Part of a series on |
Continuum mechanics |
---|
Bearing pressure is a particular case of contact mechanics often occurring in cases where a convex surface (male cylinder or sphere) contacts a concave surface (female cylinder or sphere: bore or hemispherical cup). Excessive contact pressure can lead to a typical bearing failure such as a plastic deformation similar to peening. This problem is also referred to as bearing resistance. [1]
A contact between a male part (convex) and a female part (concave) is considered when the radii of curvature are close to one another. There is no tightening and the joint slides with no friction therefore, the contact forces are normal to the tangent of the contact surface.
Moreover, bearing pressure is restricted to the case where the charge can be described by a radial force pointing towards the center of the joint.
In the case of a revolute joint or of a hinge joint, there is a contact between a male cylinder and a female cylinder. The complexity depends on the situation, and three cases are distinguished:
By "negligible clearance", H7/g6 fit is typically meant.
The axes of the cylinders are along the z-axis, and two external forces apply to the male cylinder:
The main concern is the contact pressure with the bore, which is uniformly distributed along the z-axis.
Notation:
In this first modeling, the pressure is uniform. It is equal to [3] · [4] ·: [5]
There are two ways to obtain this result.
First, we can consider a hemicylinder in a fluid, with a uniform hydrostatic pressure. The equilibrium is achieved when the resulting force on the flat surface is equal to the resulting force on the curved one. The flat surface is a D × L rectangle, therefore
q.e.d.
Second, we can integrate the pressure elementary forces. Consider a small surface dS on the cylindrical part, parallel to a generating line; its length is L, and it is bound by the angles θ and θ + dθ. This small surface element can be considered as a flat rectangle which dimensions are L × (dθ × D/2). The pressure force on the surface is equal to
The (y, z) plane is a plane of reflection symmetry, so the x compound of this force is annihilated by the force on the symmetrical surface element. The y compound of this force is equal to:
The resulting force is equal to
q.e.d.
This calculation is similar to the case of a cylindrical vessel under pressure.
If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and transforms to a sinusoidal repartition [6] ·: [7] [8]
with
This is a particular case of the following section (θ0 = π/2).
The maximum pressure is 4/π ≃ 1.27 times bigger than the case of uniform pressure.
In cases where the clearance can not be neglected, the contact between the male part is no longer the whole half-cylinder surface but is limited to a 2θ0 angle. The pressure follows Hooke's law: [9]
where
The pressure varies as:
where A and B are positive real number. The maximum pressure is:
the angle θ0 is in radians.
The rigidity coefficient K and the half contact angle θ0 can not be derived from the theory. They must be measured. For a given system — given diameters and materials —, thus for given K and clearance j values, it is possible to obtain a curve θ0 = ƒ(F/(DL)).
Relationship between pressure, clearance and contact angle
The part no. 1 is the containing cylinder (female, concave), the part no. 2 is the contained cylinder (male, convex); the center of the cylinder i is Oi, and its radius is Ri.
The reference position is an ideal situation where both cylinders are concentric. The clearing, expressed as a radius (not diameter), is:
Under the load, the part 2 gets in contact with the part 1, the he surfaces deform. we suppose that the cylinder 2 is rigid (no deformation), and that the cylinder 1 is an elastic body. The indentation of 2 into 1 has a depth of δmax; the cylinder movement is e (excentration):
We considere the frame at the center of the cylinder 1 (O1, x, y). Let M be a point on the contact surface; θ is the angle (-y, O1M). The displacement of the surface, δ, is:
with δ(0) = δmax. The coordinates of M are:
and the coordinates of O2:
Consider the frame (O1, u, v), where the axis u is (O1M). In this frame, the coordinates are:
We know that
thus
then we use the expression of e and R1 = j + R2:
The deformations are small, as we are in the elastic domain. Thus, δmax ≪ R1 and therefore |φ| ≪ 1, i.e.
thus
and
At θ = θ0, δ(0) = 0 and the first equation is
and thus
If we use the law of elasticity for a metal (α = 1):
The pressure is an affine function of cos θ:
with A = K⋅j/cos θ0 and B = A⋅cos θ0.
Case where the clearance can be neglected
If j ≃ 0 (R1 ≃ R2), then the contact is on the whole half-perimeter: 2θ0 ≃ π and cos θ0 ≃ 0. The value of 1/cos θ0 rise towards infinity, thus
As j and cos θ0 both tend towards 0, the ratio j/cos θ0 is not defined when j goes to 0. In mechanical engineering, j = 0 is an uncertain fit, it is a nonsense, both mathematically and mechanically. We are looking for a limit function
So, the pressure is a sinusoid function of θ:
thus
with
Consider an infinitesimal element of surface dS bound by θ and θ + dθ. As in the case of the uniform pressure, we have
When we integrate between -π/2 and π/2, the result is:
We know that (e.g. using the Euler's formula):
therefore
and thus
q.e.d.
Case where the clearance can not be neglected
The force on an infinitesimal element of surface is:
thus
We recognise the trigonometric identity sin 2θ = 2 sin θ cos θ :
thus
and therefore:
q.e.d.
A sphere-sphere contact corresponds to a spherical joint (socket/ball), such as a ball jointed cylinder saddle. It can also describe the situation of bearing balls.
The case is similar as above: when the parts are considered as rigid bodies and the clearance can be neglected, then the pressure is supposed to be uniform. It can also be calculated considering the projected area [3] · [10] ·: [11]
As in the case of cylinder-cylinder contact, when the parts are modeled as elastic bodies with a negligible clearance, then the pressure can be modeled with a sinusoidal repartition [6] ·: [12]
with
When the clearance can not be neglected, it is then necessary to know the value of the half contact angle θ0 , which can not be determined in a simple way and must be measured. When this value is not available, the Hertz contact theory can be used.
The Hertz theory is normally only valid when the surfaces can not conform, or in other terms, can not fit each other by elastic deformation; one surface must be convex, the other one must be also convex plane. This is not the case here, as the outer cylinder is concave, so the results must be considered with great care. The approximation is only valid when the inner radius of the container R1 is far greater than the outer radius of the content R2, in which case the surface container is then seen as flat by the content. However, in all cases, the pressure that is calculated with the Hertz theory is greater than the actual pressure (because the contact surface of the model is smaller than the real contact surface), which affords designers with a safety margin for their design.
In this theory, the radius of the female part (concave) is negative. [13]
A relative diameter of curvature is defined:
where d1 is the diameter of the female part (negative) and d2 is the diameter of the male part (positive). An equivalent module of elasticity is also defined:
where νi is the Poisson's ratio of the material of the part i and Ei its Young's modulus.
For a cylinder-cylinder contact, the width of the contact surface is:
and the maximal pressure is in the middle:
In case of a sphere-sphere contact, the contact surface is a disk whose radius is:
and the maximal pressure is in the middle:
In a bolted connection, the role of the bolts is normally to press one parts on the other; the adherence (friction) is opposed to the tangent forces and prevents the parts from sliding apart. In some cases however, the adherence is not sufficient. The bolts then play the role of stops: the screws endure shear stress whereas the hole endure bearing pressure.
In order to increase the bearing pressure of a material, there are several factors that can be considered. One of the most effective methods is to increase the surface area of the material. By increasing the surface area, the load is distributed over a larger area, reducing the bearing pressure.
In good design practice, the threaded part of the screw should be small and only the smooth part should be in contact with the plates; in the case of a shoulder screw, the clearance between the screw and the hole is very small ( a case of rigid bodies with negligible clearance). If the acceptable pressure limit Plim of the material is known, the thickness t of the part and the diameter d of the screw, then the maximum acceptable tangent force for one bolt Fb, Rd (design bearing resistance per bolt) is:
In this case, the acceptable pressure limit is calculated from the ultimate tensile stress fu and factors of safety, according to the Eurocode 3 standard [1] ·. [14] In the case of two plates with a single overlap and one row of bolts, the formula is:
where
In more complex situations, the formula is:
where
fasteners,
Steel grades (EN standard) | S235 | S275 | S355 |
---|---|---|---|
Ultimate tensile stress fu (MPa) | 360 | 430 | 510 |
When the parts are in wood, the acceptable limit pressure is about 4 to 8.5 MPa. [15]
In plain bearings, the shaft is usually in contact with a bushing (sleeve or flanged) to reduce friction. When the rotation is slow and the load is radial, the model of uniform pressure can be used (small deformations and clearance).
The product of the bearing pressure times the circumferential sliding speed, called load factor PV, is an estimation of the resistance capacity of the material against the frictional heating [16] · [17] ·. [18]
Type of bushing Maximal circumferential sliding speed | Acceptable bearing pressure (MPa) |
---|---|
Self-lubricating bushels 7 to 8 m/s 13 m/s for graphite | graphite: 5 lead bronze: 20 to 30 tin bronze: 7 to 35 |
Composite bushing, Glacier 2 to 3 m/s | acetal: 70 PTFE: 50 |
Polymer bushing 2 to 3 m/s | 7 to 10 |
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin; its polar angle measured from a fixed polar axis or zenith direction; and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal; I = I0 cos θ. The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.
The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).
In particle physics, bremsstrahlung is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.
The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive and negligible. The curved path of objects in projectile motion was shown by Galileo to be a parabola, but may also be a straight line in the special case when it is thrown directly upward or downward. The study of such motions is called ballistics, and such a trajectory is a ballistic trajectory. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward acceleration towards the Earth’s center of mass. Because of the object's inertia, no external force is needed to maintain the horizontal velocity component of the object's motion. Taking other forces into account, such as aerodynamic drag or internal propulsion, requires additional analysis. A ballistic missile is a missile only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of classical mechanics.
In physics and engineering, a phasor is a complex number representing a sinusoidal function whose amplitude, angular frequency, and initial phase are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, and sinor or even complexor.
In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.
A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle.
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.