Amplitude
and Phase: Second Order -- Help |
At left a spring/mass/dashpot system is shown, driven by a piston at the top. 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 plunger (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. A grey vertical line measured by a red segment indicates the time lag t_{0} (which is also read out in red at the bottom of the screen, below a readout of the period P in cyan). Rolling the cursor over the graphing window produces crosshairs and a readout of the values of t and x. Grab the [t] slider to set the time t, press the [>>] key to animate the system, or press the [>] or [<] key to increase or decrease t by 0.1. Grab the [b], [k], or [ω] slider to vary those parameters. The [Bode and Nyquist Plots] key toggles display of three windows on the right side of the screen. The top window displays the amplitude A as a function of ω. The midddle window displays the negative of the phase lag φ as a function of ω. The bottom window displays the complex number k/p(iω) (where p(s) = s^{2} + bs + k is the characteristic polynomial of the operator). The magnitude of this complex number, indicated by a yellow radial segment, is the amplitude A , and its angle, indicated by a green arc, is -φ. Roll the cursor over the amplitude window to cause a horiziontal yellow line to appear relating the amplitudes in the three top windows, along with a readout of the amplitude. 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.
c 2001 H. Hohn |