Assignment Brief and Guidance Scenario A support beam 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: Part 1: a. State the amplitude, phase, frequency and periodic time of each of these waves. b. Calculate the time each machine will reach maximum displacement, after being switched on. c. Calculate the time from the O time point of the waveform, (on the x axis), after which each vibration will first reach a value of -2mm. Show all working out. x₁ = 3.75 sin(100nt + 2π/9) x₂ = 4.42 sin(100nt - 2n/5) d. Plot both vibration waveforms using Excel, between -5ms and 20ms and deduct from the plots the time for each waveform to reach -2mm after point 0 on the time axis. How do these times compare with those obtained above? Part 2: a. Use the compound angle formulae to expand x₁ and x₂ into the form Asin 100rt+Bcos 100nt, where A and B are numbers to be found. b. Using your answers from above, express x₁ + x₂ in a similar form. Convert this expression into the equivalent form Rsin(100πt+a), using trigonometric identities. Part 3: a. Using appropriate spreadsheet software, copy and complete the following table of values: 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 t X₁ X₂ b. Plot the graphs of x₁ and x₂ on the same axes using any suitable computer package. Extend your table to include x₁ + x₂ and plot this graph on the same axes as the previous two. State the amplitude and frequency of the new wave. c. Plot a graph using the equation obtained in Part b above. d. Using your answers from task 2 b, what conclusions can be drawn about x₁ + x₂ and the two methods that were used to obtain this information? e. Compare the graphs obtained using the two methods above.
Assignment Brief and Guidance Scenario A support beam 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: Part 1: a. State the amplitude, phase, frequency and periodic time of each of these waves. b. Calculate the time each machine will reach maximum displacement, after being switched on. c. Calculate the time from the O time point of the waveform, (on the x axis), after which each vibration will first reach a value of -2mm. Show all working out. x₁ = 3.75 sin(100nt + 2π/9) x₂ = 4.42 sin(100nt - 2n/5) d. Plot both vibration waveforms using Excel, between -5ms and 20ms and deduct from the plots the time for each waveform to reach -2mm after point 0 on the time axis. How do these times compare with those obtained above? Part 2: a. Use the compound angle formulae to expand x₁ and x₂ into the form Asin 100rt+Bcos 100nt, where A and B are numbers to be found. b. Using your answers from above, express x₁ + x₂ in a similar form. Convert this expression into the equivalent form Rsin(100πt+a), using trigonometric identities. Part 3: a. Using appropriate spreadsheet software, copy and complete the following table of values: 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 t X₁ X₂ b. Plot the graphs of x₁ and x₂ on the same axes using any suitable computer package. Extend your table to include x₁ + x₂ and plot this graph on the same axes as the previous two. State the amplitude and frequency of the new wave. c. Plot a graph using the equation obtained in Part b above. d. Using your answers from task 2 b, what conclusions can be drawn about x₁ + x₂ and the two methods that were used to obtain this information? e. Compare the graphs obtained using the two methods above.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Assignment Brief and Guidance
Scenario
A support beam 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:
Part 1:
a. State the amplitude, phase, frequency and periodic time of each of these waves.
b. Calculate the time each machine will reach maximum displacement, after being switched on.
c. Calculate the time from the O time point of the waveform, (on the x axis), after which each vibration will first
reach a value of -2mm. Show all working out.
x₁ = 3.75 sin(100nt + 2π/9)
x₂ = 4.42 sin(100nt - 2n/5)
d.
both vibration
using
and 20ms and deduct from plots the time for
each waveform to reach -2mm after point 0 on the time axis. How do these times compare with those obtained.
above?
Part 2:
a. Use the compound angle formulae to expand x, and x₂ into the form Asin100rt+Bcos100nt, where A and B
are numbers to be found.
b. Using your answers from above, express x₁ + x₂ in a similar form. Convert this expression into the equivalent
form Rsin(100πt+a), using trigonometric identities.
t
etween-
Part 3:
a. Using appropriate spreadsheet software, copy and complete the following table of values:
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
X1
X2
b. Plot the graphs of x₁ and x₂ on the same axes using any suitable computer package. Extend your table to
include x₁ + x₂ and plot this graph on the same axes as the previous two. State the amplitude and frequency of
the new wave.
c. Plot a graph using the equation obtained in Part b above.
d. Using your answers from task 2 b, what conclusions can be drawn about x₁ + x₂ and the two methods that
were used to obtain this information?
e. Compare the graphs obtained using the two methods above.
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