声学基础课件(许肖梅)fundamentals of acoustics 07-4共24页PPT资料

WM
1 T
T 0
Fm2 cos(t )costdt
Zm
1gFm2 2 Zm
cos
1 2
vm2 Rm
cos Rm Zm
vm Fm Z m
v x m
m
WM 122xm2Rm
Mechanical Resonance In the steady state, the displacement is equal to :
v(t) Fm cos(t)
Zm
The (angular) frequency of mechanical resonance is defined as that at which the mechanical reactance Xm vanishes, this is the frequency at which a driving force will supply maximum power to the oscillator. It was also found to be the frequency of free oscillation of a similar undamped oscillator.
dA F dx Fv dt dt
Substituting the appropriate real expressions for force and speed
v dx Fm cos(t )
dt Zm
F Fm cost
dA Fm2 cos(t)cost
dt Zm
In most situations the average power is more significance than the instantaneous power.
• When
0,
W MW M m ax
WM max
1 2
Fm2 Rm
At this frequency the mechanical impedance has its minimum value of Zm=Rm It is also the frequency of maximum speed amplitude.
Note that This frequency does not give the maximum displacement amplitude.
The transient term is obtained by setting F equal to zero. The arbitrary constants are determined by applying the initial conditions to the total solution.
After a sufficient time interval, The damping term makes this portion of the solution negligible. Leaving only the steady –state term whose angular frequency ω is that of the driving force
Zm is called the complex mechanical impedance; Rm is called the mechanical resistance; Xm is called the mechanical reactance
The mechanical impedance Zm Zm ej
For the case of a sinusoidal driving force f(t)=Fmcos(ωt) applied to the oscillator at some initial time, the solution of (1-3) is the sum of two parts –a transient term containing two arbitrary constants and a steady-state term which depends of F and ω but does not contain any arbitrary constants.
For a force oscillations, the general solution
x(t)A 0e tcos(0t)F Z m msin(t)
The solution of equation is the sum of two parts: • a transient term • a steady-state term
x(t) F Z m mc o t s 2 F Z m msi n t ()
Speed gives:
v(t) Fm cos(t)
Zm
If the speed as given by above is plotted as a function of the frequency of a driving force , a curve is obtained as follow
If the average power supplied to the system
as given by:
Wm
1 2
vm2 Rm
Therefore
vm Fm Zm
W m1 2F R m m 21m 1 R m D 22 1 2F R m m 21(m R m 0)21 ( 0m D 0)2
has magnitude:
Zm
Rm2
源自文库
(m
D
)2
and phase angle:
tg1 X m
Rm
tg
1
m
Rm
D
Energy Relation
The instantaneous power, supplied to the system in the steady state is equal the product of the instantaneous driving force and the resulting instantaneous speed.