Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 39

The motion is seen to repeat itself every time the vector OP rotates once (in Figure 2.1) or after time T seconds (in Figures 2.2 and 2.3). When the motion has repeated itself, the displacement y is said to have gone through one cycle. The number of cycles that occur per second is called the frequency f. Frequency may be expressed in cycles per second or, equivalently in hertz, or as abbreviated, Hz. The use of hertz or Hz is preferable because this has become internationally agreed upon as the unit of frequency. (Note cycles per second = hertz.) Thus

(2.1)

Figure 2.3 Simple harmonic motion with initial phase angle ϕ.

The time T is known as the period and is usually measured in seconds. From Figures 2.2 and 2.3, we see that the motion repeats itself every time ωt increases by 2π, since sin(0) = sin(2π) = sin(4π) = 0, and so on. Thus ωT = 2π and from Eq. (2.1),

(2.2)

The angular frequency, ω, is expressed in radians per second (rad/s).

The motion described by the displacement y in Figure 2.2 or the projection OP on the X‐ or Y‐axes in Figure 2.2 is said to be simple harmonic. We must now discuss something called the initial phase angle, which is sometimes just called phase. For the case we have chosen in Figure 2.2, the phase angle is zero. If, instead, we start counting time from when the vector points in the direction OP₁, as shown in Figure 2.3, and we let the angle XOP₁ = ϕ, this is equivalent to moving the time origin t seconds to the right in Figure 2.2. Time is started when P is at P₁ and thus the initial displacement is Asin(ϕ). The initial phase angle is ϕ. After time t, P₁ has moved to P₂ and the displacement

(2.3)

If the initial phase angle ϕ = 0°, then y = A sin(ωt); if the phase angle ϕ = 90°, then y = A sin(ωt + π/2) = A cos(ωt). For mathematical convenience, complex exponential notation is often used. If the displacement is written as

(2.3a)

and we remember that A e^jωt = A[cos(ωt) + j sin(ωt)], we see in Figure 2.1 that the real part of Eq. (2.3a) is represented by the projection of the point P onto the x‐axis, A cos(ωt), and of the point P onto the Y‐ (or imaginary axis), A sin(ωt). Simple harmonic motion, then, is often written as the real part of A e^jωt, or in the more general form y = Ae^{j(ωt + ϕ)}. If the constant A is made complex, then the displacement can be written as the real part of y = A e^jωt, where A = Ae^jϕ.