Chapter 4 Oscillations 4.1 Harmonic oscillations The general form of a harmonic oscillation is: Ψ ( t )= ˆ Ψ e i ( ω t ± ϕ ) ≡ ˆ Ψ cos( ω t ± ϕ ) , where ˆ Ψ is the amplitude . A superposition of several harmonic oscillations with the same frequency results in another harmonic oscillation: ± i ˆ Ψ i cos( α i ± ω t )= ˆ Φ cos( β ± ω t ) with: tan( β )= ∑ i ˆ Ψ i sin( α i ) ∑ i ˆ Ψ i cos( α i ) and ˆ Φ 2 = ± i ˆ Ψ 2 i +2 ± j>i ± i ˆ Ψ i ˆ Ψ j cos( α i-α j ) For harmonic oscillations holds: ² x ( t ) dt = x ( t ) i ω and d n x ( t ) dt n =( i ω ) n x ( t ) . 4.2 Mechanic oscillations For a construction with a spring with constant C parallel to a damping k which is connected to a mass M , to which a periodic force F ( t )= ˆ F cos( ω t ) is applied holds the equation of motion m ¨ x = F ( t )-k ˙ x-Cx . With complex amplitudes, this becomes-m ω 2 x = F-Cx-ik ω x . With ω 20 = C/m follows: x = F m ( ω 20-ω 2 )+ ik ω , and for the velocity holds:
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.