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Damped harmonic oscillation.
The system looses energy by a drag force F D = – r× u, or a voltage drope on the active resistor uR= R× i, therefore the amplitude has exponential decay on time A (t) =А 0 × е - b × t. The equation of damped oscillations has a mode
where x(t) – physical quantity, which oscillates; А 0 – initial amplitude of oscillations; b – damping coefficient; j0 – initial phase (phase constant). Cyclic frequency of damped oscillations (is written with no index) less then eigenfrequency:
Parameters of linear damping oscillations:
1) Relaxation of vibrations – lessening of amplitude in е =2, 71 times. The time of relaxation: t = 1 / b. (5) 2) As far as amplitude of damped oscillations uninterruptedly decreases A (t) =А0× е - b× t, then the value of oscillating quantity x(t) will never repeat. That’s why, the quantity
is called conventional period – minimal time, during which the value of oscillating quantity x(t) will be equal peak magnitude (amplitude).
3) Decay decrement is a relation of two neibouring amplitude: D = At / At+T = e – b T. (7)
4) Logarithmic decay decrement (damping constant): d = ln D =b T CONV, (8)
5) Quality factor of system Q = w0 / 2b. (9)
In equation of oscillations (2) it is described both mechanical, and electromagnetic oscillations, therefore it is possible to set up correspondence of mechanical and electrical oscillations’ parameters:
EXAMPLE OF PROBLEM SOLUTION Example 2. The oscillating RLC- circuit consists of capacitor, and coil of inductance of 2 mH and resistor. At the initial moment of time charge on the capacitor plates is maximal and equals q 0= Q 0=2m C. Conventional period of oscillations 1 ms, logarithmic decay decrement is 0, 8. 1) To write down the equation of oscillations of charge with numerical coefficients. 2) To define the capacity of capacitor and the resistance of resistor.
Solution: 1) Oscillations in circuit will be damping. Let’s write the equation of damped oscillations of charge in a general view:
where Q 0 – the initial amplitude of charge, β – damping coefficient; ω – cyclic frequency of damped oscillations; j0 – initial phase. From a definition of the conventional period T CONV = 2p / w we express cyclic frequency of damped oscillations: w = 2p / T CONV. (2.2) From a definition of initial value of oscillating quantity q 0= Q × cos(j0) and considering that the oscillations beginning from the position of maximal charge on the capacitor Q, we find the initial phase of oscillations: j0= arc cos(q 0 / Q)= arc cos(1)=0. (2.3) From a definition of the logarithmic decay decrement is d=b T CONV, whence the coefficient of damping b=d / T CONV. (2.4) Let’s check, whether the right part of the formula (2.2) gives the unit of cyclic frequency [ rad/s ], and the left part of the formula (2.4) – measurement unit of damping coefficient [1 /s ]: [w]= rad / s; [b]= 1 / s. Let’s substitute the numerical values in the formulas (2.2) and (2.4)
Let’ write down the equation of oscillation of charge with numerical coefficients
2) Let’s substitute the equation of eigenfrequency
and find expression of capacity of capacitor:
From a definition of the damping coefficient b= R/ 2 L we obtain resistance of resistor R: R = 2β L. (2.8) Let’s check, whether the right part of the formula (2.7) gives the unit of electrocapacity [ F ], and the left part of formula (2.8) – the unit of resistance [ Ω ]:
Let’s make the calculations:
Results: 1) 2) C= 1, 25× 10-5 F , R = 3, 2 Ω.
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