Canonical Form of Underdamped Oscillatory System
Theorem
Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
- $(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the value of $x$ can be expressed in the form:
- $x = \dfrac {x_0 a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
- $\alpha = \sqrt {a^2 - b^2}$
- $\theta = \map \arctan {\dfrac b \alpha}$
This can be referred to as the canonical form of the solution of $(1)$.
Proof
From Solution of Constant Coefficient Homogeneous LSOODE: Complex Roots of Auxiliary Equation, the general solution of $(1)$ is:
- $x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$
where:
- $\alpha = \sqrt {a^2 - b^2}$
This is a homogeneous linear second order ODE with constant coefficients.
Let $m_1$ and $m_2$ be the roots of the auxiliary equation:
- $m^2 + 2 b + a^2 = 0$
From Solution to Quadratic Equation with Real Coefficients:
\(\ds m_1\) | \(=\) | \(\ds -b + i \sqrt {a^2 - b^2}\) | ||||||||||||
\(\ds m_1, m_2\) | \(=\) | \(\ds -b - i \sqrt {a^2 - b^2}\) |
So from Solution of Constant Coefficient Homogeneous LSOODE: Complex Roots of Auxiliary Equation:
- $x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$
where:
- $\alpha = \sqrt {a^2 - b^2}$
The following assumptions are made:
- We may label a particular point in time $t = 0$ at which the derivative of $x$ with respect to $t$ is itself zero.
- We allow that at this arbitrary $t = 0$, the value of $x$ is assigned the value $x = x_0$.
This corresponds, for example, with a physical system in which the moving body is pulled from its equilibrium position and released from stationary at time zero.
Differentiating $(1)$ with respect to $t$ gives:
- $\quad x' = -b e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t} + e^{-b t} \paren {-\alpha C_1 \sin \alpha t + \alpha C_2 \cos \alpha t}$
Setting the initial condition $x = x_0$ when $t = 0$ in $(1)$:
\(\ds x_0\) | \(=\) | \(\ds e^0 \paren {C_1 \cos 0 + C_2 \sin 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds C_1\) |
Setting the initial condition $x' = 0$ when $t = 0$:
\(\ds 0\) | \(=\) | \(\ds -b e^0 \paren {C_1 \cos 0 + C_2 \sin 0} + e^0 \paren {-\alpha C_1 \sin 0 + \alpha C_2 \cos 0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -b C_1 + \alpha C_2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C_2\) | \(=\) | \(\ds \frac {b C_1} \alpha\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {b x_0} \alpha\) |
Hence:
\(\ds x\) | \(=\) | \(\ds e^{-b t} \paren {x_0 \cos \alpha t + \frac {b x_0} \alpha \sin \alpha t}\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds e^{-b t} \dfrac {x_0} \alpha \paren {\alpha \cos \alpha t + b \sin \alpha t}\) |
From Multiple of Sine plus Multiple of Cosine: Cosine Form, $(2)$ can be expressed as:
\(\ds x\) | \(=\) | \(\ds \dfrac {x_0} \alpha e^{-b t} \paren {\sqrt {\alpha^2 + b^2} \map \cos {\alpha t + \arctan \dfrac {-b} \alpha} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x_0 \sqrt {\sqrt{a^2 - b^2}^2 + b^2} } \alpha e^{-b t} \map \cos {\alpha t + \arctan \dfrac {-b} \alpha}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x_0 \sqrt {a^2 - b^2 + b^2} } \alpha e^{-b t} \map \cos {\alpha t + \arctan \dfrac {-b} \alpha}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x_0 a} \alpha e^{-b t} \cos {\alpha t + \arctan \dfrac {-b} \alpha}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x_0 a} \alpha e^{-b t} \cos {\alpha t - \arctan \dfrac b \alpha}\) | Tangent Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x_0 a} \alpha e^{-b t} \cos {\alpha t - \theta}\) | where $\theta = \arctan \dfrac b \alpha$ |
$\blacksquare$
Also presented as
This can also be seen presented as:
- $x = \dfrac {x_0 \sqrt {\alpha^2 + b^2} } \alpha e^{-b t} \cos {\alpha t - \theta}$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems: $(20)$