The Phasor Analysis Method For Harmonically Forced Linear Systems Daniel S. Stutts, Ph.D. April 4, 1999 Revised: 10-15-010, 9-1-011 1 Introduction One of the most common tasks in vibration analysis is the determination of the steady-state response of a system to harmonic forcing. This is also true in AC electrical circuit analysis. The most common case encountered in vibration engineering is that of the forced underdamped system. In this case, the transient portion of the response vanishes exponentially as e ζωt, where ζ is the so-called dimensionless damping constant, and for the underdamped case, 0 < ζ < 1. The prototypical mass-spring-damper system is shown in Figure 1, where in the harmonically forced case, we shall assume that F (t) = sin ωt, where ω is the forcing frequency in radians second. x M f(t) C Figure 1. Prototypical forced mass-spring-damper system In the following, we shall determine the steady-state response to harmonic forcing via two methods: (1) the conventional approach via the method of undetermined coefficients, and () the Phasor method. First, we note that the equation of motion is given by Equation (1) may be recast as Mẍ + Cẋ + x = sin ωt. (1) ẍ + ζω n ẋ + ωnx = sin ωt. M () Equation () is the so-called canonical form of the equation of motion. Assuming the underdamped case (0 < ζ < 1), we may find the homogeneous or transient solution is given by x h (t) = e ζωnt (A cos ω d t + B sin ω d t), (3) where the undamped natural frequency ω n = M, and the damped natural frequency ω d = ω n 1 ζ 1
The Forced Response Via the Method of Undetermined Coefficients The forced or particular solution is given by x p (t) = C 1 cos ωt + C sin ωt. (4) Substitution of Equation (4) into Equation () yields ( ω C 1 cos ωt ω C sin ωt ) + ζω n ω ( C 1 sin ωt + C cos ωt) (5) Collecting the coefficients of sin and cosine in Equation (6) yields +ωn (C 1 cos ωt + C sin ωt) = sin ωt. M ζω n ωc 1 + ( ω n ω ) C = M, (6) and ( ω n ω ) C 1 + ζω n ωc = 0. (7) Dividing Equations (6) and (7) by ωn, and recasting them in matrix form yields [ ( ) ] ( ) ( ) ζr 1 r F0 ( ) C1 = 1 r ζr 0 where r = ω ω n. Solving for C 1 and C yields and Hence, the total solution is given by C ζ r F0 C 1 = (ζr) + (1 r ), (9) C = where A and B depend on the initial conditions, and ( 1 r ) (ζr) + (1 r ). (10) x(t) = R ( ζr cos ωt + ( 1 r ) sin ωt ) +e ζωnt (A cos ω d t + B sin ω d t), (11) R = (ζr) + (1 r ). (1) Clearly, as t, x(t) x p (t), where, x p (t) is the particular, or steady-state solution to Equation (). Thus, at steady-state, x(t) = x p (t), and we have where, and x(t) = (sin φ cos ωt cos φ sin ωt), (13) R ζr R = sin φ, (14) 1 r = cos φ. (15) R From trigonometry 1, we have sin(θ φ) = sin θ cos φ cos θ sin φ. Hence, we may rewrite Equation (13) as x(t) = sin(ωt φ), (16) R 1 See Equation (7) in Some Useful Complex Algebra and Trigonometric Relationships, D. S. Stutts, March 10, 004. (8)
where, Thus, or tan φ = sin φ cos φ = ζr 1 r. (17) ( ) ζr φ = tan 1 1 r ( ) ζr φ = π + tan 1 1 r for r 1, (18) for r > 1. (19) Equation (16) reveals that the steady-state response of the system may be written in terms of a magnitude and a phase shift. x(t) = X sin (ωt φ), (0) where Examination of Equation (0) leads to a much quicker route of analysis. X = R =. (1) (ζr) + (1 r ) From Equations (13) through (17), we may write the displacement X as a complex vector in the following manner X = [cos φ j sin φ]. () R Let us now define a mechanical impedance Z(r)(= Z(ω)) analogous to electrical impedance except that in this case, current and voltage are replaced by force and displacement respectively given by Hence, Z(r) = X = 1 [cos φ j sin φ], (3) R so X = Z(r). (4) From Equations (14) and (15), the impedance may be recast in polar form (magnitude and phase) as Z(r) = X = e jφ = e jφ (ζr) + (1 r ) R. (5) In equation (4), is the magnitude of the harmonic input force, and X provides information about the magnitude as well as the direction of the response. Defining the real and imaginary parts of Z as and R 1 = cos θ R R = sin θ R the mechanical impedance for the system is shown graphically in Figure. (6) (7) If we consider a complex forcing function given by Then, we may compute the response r(t) from Equations (3) and (4) f(t) = e jωt = (cos ωt + j sin ωt), (8) r(t) = Z(r)f(t) = [cos φ j sin φ] (cos ωt + j sin ωt). (9) R Although the the phase is actually negative, it is shown here in the first quadrant for clarity. 3
Im R Z φ R 1 Re Figure. Polar (phasor) representation of the mechanical impedance in the complex plane. Completing the multiplication in Equation (9), and grouping real and imaginary terms yields or r(t) = [cos φ cos ωt + sin φ sin ωt + j (cos φ cos ωt sin φ sin ωt)]. (30) R r(t) = R cos (ωt φ) + j sin (ωt φ). (31) R Defining two complex vector operations Re {r(t)}, and Im {r(t)} to take the real and imaginary parts of r(t) respectively, Equation (31) may be written as r(t) = F [ { 0 Re e (ωt φ)} { + jim e (ωt φ)}]. (3) R Im f(t) R sin t φ ωt r(t) R cos t Re Figure 3. Polar (phasor) representation of the system response to harmonic forcing Hence, we see that the steady-state system response is related to the harmonic forcing as shown in Figure 3. Thus, the steady-state response of a system to a harmonic forcing function, f(t) = sin ωt, or f(t) = cos ωt, may be computed by assuming f(t) = e jωt, and taking the imaginary or real part of the result respectively. It is 4
clear from Figure 3 that the steady-state response of a system to either f(t) = sin ωt, or f(t) = cos ωt may be thought of as a projection of the response due to f(t) = e jωt on the imaginary or real axes in the complex plane. 3 The Phasor Method Given the result shown in Equations (31) and (3), we may approach the solution of Equation () for the steady- -state response in a completely different manner. Letting x(t) = XIm { e jωt}, and substituting into Equation (), yields 3 [ ω n ω + jζω n ω ] XIm { e jωt} = M Im { e jωt} = ωn Im { e jωt}. (33) Canceling the phasors on either side of Equation (33),dividing by ω n, and solving for X, yields X = [1 r + jζr] = e jφ, (34) (1 r ) + (ζr) where,as before, r = ω ω n, and φ is given by Equation (18). Thus, the steady-state solution is given by e jφ x(t) = Im (1 { e jωt } F { 0 = Im e r ) + (ζr) (1 j(ωt φ)}, (35) r ) + (ζr) or, x(t) = sin (ωt φ) = X sin (ωt φ). (36) (1 r ) + (ζr) Examination of Equations (1) and (1), reveals that the result given in Equation (36) is identical to that obtained in Equation (0), but using the phasor method, required far fewer algebraic steps. The approach is identical for f(t) = cos ωt, except that a solution of the form x(t) = Re { e jωt} is assumed instead. Im M ω X c ωx F(t) φ ωt X X Re Figure 4. Force vector relationships for a harmonically forced spring-mass-damper system. The phase relationship between the individual force components of the system as described by Equation (1) are shown graphically in Figure 4, where it may be seen that the damper force and inertial forces are 90 and 180 out of phase with the spring force respectively. 3 The Real and Imaginary components yield two scalar equations. 5
The phasor method is credited to Charles Proteus Steinmetz who was an engineer at General Electric around the turn of the century. With the growing importance of AC circuitry, Steinmetz was compelled to develop a quicker method of analysis because determining the steady-state response of even relatively simple circuits to harmonic voltage excitation is extremely tedious using conventional methods. 6