Tuesday, March 3, 2015

Simple Machines

Simple Machines

Machines

A machine is a device that can change the magnitude or line of action, or both magnitude and line of action of a force. A simple machine usually amplifies an input force, called the effort, to give a larger output force, called the load. Some typical examples of simple machines include pulley systems, screw- jacks, gear systems and lever systems.

Force Ratio, Movement Ratio and Efficiency

The force ratio or mechanical advantage is defined as the ratio of load to effort, i.e.

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Since both load and effort are measured in newtons, force ratio is a ratio of the same units and thus is a dimension-less quantity.

The movement ratio or velocity ratio is defined as the ratio of the distance moved by the effort to the distance moved by the load, i.e.

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Since the numerator and denominator are both measured in metres, movement ratio is a ratio of the same units and thus is a dimension-less quantity.

The efficiency of a simple machine is defined as the ratio of the force ratio to the movement ratio, i.e.

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Since the numerator and denominator are both dimension-less quantities, efficiency is a imension-less quantity. It is usually expressed as a percent- age, thus:

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Due to the effects of friction and inertia associated with the movement of any object, some of the input energy to a machine is converted into heat and losses occur. Since losses occur, the energy output of a machine is less than the energy input, thus the mechanical efficiency of any machine cannot reach 100%.

For example, a simple machine raises a load of 160 kg through a distance of 1.6 m. The effort applied to the machine is 200 N and moves through a distance of 16 m.

From equation (1),

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Pulleys

A pulley system is a simple machine. A single-pulley system, shown in Figure 22.1(a), changes the line of action of the effort, but does not change the magnitude of the force.

A two-pulley system, shown in Figure 22.1(b), changes both the line of action and the magnitude of the force. Theoretically, each of the ropes marked (i) and (ii) share the load equally, thus the theoretical effort is only half of the load, i.e. the theoretical force ratio is 2. In practice the actual force ratio is less than 2 due to losses.

A three-pulley system is shown in Figure 22.1(c). Each of the ropes marked (i), (ii) and (iii) carry one-third of the load, thus the theoretical force ratio is 3. In general, for a multiple pulley system having a total of n pulleys, the theoretical force ratio is n. Since the theoretical efficiency of a pulley system (neglecting losses) is 100 and since from equation (3),

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The Screw-jack

A simple screw-jack is shown in Figure 22.2 and is a simple machine since it changes both the magnitude and the line of action of a force.

The screw of the table of the jack is located in a fixed nut in the body of the jack. As the table is rotated by means of a bar, it raises or lowers a load placed on the table. For a single-start thread, as shown, for one complete revolution of the table, the effort moves through a distance 2nr, and the load

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moves through a distance equal to the lead of the screw, say, l

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For example, a screw-jack is used to support the axle of a car, the load on it being 2.4 kN. The screw jack has an effort of effective radius 200 mm and a single-start square thread, having a lead of 5 mm. If an effort of 60 N is required to raise the car axle:

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Gear Trains

A simple gear train is used to transmit rotary motion and can change both the magnitude and the line of action of a force, hence is a simple machine. The gear train shown in Figure 22.3 consists of spur gears and has an effort applied to one gear, called the driver, and a load applied to the other gear, called the follower.

In such a system, the teeth on the wheels are so spaced that they exactly fill the circumference with a whole number of identical teeth, and the teeth on the driver and follower mesh without interference. Under these conditions, the number of teeth on the driver and follower are in direct proportion to the circumference of these wheels, i.e.

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If there are, say, 40 teeth on the driver and 20 teeth on the follower then the follower makes two revolutions for each revolution of the driver. In general:

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It follows from equation (6) that the speeds of the wheels in a gear train are inversely proportional to the number of teeth. The ratio of the speed of the driver wheel to that of the follower is the movement ratio, i.e.

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When the same direction of rotation is required on both the driver and the follower an idler wheel is used as shown in Figure 22.4

Let the driver, idler, and follower be A, B and C, respectively, and let N be the speed of rotation and T be the number of teeth. Then from equation (7),

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This shows that the movement ratio is independent of the idler, only the

direction of the follower being altered.

A compound gear train is shown in Figure 22.5, in which gear wheels B and C are fixed to the same shaft and hence NB = NC

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For example, a compound gear train consists of a driver gear A, having 40 teeth, engaging with gear B, having 160 teeth. Attached to the same shaft as B, gear C has 48 teeth and meshes with gear D on the output shaft, having 96 teeth.

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Levers

A lever can alter both the magnitude and the line of action of a force and is thus classed as a simple machine. There are three types or orders of levers, as shown in Figure 22.6

A lever of the first order has the fulcrum placed between the effort and the load, as shown in Figure 22.6(a).

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A lever of the second order has the load placed between the effort and the fulcrum, as shown in Figure 22.6(b).

A lever of the third order has the effort applied between the load and the fulcrum, as shown in Figure 22.6(c).

Problems on levers can largely be solved by applying the principle of moments (see Chapter 12). Thus for the lever shown in Figure 22.6(a), when the lever is in equilibrium,

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