AMPLITUDE AND PHASE: SECOND ORDER IV

At left a spring/mass/dashpot system is shown, driven by a force applied to the mass.

The equation governing this system is displayed in yellow at the top. The mass is denoted by $$m$$; $$b$$ is the damping constant, $$k$$ is the spring constant, and $$\omega$$ is the circular frequency of the sinusoidal motion of the piston.

To the right, the value of the force (the input signal $$\cos(\omega t)$$) is graphed in cyan, and the position of the mass (the system response $$x$$) is graphed in yellow. Diamonds indicate the current values $$\cos(\omega t)$$ 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.

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 $$[m], [b],$$ or $$[k]$$ 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 $$\omega$$. The window below it displays the negative of the phase lag $$\phi$$ as a function of $$\omega$$.

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 $$\frac{1}{p(i\omega)}$$ (where $$p(s) = ms^2 + bs + k$$ is the characteristic polynomial) as $$\omega$$ varies over positive values. A yellow diamond marks the value of this complex number for the chosen value of $$\omega$$. 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 green arc, is $$-\phi$$.

Roll the cursor over the amplitude window to cause a horizontal 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.

If the amplitude of the system response exceeds 4, the mechanical system breaks and is no longer shown.

Note: These are not quite truly Bode or Nyquist plots. A Bode plot graphs $$\log(A)$$ vs $$\log(\omega)$$ or $$-\phi$$ vs $$\log(\omega)$$. A Nyquist plot displays $$\frac{k}{p(i\omega)}$$ as $$\omega$$ ranges from $$-\infty$$ to $$+\infty$$ it has a portion above the real axis which is symmetric with what is drawn.

© 2001 H. Hohn and H. Miller