The displacement caused by the vibrations can be modelled by the following equations. x1 = 3.75 sin (100πt + 2?/9) ?2 = 4.42 sin (100?? − 2?/ 5) i. State the amplitude, phase, frequency and periodic time of each of these waves. ii. When both machines are switched on, how many seconds does it take for each machine to produce its maximum displacement? iii. At what time does each vibration first reach a displacement of −2???
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
Use analytical and computational methods for solving problems by relating sinusoidal wave and
vector functions to their respective construction applications
A support beam, within an industrial building, is subjected to vibrations along its length; emanating from two
machines situated at opposite ends of the beam. The displacement caused by the vibrations can be modelled
by the following equations.
x1 = 3.75 sin (100πt + 2?/9)
?2 = 4.42 sin (100?? − 2?/ 5)
i. State the amplitude, phase, frequency and periodic time of each of these waves.
ii. When both machines are switched on, how many seconds does it take for each machine to produce
its maximum displacement?
iii. At what time does each vibration first reach a displacement of −2???
iv. Use the compound angle formulae to expand ?1and ?2 into the form ? sin 100?? ± ? cos 100??,
where A and B are numbers to be found.
v. Using your answers from part iv, express x1 + x2 in a similar form. Convert this expression into the
equivalent form Rsin(100πt + ∝).
vi. Model x1+x2 waves graphically and analyse the variation between graphical and analytical
methods
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