3 Theoretical power spectral density for a damped harmonic oscillator

Our cantilever can be approximated as a damped harmonic oscillator

$$m\ddt{x} + \gamma \dt{x} + k x = F(t), % DHO for Damped Harmonic Oscillator$$ (28)

where $x$ is the displacement from equilibrium, $m$ is the effective mass, $\gamma$ is the effective drag coefficient, $k$ is the spring constant, and $F(t)$ is the external driving force. During the non-contact phase of calibration, $F(t)$ comes from random thermal noise.

In the following analysis, we use the unitary, angular frequency Fourier transform normalization

$$\ensuremath{{\mathcal F}\left\{ {x(t)} \right\}} \equiv \frac{1}{\sqrt{2\pi}} \ensuremath{\int_{-\infty}^{\infty} {x(t) e^{-i \omega t}} \dd{t}}$$ (29)

We also use the following theorems (proved elsewhere):

$$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1}{2}[1+\cos(\theta)]}$$ [10], (30)

$$\ensuremath{{\mathcal F}\left\{ {\nderiv{n}{t}{x(t)}} \right\}} = (i \omega)^n x(\omega)$$ [11], (31)

$$\ensuremath{\int_{-\infty}^{\infty} {\ensuremath{\left\vert{x(t)}\right\vert^2}} \dd{t}} = \ensuremath{\int_{-\infty}^{\infty} {\ensuremath{\left\vert{x(w)}\right\vert^2}} \dd{\omega}}$$ (Parseval's)[12]. (32)

As a corollary to Parseval's theorem, we note that the one sided power spectral density per unit time ( $\operatorname{PSD}$) defined by

$$\ensuremath{\operatorname{PSD}}(x, \omega) \equiv \ensuremath{\lim_{{t_T}\rightarrow \infty}}\frac{1}{t_T} 2 \left\vert x(\omega) \right\vert^2$$ [13] (33)

relates to the variance by

$$\avg{x(t)^2} = \ensuremath{\lim_{{t_T}\rightarrow \infty}}\frac{1}{t_T} \ensu... ...ath{\int_{0}^{\infty} {\ensuremath{\operatorname{PSD}}(x,\omega)} \dd{\omega}},$$ (34)

where $t_T$ is the total time over which data has been aquired.

We also use the Wiener-Khinchin theorem, which relates the two sided power spectral density $S_{xx}(\omega)$ to the autocorrelation function $r_{xx}(t)$ via

$$S_{xx}(\omega) = \ensuremath{{\mathcal F}\left\{ { r_{xx}(t) } \right\}}$$ (Wiener-Khinchin)[14], (35)

where $r_{xx}(t)$ is defined in terms of the expectation value

$$r_{xx}(t) \equiv \avg{x(\tau)\ensuremath{\overline{x}}(\tau-t)}$$ [15] (36)

and $\ensuremath{\overline{x}}$ represents the complex conjugate of $x$.

3.1 Highly damped case

For highly damped systems, the inertial term becomes insignificant ( $m \rightarrow 0$). This model is commonly used for optically trapped beads. Because it is simpler and solutions are more easily available, we'll use it outline the general approach before diving into the general case.

Fourier transforming eq. 28 with $m=0$ and applying 31 we have

$$(i \gamma \omega + k) x(\omega) = F(\omega)$$ (37)

$$\vert x(\omega)\vert^2 = \frac{\vert F(\omega)\vert^2}{k^2 + \gamma^2 \omega^2}.$$ (38)

We compute the $\operatorname{PSD}$ by plugging eq. 38 into 33

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \ensuremath{\lim_{{t... ...frac{2\ensuremath{\left\vert{F(\omega)}\right\vert^2}}{k^2 + \gamma^2\omega^2}.$$ (39)

Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem), we can denote the one sided thermal power spectral density per unit time by

$$\ensuremath{\operatorname{PSD}}(F, \omega) = G_0 = \ensuremath{\l... ...ac{1}{t_T} 2 \ensuremath{\left\vert{F(\omega)}\right\vert^2}% label O != zero$$ (40)

Plugging eq. 40 into 39 we have

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \frac{G_0}{k^2 + \gamma^2\omega^2}.$$ (41)

This is the formula we would use to fit our measured $\operatorname{PSD}$, but let us go a bit farther to find the expected $\operatorname{PSD}$ and thermal noise given $m$, $\gamma$ and $k$.

Integrating over positive $\omega$ to find the total power per unit time yields

$$\ensuremath{\int_{0}^{\infty} {\ensuremath{\operatorname{PSD}}(x, \omega)} \dd{\omega}} = \ensuremath{\int_{0}^{\infty} {\frac{G_0}{k^2 + \gamma^2\omega^2}} \dd{\omega}}$$ (42)

$$= \frac{G_0}{\gamma}\ensuremath{\int_{0}^{\infty} {\frac{1}{k^2 + z^2}} \dd{z}}$$ (43)

$$= \frac{G_0 \pi}{2 \gamma k},$$ (44)

where the integral is solved in Section 5.

Plugging into our corollary to Parseval's theorem (eq. 34),

$$\avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma k}$$ (45)

Plugging eq. 45 into eqn. 1 we have

$$k \frac{G_0 \pi}{2 \gamma k} = k_BT$$ (46)

$$G_0 = \frac{2 \gamma k_BT}{\pi}.$$ (47)

So we expect $X(t)$ to have a power spectral density per unit time given by

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \frac{2}{\pi} \cdot \frac{\gamma k_BT}{k^2 + \gamma^2\omega^2}.$$ (48)

3.2 General form

The procedure here is exactly the same as the previous section. The integral normalizing $G_0$ just become a little more complicated...

Fourier transforming eq. 28 and applying 31 we have

$$(-m\omega^2 + i \gamma \omega + k) x(\omega) = F(\omega)$$ (49)

$$(\omega_0^2-\omega^2 + i \beta \omega) x(\omega) = \frac{F(\omega)}{m}$$ (50)

$$\vert x(\omega)\vert^2 = \frac{\vert F(\omega)\vert^2/m^2} {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2} ,$$ (51)

where $\omega_0 \equiv \sqrt{k/m}$ is the resonant angular frequency and $\beta \equiv \gamma / m$ is the drag-acceleration coefficient.

We compute the $\operatorname{PSD}$ by plugging eq. 51 into 33

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \ensuremath{\lim_{{t... ...frac{2 \vert F(\omega)\vert^2/m^2} {(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}.$$ (52)

Plugging eq. 40 into 52 we have

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}.$$ (53)

Integrating over positive $\omega$ to find the total power per unit time yields

$$\ensuremath{\int_{0}^{\infty} {\ensuremath{\operatorname{PSD}}(x, \omega)} \dd{\omega}} = \frac{G_0}{2m^2} \ensuremath{\int_{-\infty}^{\infty} {\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}} \dd{\omega}}$$ (54)

$$= \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2}$$ (55)

$$= \frac{G_0 \pi}{2m^2\beta\omega_0^2}$$ (56)

$$= \frac{G_0 \pi}{2m^2\beta \frac{k}{m}}$$ (57)

$$= \frac{G_0 \pi}{2m \beta k}$$ (58)

The integration is detailed in Section 5. By our corollary to Parseval's theorem (eq. 34), we have

$$\avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2}$$ (59)

Plugging eq. 59 into the equipartition theorem (eqn. 1) we have

$$k \frac{G_0 \pi}{2m \beta k} = k_BT$$ (60)

$$G_0 = \frac{2}{\pi} k_BT m \beta.$$ (61)

So we expect $x(t)$ to have a power spectral density per unit time given by

$$\ensuremath{\operatorname{PSD}}(x, \omega) = \frac{2 k_BT \beta} {\pi m \left[ (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 \right] }$$ (62)

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3 Theoretical power spectral density for a damped harmonic oscillator
Copyright © 2009-10-12, W. Trevor King (contact)
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Drexel Physics