1. Engineering Mathematics Class #6 Second-Order Linear ODEs (Part2)
Sheng-Fang Huang

2. 2.4 Modeling: Free Oscillations (Mass–Spring System)
If we pull the body down a certain distance and then release it, it starts moving. We assume that it moves strictly vertically. m

3. Physical Information
Let y(t) be the displacement of the body at time t. According to Newton’s second law
Mass × Acceleration = my" = Force
(牛頓(nt)為Force單位, 1 kg = 9.8nt)
where “Force” is the resultant of all the forces acting on the body.
Downward forces: positive.
Upward forces: negative.

4. Physical Information
Let y = 0 in Fig. 32 (b). According to Hooke’s law, the force F0 that cause the stretch s0 in the spring is,
(2) F0 = –ks0
k (> 0) is called the spring constant.
The minus sign indicates that F0 points upward.
Stiff springs have large k.

5. Physical Information
F0 balances the weight W = mg of the body (where g = 980 cm/sec2). Hence,
F0 + W = –ks0 + mg = 0.
These forces will not affect the motion. Spring and body are again at rest. This is called the static equilibrium of the system (Fig. 32b).

6. Physical Information
From the position y = 0 we pull the body downward. This further stretches the spring by some amount y > 0.
By Hooke’s law this causes an (additional) upward force F1 in the spring,
F1 = –ky.
F1 is a restoring force which has the tendency to restore the system (y = 0).

7. Undamped System: ODE and Solution
Every system has damping—otherwise it would keep moving forever.
If the effect of damping is negligible, then F1 is the only force in (1) causing the motion. Hence (1) gives the model:
(3) my" + ky = 0.
We obtain as a general solution
(4) y(t) = A cos ω0 t+B sin ω0 t,
The corresponding motion is called a harmonic oscillation.

8. Harmonic Oscillation
Since the trigonometric functions in (4) have the period 2π/ω0.
ω0/2π cycles per second ( called the frequency of the oscillation)
Another name for cycles/sec is hertz (Hz).
The sum in (4) can be combined into a phase-shifted cosine with amplitude and phase angle δ = arctan (B/A),
(4*) y(t) = C cos (ω0t – δ).

9. Fig. 33. Harmonic oscillations
To find A:
y(0) = A
To find B:
y’(0) = Bω0
B = y’(0) / ω0

10. Example 1 Undamped Motion.
If an iron ball of weight W = 98 nt stretches a spring 1.09 m, how many cycles per minute will this mass–spring system execute? What will its motion be if we pull down the weight an additional 16 cm (about 6 in.) and let it start with zero initial velocity?

11. Solution:

12. Fig. 34. Harmonic oscillation in Example 1

13. Damped System: ODE and Solutions
We now add a damping force
F2 = –cy'
where c > 0 is called damping constant
Physically this can be done by connecting the body to a dashpot (闊口盤). Assume this new force is proportional to the velocity y' = dy/dt.
To our model my" = –ky, so that we have my" = –ky – cy' , or
(5) my" + cy' + ky = 0.

14. Fig Damped system F2 F1
my’’
continued

15. Damped System: ODE and Solutions
The ODE (5) is homogeneous linear and has constant coefficients. The characteristic equation is (divide (5) by m)
The roots of the equation is,
(6)

16. Discussion of the Three Class
It is now most interesting that depending on the amount of damping (much, medium, or little) there will be three types of motion corresponding to the three Cases I, II, II in Sec. 2.2:

17. Case I. Overdamping
If the damping constant c is so large that c2 > 4mk, then λ1 and λ2 are distinct real roots. In this case the corresponding general solution of (5) is
(7) y(t) = c1e-(α-β)t + c2e-(α+β)t.
Damping takes out energy so quickly that the body does not oscillate.
For t > 0 both exponents in (7) are negative because α > 0, β > 0, and β2 = α2 – k/m < α2.

18. Case II. Critical Damping
Critical damping is the border case between nonoscillatory motions (Case I) and oscillations (Case III).
It occurs when c2 = 4mk, so that β = 0, λ1 = λ2 = –α. Then the corresponding general solution of (5) is
(8) y(t) = (c1 + c2t)e-αt.
Because e-αt is never zero, c1 + c2t can have at most one positive zero.

19. Case III. Underdamping
The most interesting case which occurs when c2 < 4mk (c is small). Then in (6) is no longer real but pure imaginary, say,
(9) β = iω*
The roots of the characteristic equation are now complex conjugate,
λ1 = –α+ iω*, λ2 = –α– iω*

20. Damped Oscillations The corresponding general solution is
(10) y(t) = e-αt(A cos ω*t + B sin ω*t)
= Ce-αt cos (ω*t – δ)
where C2 = A2 + B2 and tan δ = B/A, as in (4*).
This represents damped oscillations.
Their curve lies between the dashed curves y = Ce-αt and y = –Ce-αt, touching them when ω*t –δ is an integer multiple of π because these are the points at which cos (ω*t –δ) equals 1 or –1.

21. Damped Oscillations The frequency is ω*/(2π) Hz (hertz, cycles/sec).
From (9) we see that the smaller c (> 0) is, the larger is ω* and the more rapid the oscillations become.
If c approaches 0, then ω* approaches ω0=(k/m)1/2, giving the harmonic oscillation.

22. Fig. 38. Damped oscillation in Case III [see (10)]

23. Example 2 The Three Cases of Damped Motion
How does the motion in Example 1 change if we change the damping constant c to one of the following three values, with y(0) = 0.16 and y'(0) = 0 as before?
(I) c = 100 kg/sec, (II) c = 60 kg/sec, (III) c = 10 kg/sec.

24. Fig. 39. The three solutions in Example 2