1.4: Second-Order ODE Models

A physical system that contains two energy storage elements is described by a second-order system model. Examples of second-order systems include an RLC circuit and an inertial mass with position output. The following examples illustrate second-order system models.

Example 1.6: Series RLC circuit

A series RLC circuit with voltage input \(V_s(t)\) and current output \(i(t)\) has the following governing relationship obtained by applying Kirchoff’s voltage law to the mesh:

image4 \(L\frac{\rm di(t)}{\rm dt} +Ri(t)+\frac{1}{C} \int i(t){\rm dt=V_{\rm s} (t)\)

image6 The integro-differential equation can by converted into a second-order ODE by expressing it in terms of the electric charge, \(q(t)\), as: image7 \(L\frac

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(t)\)

Figure 7: A series RLC circuit. Alternatively, the series RLC circuit can be described in terms of two first-order ODE’s involving dual variables, the current, \(i(t)\), and the capacitor voltage, \(V_c(t)\), as:

image10

\(L\frac{\rm di(t)}{\rm dt} +Ri(t)+V_{c} (t)=V_

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(t)\), \(C\frac{dV_c}{dt}=i(t)\)

Figure 7: A series RLC circuit.

Example 1.7: Inertial mass with position output

An inertial mass in a constant gravitational field has both kinetic and potential energies, modeled by a second-order ODE. The vertical motion of a mass element of weight, \(mg\), that is pulled upward by a force, \(f(t)\), is described using position output, \(y(t)\), by a second-order ODE: \[m\frac{{\rm d}^{2} y(t)}{{\rm d}t^{2} } +mg=f(t).\]

Figure 8: Motion of an inertial mass under gravity.

Example 1.8: A mass–spring–damper system

A mass–spring–damper system includes a mass affected by an applied force, \(f(t)\), when its motion is restrained by a combination of a spring and a damper (Figure 1.9). Let \(x(t)\) denote the displacement of the mass from a fixed reference; then, the dynamic equation of the system obtained by using Newton’s second law of motion takes a familiar form, given as: \[m\frac{{\rm d}^{2} x(t)}{{\rm d}t^{2} } +b\frac{{\rm d}x(t)}{{\rm d}t} +kx(t)=f(t).\]

The left hand side in the above equation represents the sum of applied (inertial, damping, and spring) forces. In compact notation, we may express the ODE as: \[m\ddot{x}\; +\; b\dot{x}\; +\; kx=f\] where the dots above the variable represent time derivative, i.e., \(\dot{x}\left(t\right)=\frac{dx\left(t\right)}{dt}\), \(\ddot{x}\left(t\right)=\frac{d^2x\left(t\right)}{dt^2}\).

In the absence of damping, the dynamic equation of the mass-spring system reduces to: \[m\frac{d^{\mathrm{2}}x\left(t\right)}{dt^{\mathrm{2}}}\mathrm{+}kx\left(t\right)\mathrm{=}f\mathrm{(}t\mathrm{)}.\] We may recognize that this equation models simple harmonic motion (SHM). Let \({\omega }^2_0=k/m\); then, it can be verified by substitution that the general solution to the equation is given as:

image11 \(x\left(t\right)=A{\mathrm{cos} {\omega }_0\ }t+B{\mathrm{sin} {\omega }_0t\ }\).

Figure 9: A mass–spring–damper system.