4.8: Applications

View Nondimensionalization on YouTube

RLC Circuit

View $RLC$ circuit on YouTube

Figure $\PageIndex{1}$: RLC circuit diagram.

Consider a resister $R$, an inductor $L$, and a capacitor $C$ connected in series as shown in Fig. $\PageIndex{1}$. An AC generator provides a time-varying electromotive force (emf), $\mathcal{E}(t)$, to the circuit. Here, we determine the differential equation satisfied by the charge on the capacitor.

The constitutive equations for the voltage drops across a capacitor, a resister, and an inductor are given by $$\label{eq:1}V_C=q/C,\quad V_R=iR,\quad V_L=\frac{di}{dt}L,$$ where $C$ is the capacitance, $R$ is the resistance, and $L$ is the inductance. The charge $q$ and the current $i$ are related by $$\label{eq:2}i=\frac{dq}{dt}.$$

Kirchhoff’s voltage law states that the emf $\mathcal{E}$ applied to any closed loop is equal to the sum of the voltage drops in that loop. Applying Kirchhoff’s voltage law to the $RLC$ circuit results in $$\label{eq:3}V_L+V_R+V_C=\mathcal{E}(t);$$ or using $\eqref{eq:1}$ and $\eqref{eq:2}$, $$L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{1}{C}q=\mathcal{E}(t).\nonumber$$

The equation for the $RLC$ circuit is a second-order linear inhomogeneous differential equation with constant coefficients.

The AC voltage can be written as $\mathcal{E}(t) = \mathcal{E}_0 \cos \omega t$, and the governing equation for $q = q(t)$ can be written as $$\label{eq:4}\frac{d^2q}{dt^2}+\frac{R}{L}\frac{dq}{dt}+\frac{1}{LC}q=\frac{\mathcal{E}_0}{L}\cos\omega t.$$

Nondimensionalization of this equation will be shown to reduce the number of free parameters.

To construct dimensionless variables, we first define the natural frequency of oscillation of a system to be the frequency of oscillation in the absence of any driving or damping forces. The iconic example is the simple harmonic oscillator, with equation given by $$\overset{..}{x}+\omega_0^2x=0,\nonumber$$ and general solution given by $x(t) = A \cos \omega_0 t + B \sin\omega_0 t$. Here, the natural frequency of oscillation is $\omega_0$, and the period of oscillation is $T = 2π/\omega_0$. For the $RLC$ circuit, the natural frequency of oscillation is found from the coefficient of the $q$ term, and is given by $$\omega_0=\frac{1}{\sqrt{LC}}.\nonumber$$

Making use of $\omega_0$, with units of one over time, we can define a dimensionless time $\tau$ and a dimensionless charge $Q$ by $$\tau =\omega_0t,\quad Q=\frac{\omega_0^2L}{\mathcal{E}_0}q.\nonumber$$

The resulting dimensionless equation for the $RLC$ circuit is then given by $$\label{eq:5}\frac{d^2Q}{d\tau ^2}+\alpha\frac{dQ}{d\tau}+Q=\cos\beta\tau ,$$ where $\alpha$ and $\beta$ are dimensionless parameters given by $$\alpha=\frac{R}{L\omega_0},\quad\beta=\frac{\omega}{\omega_0}.\nonumber$$

Notice that the original five parameters in $\eqref{eq:4}$, namely $R,\: L,\: C,\: \mathcal{E}_0$ and $\omega$, have been reduced to the two dimensionless parameters $\alpha$ and $\beta$. We will return later to solve $\eqref{eq:5}$ after visiting two more applications that will be shown to be governed by the same dimensionless equation.

Figure $\PageIndex{2}$: Mass-spring system (top view).

Mass on a Spring

View Mass on a Spring on YouTube

Consider a mass connected by a spring to a fixed wall, with top view shown in Fig. $\PageIndex{2}$. The spring force is modeled by Hooke’s law, $F_s = −kx$, and sliding friction is modeled as $F_f = −cdx/dt$. An external force is applied and is assumed to be sinusoidal, with $F_e = F_0 \cos\omega t$. Newton’s equation, $F = ma$, results in $$m\frac{d^2x}{dt^2}=-kx-c\frac{dx}{dt}+F_0\cos\omega t.\nonumber$$

Rearranging terms, we obtain $$\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x=\frac{F_0}{m}\cos\omega t.\nonumber$$

Here, the natural frequency of oscillation is given by $$\omega_0=\sqrt{\frac{k}{m}},\nonumber$$ and we define a dimensionless time $\tau$ and a dimensionless position $X$ by $$\tau =\omega_0t,\quad X=\frac{m\omega_0^2}{F_0}x.\nonumber$$

The resulting dimensionless equation is given by $$\label{eq:6}\frac{d^2X}{d\tau ^2}+\alpha\frac{dX}{d\tau }+X=\cos\beta\tau,$$ where here, $\alpha$ and $\beta$ are dimensionless parameters given by $$\alpha=\frac{c}{m\omega_0},\quad\beta =\frac{\omega}{\omega_0}.\nonumber$$

Notice that even though the physical problem is different, and the dimensionless variables are defined differently, the resulting dimensionless equation $\eqref{eq:6}$ for the mass-spring system is the same as that for the $RLC$ circuit $\eqref{eq:5}$.

Figure $\PageIndex{3}$: The pendulum.

Pendulum

View Pendulum on YouTube

Here, we consider a mass that is attached to a massless rigid rod and is constrained to move along an arc of a circle centered at the pivot point (see Fig. $\PageIndex{3}$). Suppose $l$ is the fixed length of the connecting rod, and $θ$ is the angle it makes with the vertical.

We can apply Newton’s equation, $F = ma$, to the mass with origin at the bottom and axis along the arc with positive direction to the right. The position $s$ of the mass along the arc is given by $s = lθ$. The relevant gravitational force on the pendulum is the component along the arc, and from Fig. $\PageIndex{3}$ is observed to be $$F_g=-mg\sin\theta.\nonumber$$

We model friction to be proportional to the velocity of the pendulum along the arc, that is $$F-f=-c\overset{.}{s}=-cl\overset{.}{\theta}.\nonumber$$

With a sinusoidal external force, $F_e = F_0 \cos\omega t$, Newton’s equation $m\overset{..}{s}= F_g + F_f + F_e$ results in $$ml\overset{..}{\theta}=-mg\sin\theta -cl\overset{.}{\theta}+F_0\cos\omega t.\nonumber$$

Rewriting, we have $$\overset{..}{\theta}+\frac{c}{m}\overset{.}{\theta}+\frac{g}{l}\sin\theta=\frac{F_0}{ml}\cos\omega t.\nonumber$$

At small amplitudes of oscillation, we can approximate $\sin θ\approx θ$, and the natural frequency of oscillation $\omega_0$ of the mass is given by $$\omega_0=\sqrt{\frac{g}{l}}.\nonumber$$

Nondimensionalizing time as $\tau = \omega_0t$, the dimensionless pendulum equation becomes $$\frac{d^2\theta}{d\tau ^2}+\alpha\frac{d\theta}{d\tau}+\sin\theta=\gamma\cos\beta\tau ,\nonumber$$ where $\alpha$, $\beta$, and $\gamma$ are dimensionless parameters given by $$\alpha=\frac{c}{m\omega_0},\quad\beta=\frac{\omega}{\omega_0},\quad\gamma=\frac{F_0}{ml\omega_0^2}.\nonumber$$

The nonlinearity of the pendulum equation, with the term $\sin \theta$, results in the additional dimensionless parameter $\gamma$. For small amplitude of oscillation, however, we can scale $\theta$ by $\theta =\gamma\Theta$, and the small amplitude dimensionless equation becomes $$\label{eq:7}\frac{d^2\Theta}{d\tau ^2}+\alpha\frac{d\Theta}{d\tau}+\Theta=\cos\beta\tau ,$$ the same equation as $\eqref{eq:5}$ and $\eqref{eq:6}$.