At
left a spring/mass/dashpot system is shown, driven by a dashpot at the
bottom.
The
equation governing this system is displayed in yellow at the top. Mass
is set to 1; b is the damping constant, k is the spring constant, and
ω is the circular frequency of the sinusoidal motion of the piston.
To
the right, the position of the top of the dashpot (the input signal
cos(ωt)) is graphed in cyan, and the position of the mass (the
system response x ) is graphed in yellow. Diamonds indicate the current
values cos(ωt) and of x , and a vertical white line between them
indicates the extension of the spring. The time lag t0 is
read out in red at the bottom of the screen, below a readout of the
period P in cyan. If t0> 0 , it is represented by a grey
vertical line measured by a red segment.
Rolling
the cursor over the graphing window produces crosshairs and a readout
of the values of t and x. The horizontal crosshair line is continued
in the window displaying the system.
The time value is set using a slider under the window. The [>>] key starts an
animation. The [<<] key resets t to
t = 0.
Grab
the [b], [k], or [ω] slider to vary those parameters.
The
[Bode plots] key toggles display of two windows on the right side of
the screen. The top window displays the amplitude A of the sinusodial
response as a function of ω. The window below it displays the
negative of the phase lag φ as a function of ω.
The
[Nyquist plot] key toggles display of a window at lower right, showing
a portion of the complex plane. On it, a grey curve traces the path
traversed by the complex gain biω / p(iω) (where p(s) =
s2 + bs + k is the characteristic polynomial) as ω
varies over positive values. A yellow diamond marks the value of this
complex number for the chosen value of ω. A yellow line segment
connects it to the origin. The length of this segment is the amplitude
A, and the angle up from the positive real axis, marked by a a green
arc, is -φ.
Roll
the cursor over the amplitude window to cause a horizontal yellow line
to appear in that window and in the graphing window, marking the maximal
displacement, and a readout of that maximal value.
Roll
the cursor over the phase shift window to cause a readout of the phase
shift.
Note:
These are not quite truly Bode or Nyquist plots. A Bode plot graphs
log(A) vs log(ω) or -φ vs log(ω). A Nyquist plot displays
k/p(i ω) as omega ranges from -∞ to +∞ it has a portion
above the real axis which is symmetric with what is drawn.
©
2006 H. Miller and J.-M. Claus