Simple machine

The idea of a simple machine originated with the Greek philosopher Archimedes around the 3rd century BC, who studied the Archimedean simple machines: lever, pulley, and screw. He discovered the principle of mechanical advantage in the lever. Archimedes' famous remark with regard to the lever: "Give me a place to stand on, and I will move the Earth," () expresses his realization that there was no limit to the degree of force amplification that could be achieved by using mechanical advantage. Later Greek philosophers defined the classic five simple machines (excluding the inclined plane) and were able to calculate their (ideal) mechanical advantage. However the Greeks' understanding was limited to the statics of simple machines (the balance of forces), and did not include dynamics, the tradeoff between force and distance, or the concept of work. During the Renaissance the dynamics of the mechanical powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, 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 (On Mechanics), in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain that simple machines do not create energy, only transform it. ==Ideal simple machine==

If a simple machine does not dissipate energy through friction, wear or deformation, then energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, and the mechanical advantage can be calculated from its geometric dimensions. Although each machine works differently mechanically, the way they function is similar mathematically. In each machine, a force F_\text{in} is applied to the device at one point, and it does work moving a load F_\text{out} at another point. Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply the magnitude of the force by a factor, the mechanical advantage \mathrm{MA} = {F_\text{out} \over F_\text{in}} that can be calculated from the machine's geometry and friction. Simple machines do not contain a source of energy, so they cannot do more work than they receive from the input force. Due to conservation of energy, in an ideal simple machine, the power output (rate of energy output) at any time P_\text{out} is equal to the power input P_\text{in} P_\text{out} = P_\text{in}\! The power output equals the velocity of the load v_\text{out}\, multiplied by the load force P_\text{out} = F_\text{out} v_\text{out}\,. Similarly the power input from the applied force is equal to the velocity of the input point v_\text{in}\, multiplied by the applied force P_\text{in} = F_\text{in} v_\text{in}\!. Therefore, F_\text{out}v_\text{out} = F_\text{in}v_\text{in}\, So the mechanical advantage of an ideal machine \mathrm{MA}_\text{ideal}\, is equal to the velocity ratio, the ratio of input velocity to output velocity \mathrm{MA}_\text{ideal} = {F_\text{out} \over F_\text{in}} = {v_\text{in} \over v_\text{out}}\, The velocity ratio is also equal to the ratio of the distances covered in any given period of time {v_\text{out} \over v_\text{in}} = {d_\text{out} \over d_\text{in}} Therefore, the mechanical advantage of an ideal machine is also equal to the distance ratio, the ratio of input distance moved to output distance moved This can be calculated from the geometry of the machine. For example, the mechanical advantage and distance ratio of the lever is equal to the ratio of its lever arms. The mechanical advantage can be greater or less than one: • If \mathrm{MA} > 1\,, the output force is greater than the input, the machine acts as a force amplifier, but the distance moved by the load d_\text{out} is less than the distance moved by the input force d_\text{in}\,. • If \mathrm{MA} , the output force is less than the input, but the distance moved by the load is greater than the distance moved by the input force. In the screw, which uses rotational motion, the input force should be replaced by the torque, and the velocity by the angular velocity the shaft is turned. ==Friction and efficiency==

All real machines have friction, which causes some of the input power to be dissipated as heat. If P_\text{fric}\, is the power lost to friction, from conservation of energy P_\text{in} = P_\text{out} + P_\text{fric} The mechanical efficiency \eta of a machine (where 0 ) is defined as the ratio of power out to the power in, and is a measure of the frictional energy losses \begin{align} \eta & \equiv {P_\text{out} \over P_\text{in}} \\ P_\text{out} & = \eta P_\text{in} \end{align} As above, the power is equal to the product of force and velocity, so F_\text{out} v_\text{out} = \eta F_\text{in} v_\text{in} Therefore, So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiency \eta. So a machine that includes friction will not be able to move as large a load as a corresponding ideal machine using the same input force. ==Compound machines==

A compound machine is a machine formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example, a bench vise consists of a lever (the vise's handle) in series with a screw, and a simple gear train consists of a number of gears (wheels and axles) connected in series. The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is \mathrm{MA}_\text{compound} = {F_{\text{out}N} \over F_\text{in1}} Because the output force of each machine is the input of the next, F_\text{out1} = F_\text{in2}, \; F_\text{out2} = F_\text{in3},\, \ldots \; F_{\text{out}K} = F_{\text{in}K+1}, this mechanical advantage is also given by \mathrm{MA}_\text{compound} = {F_\text{out1} \over F_\text{in1}} {F_\text{out2} \over F_\text{in2}} {F_\text{out3} \over F_\text{in3}}\ldots {F_{\text{out}N} \over F_{\text{in}N}} \, Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it \mathrm{MA}_\text{compound} = \mathrm{MA}_1 \mathrm{MA}_2 \ldots \mathrm{MA}_N Similarly, the efficiency of a compound machine is also the product of the efficiencies of the series of simple machines that form it \eta_\text{compound} = \eta_1 \eta_2 \ldots \; \eta_N. ==Self-locking machines==

's self-locking property is the reason for its wide use in threaded fasteners like bolts and wood screws In many simple machines, if the load force F_{\textrm{out}} on the machine is high enough in relation to the input force F_{\textrm{in}}, the machine will move backwards, with the load force doing work on the input force. So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are called reversible, non-locking or overhauling machines, and the backward motion is called overhauling. However, in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called a self-locking, nonreversible, or non-overhauling machine. the input work W_\text{1,2} is equal to the sum of the work done on the load force W_\text{load} and the work lost to friction W_\text{fric} {{NumBlk2|:|W_\text{1,2} = W_\text{load} + W_\text{fric}|Eq. 1}} If the efficiency is below 50% {{nowrap|(\eta = W_\text{load}/W_\text{1,2} ):}} 2W_\text{load} From \begin{align} 2W_\text{load} & When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction W_\text{fric} is the same W_\text{load} = W_\text{2,1} + W_\text{fric} So the output work is W_\text{2,1} = W_\text{load} - W_\text{fric} Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force. ==Modern machine theory==

Machines are studied as mechanical systems consisting of actuators and mechanisms that transmit forces and movement, monitored by sensors and controllers. The components of actuators and mechanisms consist of links and joints that form kinematic chains. Kinematic chains Simple machines are elementary examples of kinematic chains 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. 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. 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 revolute 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. Kinematic synthesis The design of mechanisms to perform required movement and force transmission is known as kinematic synthesis. This is a collection of geometric techniques for the mechanical design of linkages, cam and follower mechanisms and gears and gear trains. ==See also==