ADS Homework 4 - Homework 4: = c/ccr > 1 Over...

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Unformatted text preview: Homework 4: = c/ccr > 1 Over Critically Damped System Determine the constants (G1 and G2) of the solution v(t). Solution: s 2 + 2s + 2 = 0 s1, 2 = - 2 (2) 2 - 4 2 = - () 2 - 2 = - 2 - 1 2 = 2 - 1 s1, 2 = - : Damping coefficient : Circular frequency of undamped system : Circular frequency of over critically damped system The solution of the differential equation is; v (t ) = G1e s1t + G2 e s2t v(t ) = G1e ( -+ )t + G2 e ( -- ) t = e -t (G1et + G2 e -t ) The initial conditions are; v (t = 0) = v(0) ; v(t = 0) = v(0) v (t = 0) = v(0) = G1 + G2 v (t ) = -e -t (G1et + G2 e -t ) + e -t (G1et - G2 e -t ) (1) v (t = 0) = -(G1 + G2 ) + (G1 - G2 ) v(0) = -v (0) + (G1 - G2 ) G1 - G2 = v(0) + v (0) (2) By using Equations 1 and 2, G1 and G2 can be obtained. G1 = G2 = v(0) + v(0) + v(0) v(0)( + ) + v(0) G1 = 2 v(0) - v(0) - v (0) v(0)( - ) - v(0) G2 = 2 The solution is; v(t ) = G1e ( -+ ) t + G 2 e ( -- ) t = v(0)( + ) + v (0) ( -+ ) t v(0)( - ) - v (0) ( -- )t e + e 2 2 1 v(0)( + ) + v(0) t v(0)( - ) - v(0) -t v(t ) = =e -t (G1et + G2 e -t ) = e -t e + e 2 2 The grafic expression for the free vibration cases are given in Figure 1. Free Vibration Cases 1.50 1.00 0.50 v(t) 0.00 -1 -0.50 -1.00 -1.50 t(s) 1 3 5 7 Undamped v(t) Critically Damped v(t) Under Critically Damped v(t) Over Critically Damped v(t) Figure 1. Free Vibration Cases 2 ...
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ADS Homework 4 - Homework 4: = c/ccr > 1 Over...

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