Mathematics and Vibrations for AVT Dr Ian Drumm

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Mathematics and Vibrations for AVTDr Ian Drumm

• The Exponential Function
• Hyperbolic Functions

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e is a magic number

• e2.7182818 is an irrational number
• Precise definition
• Or from series

Estimate of e (1000001/1000000)10000002.71828
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The Exponential Function

• Growth
• Decay

as
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Consider Growth (or decay)
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Exponential Growth

• A population grows at a rate proportional to its
  size
• is constant of proportionality
• (e.g. average number offspring per individual)
• is the initial size of the population at
  t0

Solution to
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Exponential function and vibration

• Rate of decay proportional to amplitude of
  vibration
• Used to describe vibration ( e.g. loudspeakers,
  microphones, musical instruments and even
  electronic oscillators).

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Exponential function and electronics

• The rate of charge and discharge of a capacitor
  is a function of the charge at the capacitor

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Remember laws of indices

• Note

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Natural Log Function

• The index of the exponential term can be found
  using the natural log function
• For example given
• Log both sides to give

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Example

• Function grows such that f(0)1 and f(2)4, find
  the exponential curve that fits these points.
• Assume
• Could try hard way testing values of k
• Better method

where
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Hyperbolic Functions

• Trigonometric functions cosh, shine, than
• Can be expressed as combinations of exponential
  functions

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• Can think of as odd and even parts of exponential

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Hyperbolic Identities
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Related Functions