Download In this section, we will consider several dynamic physical systems in

3.8 Linear Models: IVP
2015年11月15日
下午 12:32
In this section, we will consider several dynamic physical systems in which
the mathematical model is a linear 2nd order ODE with constant
coefficients along with initial conditions specified at time t0.
or we can also write:
*Note: the textbook define
, but this can be mistaken as the
solution to a homogeneous ODE. Therefore, we define
• 3.8.1 Spring/Mass system: Free Undamped Motion
For a spring/mass system shown in Fig. 3.8.1, we can use Newton's
second law to describe the change in the position (x) of spring/mass with
time (t). We define positive direction below the equilibrium position
(Fig. 3.8.2)
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 Example: A mass weighing 2 pounds stretches a spring 6 inches. At
t = 0, the mass is released from a 8 inches below the equilibrium
position with an upward velocity
ft/s. Determine the equation of
motion.
Solution:
Unit convert:
6 in =
m =
Alternative form of x(t)
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ft; 8 in =
=
=
slug
ft,
○ Alternative form of x(t)
We can get the period and frequency of the system from above
equation of motion x(t)=c1cos(wt) + c2sin(wt). However, the amplitude
(maximum upward or downward position) of the motion cannot be
straightforwardly observed. Therefore, we can write the equation of
motion in another form x=Asin(wt+Ф) or x=Acos(wt-Ф). In these
forms, the amplitude A and phase angle Ф are easily obtained.
x=A sin(wt+Ф)
x=A cos(wt-Ф)
Where sinФ=
Where sinФ=
cosФ=
cosФ=
A=
A=
Example: let us rewrite the equation of motion in the last example.
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 Example: let us rewrite the equation of motion in the last example.
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• 3.8.2 Spring/Mass system: Free Damped Motion
The concept of free harmonic motion is somewhat unrealistic since the
motion described in the last session assumes there is no retarding forces
such as friction. Unless the mass is suspended in vacuum, there must be
at least a resisting force due to the surrounding medium (eg. air). As Fig
3.8.5 shows, the mass could be suspended in a viscous medium (a) or
connected to a dashpot damping device (b).
The resisting force imposed by
the damping devices is often
defined as
Where β is the damping constant
The equation of motion with damping becomes:
Similar as we discussed in 3.3, this leads to three possible solutions:
○ Case I: overdamped, λ2-w2 >0
In this case, we have two distinct and real solution, therefore, the motion
of the mass (solution to the ODE) is:
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○ Case II: critically damped, λ2-w2 =0
We have double real roots; therefore,
the solution is:
○ Case III: underdamped, λ2-w2 <0
We have two conjugate complex roots.
The solution is:
 Please see this video to help you understand the difference
between overdamped and critically damped motion.
http://lms.nthu.edu.tw/sys/read_attach.php?id=743052
 Please practice with example 3,4,5 on page 155.
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• 3.8.3 Spring/Mass system: Driven Motion
We now take into consideration an external force g(t) acting on a
vibrating mass on a spring. For example, g(t) could represent a driving
force casing an oscillatory vertical motion of the support of the spring as
shown in Fig 3.8.11. The ODE of the driven or forced motion becomes:
To solve this equation, we can use
either undetermined coefficient (3.4) or
variation of parameters (3.5).
 Example: Solve and interpret the IVP.
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– Transient and Steady-state terms
○ Driven motion without damping
With a periodic impressed force and no damping force, there is not
transient term in the solution. We will also see that the free
undamped vibration can cause a severe problem in a oscillatory
mechanical system.
For this kind of motion, we can write a 2nd order ODE if the
external force is F0sin(γt):
Damped:
Undamped:
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 Please Practice with example
6~8 on page 157~158.
– Pure resonance
The above solution cannot be applied when γ=w. We shall solve the
ODE again using the when γ=w.
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