Assignment activity and guidance

Activity 1

A wing spar beam is subjected to vibrations along its length, emerging from two fuel pumps

situated at opposite ends of the beam. The displacement (in mm) caused by the vibrations can be

modelled by the following equations,

d₁ = 2.34 sin (100nt +

dz

+57)

(1

= 3.46 sin 100nt

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2.T

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To understand the best type, size and location of fasteners to restrain the beam (for minimum

vibration), you must analyse and calculate the effect of the vibrations along its length.

Considering the above displacement equations:

i) State the amplitude, phase, frequency and periodic time of each of the above equations.

ii)

Determine how long it takes for each machine to first produce its maximum displacement

(i.e. the first instance after t = 0 ms in either the positive or negative y-direction).

iii) Use the compound angle formulae to expand d₁ and d₂ into the form

a sin(100πt) ± ß cos (100mt), where a and ß are numbers to be found.

iv) Using your answers from the part (iii), express d₁ + d₂ in a similar form.

v) Analytically, convert your answer from part (iv) into the equivalent form

R sin(100nt + Ø).

vi) Plot the graphs of d₁ and d₂ on the same axes using any suitable computer package. By

using the software, also plot a graph of the sum of d, and d₂ on the same axes. Ensure

your axes are in appropriate units and have labels.

vii) State the amplitude and phase of the new wave.

viii) Using your answers from parts (v) and (vi), what conclusions can be drawn about d₁ + d₂

and the two methods that were used to obtain this information?

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