General Solution To Second Order Homogeneous LTI System

General Solution to Second-Order Homogeneous LTI System

We consider the zero-input response of a standard second-order linear time-invariant (LTI) system. The homogeneous second-order differential equation in standard form is given by the following relation:

To find the general homogeneous solution, assume a trial solution of exponential form.

x_h(t) = e^λt

Differentiating the assumed solution with respect to time gives the first and second derivatives required to substitute into the differential equation.

Substituting these derivatives into the homogeneous differential equation yields an algebraic equation in λ (because e^λt is never zero for all t, it can be divided out).

The resulting algebraic equation is the characteristic equation. Its roots determine the behaviour of the zero-input response.

The roots of the characteristic equation, obtained from the quadratic formula, are

The nature of these roots depends on the damping ratio ζ. The sign of ζ² - 1 leads to three distinct cases: (1) 0 ≤ ζ < 1 (underdamped), (2) ζ = 1 (critically damped), and (3) ζ > 1 (overdamped). Each case is treated below.

Case 1: 0 ≤ ζ < 1 - Underdamping

When 0 ≤ ζ < 1 the system is underdamped and ζ² - 1 < 0, so the characteristic roots are complex conjugates.

The complex conjugate roots can be written in the form -ζω_n ± jω_d, where ω_d is the damped natural frequency.

The general form of the real zero-input response corresponding to the complex roots is

x_h(t) = e^-ζω_nt(c₁ cos ω_dt + c₂ sin ω_dt)

The damped natural frequency ω_d is given by

To determine the constants c₁ and c₂, use the initial conditions x(0) = x₀ and ẋ(0) = ẋ₀.

Apply x(0) = x₀ to the general solution.

x_h(0) = x₀ = c₁

Differentiate the general solution to obtain ẋ_h(t).

Apply the initial velocity ẋ(0) = ẋ₀.

Substituting c₁ = x₀ into the expression for ẋ(0) gives an equation for c₂.

Solving for c₂ yields

Thus the zero-input response for the underdamped case can be written as

The response is an exponentially decaying sinusoid with envelope determined by e^-ζω_nt. A schematic illustrating typical underdamped responses for different ζ in 0 ≤ ζ < 1 is shown below.

Figure 1-2 Schematic of the zero input response of an underdamped second-order linear time-invariant system.

Case 2: ζ = 1 - Critical Damping

When ζ = 1 the system is critically damped. In this case ζ² - 1 = 0, so the two roots coincide and are real and equal.

The repeated root is

λ₁ = λ₂ = -ζω_n = -ω_n

For a repeated real root, the general homogeneous solution is

x_h(t) = e^-ω_nt(c₁ + c₂ t)

Use the initial conditions x(0) = x₀ and ẋ(0) = ẋ₀ to find c₁ and c₂.

Apply x(0) = x₀ to the general solution.

x_h(0) = x₀ = c₁

Differentiate x_h(t) to get ẋ_h(t).

Apply ẋ(0) = ẋ₀.

Substituting c₁ = x₀ into the derivative equation produces a linear equation for c₂.

Solving that equation for c₂ gives

Therefore the critically damped zero-input response becomes

A schematic of the critically damped response is shown below.

Figure 1-3 Schematic of the zero input response of a critically damped second-order linear time-invariant system.

Case 3: ζ > 1 - Overdamping

When ζ > 1 the system is overdamped. In this case ζ² - 1 > 0 and the characteristic equation has two distinct real roots.

Let these distinct real roots be λ₁ and λ₂. The general homogeneous solution is

x_h(t) = c₁ e^{λ₁ t} + c₂ e^{λ₂ t}

Determine c₁ and c₂ from the initial conditions x(0) = x₀ and ẋ(0) = ẋ₀.

Apply x(0) = x₀ to the general solution.

x_h(0) = x₀ = c₁ + c₂

Differentiate to obtain ẋ_h(t).

Apply the initial velocity condition ẋ(0) = ẋ₀.

Solve the two linear simultaneous equations for c₁ and c₂.

Thus the overdamped zero-input response can be expressed explicitly as

A schematic showing overdamped responses for various ζ > 1 is provided below.

Figure 1-4 Schematic of the zero input response of an overdamped second-order linear time-invariant system.

Remarks and applications

The three canonical forms - underdamped, critically damped, and overdamped - exhaust the possible zero-input responses of a second-order homogeneous LTI system. The damping ratio ζ and natural frequency ω_n are key design parameters in mechanical and aerospace applications: they determine transient settling time, overshoot, and whether oscillations occur. Typical uses include vibration analysis of single-degree-of-freedom systems, suspension design, and transient analysis of control systems.