General Solution To Second Order Homogeneous LTI System | Theory of Machines (TOM) - Mechanical Engineering PDF Download

General Solution to Second-Order Homogeneous LTI System

We now focus on the zero input response of the second-order LTI system of Eq. (1–17), i.e., we focus on the system

Suppose that we guess the solution to Eq. (1–26) as
 

x_h(t) = e^λt                                                                                                           (1–27)

where λ is constant that has yet to be determined. Differentiating the assumed solution of Eq. (1–27) twice, we have



Substituting the results of Eqs. (1–28) and (1–29) into (1–26), we obtain


Then, because e^λt  is not zero as a function of time, it can be dropped from Eq. (1–30) to give

Equation (1–31) is called the characteristic equation whose roots give the behavior of the zero input response of Eq. (1–17). Using the quadratic formula, the roots of Eq. (1–31) are given as

It can be seen that the types of roots admitted by Eq. (1–31) depend upon the value of ζ. In particular, the types of roots are governed by the quantity ζ² − 1. We have three cases to consider: (1) 0 ≤ ζ < 1, (2) ζ = 1, and (3) ζ > 1. We now consider each of these cases in turn.

Case 1: 0 ≤ ζ < 1 (Underdamping)

When 0 ≤ ζ < 1 the zero input response is said to be underdamped. For an underdamped system the quantity ζ² − 1 < 0 which implies that  The roots of the characteristic equation for an underdamped system are then given as

It is seen from Eq. (1–33) that the roots of the characteristic equation for an underdamped system are complex. Furthermore, the general zero input response for an underdamped system is given as

Eq. (1–34) can be written as
 

x_h(t) = e ^−ζωnt (c₁ cos ω_dt + c₂ sinω_dt)                                                                  (1–35) 

where the quantity ω_d = ω_n   is called the damped natural frequency of the system. The constants c₁ and c₂ can be solved for by using the initial conditions  as follows. First, substituting the initial condition x(0) = x₀ into Eq. (1–35), we obtain c₁ as

x_h(0) = x₀ = c1                                                                                                     (1–36)
 

Next, differentiating xh(t) in Eq. (1–35), we obtain

Applying the initial condition  we obtain

                                                                         (1–38)

Substituting the result for c₁ from Eq. (1–36) into Eq. (1–38), we obtain

                                                                                       (1–39)

Solving for c₂ we have

The zero input response for an underdamped system is then given as


A schematic of the underdamped zero input response for various values of 0 ≤ ζ < is shown in Fig. 1–2.

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 zero input response is said to be critically damped. For critically damped system the quantity ζ² − 1 = 0 which implies that  = 0. The roots of the characteristic equation for an underdamped system are then given as

λ_1,2 = −ζω_n = −ω_n                                                                        (1–42)

It is seen from Eq. (1–42) that the roots of the characteristic equation for a critically damped system are real and repeated (i.e., the two roots are the same). Furthermore, the general zero input response for a critically damped system is given as

x_h(t) = e ^−ωnt (c₁ + c₂t)                                                                (1–43)

The constants c₁ and c₂ can be solved for by using the initial conditions  as follows. First, applying the initial condition x(0) = x₀ into Eq. (1–43), we have

x_h(0) = x₀ = c₁                                                                               (1–44)

Next, differentiating Eq. (1–43), we obtain

                                                (1–45)

Applying the initial condition  we obtain

                                                                (1–46)

Substituting the result for c₁ from Eq. (1–44), we have

                                                                   (1–47)
 

Solving Eq. (1–47) for c₂ gives

                                                                          (1–48)

The zero input response for an critically damped system is then given as
 

                                          (1–49)
 

A schematic of a critically damped zero input response is shown in Fig. 1–3.

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 zero input response is said to be overdamped. For an overdamped system the quantity ζ² −1 > 0 which implies that   The roots of the characteristic equation for an underdamped system are then given as

It is seen from Eq. (1–50) that the roots of an overdamped system are real and distinct. Furthermore, the general zero input response for an overdamped system is given as

x_h(t) = c₁e^λ1t + c₂e^λ2t                                                                                          (1–51) 

The constants c₁ and c₂ can be solved for by using the initial conditions  as follows. First, applying the initial condition x(0) = x0, we obtain

x_h(0) = x₀ = c₁ + c₂                                                                                             (1–52)

Next, differentiating Eq. (1–51) gives

                                                                        (1–53)

Then, applying the initial condition x( ˙ 0) = x˙0, we obtain

                                                                               (1–54)

Equations (1–52) and (1–54) can then be solved simultaneously for c1 and c2 to give

The general zero input response for an overdamped system is then given as



A schematic of an overdamped zero input response for various values of ζ > 1 is shown in Fig. 1–4.


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