SPRING – MASS SYSTEMS

A spring is an elastic object. When stretched, it exerts a restoring force and tends to revert to it’s original length. This restoring force is proportional to the amount of stretch in accordance with Hookes Law.

kx

F_spring  where k is the spring constant.

When the spring is stretched it has elastic potential energy which is equal to the work done in stretching the spring. The work done is equal to: 2

2 1

kx

JAR 66 CATEGORY B1 MODULE 2

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If the mass is displaced from its original position and released, the force in the spring will act on the mass so as to return it to that position. It behaves like the pendulum, in that it will continue to move up and down.

The resulting motion, up and down, can be plotted against time and will result in a typical graph, which is sinusoidal.

Vibration Theory is based on the detailed analysis of vibrations and is essentially mathematical, relying heavily on trigonometry and calculus, involving sinusoidal functions and differential equations.

The simple pendulum or spring-mass would according to basic theory, continue to vibrate at constant frequency and amplitude, once the vibration had been started. In fact, the vibrations die away, due to other forces associated with motion, such as friction, air resistance etc. This is termed a Damped vibration. If a disturbing force is re-applied periodically the vibrations can be maintained indefinitely. The frequency (and to a lesser extent, the magnitude) of this disturbing force now becomes critical.

The diagram above shows a vibration in which the displacement is constant, but depending on the frequency of the disturbing force, the amplitude of vibration may decay rapidly (a damping effect) or may grow significantly.

A large increase in amplitude usually occurs when the frequency of the disturbing force coincides with the natural frequency of the vibration of the system (or some harmonic). This is known as the Resonant Frequency. Designers carry out tests to determine these frequencies, so that they can be avoided or eliminated, as they can be very damaging. If an aircraft component starts to vibrate at it’s resonant frequency it may shake itself to pieces. For example at certain constant engine RPM an engine may vibrate to destruction.

Issue 1 – 20 August 2001 Page 3-11 JAR 66 CATEGORY B1 MODULE 2 PHYSICS

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In scientific terms, machines are devices used to enable heavy loads to be moved by smaller loads. There are many examples of these machines; some of which are inclined planes, levers, pulleys, gears and screws. We shall briefly describe the lever as an example of a typical machine.

3.4.1 LEVERS

A lever is a device used to gain a mechanical advantage. In its most basic form, the lever is a beam that has a weight at each end. The weight on one end of the beam tends to rotate the beam anti-clockwise, whilst the weight on the other end tends to rotate the beam clockwise, viewed from the side.

Each weight produces a moment or turning force. The moment of an object is calculated by multiplying the object's weight by the distance the object is from the balance point or fulcrum.

A lever is in balance when the algebraic sum of the moments is zero. In other words, a 20 kg weight located 1 m to the left of the fulcrum (B) has a moment of negative, (anti-clockwise), 20 kilogram metres. A 10 kg weight located 2m to the right of the fulcrum has a positive, (clockwise), moment of 20 kilogram metres. Since the sum of the moments is zero, the lever is balanced. There are different categories or classes of lever as follows:

3.4.1.1 First Class Lever

This lever has the fulcrum between the load and the effort. An example might be using a long armed lever to lift a heavy crate with the fulcrum very close to the crate. In the example below, the effort 'E' is applied a distance 'L' from the fulcrum. The load, (resistance), 'R' acts at a distance 'I' from the fulcrum. The calculation is carried out

using the formula:

E R I L

JAR 66 CATEGORY B1 MODULE 2

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In the diagram an effort of 100N is required to lift a load or reaction of 200N. It follows that the distance between the fulcrum B and the effort must be twice the distance from the fulcrum and the reaction.

E R I L

 or LERI

Although less effort is required to lift the load (resistance), the lever does not reduce the amount of work done. Work is the result of force and distance and, if the two items from both sides are multiplied together, they are always equal.

3.4.1.2 Second Class Lever

Unlike the first-class lever, the second-class lever has the fulcrum at one end of the lever and effort is applied to the opposite end. The resistance, or weight, is typically placed near the fulcrum between the two ends.

A typical example of this lever arrangement is the wheel-barrow, which is

illustrated below, using the same terminology as before. Calculations are carried out using the same formula as for the first class-class lever although, in this case, the load and the effort move in the same direction.

Issue 1 – 20 August 2001 Page 3-13 JAR 66 CATEGORY B1 MODULE 2 PHYSICS

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3.4.1.3 Third Class Lever

In aviation, the third-class lever is primarily used to move the load (resistance) a greater distance than the effort applied. This is accomplished by applying the effort between the fulcrum and the resistance. The disadvantage of doing this, is that a much greater effort is required to produce movement. A good example of a third-class lever is a landing gear retraction mechanism, where the effort is applied close to the fulcrum, whilst the load, (the wheel/brake assembly) is at the end of the lever. This is illustrated below.