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
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 :
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.
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.
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.
has magnitude:
Zm
Rm2
(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.
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
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.
• When
0,
W MW M m ax
WM max
1 2
Fm2 Rm
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.
Zm is called the complex mechanical impedance; Rm is called the mechanical resistance; Xm is called the mechanical reactance
The mechanical impedance Zm nt 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
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