|
Elastic Instability
Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads. There are a lot of ways to study this kind of instability. One of them is to use the method of incremental deformations based on superposing a small perturbation on an equilibrium solution. Single degree of freedom-systems Consider as a simple example a rigid beam of length ''L'', hinged in one end and free in the other, and having an angular spring attached to the hinged end. The beam is loaded in the free end by a force ''F'' acting in the compressive axial direction of the beam, see the figure to the right. Moment equilibrium condition Assuming a clockwise angular deflection \theta, the clockwise moment exerted by the force becomes M_F = F L \sin\theta. The moment equilibrium equation is given by F L \sin \theta = k_\theta \theta where k_\theta is the spring constant of the angular spring (Nm/radian). Assuming \theta i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic Instability Curve
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects together * Bungee cord, a cord composed of an elastic core covered in a sheath * Chinese jump rope, a children's game resembling hopscotch and jump rope As a proper name * ''Elastic'' (album), a 2002 album by jazz saxophonist Joshua Redman *"Elastic", a 2018 single by Joey Purp *Elastic, working title of the 2012 Indian film ''Cocktail'' * Elastic NV, the company that releases the Elasticsearch search engine ** Elasticsearch, a search engine based on Apache Lucene * Amazon Elastic Compute Cloud (Amazon EC2), a web service that provides secure, resizable compute capacity in a cloud format * Elastics (orthodontics), rubber bands used in orthodontics See also * Elastic collision, a collision where kinetic energy is conserved * Elastic deformati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real number, real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as Series (mathematics), infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic function, periodic pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Timoshenko
Stepan Prokopovich Timoshenko (, ; , ; – May 29, 1972), later known as Stephen Timoshenko, was a Ukrainian and later an American engineer and academician. He is considered to be the father of modern engineering mechanics. An inventor and one of the pioneering mechanical engineers at the St. Petersburg Polytechnic University. A founding member of the Ukrainian Academy of Sciences, Timoshenko wrote seminal works in the areas of engineering mechanics, elasticity and strength of materials, many of which are still widely used today. Having started his scientific career in the Russian Empire, Timoshenko emigrated to the Kingdom of Serbs, Croats and Slovenes during the Russian Civil War and then to the United States. Biography Timoshenko was born in the village of Shpotovka, Uyezd of Konotop in the Chernigov Governorate which at that time was a territory of the Russian Empire (today in Konotop Raion, Sumy Oblast of Ukraine). He was ethnically Ukrainian. He studied at a R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drucker Stability
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. 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. Drucker's first stability criterion Drucker's first stability criterion (first proposed by Rodney Hill and also cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavitation (elastomers)
Cavitation is the unstable unhindered expansion of a microscopic void in a solid elastomer under the action of tensile hydrostatic stresses. This can occur whenever the hydrostatic tension exceeds 5/6 of Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn .... The cavitation phenomenon may manifest in any of the following situations: * imposed hydrostatic tensile stress acting on a pre-existing void * void pressurization due to gases that are generated due to chemical action (as in volatilization of low-molecular weight waxes or oils: 'blowpoint' for insufficiently cured rubber, or 'thermal blowout' for systems operating at very high temperature) * void pressurization due to gases that come out of solution (as in gases dissolved at high pressure) References Rubber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buckling
In structural engineering, buckling is the sudden change in shape (Deformation (engineering), deformation) of a structural component under Structural load, load, such as the bowing of a column under Compression (physics), compression or the wrinkling of a plate under Shearing (physics), shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have ''buckled''. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column. Buckling may occur even though the Stress (mechanics), stresses that develop in the structure are well below those needed to cause Catastrophic failure, failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mode Shape
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. The most general motion of a linear system is a superposition of its normal modes. The modes are normal in the sense that they can move independently, that is to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other. General definitions Mode In the wave theory of physics and engineering, a mode in a dynamical system is a standing wave state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nullspace
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: \ker(L) = \left\ = L^(\mathbf). Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic Instability 2DOF
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects together * Bungee cord, a cord composed of an elastic core covered in a sheath * Chinese jump rope, a children's game resembling hopscotch and jump rope As a proper name * ''Elastic'' (album), a 2002 album by jazz saxophonist Joshua Redman *"Elastic", a 2018 single by Joey Purp *Elastic, working title of the 2012 Indian film ''Cocktail'' * Elastic NV, the company that releases the Elasticsearch search engine ** Elasticsearch, a search engine based on Apache Lucene * Amazon Elastic Compute Cloud (Amazon EC2), a web service that provides secure, resizable compute capacity in a cloud format * Elastics (orthodontics), rubber bands used in orthodontics See also * Elastic collision, a collision where kinetic energy is conserved * Elastic deformati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
