Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 23

We multiply Equation (1.23) by u˙

(1.42)

The first and third term can be integrated

(1.43)

The terms in the parenthesis are kinetic and potential energy and known as constant. The expression cvu˙2 is the dissipated power, because it is the damping force times velocity.

(1.44)

And Fxu˙ is the introduced power, thus

(1.45)

So we get the power balance

(1.46)

that is fluctuating for harmonic motion but with a net power flow.

For harmonic motion the power introduced into the system by a force Fx(t)=Fxejωt that generates the velocity response vx(t)=vxejωt is

(1.47)

The first term in the bracket is constant, the second oscillating with twice the excitation frequency. The first part is called active power and the second part the reactive. All introduced energy in one half cycle comes back in the next half cycle. The time average over one period leaves only the active part

(1.48)

The velocity can be expressed by the impedance V=Z/F or vice versa, so we get

(1.49)

The power considerations further clarify the naming conventions for the real and imaginary parts of the impedance. With Equation (1.40) the power introduced into the system equals Π=12|vx|2cv. Thus, the active power is controlled by the real part or resistance whereas the reactive part is determined by the imaginary component called reactance. The energy is dissipated in the resistive damping process, but power delivered to the reactive part goes into the kinetic and potential energy of mass and spring.