General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

xh(t) = eλt                                                                                                           (1–27)


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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev
Then, because eλt  is not zero as a function of time, it can be dropped from Eq. (1–30) to give

 

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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 General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev The roots of the characteristic equation for an underdamped system are then given as

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

 

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

Eq. (1–34) can be written as
 

xh(t) = e −ζωnt (c1 cos ωdt + c2 sinωdt)                                                                  (1–35) 

 

where the quantity ωd = ω General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev is called the damped natural frequency of the system. The constants c1 and c2 can be solved for by using the initial conditions General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev as follows. First, substituting the initial condition x(0) = x0 into Eq. (1–35), we obtain c1 as

xh(0) = x= c1                                                                                                     (1–36)
 

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

Applying the initial condition General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev we obtain

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                         (1–38)

 

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                                       (1–39)

 

Solving for c2 we have

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev
A schematic of the underdamped zero input response for various values of 0 ≤ ζ < is shown in Fig. 1–2.

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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 General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev = 0. The roots of the characteristic equation for an underdamped system are then given as

λ1,2 = −ζωn = −ωn                                                                        (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

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

The constants c1 and c2 can be solved for by using the initial conditions General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev as follows. First, applying the initial condition x(0) = x0 into Eq. (1–43), we have

xh(0) = x0 = c                                                                              (1–44)

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                (1–45)

Applying the initial condition General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev we obtain

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                (1–46)

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                   (1–47)
 

Solving Eq. (1–47) for c2 gives

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                          (1–48)

 

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                          (1–49)
 

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

 

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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 General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev  The roots of the characteristic equation for an underdamped system are then given as

 

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

xh(t) = c1eλ1t + c2eλ2t                                                                                          (1–51) 

The constants c1 and c2 can be solved for by using the initial conditions General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev as follows. First, applying the initial condition x(0) = x0, we obtain

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

Next, differentiating Eq. (1–51) gives

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                        (1–53)

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev                                                                               (1–54)

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

 

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev

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

General Solution To Second Order Homogeneous LTI System Mechanical Engineering Notes | EduRev
Figure 1–4 Schematic of the zero input response of an overdamped second-order linear time-invariant system.

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