Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

normal upper Pi Subscript in Baseline equals upper F Subscript x Baseline ModifyingAbove u With dot (1.45)

So we get the power balance

normal upper Pi Subscript diss Baseline equals normal upper Pi Subscript in (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.

1.2.4 Damping

In many practical applications ζ is small and the amplitude can be estimated by linear expansion from (1.34)

ModifyingAbove u With caret Subscript r Baseline almost-equals StartFraction ModifyingAbove u With caret Subscript 0 Baseline Over 2 zeta EndFraction left-parenthesis 1 plus StartFraction zeta squared Over 2 EndFraction right-parenthesis almost-equals StartFraction ModifyingAbove u With caret Subscript 0 Baseline Over 2 zeta EndFraction (1.50)

with the corresponding phase angle

phi Subscript r Baseline almost-equals arc tangent left-parenthesis minus StartFraction 1 Over zeta EndFraction right-parenthesis (1.51)

The amplitude- and phase resonances are assumed to be equal for systems with small damping. The magnification is thus 1/2ζ, and it is called the quality factor:

StartFraction ModifyingAbove u With caret Subscript r Baseline Over ModifyingAbove u With caret Subscript 0 Baseline EndFraction equals StartFraction 1 Over 2 zeta EndFraction equals upper Q (1.52)

Figure 1.10 Half power bandwidth for harmonic oscillator. Source: Alexander Peiffer.

This factor is a measure for the sharpness of the resonance peak or the quality of the resonator. In response diagrams when the shape of the amplitude over frequency is measured the half power bandwidth is used. This is the distance of the points where the amplitude is 1/2 of the peak value u^r. Solving Equation (1.31) for u^r/u^0=1/2 the frequencies of half power can be found:

omega Subscript 1 slash 2 Baseline equals left-parenthesis 1 plus-or-minus zeta right-parenthesis omega 0 (1.53)

and therefore

upper Q equals StartFraction 1 Over 2 zeta EndFraction equals StartFraction omega 0 Over omega 2 minus omega 1 EndFraction (1.54)

Obviously the decay time is also related to damping. If Equation (1.22) is considered we get

upper Q equals StartFraction omega 0 tau Over 2 EndFraction (1.55)

We have presented several expressions that describe damping. Nevertheless, even more quantities for damping are used depending on the engineering discipline and will be shown in Section 1.2.5. Section 1.2.5.1 aims at sorting all those expressions and their relationships among each other.

1.2.5 Damping in Real Systems

Viscous damping is rare in real systems, it only exists if the surface that is connected to liquids moves so slow that no turbulent motion appears. Observation of experiments with damping normally doesn’t show damping that increases

Скачать книгу