverse ) of a rotation is also a rotation. Thus, the rotations around

4347-436: The vector itself. As dimensions increase the number of rotation vectors increases. Along a four dimensional space (a hypervolume ), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various volumes , where each intersection is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, were a 3d object can be rotated perpendicular to

4416-400: The wheel positions steady under heavy stress, and can therefore support heavy loads. Straight axles are used on trains (that is, locomotives and railway wagons), for the rear axles of commercial trucks, and on heavy-duty off-road vehicles. The axle can optionally be protected and further reinforced by enclosing the length of the axle in a housing. In split-axle designs, the wheel on each side

4485-436: The wheel to allow the wheel or gear to rotate around the axle. Sometimes, especially on bicycles, the latter type of axle is referred to as a spindle . On cars and trucks, several senses of the word axle occur in casual usage, referring to the shaft itself, its housing, or simply any transverse pair of wheels. Strictly speaking, a shaft that rotates with the wheel, being either bolted or splined in fixed relation to it,

4554-456: The wheels at each end, while the rigid outer tube maintains the position of the wheels at fixed angles relative to the axle, and controls the angle of the axle and wheels assembly to the vehicle body. The solid axles (housings) in this system must also bear the weight of the vehicle plus any cargo. A non-driving axle, such as the front beam axle in heavy-duty trucks and some two-wheel drive light trucks and vans, will have no shaft, and serves only as

4623-403: The wheels, usually via constant-velocity joints . Like a full floating axle system, the drive shafts in a front-wheel-drive independent suspension system do not support any vehicle weight. A straight axle is a single rigid shaft connecting a wheel on the left side of the vehicle to a wheel on the right side. The axis of rotation fixed by the axle is common to both wheels. Such a design can keep

4692-400: The wheels. They "float" inside an assembly that carries the vehicle's weight. Thus the only stress it must endure is torque (not lateral bending force). Full-floating axle shafts are retained by a flange bolted to the hub , while the hub and bearings are retained on the spindle by a large nut. In contrast, a semi-floating design carries the weight of the vehicle on the axle shaft itself; there

4761-406: The z axis. The speed of rotation is given by the angular frequency (rad/s) or frequency ( turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s ), caused by torque . The ratio of torque to the angular acceleration is given by the moment of inertia . The angular velocity vector (an axial vector ) also describes

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