Section 5.1 LRC Series Circuit
Damped Motion
March 28, 2011
Driven Motion
Resonance
MTH 225 Differential Equations
Notes 5.1 Linear Models: Initial­Value Problems ­ Part III
P. Seeburger
Part III: Spring/Mass Systems: Driven Motion
Suppose an external force f (t) acts on the spring­mass system. For example, the object to which the spring is attached could be made to move up and down in an oscillating motion.
Adding this force into the DE for the motion of the mass on the spring, we obtain the DE of _____________________ or ____________________ motion. That DE is:
Dividing by m, gives us the DE:
Ex. 1: Interpret and solve the IVP:
x(0) = ½, x′(0) = 0.
,
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Section 5.1 LRC Series Circuit
March 28, 2011
Transient & Steady­State Terms
When F is a periodic function, the general solution of the DE for driven motion will be the sum of a
non­periodic function _________ and a periodic function _________.
Note that as t → ∞, xc(t) → ________.
In this case, xc(t) is said to be a _________________________ term, and xp(t) is called a __________________________________ term.
This means that the effect of the initial conditions on a driven spring­mass system is transient.
If there is no damping, there will be no transient term (that is, for undamped forced motion).
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Section 5.1 LRC Series Circuit
March 28, 2011
LRC Series Circuit Analogue
Remember that Kirchoff’s 2nd Law states that the sum of the voltages drops
across the inductor, the resistor, and the capacitor is equal to the impressed
voltage on the circuit, E(t). That is:
But since
i(t) = , we can write this DE as a 2nd­order DE in q:
The terminology used for series circuits is similar to that used to describe spring/mass systems.
If E(t) = 0, the electrical vibrations of the circuit are said to be _________________.
The characteristic equation for this DE is:
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Section 5.1 LRC Series Circuit
The circuit is said to be:
March 28, 2011
Overdamped if ___________________________________.
Critically damped if _______________________________.
Underdamped if __________________________________.
In each case, as t → ∞, q(t) → ________.
In the underdamped case, the charge on the capacitor ____________________as it decays, that is it is charging and discharging as as t → ∞.
If E(t) = 0 and R = 0, the circuit is said to be undamped. The response of the circuit is simple harmonic and the electrical vibrations will never decay to 0.
Ex. 2: Find the charge q(t) on the capacitor in an LRC series circuit when L = 0.25 henry, R = 10 ohms, C = 0.001 farad, E(t) = 0, q(0) = q­0 coulombs, and i(0) = 0 amperes.
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Section 5.1 LRC Series Circuit
March 28, 2011
When there is an impressed voltage E(t) on the circuit, the electrical vibrations are said to be _______________. In the case when R ≠ 0, the complementary function qc(t) is said to be a transient solution and if E(t) is periodic or constant, then the particular solution qp(t) is a ________________
_____________________ solution.
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