Monday, September 20, 2021

09-20-2021-0633 - Mechanics drafting (resultant force, deformation, beam, lateral load, contraflexure, etc.)

In physics and engineering, a resultant forceis the single force and associated torqueobtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces.^[1] Calculating and visualizing the resultant force on a body is done through computational analysis, or (in the case of sufficiently simple systems) a free body diagram.

The point of application of the resultant force determines its associated torque. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term ' resultant force-torque.

https://en.wikipedia.org/wiki/Resultant_force

https://en.wikipedia.org/wiki/Bending_moment

https://en.wikipedia.org/wiki/Cantilever

https://en.wikipedia.org/wiki/Hogging_and_sagging

In solid mechanics, a point along a beam under a lateral load is known as a point of contraflexure if the bending moment about the point equals zero.^[1] In a bending moment diagram, it is the point at which the bending moment curve intersects with the zero line (i.e. where the bending moment reverses direction along the beam). Knowing the place of the contraflexure is especially useful when designing reinforced concrete or structural steel beams and also for designing bridges.

Flexural reinforcement may be reduced at this point. However, to omit reinforcement at the point of contraflexure entirely is inadvisable as the actual location is unlikely to realistically be defined with confidence. Additionally, an adequate quantity of reinforcement should extend beyond the point of contraflexure to develop bond strength and to facilitate shear force transfer.

Pages in category "Mathematics of rigidity"

The following 14 pages are in this category, out of 14 total. This list may not reflect recent changes (learn more).

A

Alexandrov's uniqueness theorem

B

Bricard octahedron

C

P

R

S

Categories:

https://en.wikipedia.org/wiki/Category:Mathematics_of_rigidity

https://en.wikipedia.org/wiki/Category:Mechanics

https://en.wikipedia.org/wiki/Category:Dynamical_systems

In classical mechanics, a physical system is termed a monogenic system if the force acting on the system can be modelled in a particular, especially convenient mathematical form. The systems that are typically studied in physics are monogenic. The term was introduced by Cornelius Lanczos in his book The Variational Principles of Mechanics (1970).^[1]^[2]

In Lagrangian mechanics, the property of being monogenic is a necessary condition for certain different formulations to be mathematically equivalent. If a physical system is both a holonomic system and a monogenic system, then it is possible to derive Lagrange's equations from d'Alembert's principle; it is also possible to derive Lagrange's equations from Hamilton's principle.^[3]

https://en.wikipedia.org/wiki/Monogenic_system

Stall torque is the torque produced by a mechanical device whose output rotational speed is zero. It may also mean the torque load that causes the output rotational speed of a device to become zero, i.e., to cause stalling. Electric motors, steam engines and hydrodynamic transmissions are all capable of developing torque when stalled.

https://en.wikipedia.org/wiki/Stall_torque

In statistical mechanics, the radial distribution function, (or pair correlation function) $g(r)$ in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.

If a given particle is taken to be at the origin O, and if $\rho =N/V$ is the average number density of particles, then the local time-averaged density at a distance $r$ from O is $\rho g(r)$ . This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below.

https://en.wikipedia.org/wiki/Radial_distribution_function

The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object $X$ in $n$ -dimensional space is the intersection of all hyperplanes that divide $X$ into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of $X$ . For an object of uniform composition, the centroid of a body is also its center of mass. In the case of two-dimensional objects shown below, the hyperplanes are simply lines.

https://en.wikipedia.org/wiki/List_of_centroids

Chain drive is a way of transmitting mechanical power from one place to another. It is often used to convey power to the wheels of a vehicle, particularly bicycles and motorcycles. It is also used in a wide variety of machines besides vehicles.

https://en.wikipedia.org/wiki/Chain_drive

Cyclic stress is the distribution of forces (aka stresses) that change over time in a repetitive fashion. As an example, consider one of the large wheels used to drive an aerial lift such as a ski lift. The wire cable wrapped around the wheel exerts a downward force on the wheel and the drive shaft supporting the wheel. Although the shaft, wheel, and cable move, the force remains nearly vertical relative to the ground. Thus a point on the surface of the drive shaft will undergo tension when it is pointing towards the ground and compression when it is pointing to the sky.

https://en.wikipedia.org/wiki/Cyclic_stress

In physics, Carroll's paradox arises when considering the motion of a falling rigid rod that is specially constrained. Considered one way, the angular momentum stays constant; considered in a different way, it changes. It is named after Michael M. Carroll who first published it in 1984.

https://en.wikipedia.org/wiki/Carroll%27s_paradox

Non-smooth mechanics is a modeling approach in mechanics which does not require the time evolutions of the positions and of the velocities to be smooth functions anymore. Due to possible impacts, the velocities of the mechanical system are even allowed to undergo jumps at certain time instants in order to fulfill the kinematical restrictions. Consider for example a rigid model of a ball which falls on the ground. Just before the impact between ball and ground, the ball has non-vanishing pre-impact velocity. At the impact time instant, the velocity must jump to a post-impact velocity which is at least zero, or else penetration would occur. Non-smooth mechanical models are often used in contact dynamics.

https://en.wikipedia.org/wiki/Non-smooth_mechanics

The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is struck at the center of percussion. The center of percussion is often discussed in the context of a bat, racquet, door, sword or other extended object held at one end.

The same point is called the center of oscillation for the object suspended from the pivot as a pendulum, meaning that a simple pendulum with all its mass concentrated at that point will have the same period of oscillation as the compound pendulum.

In sports, the center of percussion of a bat or racquet is related to the so-called "sweet spot", but the latter is also related to vibrational bending of the object.

https://en.wikipedia.org/wiki/Center_of_percussion

https://en.wikipedia.org/wiki/Category:Oscillation

Category:Dimensionless numbers of mechanics

From Wikipedia, the free encyclopedia

Jump to navigation Jump to search

Category for dimensionless numbers in mechanics.

Pages in category "Dimensionless numbers of mechanics"

The following 3 pages are in this category, out of 3 total. This list may not reflect recent changes (learn more).

C

Cohesion number

D

Damping

L

Love number

Categories:

https://en.wikipedia.org/wiki/Category:Dimensionless_numbers_of_mechanics

A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

https://en.wikipedia.org/wiki/Banked_turn

In mechanics, a cylinder stress is a stress distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.

Cylinder stress patterns include:

circumferential stress, or hoop stress, a normal stress in the tangential (azimuth) direction.
axial stress, a normal stress parallel to the axis of cylindrical symmetry.
radial stress, a normal stress in directions coplanar with but perpendicular to the symmetry axis.

These three principal stresses- hoop, longitudinal, and radial can be calculated analytically using a mutually perpendicular tri-axial stress system.^[1]

https://en.wikipedia.org/wiki/Cylinder_stress

Pressure drop is defined as the difference in total pressure between two points of a fluid carrying network. A pressure drop occurs when frictional forces, caused by the resistance to flow, act on a fluid as it flows through the tube. The main determinants of resistance to fluid flow are fluid velocity through the pipe and fluid viscosity. Pressure drop increases proportionally to the frictional shear forces within the piping network. A piping network containing a high relative roughness rating as well as many pipe fittings and joints, tube convergence, divergence, turns, surface roughness, and other physical properties will affect the pressure drop. High flow velocities and/or high fluid viscosities result in a larger pressure drop across a section of pipe or a valve or elbow. Low velocity will result in lower or no pressure drop. The fluid may also be biphasic as in pneumatic conveying with a gas and a solid, in this case, the friction of the solid must also be taken into consideration for calculating the pressure drop.^[1]

History[edit]

Following its introduction to Europe from the New World in the late 15th century, natural rubber (polyisoprene) was regarded mostly as a fascinating curiosity. Its most useful application was its ability to erase pencil marks on paper by rubbing, hence its name. One of its most peculiar properties is a slight (but detectable) increase in temperature that occurs when a sample of rubber is stretched. If it is allowed to quickly retract, an equal amount of cooling is observed. This phenomenon caught the attention of the English physicist John Gough. In 1805 he published some qualitative observations on this characteristic as well as how the required stretching force increased with temperature.^[1]

By the mid nineteenth century, the theory of thermodynamics was being developed and within this framework, the English mathematician and physicist Lord Kelvin^[2] showed that the change in mechanical energy required to stretch a rubber sample should be proportional to the increase in temperature. Later, this would be associated with a change in entropy. The connection to thermodynamics was firmly established in 1859 when the English physicist James Joule published the first careful measurements of the temperature increase that occurred as a rubber sample was stretched.^[3] This work confirmed the theoretical predictions of Lord Kelvin.

It was not until 1838 that the American inventor Charles Goodyear found that natural rubber's properties could be immensely improved by adding a few percent sulphur. The short sulfur chains produced chemical cross-links between adjacent polyisoprenemolecules. Before it is cross-linked, the liquid natural rubber consists of very long linear chains, containing thousands of isoprenebackbone units, connected head-to-tail. Every chain follows a random path through the liquid and is in contact with thousands of other nearby chains. When heated to about 150C, cross-linker molecules (such as sulfur or dicumyl peroxide) can decompose and the subsequent chemical reactions produce a chemical bond between adjacent chains. The result is a three dimensional molecular network. All of the original polyisoprene chains are connected together at multiple points by these chemical bonds (network nodes) to form a single giant molecule. A rubber band is a single molecule, as is a latex glove! The sections between two cross-links on the same chain are called network chains and can contain up to several hundred isoprene units. In natural rubber, each cross-link produces a network node with four chains emanating from it. The network is the sine qua non of elastomers.

Because of the enormous economic and technological importance of rubber, predicting how a molecular network responds to mechanical strains has been of enduring interest to scientists and engineers. To understand the elastic properties of rubber, theoretically, it is necessary to know both the physical mechanisms that occur at the molecular level and how the random-walk nature of the polymer chain defines the network. The physical mechanisms that occur within short sections of the polymer chains produce the elastic forces and the network morphology determines how these forces combine to produce the macroscopic stressthat we observe when a rubber sample is deformed, e.g. subjected to tensile strain.

https://en.wikipedia.org/wiki/Rubber_elasticity

Drucker stability (also called the Drucker stability postulates) refers to a set of mathematical criteria that restrict the possible nonlinear stress-strain relations that can be satisfied by a solid material.^[1] The postulates are named after Daniel C. Drucker. A material that does not satisfy these criteria is often found to be unstable in the sense that application of a load to a material point can lead to arbitrary deformations at that material point unless an additional length or time scale is specified in the constitutive relations.

The Drucker stability postulates are often invoked in nonlinear finite element analysis. Materials that satisfy these criteria are generally well-suited for numerical analysis, while materials that fail to satisfy this criterion are likely to present difficulties (i.e. non-uniqueness or singularity) during the solution process.

https://en.wikipedia.org/wiki/Drucker_stability

https://en.wikipedia.org/wiki/Rubber_elasticity

https://en.wikipedia.org/wiki/Normal_distribution

https://en.wikipedia.org/wiki/Probability_density_function

https://en.wikipedia.org/wiki/Boltzmann_distribution

https://en.wikipedia.org/wiki/Tortuosity

https://en.wikipedia.org/wiki/Dihedral_angle

https://en.wikipedia.org/wiki/Kuhn_length

https://en.wikipedia.org/wiki/Conditional_probability_distribution

https://en.wikipedia.org/wiki/Entropic_force

https://en.wikipedia.org/wiki/Neo-Hookean_solid

https://en.wikipedia.org/wiki/Hooke%27s_law

https://en.wikipedia.org/wiki/Elasticity_(physics)

https://en.wikipedia.org/wiki/Category:Thermodynamics

A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable.^[1]^[2] Such constraints are called rheonomic constraints. The opposite of rheonomous is scleronomous.^[1]^[2]

A simple pendulum

Example: simple 2D pendulum[edit]

A simple pendulum

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string has a constant length. Therefore, this system is scleronomous; it obeys the scleronomic constraint

{\sqrt {x^{2}+y^{2}}}-L=0\,\!

where $(x,\ y)\,\!$ is the position of the weight and $L\,\!$ the length of the string.

A simple pendulum with oscillating pivot point

The situation changes if the pivot point is moving, e.g. undergoing a simple harmonic motion

x_{t}=x_{0}\cos \omega t\,\!

where $x_{0}\,\!$ is the amplitude, $\omega \,\!$ the angular frequency, and $t\,\!$ time.

Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous; it obeys the rheonomic constraint

{\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!

**The four fundamental forces of nature**^[27]
Property/Interaction	Gravitation	Weak	Electromagnetic	Strong
(Electroweak)	Fundamental	Residual
Acts on:	Mass - Energy	Flavor	Electric charge	Color charge	Atomic nuclei
Particles experiencing:	All	Quarks, leptons	Electrically charged	Quarks, Gluons	Hadrons
Particles mediating:	Graviton (not yet observed)	W⁺ W⁻ Z⁰	γ	Gluons	Mesons
Strength in the scale of quarks:	10⁻⁴¹	10⁻⁴	1	60	Not applicable to quarks
Strength in the scale of protons/neutrons:	10⁻³⁶	10⁻⁷	1	Not applicable to hadrons	20

Torsion balance[edit]

Drawing of Coulomb's torsion balance. From Plate 13 of his 1785 memoir.

Torsion balance used by Paul R. Heyl in his measurements of the gravitational constant G at the U.S. National Bureau of Standards (now NIST) between 1930 and 1942.

The torsion balance, also called torsion pendulum, is a scientific apparatus for measuring very weak forces, usually credited to Charles-Augustin de Coulomb, who invented it in 1777, but independently invented by John Michell sometime before 1783.^[3] Its most well-known uses were by Coulomb to measure the electrostatic force between charges to establish Coulomb's Law, and by Henry Cavendish in 1798 in the Cavendish experiment^[4]to measure the gravitational force between two masses to calculate the density of the Earth, leading later to a value for the gravitational constant.

The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak torsion spring. If an unknown force is applied at right angles to the ends of the bar, the bar will rotate, twisting the fiber, until it reaches an equilibrium where the twisting force or torque of the fiber balances the applied force. Then the magnitude of the force is proportional to the angle of the bar. The sensitivity of the instrument comes from the weak spring constant of the fiber, so a very weak force causes a large rotation of the bar.

In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread. The ball was charged with a known charge of static electricity, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls. Determining the force for different charges and different separations between the balls, he showed that it followed an inverse-square proportionality law, now known as Coulomb's law.

To measure the unknown force, the spring constant of the torsion fiber must first be known. This is difficult to measure directly because of the smallness of the force. Cavendish accomplished this by a method widely used since: measuring the resonant vibration periodof the balance. If the free balance is twisted and released, it will oscillate slowly clockwise and counterclockwise as a harmonic oscillator, at a frequency that depends on the moment of inertia of the beam and the elasticity of the fiber. Since the inertia of the beam can be found from its mass, the spring constant can be calculated.

Coulomb first developed the theory of torsion fibers and the torsion balance in his 1785 memoir, Recherches theoriques et experimentales sur la force de torsion et sur l'elasticite des fils de metal &c. This led to its use in other scientific instruments, such as galvanometers, and the Nichols radiometer which measured the radiation pressure of light. In the early 1900s gravitational torsion balances were used in petroleum prospecting. Today torsion balances are still used in physics experiments. In 1987, gravity researcher A.H. Cook wrote:

The most important advance in experiments on gravitation and other delicate measurements was the introduction of the torsion balance by Michell and its use by Cavendish. It has been the basis of all the most significant experiments on gravitation ever since.^[5]

Torsional harmonic oscillators[edit]

Definition of terms
Term	Unit	Definition
$\theta \,$	rad	Angle of deflection from rest position
$I\,$	kg m²	Moment of inertia
$C\,$	joule s rad⁻¹	Angular damping constant
$\kappa \,$	N m rad⁻¹	Torsion spring constant
$\tau \,$	${\mathrm {N\,m}}\,$	Drive torque
$f_{n}\,$	Hz	Undamped (or natural) resonant frequency
$T_{n}\,$	s	Undamped (or natural) period of oscillation
$\omega _{n}\,$	${\mathrm {rad\,s^{{-1}}}}\,$	Undamped resonant frequency in radians
$f\,$	Hz	Damped resonant frequency
$\omega\,$	${\mathrm {rad\,s^{{-1}}}}\,$	Damped resonant frequency in radians
$\alpha \,$	${\mathrm {s^{{-1}}}}\,$	Reciprocal of damping time constant
$\phi \,$	rad	Phase angle of oscillation
$L\,$	m	Distance from axis to where force is applied

Torsion balances, torsion pendulums and balance wheels are examples of torsional harmonic oscillators that can oscillate with a rotational motion about the axis of the torsion spring, clockwise and counterclockwise, in harmonic motion. Their behavior is analogous to translational spring-mass oscillators (see Harmonic oscillator Equivalent systems). The general differential equationof motion is:

I{\frac {d^{2}\theta }{dt^{2}}}+C{\frac {d\theta }{dt}}+\kappa \theta =\tau (t)

If the damping is small, $C\ll {\sqrt {{\frac {\kappa }{I}}}}\,$ , as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system:

f_{n}={\frac {\omega _{n}}{2\pi }}={\frac {1}{2\pi }}{\sqrt {{\frac {\kappa }{I}}}}\,

Therefore, the period is represented by:

T_{n}={\frac {1}{f_{n}}}={\frac {2\pi }{\omega _{n}}}=2\pi {\sqrt {{\frac {I}{\kappa }}}}\,

The general solution in the case of no drive force ( $\tau =0\,$ ), called the transient solution, is:

\theta =Ae^{{-\alpha t}}\cos {(\omega t+\phi )}\,

where:

\alpha =C/2I\,

\omega ={\sqrt {\omega _{n}^{2}-\alpha ^{2}}}={\sqrt {\kappa /I-(C/2I)^{2}}}\,

Applications[edit]

Animation of a torsion spring oscillating

The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency $f_{n}\,$ sets the rate of the watch. The resonant frequency is regulated, first coarsely by adjusting $I\,$ with weight screws set radially into the rim of the wheel, and then more finely by adjusting $\kappa \,$ with a regulating lever that changes the length of the balance spring.

In a torsion balance the drive torque is constant and equal to the unknown force to be measured $F\,$ , times the moment arm of the balance beam $L\,$ , so $\tau (t)=FL\,$ . When the oscillatory motion of the balance dies out, the deflection will be proportional to the force:

\theta =FL/\kappa \,

To determine $F\,$ it is necessary to find the torsion spring constant $\kappa \,$ . If the damping is low, this can be obtained by measuring the natural resonant frequency of the balance, since the moment of inertia of the balance can usually be calculated from its geometry, so:

\kappa =(2\pi f_{n})^{2}I\,

In measuring instruments, such as the D'Arsonval ammeter movement, it is often desired that the oscillatory motion die out quickly so the steady state result can be read off. This is accomplished by adding damping to the system, often by attaching a vane that rotates in a fluid such as air or water (this is why magnetic compasses are filled with fluid). The value of damping that causes the oscillatory motion to settle quickest is called the critical damping $C_{c}\,$ :

C_{c}=2{\sqrt {\kappa I}}\,

External links[edit]

Wikimedia Commons has media related to Torsion springs.

Torsion balance interactive java tutorial
Torsion spring calculator
Big G measurement, description of 1999 Cavendish experiment at Univ. of Washington, showing torsion balance[link broken]
How torsion balances were used in petroleum prospecting (web archive link)
Mechanics of torsion springs. Web archive link, accessed December 8, 2016.
Solved mechanics problems involving springs (springs in series and in parallel)
Milestones in the History of Springs

https://en.wikipedia.org/wiki/Torsion_spring#Torsion_balance

A sliding pillar suspension is a form of independent front suspension for light cars. The stub axle and wheel assembly are attached to a vertical pillar or kingpin which slides up and down through a bush or bushes which are attached to the vehicle chassis, usually as part of transverse outrigger assemblies, sometimes resembling a traditional beam axle, although fixed rigidly to the chassis.^[1] Steering movement is provided by allowing this same sliding pillar to also rotate.

https://en.wikipedia.org/wiki/Sliding_pillar_suspension

Brownian motion, or pedesis (from Ancient Greek: πήδησις /pɛ̌ːdɛːsis/ "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).^[2]

This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the Equipartition theorem).

This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchellaimmersed in water. In 1905, almost eighty years later, theoretical physicist Albert Einsteinpublished a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.^[3] The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atomsand molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter".^[4]

The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).^[5]^[6]

The Roman philosopher-poet Lucretius' scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of the motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms:

Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.

Although the mingling motion of dust particles is caused largely by air currents, the glittering, tumbling motion of small dust particles is caused chiefly by true Brownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example".^[8]

While Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, the discovery of this phenomenon is often credited to the botanist Robert Brown in 1827. Brown was studying pollen grains of the plant Clarkia pulchellasuspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the stock market is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.^[9]

Albert Einstein (in one of his 1905 papers) and Marian Smoluchowski (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908.

https://en.wikipedia.org/wiki/Brownian_motion

https://en.wikipedia.org/wiki/Barometric_formula

https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet

https://en.wikipedia.org/wiki/Mean_free_path

https://en.wikipedia.org/wiki/Walther_Nernst

https://en.wikipedia.org/wiki/Molecular_diffusion

https://en.wikipedia.org/wiki/J._J._Thomson

https://en.wikipedia.org/wiki/Boltzmann_constant

https://en.wikipedia.org/wiki/Van_%27t_Hoff_factor

The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. The pressure drops approximately by 11.3 pascals per meter in first 1000 meters above sea level.

https://en.wikipedia.org/wiki/Barometric_formula

https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)

https://en.wikipedia.org/wiki/Reduced_mass

https://en.wikipedia.org/wiki/Brownian_motion

https://en.wikipedia.org/wiki/Category:Mechanics

Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells.^[1] The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called membrane forces, shear forces, and bending moment) that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells).

Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer powers the thickness coordinate z (or x₃). Stress resultants are so defined to represent the effect of stress as a membrane force N(zero power in z), bending moment M (power 1) on a beam or shell (structure). Stress resultants are necessary to eliminate the zdependency of the stress from the equations of the theory of plates and shells.

https://en.wikipedia.org/wiki/Stress_resultants

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.

Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.^[1] The error is then typically of the form $~ exp(- c /ε)$ where $ε$ is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.

See asymptotic analysis and big O notation for the notation used in this article.

https://en.wikipedia.org/wiki/Asymptotic_expansion

A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite of scleronomous is rheonomous.

https://en.wikipedia.org/wiki/Scleronomous

Sandwich theory^[1]^[2] describes the behaviour of a beam, plate, or shell which consists of three layers—two facesheets and one core. The most commonly used sandwich theory is linear and is an extension of first order beam theory. Linear sandwich theory is of importance for the design and analysis of sandwich panels, which are of use in building construction, vehicle construction, airplane construction and refrigeration engineering.

Some advantages of sandwich construction are:

Sandwich cross sections are composite. They usually consist of a low to moderate stiffness core which is connected with two stiff exterior facesheets. The composite has a considerably higher shear stiffness to weight ratio than an equivalent beam made of only the core material or the facesheet material. The composite also has a high tensile strength to weight ratio.
The high stiffness of the facesheet leads to a high bending stiffness to weight ratio for the composite.

The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section. If the radius of curvature during bending is large compared to the thickness of the sandwich beam and the strains in the component materials are small, the deformation of a sandwich composite beam can be separated into two parts

deformations due to bending moments or bending deformation, and
deformations due to transverse forces, also called shear deformation.

Sandwich beam, plate, and shell theories usually assume that the reference stress state is one of zero stress. However, during curing, differences of temperature between the facesheets persist because of the thermal separation by the core material. These temperature differences, coupled with different linear expansions of the facesheets, can lead to a bending of the sandwich beam in the direction of the warmer facesheet. If the bending is constrained during the manufacturing process, residual stresses can develop in the components of a sandwich composite. The superposition of a reference stress state on the solutions provided by sandwich theory is possible when the problem is linear. However, when large elastic deformations and rotations are expected, the initial stress state has to be incorporated directly into the sandwich theory.

https://en.wikipedia.org/wiki/Sandwich_theory

A differential pulley, also called "Weston differential pulley", sometimes "chain hoist" or colloquially "chain fall", is used to manually lift very heavy objects like car engines. It is operated by pulling upon the slack section of a continuous chain that wraps around pulleys. The relative size of two connected pulleys determines the maximum weight that can be lifted by hand. If the pulley radii are close enough the load will remain in place (and not lower under the force of gravity) until the chain is pulled.^[1]

https://en.wikipedia.org/wiki/Differential_pulley

In physics, an energy well describes a 'stable' equilibrium that is not at lowest possible energy.

In general, modern physics holds the view that the universe - and systems therein - spontaneously drives toward a state of lower energy, if possible. For example, a bowling ball pitched atop a smooth hump (which has potential energy in the presence of gravity), will tend to roll down to the lowest point it possibly can. Once there, this reduces the total potential energy of the system.

On the other hand, if the bowling ball is resting in a valley between two humps - no matter how big the drops outside the humps - it will stay there indefinitely. Even though the system could achieve a lower energy state, it cannot do so without external energy being applied: (locally) it is at its lowest energy state, and only a force from outside the system can 'push' it over one of the humps so a lower state can be achieved.

The concept of an energy well is a key part of teaching basic physics, especially quantum mechanics. Here, students often solve the one-dimensional Schrödinger Equation for an electron trapped in a potential well from which it has insufficient energy to escape. The solution to this problem is a series of sinusoidal waves of fractional integral wavelengths determined by the width of the well.

https://en.wikipedia.org/wiki/Energy_well

A mechanical or physical shock is a sudden acceleration caused, for example, by impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation.

Shock describes matter subject to extreme rates of force with respect to time. Shock is a vector that has units of an acceleration (rate of change of velocity). The unit g (or g) represents multiples of the acceleration of gravity and is conventionally used.

A shock pulse can be characterised by its peak acceleration, the duration, and the shape of the shock pulse (half sine, triangular, trapezoidal, etc.). The Shock response spectrum is a method for further evaluating a mechanical shock.^[1]

https://en.wikipedia.org/wiki/Shock_(mechanics)

https://en.wikipedia.org/wiki/Vaccum

https://en.wikipedia.org/wiki/Void

https://en.wikipedia.org/wiki/Shock_(mechanics)

https://en.wikipedia.org/wiki/Buckling

https://en.wikipedia.org/wiki/Fouling

https://en.wikipedia.org/wiki/Fracture

https://en.wikipedia.org/wiki/Mechanical_overload

https://en.wikipedia.org/wiki/Thermal_shock

https://en.wikipedia.org/wiki/Stress_corrosion_cracking

https://en.wikipedia.org/wiki/Hydrogen_embrittlement

https://en.wikipedia.org/wiki/Yield_(engineering)

https://en.wikipedia.org/wiki/Creep_(deformation)

https://en.wikipedia.org/wiki/Corrosion

https://en.wikipedia.org/wiki/Stiffening

Six degrees of freedom (6DoF) refers to the freedom of movement of a rigid body in three-dimensional space. Specifically, the body is free to change position as forward/backward (surge), up/down (heave), left/right (sway) translation in three perpendicular axes, combined with changes in orientation through rotation about three perpendicular axes, often termed yaw (normal axis), pitch (transverse axis), and roll (longitudinal axis). Three degrees of freedom (3DOF), a term often used in the context of virtual reality, refers to tracking of rotational motion only: pitch, yaw, and roll.^[1]^[2]

https://en.wikipedia.org/wiki/Six_degrees_of_freedom

Stokes' theorem,^[1] also known as Kelvin–Stokes theorem^[2]^[3] after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem,^[4] is a theorem in vector calculus on $\mathbb {R} ^{3}$ . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.

Stokes' theorem is a special case of the generalized Stokes' theorem.^[5]^[6] In particular, a vector field on $\mathbb {R} ^{3}$ can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.

https://en.wikipedia.org/wiki/Stokes%27_theorem

In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkagesor hinges.

https://en.wikipedia.org/wiki/Structural_rigidity

https://en.wikipedia.org/wiki/Linkage_(mechanical)

https://en.wikipedia.org/wiki/Hinge

In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analysis because comprehensive testing is impractical on many projects, such as bridges and buildings, but the structure's ability to carry a load must be determined to a reasonable accuracy.

Many systems are intentionally built much stronger than needed for normal usage to allow for emergency situations, unexpected loads, misuse, or degradation (reliability).

https://en.wikipedia.org/wiki/Factor_of_safety

The four-point flexural test provides values for the modulus of elasticity in bending $E_{f}$ , flexural stress $\sigma _{f}$ , flexural strain $\varepsilon_f$ and the flexural stress-strain response of the material. This test is very similar to the three-point bending flexural test. The major difference being that with the addition of a fourth bearing the portion of the beam between the two loading points is put under maximum stress, as opposed to only the material right under the central bearing in the case of three point bending.

This difference is of prime importance when studying brittle materials, where the number and severity of flaws exposed to the maximum stress is directly related to the flexural strength and crack initiation. Compared to the three-point bending flexural test, there are no shear forces in the four-point bending flexural test in the area between the two loading pins.^[1] The four-point bending test is therefore particularly suitable for brittle materials that cannot withstand shear stresses very well.

It is one of the most widely used apparatus to characterize fatigue and flexural stiffness of asphalt mixtures.^[2]

https://en.wikipedia.org/wiki/Four-point_flexural_test

In mechanics, suspension is a system of components allowing a machine (normally a vehicle) to move smoothly with reduced shock.

Types may include:

car suspension, four-wheeled motor vehicle suspension
motorcycle suspension, two-wheeled motor vehicle suspension
- Motorcycle fork, a component of motorcycle suspension system
bicycle suspension

Related concepts include:

Relation to internal pressure[edit]

Thin-walled assumption[edit]

For the thin-walled assumption to be valid, the vessel must have a wall thickness of no more than about one-tenth (often cited as Diameter / t > 20) of its radius.^[4] This allows for treating the wall as a surface, and subsequently using the Young–Laplace equationfor estimating the hoop stress created by an internal pressure on a thin-walled cylindrical pressure vessel:

\sigma _{\theta }={\dfrac {Pr}{t}}\

(for a cylinder)

\sigma _{\theta }={\dfrac {Pr}{2t}}\

(for a sphere)

where

P is the internal pressure
t is the wall thickness
r is the mean radius of the cylinder
$\sigma _{\theta }\!$ is the hoop stress.

The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure may reach several atmospheres. In practical engineering applications for cylinders (pipes and tubes), hoop stress is often re-arranged for pressure, and is called Barlow's formula.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and d are inches (in). SI units for Pare pascals (Pa), while t and d=2r are in meters (m).

When the vessel has closed ends, the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.

\sigma _{z}={\dfrac {F}{A}}={\dfrac {Pd^{2}}{(d+2t)^{2}-d^{2}}}\

Though this may be approximated to

\sigma _{z}={\dfrac {Pr}{2t}}\

There is also a radial stress $\sigma _{r}\$ that is developed perpendicular to the surface and may be estimated in thin walled cylinders as:

\sigma _{r}={-P}\

However, in the thin-walled assumption the ratio ${\dfrac {r}{t}}\$ is large, so in most cases this component is considered negligible compared to the hoop and axial stresses. ^[5]

Thick-walled vessels[edit]

When the cylinder to be studied has a $radius/thickness$ ratio of less than 10 (often cited as $diameter/thickness<20$ ) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and shear stressthrough the cross section can no longer be neglected.

These stresses and strains can be calculated using the Lamé equations, a set of equations developed by French mathematician Gabriel Lamé.

\sigma _{r}=A-{\dfrac {B}{r^{2}}}\

\sigma _{\theta }=A+{\dfrac {B}{r^{2}}}\

where:

A

and

B

are constants of integration, which may be discovered from the boundary conditions

r

is the radius at the point of interest (e.g., at the inside or outside walls)

$A$ and $B$ may be found by inspection of the boundary conditions. For example, the simplest case is a solid cylinder:

if $R_{i}=0$ then $B=0$ and a solid cylinder cannot have an internal pressure so $A=P_{o}$

Being that for thick-walled cylinders, the ratio ${\dfrac {r}{t}}\$ is less than 10, the radial stress, in proportion to the other stresses, becomes non-negligible (i.e. P is no longer much, much less than Pr/t and Pr/2t), and so the thickness of the wall becomes a major consideration for design (Harvey, 1974, pp. 57).

In pressure vessel theory, any given element of the wall is evaluated in a tri-axial stress system, with the three principal stresses being hoop, longitudinal, and radial. Therefore, by definition, there exist no shear stresses on the transverse, tangential, or radial planes.^[6]

In thick-walled cylinders, the maximum shear stress at any point is given by half of the algebraic difference between the maximum and minimum stresses, which is, therefore, equal to half the difference between the hoop and radial stresses. The shearing stress reaches a maximum at the inner surface, which is significant because it serves as a criterion for failure since it correlates well with actual rupture tests of thick cylinders (Harvey, 1974, p. 57).

https://en.wikipedia.org/wiki/Cylinder_stress#Hoop_stress

Radial stress is stress towards or away from the central axis of a component.

Pressure vessels[edit]

The walls of pressure vessels generally undergo triaxial loading. For cylindrical pressure vessels, the normal loads on a wall element are:

the longitudinal stress
the circumferential (hoop) stress
the radial stress.

The radial stress for a thick-walled cylinder is equal and opposite to the gauge pressure on the inside surface, and zero on the outside surface. The circumferential stress and longitudinal stresses are usually much larger for pressure vessels, and so for thin-walled instances, radial stress is usually neglected.

https://en.wikipedia.org/wiki/Radial_stress

The Hosford yield criterion is a function that is used to determine whether a material has undergone plastic yielding under the action of stress.

https://en.wikipedia.org/wiki/Hosford_yield_criterion

https://en.wikipedia.org/wiki/Ideal_machine

https://en.wikipedia.org/wiki/Category:Mechanics

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation.

With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made.

The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elasticmaterials like concrete and steel, since a common goal in the design of such structures is to minimize their deformation under typical loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.^[1]

https://en.wikipedia.org/wiki/Infinitesimal_strain_theory

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material.^[1] The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov^[2] and independently in 1968 by James R. Rice,^[3] who showed that an energetic contour path integral (called J) was independent of the path around a crack.

Experimental methods were developed using the integral that allowed the measurement of critical fracture properties in sample sizes that are too small for Linear Elastic Fracture Mechanics (LEFM) to be valid. ^[4] These experiments allow the determination of fracture toughness from the critical value of fracture energy J_Ic, which defines the point at which large-scale plastic yielding during propagation takes place under mode I loading.^[1]^[5]

The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading.^[6] This is generally true, under quasistatic conditions, only for linear elastic materials. For materials that experience small-scale yielding at the crack tip, J can be used to compute the energy release rate under special circumstances such as monotonic loading in mode III (antiplane shear). The strain energy release rate can also be computed from J for pure power-law hardening plastic materials that undergo small-scale yielding at the crack tip.

The quantity J is not path-independent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed that J is path-independent in plastic materials when there is no non-proportional loading. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials.

https://en.wikipedia.org/wiki/J-integral

Telescoping in mechanics describes the movement of one part sliding out from another, lengthening an object (such as a telescope or the lift arm of an aerial work platform) from its rest state. In modern equipment this can be achieved by a hydraulics, but pulleys are generally used for simpler designs such as extendable ladders & amateur radio antennas.

Showing the telescopic principle, an object collapsed (above) and extended (below), providing more reach.

https://en.wikipedia.org/wiki/Telescoping_(mechanics)

The thermal center is a concept used in applied mechanicsand engineering. When a solid body is exposed to a thermal variation, an expansion will occur, changing the dimensions and potentially the shape of the body and the position of its points. Under certain circumstances it may happen that one point belonging to the space associated to the body has no displacement at all: this point is called the thermal center (TC).^[1]

https://en.wikipedia.org/wiki/Thermal_center

https://en.wikipedia.org/wiki/Symmetry

https://en.wikipedia.org/wiki/Supersymmetry

In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. Torsion is expressed in either the pascal (Pa), an SI unit for newtons per square metre, or in pounds per square inch (psi) while torque is expressed in newton metres (N·m) or foot-pound force (ft·lbf). In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius.

In non-circular cross-sections, twisting is accompanied by a distortion called warping, in which transverse sections do not remain plane.^[1] For shafts of uniform cross-section unrestrained against warping, the torsion is:

T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}G\varphi

where:

T is the applied torque or moment of torsion in Nm.
$\tau$ (tau) is the maximum shear stress at the outer surface
J_T is the torsion constant for the section. For circular rods, and tubes with constant wall thickness, it is equal to the polar moment of inertia of the section, but for other shapes, or split sections, it can be much less. For more accuracy, finite element analysis(FEA) is the best method. Other calculation methods include membrane analogy and shear flow approximation. ^[2]
r is the perpendicular distance between the rotational axis and the farthest point in the section (at the outer surface).
ℓ is the length of the object to or over which the torque is being applied.
φ (phi) is the angle of twist in radians.
G is the shear modulus, also called the modulus of rigidity, and is usually given in gigapascals (GPa), lbf/in² (psi), or lbf/ft² or in ISO units N/mm².
The product J_TG is called the torsional rigidity w_T.

https://en.wikipedia.org/wiki/Torsion_(mechanics)

The three-point bending flexural test provides values for the modulus of elasticity in bending

E_{f}

, flexural stress

\sigma _{f}

, flexural strain

\epsilon _{f}

and the flexural stress–strain response of the material. This test is performed on a universal testing machine (tensile testing machine or tensile tester) with a three-point or four-point bend fixture.The main advantage of a three-point flexural test is the ease of the specimen preparation and testing. However, this method has also some disadvantages: the results of the testing method are sensitive to specimen and loading geometry and strain rate.

https://en.wikipedia.org/wiki/Three-point_flexural_test

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.

The mass might be a projectile or a satellite.^[1] For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.

In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence $(f^{k}(x))_{{k\in {\mathbb {N}}}}$ of values calculated by the iterated application of a mapping $f$ to an element $x$ of its source.

https://en.wikipedia.org/wiki/Trajectory

A truss is an assembly of members such as beams, connected by nodes, that creates a rigid structure.^[1]

In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so that the assemblage as a whole behaves as a single object".^[2] A "two-force member" is a structural component where force is applied to only two points. Although this rigorous definition allows the members to have any shape connected in any stable configuration, trusses typically comprise five or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes.

In this typical context, external forces and reactions to those forces are considered to act only at the nodes and result in forces in the members that are either tensile or compressive. For straight members, moments (torques) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes, as is necessary for the links to be two-force members.

A planar truss is one where all members and nodes lie within a two-dimensional plane, while a space truss has members and nodes that extend into three dimensions. The top beams in a truss are called top chords and are typically in compression, the bottom beams are called bottom chords, and are typically in tension. The interior beams are called webs, and the areas inside the webs are called panels,^[3] or from graphic statics (see Cremona diagram) polygons.^[4]

https://en.wikipedia.org/wiki/Truss

In contact mechanics, the term unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies. Constraints of this kind are omnipresent in non-smooth multibody dynamicsapplications, such as granular flows,^[1] legged robot, vehicle dynamics, particle damping, imperfect joints,^[2] or rocket landings. In these applications, the unilateral constraints result in impacts happening, therefore requiring suitable methods to deal with such constraints.

https://en.wikipedia.org/wiki/Unilateral_contact

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

https://en.wikipedia.org/wiki/Continuum_mechanics

In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system.^[1]^[2] Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as

{\mathbf {F}}_{{{\mathrm {ext}}}}+{\mathbf {v}}_{{{\mathrm {rel}}}}{\frac {{\mathrm {d}}m}{{\mathrm {d}}t}}=m{{\mathrm {d}}{\mathbf v} \over {\mathrm {d}}t}

where F_ext is the net external force on the body, v_rel is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body, and v is the velocity of the body.^[3] In astrodynamics, which deals with the mechanics of rockets, the term v_rel is often called the effective exhaust velocity and denoted v_e.^[4]

https://en.wikipedia.org/wiki/Variable-mass_system

Vis viva (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energyin an early formulation of the principle of conservation of energy.

https://en.wikipedia.org/wiki/Vis_viva

Varignon's theorem is a theorem by French mathematician Pierre Varignon (1654-1722), published in 1687 in his book Projet d'une nouvelle mécanique. The theorem states that the torque of a resultant of two concurrent forces about any point is equal to the algebraic sum of the torques of its components about the same point. In other words, "If many concurrent forces are acting on a body, then the algebraic sum of torques of all the forces about a point in the plane of the forces is equal to the torque of their resultant about the same point."^[1]

https://en.wikipedia.org/wiki/Varignon%27s_theorem_(mechanics)

In materials science, the yield strength anomaly refers to materials wherein the yield strength (i.e., the stress necessary to initiate plastic yielding) increases with temperature.^[1]^[2]^[3] For the majority of materials, the yield strength decreases with increasing temperature. In metals, this decrease in yield strength is due to the thermal activation of dislocation motion, resulting in easier plastic deformation at higher temperatures.^[4]

In some cases, a yield strength anomaly refers to a decrease in the ductility of a material with increasing temperature, which is also opposite the trend in the majority of materials. Anomalies in ductility can be more clear, as an anomalous effect on yield strength can be obscured by its typical decrease with temperature.^[5] In concert with yield strength or ductility anomalies, some materials demonstrate extrema in other temperature dependent properties, such as a minimum in ultrasonic damping, or a maximum in electrical conductivity.^[6]

The yield strength anomaly in β-brass was one of the earliest discoveries such a phenomenon,^[7] and several other ordered intermetallic alloys demonstrate this effect. Precipitation hardened superalloys exhibit a yield strength anomaly over a considerable temperature range. For these materials, the yield strength shows little variation between room temperature and several hundred degrees Celsius. Eventually, a maximum yield strength is reached. For even higher temperatures, the yield strength decreases and, eventually, drops to zero when reaching the melting temperature, where the solid material transforms into a liquid. For ordered intermetallics, the temperature of the yield strength peak is roughly 50% of the absolute melting temperature.^[8]

https://en.wikipedia.org/wiki/Yield_strength_anomaly

In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on earth their mass distribution is almost homogeneous.

https://en.wikipedia.org/wiki/Mass_distribution

Kinematic synthesis, also known as mechanism synthesis, determines the size and configuration of mechanisms that shape the flow of power through a mechanical system, or machine, to achieve a desired performance.^[1] The word synthesis refers to combining parts to form a whole.^[2] Hartenberg and Denavit describe kinematic synthesis as^[3]

...it is design, the creation of something new. Kinematically, it is the conversion of a motion idea into hardware.

The earliest machines were designed to amplify human and animal effort, later gear trains and linkage systems captured wind and flowing water to rotate millstones and pumps. Now machines use chemical and electric power to manufacture, transport, and process items of all types. And kinematic synthesis is the collection of techniques for designing those elements of these machines that achieve required output forces and movement for a given input.

Applications of kinematic synthesis include determining:

the topology and dimensions of a linkage system to achieve a specified task;^[4]
the size and shape of links of a robot to move parts and apply forces in a specified workspace;^[5]
the mechanical configuration of end-effectors, or grippers, for robotic systems;^[6]
the shape of a cam and follower to achieve a desired output movement coordinated with a specified input movement;^[7]
the shape of gear teeth to ensure a desired coordination of input and output movement;^[8]
the configuration of a system of gears, belts, and cable, or rope drives, to perform a desired power transmission;
the size and shape of fixturing systems to provide precision in part manufacture and component assembly.^[9]

Kinematic synthesis for a mechanical system is described as having three general phases, known as type synthesis, number synthesis and dimensional synthesis.^[3] Type synthesis matches the general characteristics of a mechanical system to the task at hand, selecting from an array of devices such as a cam-follower mechanism, linkage, gear train, a fixture or a robotic system for use in a required task. Number synthesis considers the various ways a particular device can be constructed, generally focussed on the number and features of the parts. Finally, dimensional synthesis determines the geometry and assembly of the components that form the device.

https://en.wikipedia.org/wiki/Kinematic_synthesis

In kinetics, König's theorem or König's decomposition is a mathematical relation derived by Johann Samuel König that assists with the calculations of angular momentum and kinetic energy of bodies and systems of particles.

https://en.wikipedia.org/wiki/König%27s_theorem_(kinetics)

Friction torque is the torque caused by the frictional force that occurs when two objects in contact move. Like all torques, it is a rotational force which may be measured in newton metres or pounds-feet.

https://en.wikipedia.org/wiki/Friction_torque

Lamé's stress ellipsoid is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse.

https://en.wikipedia.org/wiki/Lamé%27s_stress_ellipsoid

https://en.wikipedia.org/wiki/Line_of_thrust

Screw theory is the algebraic calculation of pairs of vectors, such as forces and momentsor angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies.^[1]^[2]The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics).^[3]

Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors.^[3]

https://en.wikipedia.org/wiki/Screw_theory

$q,\;$	dynamic pressure in pascals (i.e., kg/m⋅s²),
$\rho ,\;$	fluid mass density (e.g. in kg/m³, in SI units),
$u,\;$	flow speed in m/s.

Nikiya Anton Bettey 2021

Blog Archive

Monday, September 20, 2021

09-20-2021-0633 - Mechanics drafting (resultant force, deformation, beam, lateral load, contraflexure, etc.)

See also[edit]

Pages in category "Mathematics of rigidity"

A

B

C

F

G

L

P

R

S

Category:Dimensionless numbers of mechanics

Pages in category "Dimensionless numbers of mechanics"

C

D

L

See also[edit]

History[edit]

Example: simple 2D pendulum[edit]

See also[edit]

Torsion balance[edit]

Torsional harmonic oscillators[edit]

Applications[edit]

See also[edit]

External links[edit]

See also[edit]

See also[edit]

Hoop stress[edit]

Relation to internal pressure[edit]

Thin-walled assumption[edit]

Thick-walled vessels[edit]

Pressure vessels[edit]

No comments:

Post a Comment