The forced spring-mass system 4x'' + 16.8x = cos(2t) exhibits amplitude modulation, or beats. What is the beat frequency?

Answer 1

4r² + 16.8 = 0; r² = - 4.2; r = ±√4.2i; x_gh = Ccos√4.2t + Dsin√4.2t

x_p = Acos2t + Bsin2t; x_p' = - 2Asin2t + 2Bcos2t; x_p'' = - 4Acos2t - 4Bsin2t;

4Acos2t - 4Bsin2t + 16.8Acos2t + 16.8Bsin2t = cos2t; 12.8Acos2t + 12.8Bsin2t = cos2t;

B = 0; A = 1/12.8 = 5/64. So, x_p = 5/64cos2t;

x = 5/64cos2t + Ccos√4.2t + Dsin√4.2t.

Because √4.2 ≈ 2. We have x ≈ (5/64 + C)cos2t Dsin2t and frequency f ≈ 1/π