DIRSIG Scene Studies

4.1

k m

b Equilibrium y

point

Figure 4.1 Damped mass–spring oscillator

Practically all mechanical systems also experience friction, and for vibrational motion this force is usually modeled accurately by a term proportional to velocity:

(2)

where is the damping coefficientand the negative sign has the same significance as in equation (1).

The other forces on the oscillator are usually regarded as externalto the system. Although they may be gravitational, electrical, or magnetic, commonly the most important external forces are transmitted to the mass by shaking the supports holding the system. For the moment we lump all the external forces into a single,knownfunction Newton’s law then provides the differential equation for the mass–spring oscillator:

or (3)

What do mass–spring motions look like? From our everyday experience with weak auto suspensions, musical gongs, and bowls of jelly, we expect that when there is no friction or external force, the (idealized) motions would be perpetual vibrations like the ones depicted in Figure 4.2. These vibrations resemble sinusoidal functions, with their amplitude depending on the initial displacement and velocity. The frequency of the oscillations increases for stiffer springs but decreases for heavier masses.

In Section 4.3 we will show how to find these solutions. Example 1 demonstrates a quick calculation that confirms our intuitive predictions.

Verify that if b0 and , equation (3) has a solution of the form cos t and that the angular frequency vincreases with kand decreases with m.

Under the conditions stated, equation (3) simplifies to (4)

The second derivative of is and if we insert it into (4), we find

which is indeed zero if This increases with k and decreases with m, as predicted. ◆

v 2k

/

^m.

my–ky mv²cosvtkcosvt , v²cosvt, yAtB

my–ky0 .

yAtB F_extAtB0

Ab0B

my by kyFextAtB . my– kyby¿F_extAtB

F_extAtB. b A0B

F_friction bdy

dt by¿ , 154 Chapter 4 Linear Second-Order Equations

y

t y

t y

t

(a) (b) (c) Figure 4.2 (a) Sinusoidal oscillation, (b) stiffer spring, and (c) heavier mass

Example 1 Solution

When damping is present, the oscillations die out, and the motions resemble Figure 4.3. In Figure 4.3(a) the graph displays a damped oscillation; damping has slowed the frequency, and the amplitude appears to diminish exponentially with time. In Figure 4.3(b) the damping is so dominant that it has prevented the system from oscillating at all. Devices that are supposedto vibrate, like tuning forks or crystal oscillators, behave like Figure 4.3(a), and the damping effect is usually regarded as an undesirable loss mechanism. Good automotive suspension systems, on the other hand, behave like Figure 4.3(b); they exploit damping to suppressthe oscillations.

The procedures for solving (unforced) mass–spring systems with damping are also described in Section 4.3, but as Examples 2 and 3 below show, the calculations are more com-plex. Example 2 has a relatively low damping coefficient and illustrates the solutions for the “underdamped” case in Figure 4.3(a). In Example 3 the damping is more severe

, and the solution is “overdamped” as in Figure 4.3(b).

Verify that the exponentially damped sinusoid given by is a solution to

equa-tion (3) if , and .

The derivatives of yare

and insertion into (3) gives

◆

Verify that the simple exponential function is a solution to equation (3) if , and .

The derivatives of yare and insertion into (3) produces

◆ Now if a mass–spring system is driven by an external force that is sinusoidal at the angular frequency , our experiences indicate that although the initial response of the system may be somewhat erratic, eventually it will respond in “sync” with the driver and oscillate at the same frequency, as illustrated in Figure 4.4 on page 156.

my–by¿ky A1By–10y¿25y25e^5t10A5e^5tB25e^5t0 . y¿AtB 5e^5t, y–AtB25e^5t

b10 F_ext0, m1, k25

yAtBe^5t 0 .

25e^3tcos4t

7e^3tcos4t24e^3tsin4t6A3e^3tcos4t4e^3tsintB my–by¿ky ^A1By–6y¿25y

7e^3tcos4t24e^3tsin4t ,

y–AtB9e^3tcos4t12e^3tsin4t12e^3tsin4t16e^3tcos4t y¿AtB 3e^3tcos4t4e^3tsin4t ,

b6 F_ext0, m1, k25

yAtBe^3tcos4t

Ab10B

Ab6B

Section 4.1 Introduction: The Mass–Spring Oscillator 155

Example 2 Solution

Example 3 Solution

y

t y

t

(a) (b)

Figure 4.3 (a) Low damping and (b) high damping

Common examples of systems vibrating in synchronization with their drivers are sound system speakers, cyclists bicycling over railroad tracks, electronic amplifier circuits, and ocean tides (driven by the periodic pull of the moon). However, there is more to the story than is revealed above. Systems can be enormously sensitive to the particular frequency at which they are driven. Thus, accurately tuned musical notes can shatter fine crystal, wind-induced vibrations at the right (wrong?) frequency can bring down a bridge, and a dripping faucet can cause inordinate headaches. These “resonance” responses (for which the responses have maxi-mum amplitudes) may be quite destructive, and structural engineers have to be very careful to ensure that their products will not resonate with any of the vibrations likely to occur in the operating environment. Radio engineers, on the other hand,dowant their receivers to resonate selectively to the desired broadcasting channel.

The calculation of these forced solutions is the subject of Sections 4.4 and 4.5. The next example illustrates some of the features of synchronous response and resonance.

Find the synchronous response of the mass–spring oscillator with m 1,b1,k25 to the force sin t.

We seek solutions of the differential equation (5)

that are sinusoids in sync with sin t; so let’s try the form Acos tBsin t. Since

we can simply insert these forms into equation (5), collect terms, and match coefficients to obtain a solution:

so

A²25BA B0 .

A ^A225BB1

3²B A25B4 ^sin t 3²A B25A4 ^cos t , 25[A cos tB sin t]

2A cos t ²B sin t3Asin t B cos t4 sinty–y¿25y

y– ²A cos t ²B sin t , y¿ A sin t Bcos t ,

yAtB y–y¿25y sin t

156 Chapter 4 Linear Second-Order Equations

F_ext

t y

t

(a) (b)

Figure 4.4 (a) Driving force and (b) response

Example 4

Solution

Section 4.1 Introduction: The Mass–Spring Oscillator 157

–0.2 –0.15 –0.1 –0.05

0 5 10 15 20 Ω

B

A 0.1

0.15

0.05

Figure 4.5 Vibration amplitudes around resonance

We find

Figure 4.5 displays A and B as functions of the driving frequency . A resonance clearly occurs around . _◆

In most of this chapter, we are going to restrict our attention to differential equations of the form

(6)

where [or , or , etc.] is the unknown function that we seek; a,b, and care constants;

and [or ] is a known function. The proper nomenclature for (6) is the linear, second-order ordinary differential equation with constant coefficients. In Sections 4.7 and 4.8, we will generalize our focus to equations with nonconstant coefficients, as well as to nonlinear equa-tions. However, (6) is an excellent starting point because we are able to obtain explicit solu-tions and observe, in concrete form, the theoretical properties that are predicted for more gen-eral equations. For motivation of the mathematical procedures and theory for solving (6), we will consistently compare it with the mass–spring paradigm:

3^inertia4 y– 3^damping4 y¿ 3^stiffness4 yF_ext . fAxB

fAtyBAtB yAxB xAtB ay–by¿cyfAtB ,

5

A

^{2 A2}25B² , B ²25

^{2 A2}25B² .

1. Verify that for and , equation (3) has a solution of the form

where . 2. If , equation (3) becomes

For this equation, verify the following:

my–by¿ ky0 . F_extAtB0

v 2k

/

^m

yAtB cosvt,

F_extAtB0

b0 (a) If y(t) is a solution, so is cy(t), for any constant c.

(b) If and are solutions, so is their sum .

3. Show that if , and ,

then equation (3) has the “critically damped”

solu-tions and . What is the limit

of these solutions as tS q? y₂AtBte^3t y₁AtB e^3t

b6 F_extAtB0, m1, k9

y₁AtB y₂AtB y₂AtB y₁AtB

4.1 ^EXERCISES

We begin our study of the linear second-order constant-coefficient differential equation (1)

with the special case where the function f(t) is zero:

(2)

This case arises when we consider mass–spring oscillators vibrating freely—that is, without external forces applied. Equation (2) is called the homogeneous form of equation (1); is the

“nonhomogeneity” in (1). (This nomenclature is not related to the way we used the term for first-order equations in Section 2.6.)

A look at equation (2) tells us that a solution of (2) must have the property that its second derivative is expressible as a linear combination of its first and zeroth derivatives.^†This sug-gests that we try to find a solution of the form , since derivatives of are just constants times . If we substitute into (2), we obtain

e^rtAar²brcB0 . ar²e^rtbre^rtce^rt0 ,

ye^rt e^rt

e^rt ye^rt

fAtB ay–by¿cy0 .

ay–by¿cyfAtB Aa0B 4. Verify that ysin 3t2 cos 3tis a solution to the

initial value problem

Find the maximum of for .

5. Verify that the exponentially damped sinusoid is a solution to equation (3) if , and . What is the limit of this solution as ?

6. An external force F(t) 2 cos 2t is applied to a mass–spring system with m1,b0, and k4, which is initially at rest; i.e., . Verify that gives the motion of this spring. What will eventually (as tincreases) happen to the spring?

In Problems 7–9, find a synchronous solution of the form A cos t B sin t to the given forced oscillator equa-tion using the method of Example 4 to solve for A and B.

7.

8.

9.

10. Undamped oscillators that are driven at resonance y–2y¿ 4y6 cos 2t8 sin 2t, 2 y–2y¿ 5y 50 sin 5t, 5 y–2y¿4y5 sin 3t, 3

yAtB ¹₂ t sin 2t

yA0B 0, y¿^A0B 0 tS q

k12 F_extAtB 0, m1, b6

yAtB e^3tsinA23 tB

q 6 t 6 q ƒyAtBƒ

yA0B 2 , y¿^A0B3 . 2y–18y0 ;

158 Chapter 4 Linear Second-Order Equations

have unusual (and nonphysical) solutions.

(a) To investigate this, find the synchronous solu-tion Acos tBsin tto the generic forced oscillator equation

(7)

(b) Sketch graphs of the coefficients A and B, as functions of , for m1,b0.1, and k25.

(c) Now set b0 in your formulas for Aand Band resketch the graphs in part (b), with m1, and k25. What happens at 5? Notice that the amplitudes of the synchronous solutions grow without bound as approaches 5.

(d) Show directly, by substituting the form Acos t Bsin tinto equation (7), that when b0 there are nosynchronous solutions if . (e) Verify that solves equation (7)

when b 0 and . Notice that this nonsynchronous solution grows in time, without bound.

Clearly one cannot neglect damping in analyz-ing an oscillator forced at resonance, because otherwise the solutions, as shown in part (e), are nonphysical. This behavior will be studied later in this chapter.

2k

/

^m

A2m^B1t sin t

2k

/

^m

my–by¿ky cos t .

In document A Systems level characterization and tradespace evaluation of a simulated airborne fourier transform infrared spectrometer for gas detection (Page 187-200)