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Unformatted text preview: 2 MA 36600 LECTURE NOTES: FRIDAY, MARCH 6 for some constants R, ω , and δ . To see why, recall the Angle Difference Formula for Cosine: We have the equation cos (α − β ) = cos α cos β + sin α sin β . A eλt cos µt + B eλt sin µt = R eλt cos (ω t − δ ) = R eλt cos ω t cos δ + R eλt sin ω t sin δ. We choose these constants such that ω = µ, R cos δ = A, R sin δ = B. Upon squaring and adding the last two equations, we have ￿ ￿ |γ 2 − 4 m k | B ω= , R = A2 + B 2 , δ = arctan . 2m A We call ω the frequency of the oscillation, R the amplitude of the oscillation, and δ the phase of the oscillation. We also call T = 2 π /ω the period of the oscillation. Mechanical Vibrations. Say that we have a mass m attached to a spring with constant k . Also say that we have a resistive force with damping coefficient γ and an external force given by F (t) at time t. If u = u(t) denotes the displacement of the mass from equilibrium at time t, then m u￿￿ + γ u￿ + k u = F (t). Recall that the general solution is in the form u(t) = uc (t) + U (t) where U = U (t) is a particular solution to the nonhomogeneous equation, and uc = uc (t) is a solution to the homogeneous equation m u￿￿ + γ u￿ + k uc = 0. c c If γ 2 − 4 m k > 0 then we expect the solutions uc (t) to have exponential decay; this is known as overdamping. If γ 2 − 4 m k < 0 then we expect the solutions uc (t) to have oscillations; this is known as underdamping. The “dividing line” between these two cases is when γ 2 − 4 m k = 0; in this case, the system is said to be critically damped. This gives a criteria on the damping coefficient γ to decide when the solutions will be periodic. Undamped Vibrations. Say that γ = 0 i.e., the system is undamped. We wish to consider the differential equation F (t) k m u￿￿ + k u = F (t) =⇒ u￿￿ + u= . m m Since both m and k are positive constants, we denote ￿ k ω0 = m as the natural frequency of the system. All solutions can be found by studying the homogeneous equation 2 u￿￿ + ω0 uc = 0. c 2 If we guess a solution in the form uc (t) = ert , then we find the characteristic equation r2 + ω0 = 0. This has the complex roots r = ±ω0 i, so that the general solution to this homogeneous equation is uc (t) = A cos ω0 t + B sin ω0 t. Recall that we may write this solution in the form ￿ B . A We call R the amplitude of the oscillation, and δ the phase of the oscillation. We also call T = 2 π /ω0 the period of the oscillation. The solution uc (t) is called the transient solution of the nonhomogeneous equation uc (t) = R cos (ω0 t − δ ) where R= A2 + B 2 , m u￿￿ + k u = F (t). In contrast, the particular solution U (t) is called the steady-state solution. δ = arctan ...
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