Theorem 3.2.7 Abel’s Theorem) 4
3.8 Forced Periodic Vibrations
We will now investigate the situation in which a periodic external force is applied to a spring- mass system. The behavior of this simple system models that of many oscillatory systems with an external force due, for example, to a motor attached to the system. We will first consider the case in which damping is present and will look later at the idealized special case in which there is assumed to be no damping.
Forced Vibrations with Damping. The algebraic calculations can be fairly complicated in this kind of problem, so we will begin with a relatively simple example.
EXAMPLE 1
Suppose that the motion of a certain spring-mass system satisfies the differential equation u′′+u′+ 5
4u=3 cost (1)
and the initial conditions
u( 0)=2, u′( 0) =3. (2)
Find the solution of this initial value problem and describe the behavior of the solution for larget.
Solution:
The homogeneous equation corresponding to equation (1) has the characteristic equation r2+r + 5
4 = 0 with rootsr = −1
2 ±i. Thus a general solutionuc(t) of this homogeneous equation is
uc(t) =c1e−t/2cost+c2e−t/2sint. (3) A particular solution of equation (1) has the formU(t) = Acost + Bsint, where A and B are found by substitutingU(t) foru in equation (1). We haveU′(t) = −Asint+ Bcost and U′′(t)= −Acost−Bsint. Thus, from equation (1) we obtain
1 4A+B
cost+
−A+ 1 4B
sint=3 cost.
Consequently,AandBmust satisfy the equations 1
4A+B=3, −A+1 4B=0, with the result thatA= 12
17 andB= 48
17. Therefore, the particular solution is U(t) = 12
17cost+48
17sint, (4)
and the general solution of equation (1) is
u=uc(t)+U(t) =c1e−t/2cost+c2e−t/2sint+12
17cost+ 48
17sint. (5)
The remaining constants c1 and c2 are determined by the initial conditions (2). From equation (5), and its first derivative, we have
u( 0) =c1+12
17 =2, u′( 0)= −1
2c1+c2+48 17 =3, soc1 = 22
17 andc2 = 14
17. Thus we finally arrive at the solution of the given initial value problem (1), (2), namely,
u= 22
17e−t/2cost+14
17e−t/2sint+12
17cost+48
17sint. (6)
The graph of the solution (6) is shown by the green curve in Figure 3.8.1.
▼
▼ It is important to note that the solution consists of two distinct parts. The first two terms on the right-hand side of equation (6) contain the exponential factore−t/2; as a result, they rapidly approach zero. It is customary to call these terms thetransient solution. The remaining terms in equation (6) involve only sines and cosines, so they represent an oscillation that continues indefinitely. We refer to them as thesteady-state solution. The dotted red and dashed blue curves in Figure 3.8.1 show the transient and the steady-state parts of the solution, respectively. The transient part comes from the solution of the homogeneous equation corresponding to equation (1) and is needed to satisfy the initial conditions. The steady-state solution is the particular solution of the full nonhomogeneous equation. After a fairly short time, the transient solution is vanishingly small and the full solution is essentially indistinguishable from the steady state.
u
t 2
1
–1
–2
–3 3
12
8 16
4 Full solution
Steady-state solution
Transient solution
FIGURE 3.8.1 Solution of the initial value problem (1), (2):
u′′+u′+5u/4=3 cost,u( 0)=2, u′( 0) =3. The full solution (solid green) is the sum of the transient solution (dotted red) and steady-state solution (dashed blue).
The equation of motion of a general spring-mass system subject to an external forceF(t) is equation (7) in Section 3.7:
mu′′(t)+γu′(t)+ku(t)=F(t) , (7) wherem,γ, andkare the mass, damping coefficient, and spring constant of the spring-mass system. Suppose now that the external force is given by F0cos(ωt) , where F0 andω are positive constants representing the amplitude and frequency, respectively, of the force. Then equation (7) becomes
mu′′+γu′+ku=F0cos(ωt). (8) Solutions of equation (8) behave very much like the solution in the preceding example.
The general solution of equation (8) must have the form
u =c1u1(t)+c2u2(t)+Acos(ωt)+Bsin(ωt) =uc(t)+U(t). (9) The first two terms on the right-hand side of equation (9) are the general solutionuc(t) of the homogeneous equation corresponding to equation (8), and the latter two terms are a particular solutionU(t) of the full nonhomogeneous equation. The coefficientsAandBcan be found, as usual, by substituting these terms into the differential equation (8), while the arbitrary constants c1andc2are available to satisfy initial conditions, if any are prescribed. The solutionsu1(t)
andu2(t) of the homogeneous equation depend on the rootsr1 andr2 of the characteristic equationmr2+γr+k=0. Sincem,γ, andkare all positive, it follows thatr1andr2either are real and negative or are complex conjugates with a negative real part. In either case, both u1(t) andu2(t) approach zero ast → ∞. Sinceuc(t) dies out astincreases, it is called the transient solution. In many applications, it is of little importance and (depending on the value ofγ) may well be undetectable after only a few seconds.
The remaining terms in equation (9)---namely,U(t) = Acos(ωt)+Bsin(ωt) ---do not die out astincreases but persist indefinitely, or as long as the external force is applied. They represent a steady oscillation with the same frequency as the external force and are called the steady-state solutionor theforced responseof the system. The transient solution enables us to satisfy whatever initial conditions may be imposed. With increasing time, the energy put into the system by the initial displacement and velocity is dissipated through the damping force, and the motion then becomes the response of the system to the external force. Without damping, the effect of the initial conditions would persist for all time.
It is convenient to expressU(t) as a single trigonometric term rather than as a sum of two terms. Recall that we did this for other similar expressions in Section 3.7. Thus we write
U(t) =Rcos(ωt−δ). (10)
The amplitudeRand phaseδdepend directly onAandBand indirectly on the parameters in the differential equation (8). It is possible to show, by straightforward but somewhat lengthy algebraic computations, that
R= F0
∆, cosδ =m(ω20−ω2)
∆ , and sinδ= γ ω
∆ , (11)
where
∆ =
m2(ω02−ω2)2+γ2ω2 and ω20 = k
m. (12)
Recall thatω0is the natural frequency of the unforced system in the absence of damping.
We now investigate how the amplitudeRof the steady-state oscillation depends on the frequencyω of the external force. Substituting from equation (12) into the expression for R in equation (11) and executing some algebraic manipulations, we find that
Rk F0 =
⎛
⎝
1− ω
ω0 22
+Γ ω
ω0 2⎞
⎠
−1/2
whereΓ = γ2
mk. (13)
Observe that the quantityRk/F0is the ratio of the amplitudeRof the forced response toF0/k, the static displacement of the spring produced by a forceF0.
For low frequency excitation---that is, asω → 0---it follows from equation (13) that Rk/F0 → 1 or R → F0/k. At the other extreme, for very high frequency excitation, equation (13) implies thatR → 0 asω → ∞. At an intermediate value ofω the amplitude may have a maximum. To find this maximum point, we can differentiateRwith respect toω and set the result equal to zero. In this way we find that the maximum amplitude occurs when ω =ωmax, where
ω2max=ω02− γ2 2m2 =ω02
1− γ2
2mk
. (14)
Note thatωmax< ω0and thatωmaxis close toω0whenγ is small. The maximum value ofRis
Rmax= F0
γ ω0
1−(γ2/4mk) ∼= F0 γ ω0
1+ γ2
8mk
, (15)
where the last expression is an approximation that is valid whenγis small (see Problem5). If γ2
mk > 2, thenωmaxas given by equation (14) is imaginary; in this case the maximum value of R occurs forω = 0, and R is a monotone decreasing function ofω. Recall that critical damping occurs when γ2
mk =4.
For smallγ it follows from equation (15) that Rmax ∼= F0 γ ω0
. Thus, for lightly damped systems, the amplitude R of the forced response when ω is near ω0 is quite large even for relatively small external forces, and the smaller the value of γ, the more pronounced is this effect. This phenomenon is known asresonance, and it is often an important design consideration. Resonance can be either good or bad, depending on the circumstances. It must be taken very seriously in the design of structures, such as buildings and bridges, where it can produce instabilities that might lead to the catastrophic failure of the structure. On the other hand, resonance can be put to good use in the design of instruments, such as seismographs, that are intended to detect weak periodic incoming signals.
2 4 6 8 10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Γ = 2
Γ = 0.5 Rk/F0
ω ω/ Γ = 0.1
Γ = 0.015625 Γ →0
0 FIGURE 3.8.2 Forced vibration with damping: amplitude of steady-state response versus frequency of driving force for several values of the dimensionless damping parameterΓ=γ2/mk.
Figure 3.8.2 contains some representative graphs of Rk
F0 versus ωω
0 for several values of Γ = γ2
mk. We refer toΓ as a damping parameter, as the following examples will explain.
The graph corresponding toΓ = 0.015625 is included because this is the value ofΓ that occurs in Example 2 below. Note particularly the sharp peak in the curve corresponding to Γ = 0.015625 near ω
ω0 = 1. The limiting case asΓ → 0 is also shown. It follows from equation (13), or from equations (11) and (12), thatR → F0
m
ω02−ω2
asγ →0 and hence Rk
F0 is asymptotic to the vertical lineω =ω0, as shown in the figure. As the damping in the system increases, the peak response gradually diminishes.
Figure 3.8.2 also illustrates the usefulness of dimensionless variables. You can easily verify that each of the quantities Rk
F0, ω
ω0, andΓ is dimensionless (see Problem 9d). The importance of this observation is that the number of significant parameters in the problem has been reduced to three rather than the five that appear in equation (8). Thus only one family of curves, of which a few are shown in Figure 3.8.2, is needed to describe the response-versus- frequency behavior of all systems governed by equation (8).
The phase angleδ also depends in an interesting way onω. Forω near zero, it follows from equations (11) and (12) that cosδ ∼=1 and sinδ ∼=0. Thusδ ∼=0, and the response is nearly in phase with the excitation, meaning that they rise and fall together and, in particular, assume their respective maxima nearly together and their respective minima nearly together.
Forω =ω0we find that cosδ =0 and sinδ =1, soδ =π/2. In this case the response lags behind the excitation byπ/2; that is, the peaks of the response occurπ/2 later than the peaks of the excitation, and similarly for the valleys. Finally, forω very large, we have cosδ ∼= −1 and sinδ∼=0. Thusδ ∼=πso that the response is nearly out of phase with the excitation; this means that the response is minimum when the excitation is maximum, and vice versa. Figure 3.8.3 shows the graphs ofδversusω /ω0for several values ofΓ. For small damping, the phase transition from nearδ =0 to nearδ =π occurs rather abruptly, whereas for larger values of the damping parameter, the transition takes place more gradually.
Γ = 0.1
Γ = 0.5
ω/ δ
ω Γ = 2
Γ π
π 2
= 0.015625
0 FIGURE 3.8.3 Forced vibration with damping: phase of steady-state response versus frequency of driving force for several values of the dimensionless damping parameterΓ=γ2/mk.
EXAMPLE 2
Consider the initial value problem u′′+1
8u′+u=3 cos(ωt) , u( 0)=2, u′( 0) =0. (16) Show plots of the solution for different values of the forcing frequencyω, and compare them with corresponding plots of the forcing function.
Solution:
For this system we haveω0=1 andΓ =1/64=0.015625. Its unforced motion was discussed in Example 3 of Section 3.7, and Figure 3.7.7 shows the graph of the solution of the unforced problem.
Figures 3.8.4, 3.8.5, and 3.8.6 show the solution of the forced problem (16) forω =0.3,ω =1, and ω =2, respectively. The graph of the corresponding forcing function is also shown in each figure.
In this example the static displacement,F0/k, is equal to 3.
Figure 3.8.4 shows the low frequency case,ω /ω0 = 0.3. After the initial transient response is substantially damped out, the remaining steady-state response is essentially in phase with the excitation, and the amplitude of the response is somewhat larger than the static displacement. To be specific,R∼=3.2939 andδ∼=0.041185.
The resonant case,ω /ω0=1, is shown in Figure 3.8.5. Here, the amplitude of the steady-state response is eight times the static displacement, and the figure also shows the predicted phase lag of π/2 relative to the external force.
The case of comparatively high frequency excitation is shown in Figure 3.8.6. Observe that the amplitude of the steady forced response is approximately one-third the static displacement and that the phase difference between the excitation and the response is approximatelyπ. More precisely, we find thatR∼=0.99655 and thatδ ∼=3.0585.
▼
▼
–3 –2 –1 1 2 3
10 20 30 40 50 60 70 80t
u
Solution Forcing function FIGURE 3.8.4 A forced vibration with damping; the solution (solid blue) of equation (16) withω =0.3:u′′+ 18u′+u=3 cos( 0.3t) ,
u( 0) =2, u′( 0) =0. The dashed red curve is the external force:
F(t) =3 cos( 0.3t) .
–20 –10 10 20
t u
Solution Forcing function
10 20 30 40 50 60
FIGURE 3.8.5 A forced vibration with damping; the solution (solid blue) of equation (16) withω =1:u′′+18u′+u=3 cost, u( 0) =2, u′( 0) =0. The dashed red curve is the external force:F(t) =3 cost.
–3 –2 –1 1 2 3
t u
Solution Forcing function
10 20 30 40 50
FIGURE 3.8.6 A forced vibration with damping; the solution (solid blue) of equation (16) withω =2:u′′+18u′+u=3 cos( 2t) , u( 0)=2, u′( 0) =0. The dashed red curve is the external force:F(t) =3 cos( 2t) .
Forced Vibrations Without Damping. We now assume thatγ =0 in equation (8), thereby obtaining the equation of motion of an undamped forced oscillator,
mu′′+ku=F0cos(ωt). (17)
The form of the general solution of equation (17) is different, depending on whether the forcing frequencyω is different from or equal to the natural frequencyω0 =
k/mof the unforced system. First consider the caseω =ω0; then the general solution of equation (17) is
u =c1cos ω0t
+c2sin ω0t
+ F0 m(ω02−ω2)
cos(ωt). (18)
The constantsc1 andc2 are determined by the initial conditions. The resulting motion is, in general, the sum of two periodic motions of different frequencies (ω0 andω) and different amplitudes as well.
It is particularly interesting to suppose that the mass is initially at rest so that the initial conditions areu( 0) =0 andu′( 0) =0. Then the energy driving the system comes entirely from the external force, with no contribution from the initial conditions. In this case it turns out that the constantsc1andc2in equation (18) are given by
c1= − F0
m(ω02−ω2), c2=0, (19)
and the solution of equation (17) is
u= F0
m(ω20−ω2)
cos(ωt)−cos(ω0t)
. (20)
This is the sum of two periodic functions of different periods but the same amplitude. Making use of the trigonometric identities for cos(A±B) withA=1
2(ω0+ω)tandB= 1
2(ω0−ω)t, we can write equation (20) in the form
u= 2F0 m
ω02−ω2 sin
1
2 ω0−ω)t
sin 1
2(ω0+ω)t
. (21)
If |ω0 − ω| is small, then ω0 + ω is much greater than |ω0 − ω|. Consequently, sin
1
2(ω0+ω)t
is a rapidly oscillating function compared to sin 1
2(ω0−ω)t
. Thus the motion is a rapid oscillation with frequency1
2(ω0+ω) but with a slowly varying sinusoidal amplitude
2F0
m
ω02−ω2
sin 1
2 ω0−ω t
.
This type of motion, possessing a periodic variation of amplitude, exhibits what is called a beat. For example, such a phenomenon occurs in acoustics when two tuning forks of nearly equal frequency are excited simultaneously. In this case the periodic variation of amplitude is quite apparent to the unaided ear. In electronics, the variation of the amplitude with time is calledamplitude modulation.
EXAMPLE 3
Solve the initial value problem u′′+u= 1
2cos( 0.8t) , u( 0) =0, u′( 0)=0, (22) and plot the solution.
Solution:
In this caseω0=1,ω =0.8, andF0= 1
2, so from equation (21) the solution of the given problem is
u=2.778 sin( 0.1t) sin( 0.9t). (23)
▼
▼ A graph of this solution is shown in Figure 3.8.7. The amplitude variation has a slow frequency of 0.1 and a corresponding slow period of 2π/0.1=20π. Note that a half-period of 10πcorresponds to a single cycle of increasing and then decreasing amplitude. The displacement of the spring-mass system oscillates with a relatively fast frequency of 0.9, which is only slightly less than the natural frequencyω0.
Now imagine that the forcing frequency ω is increased, say, to ω = 0.9. Then the slow frequency is halved to 0.05, and the corresponding slow half-period is doubled to 20π. The multiplier 2.7778 also increases substantially, to 5.263. However, the fast frequency is only marginally increased, to 0.95. Can you visualize what happens asω takes on values closer and closer to the natural frequencyω0=1?
2
1 3
–1
–3 –2
u = 2.778 sin (0.1t)
u = – 2.77778 sin (0.1t) u = 2.778 sin (0.1t) sin (0.9t) u
10 20 30 40 50 60 t
FIGURE 3.8.7 A beat; the solution (solid blue) of equation (22):u′′+u= 1
2cos( 0.8t) , u( 0)=0,u′( 0)=0 isu=2.778 sin ( 0.1t) sin( 0.9t) . The dashed red curve is the external forceF(t)= 1
2cos( 0.8t) .
Now let us return to equation (17) and consider the case of resonance, whereω =ω0; that is, the frequency of the forcing function is the same as the natural frequency of the system.
Then the nonhomogeneous termF0cos(ωt) is a solution of the homogeneous equation. In this case the solution of equation (17) is
u=c1cosω0t+c2sinω0t+ F0
2mω0tsin(ω0t). (24) Consider the following example.
EXAMPLE 4
Solve the initial value problem u′′+u= 1
2cost, u( 0)=0, u′( 0) =0, (25) and plot the graph of the solution.
Solution:
The general solution of the differential equation is
u=c1cost+c2sint+ t 4sint,
▼
▼ and the initial conditions require thatc1 = c2 = 0. Thus the solution of the given initial value problem is
u= t
4sint. (26)
The graph of the solution is shown in Figure 3.8.8.
10
10 20 30 40
5
–5
–10 u
t sin t
u = –t 4 u = t 4
u = t 4
FIGURE 3.8.8 Resonance; the solution (solid blue) of equation (25):
u′′+u= 1
2cost,u( 0)=0,u′( 0)=0 isu= t 4sint.
Because of the termtsin(ω0t) , the solution (24) predicts that the motion will become unbounded ast → ∞ regardless of the values of c1 and c2, and Figure 3.8.8 bears this out. Of course, in reality, unbounded oscillations do not occur, because the spring cannot stretch infinitely far. Moreover, as soon asu becomes large, the mathematical model on which equation (17) is based is no longer valid, since the assumption that the spring force depends linearly on the displacement requires thatube small. As we have seen, if damping is included in the model, the predicted motion remains bounded; however, the response to the input functionF0cos(ωt) may be quite large if the damping is small andω is close toω0.
Problems
In each of Problems1through 3, write the given expression as a product of two trigonometric functions of different frequencies.
1. sin( 7t)−sin( 6t) 2. cos(πt)+cos( 2πt) 3. sin( 3t)+sin( 4t)
4. A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(t/2) N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass.
5. a. Find the solution of the initial value problem in Problem4.
b. Identify the transient and steady-state parts of the solution.
G c. Plot the graph of the steady-state solution.
N d.If the given external force is replaced by a force of 2 cos(ωt) of frequencyω, find the value ofω for which the amplitude of the forced response is maximum.
N 6. A mass that weighs 8 lb stretches a spring 6 in. The system is acted on by an external force of 8 sin( 8t) lb. If the mass is pulled down 3 in and then released, determine the position of the mass at any time. Determine the first four times at which the velocity of the mass is zero.
7. A spring is stretched 6 in by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of
1
4lb·s/ft and is acted on by an external force of 4 cos( 2t) lb.
a. Determine the steady-state response of this system.
b. If the given mass is replaced by a massm, determine the value ofmfor which the amplitude of the steady-state response is maximum.
8. A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of ( 3 cos( 3t) − 2 sin( 3t) ) N, determine the steady-state response.
Express your answer in the formRcos(ωt−δ) .
9. In this problem we ask you to supply some of the details in the analysis of a forced damped oscillator.
a. Derive equations (10), (11), and (12) for the steady-state solution of equation (8).
b. Derive the expression in equation (13) forRk/F0.
c. Show thatω2maxand Rmaxare given by equations (14) and (15), respectively.
d. Verify that Rk/F0, ω /ω0, and Γ = γ2/(mk) are all dimensionless quantities.
10. Find the velocity of the steady-state response given by equation (10). Then show that the velocity is maximum whenω =ω0. 11. Find the solution of the initial value problem
u′′+u=F(t) , u( 0)=0, u′( 0) =0, where
F(t)=
⎧
⎨
⎩
F0t, 0≤t≤π,
F0( 2π−t) , π <t≤2π,
0, 2π < t.
Hint:Treat each time interval separately, and match the solutions in the different intervals by requiringuandu′to be continuous functions oft.
N 12. A series circuit has a capacitor of 0.25×10−6F, a resistor of 5×103Ω, and an inductor of 1 H. The initial charge on the capacitor is zero. If a 12 V battery is connected to the circuit and the circuit is closed att=0, determine the charge on the capacitor att=0.001 s, att=0.01 s, and at any timet. Also determine the limiting charge as t→ ∞.
N 13. Consider the forced but undamped system described by the initial value problem
u′′+u=3 cos(ωt) , u( 0) =0, u′( 0)=0.
a. Find the solutionu(t) forω =1.
G b.Plot the solutionu(t) versustforω =0.7,ω =0.8, and ω =0.9. Describe how the responseu(t) changes asωvaries in this interval. What happens asωtakes on values closer and closer
to 1? Note that the natural frequency of the unforced system is ω0=1.
14. Consider the vibrating system described by the initial value problem
u′′+u=3 cos(ωt) , u( 0) =1, u′( 0)=1.
a. Find the solution forω =1.
G b.Plot the solutionu(t) versustforω =0.7,ω =0.8, and ω =0.9. Compare the results with those of Problem13; that is, describe the effect of the nonzero initial conditions.
G 15. For the initial value problem in Problem13, plotu′versus uforω = 0.7,ω = 0.8, andω = 0.9. (Recall that such a plot is called a phase plot.) Use at interval that is long enough so that the phase plot appears as a closed curve. Mark your curve with arrows to show the direction in which it is traversed astincreases.
Problems16through18deal with the initial value problem u′′+1
8u′+4u=F(t) , u( 0)=2, u′( 0) =0.
In each of these problems:
G a. Plot the given forcing functionF(t) versust, and also plot the solutionu(t) versuston the same set of axes. Use atinterval that is long enough so the initial transients are substantially eliminated. Observe the relation between the amplitude and phase of the forcing term and the amplitude and phase of the response. Note thatω0=
k/m=2.
G b.Draw the phase plot of the solution; that is, plotu′versus u.
16. F(t) =3 cos(t/4) 17. F(t) =3 cos( 2t) 18. F(t) =3 cos( 6t)
G 19. A spring-mass system with a hardening spring (Problem24 of Section 3.7) is acted on by a periodic external force. In the absence of damping, suppose that the displacement of the mass satisfies the initial value problem
u′′+u+1
5u3=cosωt, u( 0)=0, u′( 0) =0.
a. Letω = 1 and plot a computer-generated solution of the given problem. Does the system exhibit a beat?
b. Plot the solution for several values ofω between 1/2 and 2.
Describe how the solution changes asω increases.
References
Coddington, E. A., An Introduction to Ordinary Differential Equations(Englewood Cliffs, NJ: Prentice-Hall, 1961; New York: Dover, 1989).
There are many books on mechanical vibrations and electric circuits. One that deals with both is
Close, C. M., and Frederick, D. K., Modeling and Analysis of Dynamic Systems(3rd ed.) (New York: Wiley, 2001).
A classic book on mechanical vibrations is
Den Hartog, J. P.,Mechanical Vibrations(4th ed.) (New York:
McGraw-Hill, 1956; New York: Dover, 1985).
An intermediate-level book is
Thomson, W. T.,Theory of Vibrations with Applications(5th ed.) (Englewood Cliffs, NJ: Prentice-Hall, 1997).
An elementary book on electric circuits is
Bobrow, L. S.,Elementary Linear Circuit Analysis(New York:
Oxford University Press, 1996).