Virtual work

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

In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action.

The work of a force on a particle along a virtual displacement is known as the virtual work.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

History

The principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians as "the law of lever". The idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Descartes, Torricelli, Wallis, and Huygens, in varying degrees of generality, when solving problems in statics. Working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both rigid bodies as well as fluids. Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1715, which was later published in Varignon's second volume of Nouvelle mécanique ou Statique in 1725. This formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 D'Alembert published his Traité de Dynamique where he applied the principle of virtual work, based on Bernoulli's work, to solve various problems in dynamics. His idea was to convert a dynamical problem into static problem by introducing inertial force. In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved. A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic, essentially D'Alembert's principle, was given in his Mécanique Analytique of 1788. Although Lagrange had presented his version of least action principle prior to this work, he recognized the virtual work principle to be more fundamental mainly because it could be assumed alone as the foundation for all mechanics, unlike the modern understanding that least action does not account for non-conservative forces.

Overview

If a force acts on a particle as it moves from point ${\displaystyle A}$ to point ${\displaystyle B}$, then, for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path. The principle of virtual work, which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero (to the first order). The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, and is termed the calculus of variations.

Consider a point particle that moves along a path which is described by a function ${\displaystyle \mathbf{r}(t)}$ from point ${\displaystyle A}$, where ${\displaystyle \mathbf{r}(t=t_0)}$, to point ${\displaystyle B}$, where ${\displaystyle \mathbf{r}(t=t_1)}$. It is possible that the particle moves from ${\displaystyle A}$ to ${\displaystyle B}$ along a nearby path described by ${\displaystyle \mathbf{r}(t)+\delta \mathbf{r}(t)}$, where ${\displaystyle \delta \mathbf{r}(t)}$ is called the variation of ${\displaystyle \mathbf{r}(t)}$. The variation ${\displaystyle \delta \mathbf{r}(t)}$ satisfies the requirement ${\displaystyle \delta \mathbf{r}(t_0)=\delta \mathbf{r}(t_1)=0}$. The scalar components of the variation ${\displaystyle \delta r_1(t)}$, ${\displaystyle \delta r_2(t)}$ and ${\displaystyle \delta r_3(t)}$ are called virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates ${\displaystyle q_i}$, ${\displaystyle i=1,2,...,n}$. In which case, the variation of the trajectory ${\displaystyle q_i(t)}$ is defined by the virtual displacements ${\displaystyle \delta q_i}$, ${\displaystyle i=1,2,...,n}$.

Virtual work is the total work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements. When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero.

Mathematical treatment

Consider a particle P that moves from a point A to a point B along a trajectory r(t), while a force F(r(t)) is applied to it. The work done by the force F is given by the integral $${\displaystyle W={\int }_{\mathbf{r}(t_0)=A}^{\mathbf{r}(t_1)=B}\mathbf{F}\cdot d\mathbf{r}={\int }_{t_0}^{t_1}\mathbf{F}\cdot \frac{d\mathbf{r}}{dt}dt={\int }_{t_0}^{t_1}\mathbf{F}\cdot \mathbf{v}dt,}$$ where dr is the differential element along the curve that is the trajectory of P, and v is its velocity. It is important to notice that the value of the work W depends on the trajectory r(t).

Now consider particle P that moves from point A to point B again, but this time it moves along the nearby trajectory that differs from r(t) by the variation δr(t) = εh(t), where ε is a scaling constant that can be made as small as desired and h(t) is an arbitrary function that satisfies h(t₀) = h(t₁) = 0. Suppose the force F(r(t) + εh(t)) is the same as F(r(t)). The work done by the force is given by the integral $${\displaystyle \overline{W}={\int }_{\mathbf{r}(t_0)=A}^{\mathbf{r}(t_1)=B}\mathbf{F}\cdot d(\mathbf{r}+\epsilon \mathbf{h})={\int }_{t_0}^{t_1}\mathbf{F}\cdot \frac{d(\mathbf{r}(t)+\epsilon \mathbf{h}(t))}{dt}dt={\int }_{t_0}^{t_1}\mathbf{F}\cdot (\mathbf{v}+\epsilon \dot{\mathbf{h}})dt.}$$ The variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be $${\displaystyle \delta W=\overline{W}-W={\int }_{t_0}^{t_1}(\mathbf{F}\cdot \epsilon \dot{\mathbf{h}})dt.}$$

If there are no constraints on the motion of P, then 3 parameters are needed to completely describe P's position at any time t. If there are k (k ≤ 3) constraint forces, then n = (3 − k) parameters are needed. Hence, we can define n generalized coordinates q_i (t) (i = 1,...,n), and express r(t) and δr = εh(t) in terms of the generalized coordinates. That is, $${\displaystyle \mathbf{r}(t)=\mathbf{r}(q_1,q_2,\dots ,q_n;t),}$$ $${\displaystyle \mathbf{h}(t)=\mathbf{h}(q_1,q_2,\dots ,q_n;t).}$$ Then, the derivative of the variation δr = εh(t) is given by $${\displaystyle \frac{d}{dt}\delta \mathbf{r}=\frac{d}{dt}\epsilon \mathbf{h}=\sum _{i=1}^n\frac{\mathrm{\partial }\mathbf{h}}{\mathrm{\partial }{q}_i}\epsilon {\dot{q}}_i,}$$ then we have $${\displaystyle \delta W={\int }_{t_0}^{t_1}\left(\sum _{i=1}^n\mathbf{F}\cdot \frac{\mathrm{\partial }\mathbf{h}}{\mathrm{\partial }{q}_i}\epsilon {\dot{q}}_i\right)dt=\sum _{i=1}^n\left({\int }_{t_0}^{t_1}\mathbf{F}\cdot \frac{\mathrm{\partial }\mathbf{h}}{\mathrm{\partial }{q}_i}\epsilon {\dot{q}}_{i}dt\right).}$$

The requirement that the virtual work be zero for an arbitrary variation δr(t) = εh(t) is equivalent to the set of requirements $${\displaystyle Q_i=\mathbf{F}\cdot \frac{\mathrm{\partial }\mathbf{h}}{\mathrm{\partial }{q}_i}=0,i=1,\dots ,n.}$$ The terms Q_i are called the generalized forces associated with the virtual displacement δr.

Static equilibrium

Static equilibrium is a state in which the net force and net torque acted upon the system is zero. In other words, both linear momentum and angular momentum of the system are conserved. The principle of virtual work states that the virtual work of the applied forces is zero for all virtual movements of the system from static equilibrium. This principle can be generalized such that three dimensional rotations are included: the virtual work of the applied forces and applied moments is zero for all virtual movements of the system from static equilibrium. That is $${\displaystyle \delta W=\sum _{i=1}^m{\mathbf{F}}_i\cdot \delta {\mathbf{r}}_i+\sum _{j=1}^n{\mathbf{M}}_j\cdot \delta {\varphi }_j=0,}$$ where F_i , i = 1, 2, ..., m and M_j , j = 1, 2, ..., n are the applied forces and applied moments, respectively, and δr_i , i = 1, 2, ..., m and δφ_j, j = 1, 2, ..., n are the virtual displacements and virtual rotations, respectively.

Suppose the system consists of N particles, and it has f (f ≤ 6N) degrees of freedom. It is sufficient to use only f coordinates to give a complete description of the motion of the system, so f generalized coordinates q_k , k = 1, 2, ..., f are defined such that the virtual movements can be expressed in terms of these generalized coordinates. That is, $${\displaystyle \delta {\mathbf{r}}_i(q_1,q_2,\dots ,q_f;t),i=1,2,\dots ,m;}$$ $${\displaystyle \delta {\varphi }_j(q_1,q_2,\dots ,q_f;t),j=1,2,\dots ,n.}$$

The virtual work can then be reparametrized by the generalized coordinates: $${\displaystyle \delta W=\sum _{k=1}^f\left[\left(\sum _{i=1}^m{\mathbf{F}}_i\cdot \frac{\mathrm{\partial }{\mathbf{r}}_i}{\mathrm{\partial }{q}_k}+\sum _{j=1}^n{\mathbf{M}}_j\cdot \frac{\mathrm{\partial }{\varphi }_j}{\mathrm{\partial }{q}_k}\right)\delta q_k\right]=\sum _{k=1}^f{Q}_k\delta q_k,}$$ where the generalized forces Q_k are defined as $${\displaystyle Q_k=\sum _{i=1}^m{\mathbf{F}}_i\cdot \frac{\mathrm{\partial }{\mathbf{r}}_i}{\mathrm{\partial }{q}_k}+\sum _{j=1}^n{\mathbf{M}}_j\cdot \frac{\mathrm{\partial }{\varphi }_j}{\mathrm{\partial }{q}_k},k=1,2,\dots ,f.}$$ Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives. That is, $${\displaystyle Q_k=\sum _{i=1}^m{\mathbf{F}}_i\cdot \frac{\mathrm{\partial }{\mathbf{v}}_i}{\mathrm{\partial }{\dot{q}}_k}+\sum _{j=1}^n{\mathbf{M}}_j\cdot \frac{\mathrm{\partial }{\omega }_j}{\mathrm{\partial }{\dot{q}}_k},k=1,2,\dots ,f.}$$

The principle of virtual work requires that the virtual work done on a system by the forces F_i and moments M_j vanishes if it is in equilibrium. Therefore, the generalized forces Q_k are zero, that is $${\displaystyle \delta W=0\Rightarrow Q_k=0k=1,2,\dots ,f.}$$

Constraint forces

An important benefit of the principle of virtual work is that only forces that do work as the system moves through a virtual displacement are needed to determine the mechanics of the system. There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis. The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint.

Lanczos presents this as the postulate: "The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints." The argument is as follows. The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint forces. This means the virtual work of the constraint forces must be zero as well.

Law of the lever

A lever is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force F_A at a point A located by the coordinate vector r_A on the bar. The lever then exerts an output force F_B at the point B located by r_B. The rotation of the lever about the fulcrum P is defined by the rotation angle θ.

This is an engraving from Mechanics Magazine published in London in 1824.

Let the coordinate vector of the point P that defines the fulcrum be r_P, and introduce the lengths $${\displaystyle a=|{\mathbf{r}}_A-{\mathbf{r}}_P|,b=|{\mathbf{r}}_B-{\mathbf{r}}_P|,}$$ which are the distances from the fulcrum to the input point A and to the output point B, respectively.

Now introduce the unit vectors e_A and e_B from the fulcrum to the point A and B, so $${\displaystyle {\mathbf{r}}_A-{\mathbf{r}}_P=a{\mathbf{e}}_A,{\mathbf{r}}_B-{\mathbf{r}}_P=b{\mathbf{e}}_B.}$$ This notation allows us to define the velocity of the points A and B as $${\displaystyle {\mathbf{v}}_A=\dot{\theta }a{\mathbf{e}}_A^{\perp },{\mathbf{v}}_B=\dot{\theta }b{\mathbf{e}}_B^{\perp },}$$ where e_A^⊥ and e_B^⊥ are unit vectors perpendicular to e_A and e_B, respectively.

The angle θ is the generalized coordinate that defines the configuration of the lever, therefore using the formula above for forces applied to a one degree-of-freedom mechanism, the generalized force is given by $${\displaystyle Q={\mathbf{F}}_A\cdot \frac{\mathrm{\partial }{\mathbf{v}}_A}{\mathrm{\partial }\dot{\theta }}-{\mathbf{F}}_B\cdot \frac{\mathrm{\partial }{\mathbf{v}}_B}{\mathrm{\partial }\dot{\theta }}=a({\mathbf{F}}_A\cdot {\mathbf{e}}_A^{\perp })-b({\mathbf{F}}_B\cdot {\mathbf{e}}_B^{\perp }).}$$

Now, denote as F_A and F_B the components of the forces that are perpendicular to the radial segments PA and PB. These forces are given by $${\displaystyle F_A={\mathbf{F}}_A\cdot {\mathbf{e}}_A^{\perp },F_B={\mathbf{F}}_B\cdot {\mathbf{e}}_B^{\perp }.}$$ This notation and the principle of virtual work yield the formula for the generalized force as $${\displaystyle Q=a{F}_A-b{F}_B=0.}$$

The ratio of the output force F_B to the input force F_A is the mechanical advantage of the lever, and is obtained from the principle of virtual work as $${\displaystyle MA=\frac{F_B}{F_A}=\frac{a}{b}.}$$

This equation shows that if the distance a from the fulcrum to the point A where the input force is applied is greater than the distance b from fulcrum to the point B where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever reduces the magnitude of the input force.

This is the law of the lever, which was proven by Archimedes using geometric reasoning.

Gear train

A gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next. For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ω_A of the input gear to the angular velocity ω_B of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then $${\displaystyle \frac{{\omega }_A}{{\omega }_B}=R.}$$

The input torque T_A acting on the input gear G_A is transformed by the gear train into the output torque T_B exerted by the output gear G_B. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear, that is $${\displaystyle {\omega }_A=\omega ,{\omega }_B=\omega /R.}$$

The formula above for the principle of virtual work with applied torques yields the generalized force $${\displaystyle Q=T_A\frac{\mathrm{\partial }{\omega }_A}{\mathrm{\partial }\omega }-T_B\frac{\mathrm{\partial }{\omega }_B}{\mathrm{\partial }\omega }=T_A-T_B/R=0.}$$

The mechanical advantage of the gear train is the ratio of the output torque T_B to the input torque T_A, and the above equation yields $${\displaystyle MA=\frac{T_B}{T_A}=R.}$$

Thus, the speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.

Dynamic equilibrium for rigid bodies

If the principle of virtual work for applied forces is used on individual particles of a rigid body, the principle can be generalized for a rigid body: When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero; and conversely, if the total virtual work of all external forces acting on a rigid body is zero then the body is in equilibrium.

If a system is not in static equilibrium, D'Alembert showed that by introducing the acceleration terms of Newton's laws as inertia forces, this approach is generalized to define dynamic equilibrium. The result is D'Alembert's form of the principle of virtual work, which is used to derive the equations of motion for a mechanical system of rigid bodies.

The expression compatible displacements means that the particles remain in contact and displace together so that the work done by pairs of action/reaction inter-particle forces cancel out. Various forms of this principle have been credited to Johann (Jean) Bernoulli (1667–1748) and Daniel Bernoulli (1700–1782).

Generalized inertia forces

Let a mechanical system be constructed from n rigid bodies, B_i, i=1,...,n, and let the resultant of the applied forces on each body be the force-torque pairs, F_i and T_i, i = 1,...,n. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity V_i and angular velocities ω_i, i=1,...,n, for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one degree of freedom.

Consider a single rigid body which moves under the action of a resultant force F and torque T, with one degree of freedom defined by the generalized coordinate q. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force Q* associated with the generalized coordinate q is given by $${\displaystyle Q^{\ast }=-(M\mathbf{A})\cdot \frac{\mathrm{\partial }\mathbf{V}}{\mathrm{\partial }\dot{q}}-([I_R]\alpha +\omega \times [I_R]\omega )\cdot \frac{\mathrm{\partial }\mathit{\omega }}{\mathrm{\partial }\dot{q}}.}$$ This inertia force can be computed from the kinetic energy of the rigid body, $${\displaystyle T=\frac{1}{2}M\mathbf{V}\cdot \mathbf{V}+\frac{1}{2}\mathit{\omega }\cdot [I_R]\mathit{\omega },}$$ by using the formula $${\displaystyle Q^{\ast }=-\left(\frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }\dot{q}}-\frac{\mathrm{\partial }T}{\mathrm{\partial }q}\right).}$$

A system of n rigid bodies with m generalized coordinates has the kinetic energy $${\displaystyle T=\sum _{i=1}^n\left(\frac{1}{2}M{\mathbf{V}}_i\cdot {\mathbf{V}}_i+\frac{1}{2}{\mathit{\omega }}_i\cdot [I_R]{\mathit{\omega }}_i\right),}$$ which can be used to calculate the m generalized inertia forces $${\displaystyle Q_j^{\ast }=-\left(\frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }{\dot{q}}_j}-\frac{\mathrm{\partial }T}{\mathrm{\partial }{q}_j}\right),j=1,\dots ,m.}$$

D'Alembert's form of the principle of virtual work

D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that $${\displaystyle \delta W=(Q_1+Q_1^{\ast })\delta q_1+\cdots +(Q_m+Q_m^{\ast })\delta q_m=0,}$$ for any set of virtual displacements δq_j. This condition yields m equations, $${\displaystyle Q_j+Q_j^{\ast }=0,j=1,\dots ,m,}$$ which can also be written as $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }{\dot{q}}_j}-\frac{\mathrm{\partial }T}{\mathrm{\partial }{q}_j}=Q_j,j=1,\dots ,m.}$$ The result is a set of m equations of motion that define the dynamics of the rigid body system, known as Lagrange's equations or the generalized equations of motion.

If the generalized forces Q_j are derivable from a potential energy V(q₁,...,q_m), then these equations of motion take the form $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }{\dot{q}}_j}-\frac{\mathrm{\partial }T}{\mathrm{\partial }{q}_j}=-\frac{\mathrm{\partial }V}{\mathrm{\partial }{q}_j},j=1,\dots ,m.}$$

In this case, introduce the Lagrangian, L = T − V, so these equations of motion become $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_j}-\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_j}=0j=1,\dots ,m.}$$ These are known as the Euler-Lagrange equations for a system with m degrees of freedom, or Lagrange's equations of the second kind.

Virtual work principle for a deformable body

Consider now the free body diagram of a deformable body, which is composed of an infinite number of differential cubes. Let's define two unrelated states for the body:

• The ${\displaystyle \mathit{\sigma }}$-State : This shows external surface forces T, body forces f, and internal stresses ${\displaystyle \mathit{\sigma }}$ in equilibrium.
• The ${\displaystyle \mathit{ϵ}}$-State : This shows continuous displacements ${\displaystyle {\mathbf{u}}^{\ast }}$ and consistent strains ${\displaystyle {\mathit{ϵ}}^{\ast }}$.

The superscript * emphasizes that the two states are unrelated. Other than the above stated conditions, there is no need to specify if any of the states are real or virtual.

Imagine now that the forces and stresses in the ${\displaystyle \mathit{\sigma }}$-State undergo the displacements and deformations in the ${\displaystyle \mathit{ϵ}}$-State: We can compute the total virtual (imaginary) work done by all forces acting on the faces of all cubes in two different ways:

• First, by summing the work done by forces such as ${\displaystyle F_A}$ which act on individual common faces (Fig.c): Since the material experiences compatible displacements, such work cancels out, leaving only the virtual work done by the surface forces T (which are equal to stresses on the cubes' faces, by equilibrium).
• Second, by computing the net work done by stresses or forces such as ${\displaystyle F_A}$, ${\displaystyle F_B}$ which act on an individual cube, e.g. for the one-dimensional case in Fig.(c): $${\displaystyle F_B\left(u^{\ast }+\frac{\mathrm{\partial }{u}^{\ast }}{\mathrm{\partial }x}dx\right)-F_A{u}^{\ast }\approx \frac{\mathrm{\partial }{u}^{\ast }}{\mathrm{\partial }x}\sigma dV+u^{\ast }\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }x}dV={ϵ}^{\ast }\sigma dV-u^{\ast }fdV}$$ where the equilibrium relation ${\displaystyle \frac{\mathrm{\partial }\sigma }{\mathrm{\partial }x}+f=0}$ has been used and the second order term has been neglected.
  Integrating over the whole body gives: $${\displaystyle {\int }_V{\mathit{ϵ}}^{\ast T}\mathit{\sigma }dV}$$ – Work done by the body forces f.

Equating the two results leads to the principle of virtual work for a deformable body:

$${\displaystyle \text{Total external virtual work}={\int }_V{\mathit{ϵ}}^{\ast T}\mathit{\sigma }dV}$$

(d)

where the total external virtual work is done by T and f. Thus,

$${\displaystyle {\int }_S{\mathbf{u}}^{\ast T}\mathbf{T}dS+{\int }_V{\mathbf{u}}^{\ast T}\mathbf{f}dV={\int }_V{\mathit{ϵ}}^{\ast T}\mathit{\sigma }dV}$$

(e)

The right-hand-side of (d,e) is often called the internal virtual work. The principle of virtual work then states: External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains. It includes the principle of virtual work for rigid bodies as a special case where the internal virtual work is zero.

Proof of equivalence between the principle of virtual work and the equilibrium equation

We start by looking at the total work done by surface traction on the body going through the specified deformation: $${\displaystyle {\int }_S\mathbf{u}\cdot \mathbf{T}dS={\int }_S\mathbf{u}\cdot \mathit{\sigma }\cdot \mathbf{n}dS}$$

Applying divergence theorem to the right hand side yields: $${\displaystyle {\int }_S\mathbf{u}\cdot \mathit{\sigma }\cdot \mathbf{n}dS={\int }_V\mathrm{\nabla }\cdot \left(\mathbf{u}\cdot \mathit{\sigma }\right)dV}$$

Now switch to indicial notation for the ease of derivation. $${\displaystyle \begin{array}{rl}{\int }_V\mathrm{\nabla }\cdot \left(\mathbf{u}\cdot \mathit{\sigma }\right)dV& ={\int }_V\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left(u_i{\sigma }_{ij}\right)dV\\ & ={\int }_V\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{\sigma }_{ij}+u_i\frac{\mathrm{\partial }{\sigma }_{ij}}{\mathrm{\partial }{x}_j}\right)dV\end{array}}$$

To continue our derivation, we substitute in the equilibrium equation ${\displaystyle \frac{\mathrm{\partial }{\sigma }_{ij}}{\mathrm{\partial }{x}_j}+f_i=0}$. Then $${\displaystyle {\int }_V\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{\sigma }_{ij}+u_i\frac{\mathrm{\partial }{\sigma }_{ij}}{\mathrm{\partial }{x}_j}\right)dV={\int }_V\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{\sigma }_{ij}-u_i{f}_i\right)dV}$$

The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows: $${\displaystyle \begin{array}{rl}{\int }_V\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}{\sigma }_{ij}-u_i{f}_i\right)dV& ={\int }_V\left(\frac{1}{2}\left[\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)+\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}-\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\right]{\sigma }_{ij}-u_i{f}_i\right)dV\\ & ={\int }_V\left(\left[{ϵ}_{ij}+\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}-\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\right]{\sigma }_{ij}-u_i{f}_i\right)dV\\ & ={\int }_V\left({ϵ}_{ij}{\sigma }_{ij}-u_i{f}_i\right)dV\\ & ={\int }_V\left(\mathit{ϵ}:\mathit{\sigma }-\mathbf{u}\cdot \mathbf{f}\right)dV\end{array}}$$ where ${\displaystyle \mathit{ϵ}}$ is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.

Now recap. We have shown through the above derivation that $${\displaystyle {\int }_S\mathbf{u}\cdot \mathbf{T}dS={\int }_V\mathit{ϵ}:\mathit{\sigma }dV-{\int }_V\mathbf{u}\cdot \mathbf{f}dV}$$

Move the 2nd term on the right hand side of the equation to the left: $${\displaystyle {\int }_S\mathbf{u}\cdot \mathbf{T}dS+{\int }_V\mathbf{u}\cdot \mathbf{f}dV={\int }_V\mathit{ϵ}:\mathit{\sigma }dV}$$

The physical interpretation of the above equation is, the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains.

For practical applications:

• In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.
• In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.

These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.

Principle of virtual displacements

Depending on the purpose, we may specialize the virtual work equation. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:

• Virtual displacements and strains as variations of the real displacements and strains using variational notation such as ${\displaystyle \delta \mathbf{u}\equiv {\mathbf{u}}^{\ast }}$ and ${\displaystyle \delta \mathit{ϵ}\equiv {\mathit{ϵ}}^{\ast }}$
• Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero. There remains only external surface forces on the part ${\displaystyle S_t}$ that do work.

The virtual work equation then becomes the principle of virtual displacements:

$${\displaystyle {\int }_{S_t}\delta {\mathbf{u}}^T\mathbf{T}dS+{\int }_V\delta {\mathbf{u}}^T\mathbf{f}dV={\int }_V\delta {\mathit{ϵ}}^T\mathit{\sigma }dV}$$

(f)

This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part ${\displaystyle S_t}$ of the surface. Conversely, (f) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on ${\displaystyle S_t}$, and proceeding in the manner similar to (a) and (b).

Since virtual displacements are automatically compatible when they are expressed in terms of continuous, single-valued functions, we often mention only the need for consistency between strains and displacements. The virtual work principle is also valid for large real displacements; however, Eq.(f) would then be written using more complex measures of stresses and strains.

Principle of virtual forces

Here, we specify:

• Virtual forces and stresses as variations of the real forces and stresses.
• Virtual forces be zero on the part ${\displaystyle S_t}$ of the surface that has prescribed forces, and thus only surface (reaction) forces on ${\displaystyle S_u}$ (where displacements are prescribed) would do work.

The virtual work equation becomes the principle of virtual forces:

$${\displaystyle {\int }_{S_u}{\mathbf{u}}^T\delta \mathbf{T}dS+{\int }_V{\mathbf{u}}^T\delta \mathbf{f}dV={\int }_V{\mathit{ϵ}}^T\delta \mathit{\sigma }dV}$$

(g)

This relation is equivalent to the set of strain-compatibility equations as well as of the displacement boundary conditions on the part ${\displaystyle S_u}$. It has another name: the principle of complementary virtual work.

Alternative forms

A specialization of the principle of virtual forces is the unit dummy force method, which is very useful for computing displacements in structural systems. According to D'Alembert's principle, inclusion of inertial forces as additional body forces will give the virtual work equation applicable to dynamical systems. More generalized principles can be derived by:

• allowing variations of all quantities.
• using Lagrange multipliers to impose boundary conditions and/or to relax the conditions specified in the two states.

These are described in some of the references.

Among the many energy principles in structural mechanics, the virtual work principle deserves a special place due to its generality that leads to powerful applications in structural analysis, solid mechanics, and finite element method in structural mechanics.

Machine

From Wikipedia, the free encyclopedia

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

Assorted worker-operated machinery at the Láng Machine Factory in Budapest, Hungary in 1977

A machine is a physical system that uses power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecules, such as molecular machines. Machines can be driven by animals and people, by natural forces such as wind and water, and by chemical, thermal, or electrical power, and include a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement. They can also include computers and sensors that monitor performance and plan movement, often called mechanical systems.

Renaissance natural philosophers identified six simple machines which were the elementary devices that put a load into motion, and calculated the ratio of output force to input force, known today as mechanical advantage.

Modern machines are complex systems that consist of structural elements, mechanisms and control components and include interfaces for convenient use. Examples include: a wide range of vehicles, such as trains, automobiles, boats and airplanes; appliances in the home and office, including computers, building air handling and water handling systems; as well as farm machinery, machine tools and factory automation systems and robots.

Etymology

The English word machine comes through Middle French from Latin machina, which in turn derives from the Greek (Doric μαχανά makhana, Ionic μηχανή mekhane 'contrivance, machine, engine', a derivation from μῆχος mekhos 'means, expedient, remedy'). The word mechanical (Greek: μηχανικός) comes from the same Greek roots. A wider meaning of 'fabric, structure' is found in classical Latin, but not in Greek usage. This meaning is found in late medieval French, and is adopted from the French into English in the mid-16th century.

In the 17th century, the word machine could also mean a scheme or plot, a meaning now expressed by the derived machination. The modern meaning develops out of specialized application of the term to stage engines used in theater and to military siege engines, both in the late 16th and early 17th centuries. The OED traces the formal, modern meaning to John Harris' Lexicon Technicum (1704), which has:

Machine, or Engine, in Mechanicks, is whatsoever hath Force sufficient either to raise or stop the Motion of a Body. Simple Machines are commonly reckoned to be Six in Number, viz. the Ballance, Leaver, Pulley, Wheel, Wedge, and Screw. Compound Machines, or Engines, are innumerable.

The word engine used as a (near-) synonym both by Harris and in later language derives ultimately (via Old French) from Latin ingenium 'ingenuity, an invention'.

History

A flint hand axe was found in Winchester.

The hand axe, made by chipping flint to form a wedge, in the hands of a human transforms force and movement of the tool into a transverse splitting forces and movement of the workpiece. The hand axe is the first example of a wedge, the oldest of the six classic simple machines, from which most machines are based. The second oldest simple machine was the inclined plane (ramp), which has been used since prehistoric times to move heavy objects.

The other four simple machines were invented in the ancient Near East. The wheel, along with the wheel and axle mechanism, was invented in Mesopotamia (modern Iraq) during the 5th millennium BC. The lever mechanism first appeared around 5,000 years ago in the Near East, where it was used in a simple balance scale, and to move large objects in ancient Egyptian technology. The lever was also used in the shadoof water-lifting device, the first crane machine, which appeared in Mesopotamia c. 3000 BC, and then in ancient Egyptian technology c. 2000 BC. The earliest evidence of pulleys date back to Mesopotamia in the early 2nd millennium BC, and ancient Egypt during the Twelfth Dynasty (1991-1802 BC). The screw, the last of the simple machines to be invented, first appeared in Mesopotamia during the Neo-Assyrian period (911–609) BC. The Egyptian pyramids were built using three of the six simple machines, the inclined plane, the wedge, and the lever.

Three of the simple machines were studied and described by Greek philosopher Archimedes around the 3rd century BC: the lever, pulley and screw. Archimedes discovered the principle of mechanical advantage in the lever. Later Greek philosophers defined the classic five simple machines (excluding the inclined plane) and were able to roughly calculate their mechanical advantage. Hero of Alexandria (c. 10–75 AD) in his work Mechanics lists five mechanisms that can "set a load in motion"; lever, windlass, pulley, wedge, and screw, and describes their fabrication and uses. However, the Greeks' understanding was limited to statics (the balance of forces) and did not include dynamics (the tradeoff between force and distance) or the concept of work.

This ore crushing machine is powered by a water wheel.

The earliest practical wind-powered machines, the windmill and wind pump, first appeared in the Muslim world during the Islamic Golden Age, in what are now Iran, Afghanistan, and Pakistan, by the 9th century AD. The earliest practical steam-powered machine was a steam jack driven by a steam turbine, described in 1551 by Taqi ad-Din Muhammad ibn Ma'ruf in Ottoman Egypt.

The cotton gin was invented in India by the 6th century AD, and the spinning wheel was invented in the Islamic world by the early 11th century, both of which were fundamental to the growth of the cotton industry. The spinning wheel was also a precursor to the spinning jenny.

The earliest programmable machines were developed in the Muslim world. A music sequencer, a programmable musical instrument, was the earliest type of programmable machine. The first music sequencer was an automated flute player invented by the Banu Musa brothers, described in their Book of Ingenious Devices, in the 9th century. In 1206, Al-Jazari invented programmable automata/robots. He described four automaton musicians, including drummers operated by a programmable drum machine, where they could be made to play different rhythms and different drum patterns.

During the Renaissance, the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how much useful work they could perform, leading eventually to the new concept of mechanical work. In 1586 Flemish engineer Simon Stevin derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche ("On Mechanics"). He was the first to understand that simple machines do not create energy, they merely transform it.

The classic rules of sliding friction in machines were discovered by Leonardo da Vinci (1452–1519), but remained unpublished in his notebooks. They were rediscovered by Guillaume Amontons (1699) and were further developed by Charles-Augustin de Coulomb (1785).

James Watt patented his parallel motion linkage in 1782, which made the double acting steam engine practical. The Boulton and Watt steam engine and later designs powered steam locomotives, steam ships, and factories.

James Albert Bonsack's cigarette rolling machine was invented in 1880 and patented in 1881.

The Industrial Revolution was a period from 1750 to 1850 where changes in agriculture, manufacturing, mining, transportation, and technology had a profound effect on the social, economic and cultural conditions of the times. It began in the United Kingdom, then subsequently spread throughout Western Europe, North America, Japan, and eventually the rest of the world.

Starting in the later part of the 18th century, there began a transition in parts of Great Britain's previously manual labour and draft-animal-based economy towards machine-based manufacturing. It started with the mechanisation of the textile industries, the development of iron-making techniques and the increased use of refined coal.

Simple machines

Main article: Simple machine

Chambers' Cyclopædia (1728) has a table of simple mechanisms. Simple machines provide a "vocabulary" for understanding more complex machines.

The idea that a machine can be decomposed into simple movable elements led Archimedes to define the lever, pulley and screw as simple machines. By the time of the Renaissance this list increased to include the wheel and axle, wedge and inclined plane. The modern approach to characterizing machines focusses on the components that allow movement, known as joints.

Wedge (hand axe): Perhaps the first example of a device designed to manage power is the hand axe, also called biface and Olorgesailie. A hand axe is made by chipping stone, generally flint, to form a bifacial edge, or wedge. A wedge is a simple machine that transforms lateral force and movement of the tool into a transverse splitting force and movement of the workpiece. The available power is limited by the effort of the person using the tool, but because power is the product of force and movement, the wedge amplifies the force by reducing the movement. This amplification, or mechanical advantage is the ratio of the input speed to output speed. For a wedge this is given by 1/tanα, where α is the tip angle. The faces of a wedge are modeled as straight lines to form a sliding or prismatic joint.

Lever: The lever is another important and simple device for managing power. This is a body that pivots on a fulcrum. Because the velocity of a point farther from the pivot is greater than the velocity of a point near the pivot, forces applied far from the pivot are amplified near the pivot by the associated decrease in speed. If a is the distance from the pivot to the point where the input force is applied and b is the distance to the point where the output force is applied, then a/b is the mechanical advantage of the lever. The fulcrum of a lever is modeled as a hinged or revolute joint.

Wheel: The wheel is an important early machine, such as the chariot. A wheel uses the law of the lever to reduce the force needed to overcome friction when pulling a load. To see this notice that the friction associated with pulling a load on the ground is approximately the same as the friction in a simple bearing that supports the load on the axle of a wheel. However, the wheel forms a lever that magnifies the pulling force so that it overcomes the frictional resistance in the bearing.

The Kinematics of Machinery (1876) has an illustration of a four-bar linkage.

The classification of simple machines to provide a strategy for the design of new machines was developed by Franz Reuleaux, who collected and studied over 800 elementary machines. He recognized that the classical simple machines can be separated into the lever, pulley and wheel and axle that are formed by a body rotating about a hinge, and the inclined plane, wedge and screw that are similarly a block sliding on a flat surface.

Simple machines are elementary examples of kinematic chains or linkages that are used to model mechanical systems ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of a kinematic pair called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint.

This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, the rotary joint, sliding joint, cam joint and gear joint, and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints called a mechanism .

Two levers, or cranks, are combined into a planar four-bar linkage by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form a six-bar linkage or in series to form a robot.

Mechanical systems

The Boulton & Watt Steam Engine, 1784

A mechanical system manages power to accomplish a task that involves forces and movement. Modern machines are systems consisting of (i) a power source and actuators that generate forces and movement, (ii) a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement, (iii) a controller with sensors that compare the output to a performance goal and then directs the actuator input, and (iv) an interface to an operator consisting of levers, switches, and displays. This can be seen in Watt's steam engine in which the power is provided by steam expanding to drive the piston. The walking beam, coupler and crank transform the linear movement of the piston into rotation of the output pulley. Finally, the pulley rotation drives the flyball governor which controls the valve for the steam input to the piston cylinder.

The adjective "mechanical" refers to skill in the practical application of an art or science, as well as relating to or caused by movement, physical forces, properties or agents such as is dealt with by mechanics. Similarly Merriam-Webster Dictionary defines "mechanical" as relating to machinery or tools.

Power flow through a machine provides a way to understand the performance of devices ranging from levers and gear trains to automobiles and robotic systems. The German mechanician Franz Reuleaux wrote, "a machine is a combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motion." Notice that forces and motion combine to define power.

More recently, Uicker et al. stated that a machine is "a device for applying power or changing its direction."McCarthy and Soh describe a machine as a system that "generally consists of a power source and a mechanism for the controlled use of this power."

Power sources

Diesel engine, friction clutch and gear transmission of an automobile

Early Ganz Electric Generator in Zwevegem, West Flanders, Belgium

Human and animal effort were the original power sources for early machines.

Waterwheel: Waterwheels appeared around the world around 300 BC to use flowing water to generate rotary motion, which was applied to milling grain, and powering lumber, machining and textile operations. Modern water turbines use water flowing through a dam to drive an electric generator.

Windmill: Early windmills captured wind power to generate rotary motion for milling operations. Modern wind turbines also drives a generator. This electricity in turn is used to drive motors forming the actuators of mechanical systems.

Engine: The word engine derives from "ingenuity" and originally referred to contrivances that may or may not be physical devices. A steam engine uses heat to boil water contained in a pressure vessel; the expanding steam drives a piston or a turbine. This principle can be seen in the aeolipile of Hero of Alexandria. This is called an external combustion engine.

An automobile engine is called an internal combustion engine because it burns fuel (an exothermic chemical reaction) inside a cylinder and uses the expanding gases to drive a piston. A jet engine uses a turbine to compress air which is burned with fuel so that it expands through a nozzle to provide thrust to an aircraft, and so is also an "internal combustion engine." 

Power plant: The heat from coal and natural gas combustion in a boiler generates steam that drives a steam turbine to rotate an electric generator. A nuclear power plant uses heat from a nuclear reactor to generate steam and electric power. This power is distributed through a network of transmission lines for industrial and individual use.

Motors: Electric motors use either AC or DC electric current to generate rotational movement. Electric servomotors are the actuators for mechanical systems ranging from robotic systems to modern aircraft.

Fluid Power: Hydraulic and pneumatic systems use electrically driven pumps to drive water or air respectively into cylinders to power linear movement.

Electrochemical: Chemicals and materials can also be sources of power. They may chemically deplete or need re-charging, as is the case with batteries, or they may produce power without changing their state, which is the case for solar cells and thermoelectric generators. All of these, however, still require their energy to come from elsewhere. With batteries, it is the already existing chemical potential energy inside. In solar cells and thermoelectrics, the energy source is light and heat respectively.

Mechanisms

The mechanism of a mechanical system is assembled from components called machine elements. These elements provide structure for the system and control its movement.

The structural components are, generally, the frame members, bearings, splines, springs, seals, fasteners and covers. The shape, texture and color of covers provide a styling and operational interface between the mechanical system and its users.

The assemblies that control movement are also called "mechanisms." Mechanisms are generally classified as gears and gear trains, which includes belt drives and chain drives, cam and follower mechanisms, and linkages, though there are other special mechanisms such as clamping linkages, indexing mechanisms, escapements and friction devices such as brakes and clutches.

The number of degrees of freedom of a mechanism, or its mobility, depends on the number of links and joints and the types of joints used to construct the mechanism. The general mobility of a mechanism is the difference between the unconstrained freedom of the links and the number of constraints imposed by the joints. It is described by the Chebychev–Grübler–Kutzbach criterion.

Gears and gear trains

The Antikythera mechanism (main fragment)

The transmission of rotation between contacting toothed wheels can be traced back to the Antikythera mechanism of Greece and the south-pointing chariot of China. Illustrations by the renaissance scientist Georgius Agricola show gear trains with cylindrical teeth. The implementation of the involute tooth yielded a standard gear design that provides a constant speed ratio. Some important features of gears and gear trains are:

• The ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the gear set.
• A planetary gear train provides high gear reduction in a compact package.
• It is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly.
• The speed ratios of chain and belt drives are computed in the same way as gear ratios. See bicycle gearing.

Cam and follower mechanisms

A cam and follower is formed by the direct contact of two specially shaped links. The driving link is called the cam (also see cam shaft) and the link that is driven through the direct contact of their surfaces is called the follower. The shape of the contacting surfaces of the cam and follower determines the movement of the mechanism.

Linkages

Schematic of the actuator and four-bar linkage that position an aircraft landing gear

A linkage is a collection of links connected by joints. Generally, the links are the structural elements and the joints allow movement. Perhaps the single most useful example is the planar four-bar linkage. However, there are many more special linkages:

• Watt's linkage is a four-bar linkage that generates an approximate straight line. It was critical to the operation of his design for the steam engine. This linkage also appears in vehicle suspensions to prevent side-to-side movement of the body relative to the wheels. Also see the article Parallel motion.
• The success of Watt's linkage lead to the design of similar approximate straight-line linkages, such as Hoeken's linkage and Chebyshev's linkage.
• The Peaucellier linkage generates a true straight-line output from a rotary input.
• The Sarrus linkage is a spatial linkage that generates straight-line movement from a rotary input.
• The Klann linkage and the Jansen linkage are recent inventions that provide interesting walking movements. They are respectively a six-bar and an eight-bar linkage.

Planar mechanism

A planar mechanism is a mechanical system that is constrained so the trajectories of points in all the bodies of the system lie on planes parallel to a ground plane. The rotational axes of hinged joints that connect the bodies in the system are perpendicular to this ground plane.

Spherical mechanism

A spherical mechanism is a mechanical system in which the bodies move in a way that the trajectories of points in the system lie on concentric spheres. The rotational axes of hinged joints that connect the bodies in the system pass through the center of these circle.

Spatial mechanism

A spatial mechanism is a mechanical system that has at least one body that moves in a way that its point trajectories are general space curves. The rotational axes of hinged joints that connect the bodies in the system form lines in space that do not intersect and have distinct common normals.

Flexure mechanisms

A flexure mechanism consists of a series of rigid bodies connected by compliant elements (also known as flexure joints) that is designed to produce a geometrically well-defined motion upon application of a force.

Machine elements

The elementary mechanical components of a machine are termed machine elements. These elements consist of three basic types (i) structural components such as frame members, bearings, axles, splines, fasteners, seals, and lubricants, (ii) mechanisms that control movement in various ways such as gear trains, belt or chain drives, linkages, cam and follower systems, including brakes and clutches, and (iii) control components such as buttons, switches, indicators, sensors, actuators and computer controllers. While generally not considered to be a machine element, the shape, texture and color of covers are an important part of a machine that provide a styling and operational interface between the mechanical components of a machine and its users.

Structural components

A number of machine elements provide important structural functions such as the frame, bearings, splines, spring and seals.

• The recognition that the frame of a mechanism is an important machine element changed the name three-bar linkage into four-bar linkage. Frames are generally assembled from truss or beam elements.
• Bearings are components designed to manage the interface between moving elements and are the source of friction in machines. In general, bearings are designed for pure rotation or straight line movement.
• Splines and keys are two ways to reliably mount an axle to a wheel, pulley or gear so that torque can be transferred through the connection.
• Springs provides forces that can either hold components of a machine in place or acts as a suspension to support part of a machine.
• Seals are used between mating parts of a machine to ensure fluids, such as water, hot gases, or lubricant do not leak between the mating surfaces.
• Fasteners such as screws, bolts, spring clips, and rivets are critical to the assembly of components of a machine. Fasteners are generally considered to be removable. In contrast, joining methods, such as welding, soldering, crimping and the application of adhesives, usually require cutting the parts to disassemble the components

Controllers

Controllers combine sensors, logic, and actuators to maintain the performance of components of a machine. Perhaps the best known is the flyball governor for a steam engine. Examples of these devices range from a thermostat that as temperature rises opens a valve to cooling water to speed controllers such as the cruise control system in an automobile. The programmable logic controller replaced relays and specialized control mechanisms with a programmable computer. Servomotors that accurately position a shaft in response to an electrical command are the actuators that make robotic systems possible.

Computing machines

The arithmometre was designed by Charles Xavier Thomas, c. 1820, for the four rules of arithmetic. It was manufactured 1866-1870 AD and exhibited in the Tekniska museet, Stockholm, Sweden.

Charles Babbage designed machines to tabulate logarithms and other functions in 1837. His Difference engine can be considered an advanced mechanical calculator and his Analytical Engine a forerunner of the modern computer, though none of the larger designs were completed in Babbage's lifetime.

The Arithmometer and the Comptometer are mechanical computers that are precursors to modern digital computers. Models used to study modern computers are termed State machine and Turing machine.

Molecular machines

A ribosome is a biological machine that utilizes protein dynamics.

The biological molecule myosin reacts to ATP and ADP to alternately engage with an actin filament and change its shape in a way that exerts a force, and then disengage to reset its shape, or conformation. This acts as the molecular drive that causes muscle contraction. Similarly the biological molecule kinesin has two sections that alternately engage and disengage with microtubules causing the molecule to move along the microtubule and transport vesicles within the cell, and dynein, which moves cargo inside cells towards the nucleus and produces the axonemal beating of motile cilia and flagella. "In effect, the motile cilium is a nanomachine composed of perhaps over 600 proteins in molecular complexes, many of which also function independently as nanomachines. Flexible linkers allow the mobile protein domains connected by them to recruit their binding partners and induce long-range allostery via protein domain dynamics. " Other biological machines are responsible for energy production, for example ATP synthase which harnesses energy from proton gradients across membranes to drive a turbine-like motion used to synthesise ATP, the energy currency of a cell. Still other machines are responsible for gene expression, including DNA polymerases for replicating DNA, RNA polymerases for producing mRNA, the spliceosome for removing introns, and the ribosome for synthesising proteins. These machines and their nanoscale dynamics are far more complex than any molecular machines that have yet been artificially constructed. These molecules are increasingly considered to be nanomachines.

Researchers have used DNA to construct nano-dimensioned four-bar linkages.

Impact

Mechanization and automation

Main articles: Mechanization and Automation

This water-powered mine hoist was used for raising ore. This woodblock is from De re metallica by Georg Bauer (Latinized name Georgius Agricola, c. 1555), an early mining textbook that contains numerous drawings and descriptions of mining equipment.

Mechanization (or mechanisation in BE) is providing human operators with machinery that assists them with the muscular requirements of work or displaces muscular work. In some fields, mechanization includes the use of hand tools. In modern usage, such as in engineering or economics, mechanization implies machinery more complex than hand tools and would not include simple devices such as an un-geared horse or donkey mill. Devices that cause speed changes or changes to or from reciprocating to rotary motion, using means such as gears, pulleys or sheaves and belts, shafts, cams and cranks, usually are considered machines. After electrification, when most small machinery was no longer hand powered, mechanization was synonymous with motorized machines.

Automation is the use of control systems and information technologies to reduce the need for human work in the production of goods and services. In the scope of industrialization, automation is a step beyond mechanization. Whereas mechanization provides human operators with machinery to assist them with the muscular requirements of work, automation greatly decreases the need for human sensory and mental requirements as well. Automation plays an increasingly important role in the world economy and in daily experience.

Automata

Main article: Automaton

An automaton (plural: automata or automatons) is a self-operating machine. The word is sometimes used to describe a robot, more specifically an autonomous robot. A Toy Automaton was patented in 1863.

Mechanics

Usher reports that Hero of Alexandria's treatise on Mechanics focussed on the study of lifting heavy weights. Today mechanics refers to the mathematical analysis of the forces and movement of a mechanical system, and consists of the study of the kinematics and dynamics of these systems.

Dynamics of machines

The dynamic analysis of machines begins with a rigid-body model to determine reactions at the bearings, at which point the elasticity effects are included. The rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body.

The dynamics of a rigid body system is defined by its equations of motion, which are derived using either Newtons laws of motion or Lagrangian mechanics. The solution of these equations of motion defines how the configuration of the system of rigid bodies changes as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.

Kinematics of machines

The dynamic analysis of a machine requires the determination of the movement, or kinematics, of its component parts, known as kinematic analysis. The assumption that the system is an assembly of rigid components allows rotational and translational movement to be modeled mathematically as Euclidean, or rigid, transformations. This allows the position, velocity and acceleration of all points in a component to be determined from these properties for a reference point, and the angular position, angular velocity and angular acceleration of the component.

Machine design

Machine design refers to the procedures and techniques used to address the three phases of a machine's lifecycle:

1. invention, which involves the identification of a need, development of requirements, concept generation, prototype development, manufacturing, and verification testing;
2. performance engineering involves enhancing manufacturing efficiency, reducing service and maintenance demands, adding features and improving effectiveness, and validation testing;
3. recycle is the decommissioning and disposal phase and includes recovery and reuse of materials and components.

Computational archaeology

From Wikipedia, the free encyclopedia

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

Computational archaeology describes computer-based analytical methods for the study of long-term human behaviour and behavioural evolution. As with other sub-disciplines that have prefixed 'computational' to their name (e.g., computational biology, computational physics and computational sociology), the term is reserved for (generally mathematical) methods that could not realistically be performed without the aid of a computer.

Computational archaeology may include the use of geographical information systems (GIS), especially when applied to spatial analyses such as viewshed analysis and least-cost path analysis as these approaches are sufficiently computationally complex that they are extremely difficult if not impossible to implement without the processing power of a computer. Likewise, some forms of statistical and mathematical modelling, and the computer simulation of human behaviour and behavioural evolution using software tools such as Swarm or Repast would also be impossible to calculate without computational aid. The application of a variety of other forms of complex and bespoke software to solve archaeological problems, such as human perception and movement within built environments using software such as University College London's Space Syntax program, also falls under the term 'computational archaeology'.

The acquisition, documentation and analysis of archaeological finds at excavations and in museums is an important field having pottery analysis as one of the major topics. In this area 3D-acquisition techniques like structured light scanning (SLS), photogrammetric methods like "structure from motion" (SfM), computed tomography as well as their combinations provide large data-sets of numerous objects for digital pottery research. These techniques are increasingly integrated into the in-situ workflow of excavations. The Austrian subproject of the Corpus vasorum antiquorum (CVA) is seminal for digital research on finds within museums.

Computational archaeology is also known as "archaeological informatics" (Burenhult 2002, Huggett and Ross 2004) or "archaeoinformatics" (sometimes abbreviated as "AI", but not to be confused with artificial intelligence).

Origins and objectives

In recent years, it has become clear that archaeologists will only be able to harvest the full potential of quantitative methods and computer technology if they become aware of the specific pitfalls and potentials inherent in the archaeological data and research process. AI science is an emerging discipline that attempts to uncover, quantitatively represent and explore specific properties and patterns of archaeological information. Fundamental research on data and methods for a self-sufficient archaeological approach to information processing produces quantitative methods and computer software specifically geared towards archaeological problem solving and understanding.

AI science is capable of complementing and enhancing almost any area of scientific archaeological research. It incorporates a large part of the methods and theories developed in quantitative archaeology since the 1960s but goes beyond former attempts at quantifying archaeology by exploring ways to represent general archaeological information and problem structures as computer algorithms and data structures. This opens archaeological analysis to a wide range of computer-based information processing methods fit to solve problems of great complexity. It also promotes a formalized understanding of the discipline's research objects and creates links between archaeology and other quantitative disciplines, both in methods and software technology. Its agenda can be split up in two major research themes that complement each other:

1. Fundamental research (theoretical AI science) on the structure, properties and possibilities of archaeological data, inference and knowledge building. This includes modeling and managing fuzziness and uncertainty in archaeological data, scale effects, optimal sampling strategies and spatio-temporal effects.
2. Development of computer algorithms and software (applied AI science) that make this theoretical knowledge available to the user.

There is already a large body of literature on the use of quantitative methods and computer-based analysis in archaeology. The development of methods and applications is best reflected in the annual publications of the CAA conference (see external links section at bottom). At least two journals, the Italian Archeologia e Calcolatori and the British Archaeological Computing Newsletter, are dedicated to archaeological computing methods. AI Science contributes to many fundamental research topics, including but not limited to:

• advanced statistics in archaeology, spatial and temporal archaeological data analysis
• bayesian analysis and advanced probability models, fuzziness and uncertainty in archaeological data
• scale-related phenomena and scale transgressions
• intrasite analysis (representations of stratigraphy, 3D analysis, artefact distributions)
• landscape analysis (territorial modeling, visibility analysis)
• optimal survey and sampling strategies
• process-based modeling and simulation models
• archaeological predictive modeling and heritage management applications
• supervised and unsupervised classification and typology, artificial intelligence applications
• digital excavations and virtual reality
• computational reproducibility of archaeological research
• archaeological software development, electronic data sharing and publishing

AI science advocates a formalized approach to archaeological inference and knowledge building. It is interdisciplinary in nature, borrowing, adapting and enhancing method and theory from numerous other disciplines such as computer science (e.g. algorithm and software design, database design and theory), geoinformation science (spatial statistics and modeling, geographic information systems), artificial intelligence research (supervised classification, fuzzy logic), ecology (point pattern analysis), applied mathematics (graph theory, probability theory) and statistics.

Training and research

Scientific progress in archaeology, as in any other discipline, requires building abstract, generalized and transferable knowledge about the processes that underlie past human actions and their manifestations. Quantification provides the ultimate known way of abstracting and extending our scientific abilities past the limits of intuitive cognition. Quantitative approaches to archaeological information handling and inference constitute a critical body of scientific methods in archaeological research. They provide the tools, algebra, statistics and computer algorithms, to process information too voluminous or complex for purely cognitive, informal inference. They also build a bridge between archaeology and numerous quantitative sciences such as geophysics, geoinformation sciences and applied statistics. And they allow archaeological scientists to design and carry out research in a formal, transparent and comprehensible way.

Being an emerging field of research, AI science is currently a rather dispersed discipline in need of stronger, well-funded and institutionalized embedding, especially in academic teaching. Despite its evident progress and usefulness, today's quantitative archaeology is often inadequately represented in archaeological training and education. Part of this problem may be misconceptions about the seeming conflict between mathematics and humanistic archaeology.

Nevertheless, digital excavation technology, modern heritage management and complex research issues require skilled students and researchers to develop new, efficient and reliable means of processing an ever-growing mass of untackled archaeological data and research problems. Thus, providing students of archaeology with a solid background in quantitative sciences such as mathematics, statistics and computer sciences seems today more important than ever.

Currently, universities based in the UK provide the largest share of study programmes for prospective quantitative archaeologists, with more institutes in Italy, Germany and the Netherlands developing a strong profile quickly. In Germany, the country's first lecturer's position in AI science ("Archäoinformatik") was established in 2005 at the University of Kiel. In April 2016 the first full professorship in Archaeoinformatics has been established at the University of Cologne (Institute of Archaeology).

The most important platform for students and researchers in quantitative archaeology and AI science is the international conference on Computer Applications and Quantitative Methods in Archaeology (CAA) which has been in existence for more than 30 years now and is held in a different city of Europe each year. Vienna's city archaeology unit also hosts an annual event that is quickly growing in international importance (see links at bottom).