# 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 $$L\frac ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.04:_Second-Order_ODE_Models), /content/body/p/span/span, line 1, column 1  (t)$$

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

$$L\frac{\rm di(t)}{\rm dt} +Ri(t)+V_{c} (t)=V_ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.04:_Second-Order_ODE_Models), /content/body/p/span/span, line 1, column 1  (t)$$, $$C\frac{dV_c}{dt}=i(t)$$

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.