Even though the scope of the treatment is restricted to second-order linear differential equation, we start off by presenting an n-th order linear differential equation.
In a linear differential equation, all the coefficients, a0 to an-1 and b0 to bm, are constants. Equation (1) presents an n-th order linear differential equation. We consider systems, where n is greater than or equal to m. If m > n, the system performance contains an undesirable proportion of noise, since the system acts as a differentiator, and we have the constraint on the value of m.
We can use the operational notation,as shown below. The D-operator is defined by equation (2).
Using the D-operator, the derivatives can be replaced by D raised to the appropriate power, as shown by equation (3). We can define polynomials Q(D) and P(D), as shown by equations (4) and (5). Then we can express the differential equation defined by equation (1) in a short form, as shown by equation (6).
The solution to a differential equation is the sum of zero-input response and the zero-state solution.
In this page, the scope is restricted to getting the zero-input response. The zero-input response is the response of the system due to internal initial conditions and no external input. As the name suggests, there is zero-input to the system, when the zero-input response is obtained. The steps involved in getting the zero-input response are as follows.
The first step is to get the homogeneous differential equation. It is the differential equation of the system when the external input is zero. Let external input f(t) defined in equations (1) and (6) be zero. Then we get the homogeneous differential equation.
Equation (7) is the homogeneous differential equation. Equation (8) is the same equation, wherein the polynomial Q(D) replaces the derivatives.
It is possible to obtain the solution of equation (7) using heuristic reasoning. However, we do not try to get the solution of n-th order equation first. We will get the solution to first-order and the second-order systems first, before we take up the n-th order system.
The homogeneous equation reflects the behaviour of the system with no external signal applied. Whatever response is obtained, it is due to the initial energy stored in the system. First, it is shown how the homogeneous equation is obtained. Let the response y(t) of a first-order system due to its initial state be expressed as:
Equation (9) is the differential equation of the first-order system, with input f(t). Let the system have zero input, and then we get equation (10), which is the homogeneous equation. As stated by equation, let a0 be replaced by k. Then we equation (12). Given the homogeneous equation of the first-order system, we get equation (13) from equation (12). We can transpose dt and y, and then obtain equation (14) from equation (13). Integrate equation (14) with respect to time. Then c on the right-hand side of equation (15) represents the constant of integration. Taking the exponent of both sides, we get equation (16) wherein exp(c) has been set equal to the value of y(t) at t = 0. Equation (16) expresses the zero-input response of the first-order system.
The technique outlined for getting the zero-input response works out well for first-order system, but it is difficult to extend it to higher order systems. The technique is to assume a solution. Equation (12) states that a linear combination of y(t) and its derivative is zero, for all values of t. This means that both y(t) and its derivative should have the same form. We know that an exponential function such as exp(l t) has this property. Hence the complementary solution is assumed to be an exponential function, as shown below.
Equation (17) expresses the assumed complementary solution. Get the derivative of the assumed complementary solution, and we can substitute yC(t) and its derivative in terms of the assumed complementary solution. The resultant equation is expressed by equation (18). The validity for assuming the complementary solution to be an exponential function can be seen from equations (18) and (19). Since both yC(t) and its derivative are exponential functions of the same form, the exponential function, common to both yC(t) and its derivative, can be expressed outside the brackets as the common multiplying factor. Since the exponential function is not equal to zero, we can state that the sum of l and k is zero. In other words, l equals - k, as shown by equation (20). Equation (20) expresses the complementary solution. The value of A is the initial value of yC(t), as expressed by equation (21).
Equation (19) is known as the characteristic equation or the auxiliary equation. We also call l as the characteristic root. The other terms used are the characteristic value, the eigenvalue and the natural frequency. The complementary solution expresses the characteristic mode or the natural mode of response of a first-order system.
It is possible to obtain the same complementary solution using an integrating factor. Technically, it is the same as assuming an exponential function for the complementary solution. Only then both yC(t) and its derivative have the same form, and only then the use of a common integrating factor is valid.
We can multiply the homogeneous differential equation by the integrating factor, as shown by equation (22). Then we can get equation (23), by observing that we get equation (22) from equation (23) by differentiating equation (22) by parts. From equation (23), we get equations (24) and (25). The concept of integrating factor is useful, when the Heaviside operator is described.
A numerical example is presented above. Equation (26) presents the homogeneous differential equation. Equations (27), (28), (29) and (30) outline the steps involved in getting the zero-input response expressed by equation (31).
In the case of a second-order differential equation, the nature of roots of the auxiliary equation can be different depending on the coefficients of the homogeneous differential equation.
We consider each of these cases separately.
The homogeneous differential equation of a second-order system is presented by equation (32). As before, the complementary solution is assumed to be an exponential function, as shown by equation (33). Replace y(t) and its derivatives in equation (32) by the assumed complementary solution and its derivatives, as shown by equation (34). Since the exponential function is not zero, we get that the characteristic or the auxiliary equation expressed by equation (35) is zero.
The condition for existence of distinct roots is specified by equation (36). When the roots of the second-order characteristic equation are distinct, the system is said to be over-damped.
Given the auxiliary equation, re-stated by equation (37) and the condition for existence of distinct roots, we can express the characteristic equation by equation (38). It is seen that the roots are at - k and - m, and they can be expressed in terms of c1 and c2.
Equation (39) shows the first step in getting the roots of the over-damped system. Given that the system is over-damped, we get equation (40) from equation (39) by using equation (41). The values of k and m are stated by equation (42).
Once the values of the two roots are known, we can express the complementary solution, as shown by equation (43). The values of A and B are determined from the two initial conditions. A second-order system has two initial conditions.
It is possible to express c1 and c2 in terms of k and m, as shown by equations (44), (45) and (46).
We can verify that the complementary solution expressed by equation (43) is correct as follows. Let y1 and y2 be defined as expressed by equation (47). Get the first and the second derivative of y1. Replace y and its derivatives by y1 and its derivatives, as shown by equation (49). It is seen that the homogeneous differential equation is satisfied. Repeat the exercise with y2 and find that the homogeneous differential equation is satisfied, as shown by equation (50).
Now a numerical example is presented. A second-order homogeneous equation with initial conditions is presented by equation (51). Equation (52) presents the auxiliary equation. It is expressed as the product of two terms in equation (53). The values of roots are expressed by equation (54).
The complementary solution is expressed by equation (43). The values of A and B are determined from the two initial conditions, as shown by equations (52) and (53). The values of A and B can then be expressed in terms of the initial conditions, as shown by equation (53).
A numerical example is presented above. Equation (54a) presents the homogeneous differential equation. The auxiliary equation is expressed by equation (54b). We can form equations (54c) and (54d) and then determine that the roots are at - 1 and -3.
The complementary solution is expressed by equation (55). The values of A and B are determined from the two initial conditions, as shown by equations (56) and (57). The values of A and B can then be expressed in terms of the initial conditions, as shown by equation (58). The zero-input response is expressed by equation (59). The plot of the response is shown below.
In the case of a critically-damped second order system, both the roots are coincident. In other words, the roots are repeated at the same location. We can state that the multiplicity of roots is two. The second-order homogeneous differential equation is presented by equation (60). This system is critically-damped when c1 and c2 are related to each other, as shown.
Equation (61) presents the auxiliary equation. When the system is critically damped, we get equation (62), which states that both the roots are at the same location.
The auxiliary equation can be presented, as shown by equation (63). It is seen that c1= 2k. Then the homogeneous differential equation can be presented by equation (64). The assumed complementary solution is presented by equation (65). It is seen that the exponential function in the complementary solution is multiplied by a first-order polynomial in t. In general, if the multiplicity of roots is n, then the exponential function in the complementary solution is multiplied by a polynomial in t of order (n- 1).
The assumed complementary function can be split into two functions, as shown by equation (66). Obtain the derivatives of y1(t), as shown by equation (67). It is seen from equation (68) that y1(t) satisfies the homogeneous differential equation and hence y1(t) is a valid solution for the differential equation. It is seen from equation (69) that y2(t) also satisfies the homogeneous differential equation
The values of A and B are obtained from the initial conditions, as shown by equations (70), (71) and (72). The zero-input response is expressed by equation (73).
A numerical example is presented now. Equation (74) specifies the homogeneous differential equation of a critically-damped system along with the initial conditions. Equation (75) presents the auxiliary equation, and equations (76) presents the complementary solution.
Equations (77), (78) and (79) show how the values of A and B can be obtained from the initial conditions. The zero-input response is expressed by equation (80). The plot of the response is shown below. It can be noticed that the response looks similar to that of an over-damped system. It is difficult to distinguish between the plots of the over-damped system and the critically-damped system.
When the condition in equation (81) is satisfied, the system is under-damped. A second-order under-damped system can have a pair of complex conjugate roots.,
Equation (82) expresses the characteristic equation of the under-damped system. It can be manipulated using expressions in equation (83) and we get equations (84) and (85). It can be seen that the roots form a complex conjugate pair.
Based on the location of roots, the complementary solution is obtained, as shown by equation (86). Using Euler's identity, we get equation (87). Equation (88) is the same as the equation (87), excepting that the coefficients are named as A and B.
Equation (89) expresses the complementary solution, as the sum of two functions. Equation (90) specifies these two functions. Each of these functions should satisfy the homogeneous differential equation for the under-damped system, stated as equation (94). The derivatives of function y1(t) can be obtained, as shown by equations (91) and (92). Replace y(t) and its derivatives in equation (94) by y1(t) and its derivatives, and it is verified that equation (93) is valid. Similar exercise can be carried with function y2(t) and it can be verified that it satisfies the differential equation. Hence we can conclude that the assumed functions stated by equation (90) form the complementary solution of the homogeneous differential equation.
Equation (94) is the homogeneous differential equation of the under-damped system. This equation uses symbols for various parameters. We call k as the neper frequency, and it reflects the real part of the complex frequency associated with the under-damped system. The unit of neper frequency is second. Then wn is the undamped natural frequency and its unit is radians per second. We call wd is the damped natural frequency and its unit is radians per second. The neper frequency k can be expressed as the product of the damping coefficient x and undamped natural frequency wn. The relationship between wn and wd is expressed by equation (96). As the damping of the system reduces, wd gets closer to wn.
Equation (97) expresses the assumed complementary solution for the under-damped system. The values A and B can be obtained from the initial conditions, as shown by equation (98). Equation (99) presents the zero-input response of the under-damped second-order system. It is seen that the neper frequency dictates how fast the the zero-input response decays. The higher the value of k is, the faster the decay is.
A numerical example is presented now. Equation (100) presents the homogeneous differential equation of a system. Equation (101) presents the characteristic equation. Equation (102) presents the assumed complementary solution. From the initial conditions, we get the values of A and B. Equation (102) presents the zero-input response of the under-damped second-order system. The plot of the response is shown below. It can be seen that the response is oscillatory. The frequency of oscillation is , wd which is equal to 10 radians per second. The envelope of response decays because of the neper frequency, which is equal to 1 second.
When the second-order system has no damping, its homogeneous differential equation is of the form, shown by equation (103). The auxiliary equation is expressed by equation (104). It is seen that the roots are a conjugate pair, with imaginary values. The assumed complementary solution is expressed by equation (105). It can be derived from the assumed complementary solution of the under-damped system, as shown by equation (97). Just set the neper frequency to be zero. In this case, the frequency of oscillation is its undamped frequency itself.
Equations (106), (107) and (108) are presented to prove that the assumed complementary solution satisfies the given differential equation, and thus we can state that the assumed complementary solution is the valid, correct solution.
The values A and B can be obtained from the initial conditions, as shown by equation (109). Equation (110) presents the zero-input response of the undamped second-order system.
A numerical example is presented now. Equation (111) presents the homogeneous differential equation of a system. Equation (112) presents the assumed complementary solution. From the initial conditions, we get the values of A and B, as shown by equation (113). Equation (114) presents the zero-input response of the undamped second-order system. The plot of the response is shown below. It can be seen that the response is oscillatory. The frequency of oscillation is equal to 10 radians per second. The envelope of response does not decay because the system has no damping.
This page has presented how the zero-input response of the first-order and second-order systems can be obtained. The next page presents briefly the signals used for excitation. The subsequent page how the particular solution is obtained.