Tony Hau
- 107
- 30
The formula for general oscillation is:
The formula for underdamping oscillation is:
where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other.
After some operations, we get:
x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus of A+ and A-.
The notes of my teacher give the answer as
![Handwritten line of math: x = e^(-γx)[A] cos(ωt + θ), with the cosine term's argument partly unclear.](https://www.physicsforums.com/attachments/1586875556072-png.260642/ "1586875556072.png")
Obviously, the complex part of e^i(θ+ωx) and e^-i(θ+ωx) cancells each other. However, the real part of the two expressions is the same. So, why isn't the final answer of x(t) Ae^(-γx)2cos(θ+ωx)?
The formula for underdamping oscillation is:
where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other.
After some operations, we get:
x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus of A+ and A-.
The notes of my teacher give the answer as
Obviously, the complex part of e^i(θ+ωx) and e^-i(θ+ωx) cancells each other. However, the real part of the two expressions is the same. So, why isn't the final answer of x(t) Ae^(-γx)2cos(θ+ωx)?
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