Second Order System Response

System Transfer Function

The transfer function $H(s)$ for a second-order system is:

\[H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]

where:

$\zeta$ is the damping ratio
$\omega_n$ is the natural frequency

Response Types

The system exhibits three characteristic behaviors:

Underdamped ($\zeta < 1$)
- System oscillates with decreasing amplitude
- Common in systems with insufficient damping
Critically Damped ($\zeta = 1$)
- Fastest return to steady state without oscillation
- Optimal for many control applications
Overdamped ($\zeta > 1$)
- Returns to steady state without oscillation
- Slower response than critically damped

Interactive Plot

Code

viewof dampingRatio = Inputs.range([0, 2], {
  value: 1.2,
  step: 0.1,
  label: "Damping ratio (ζ)"
})

viewof naturalFreq = Inputs.range([0.1, 5], {
  value: 3.4,
  step: 0.1,
  label: "Natural frequency (ωn)"
})

data = {
  const t = Array.from({length: 1000}, (_, i) => i * 0.01);
  return t.map(t => ({
    t,
    y: Math.exp(-dampingRatio * naturalFreq * t) * 
       Math.cos(naturalFreq * Math.sqrt(Math.max(0, 1 - dampingRatio**2)) * t)
  }));
}

Plot.plot({
  width: 800,
  height: 500,
  grid: true,
  x: {
    label: "Time (s)",
    domain: [0, 10]
  },
  y: {
    label: "Amplitude",
    domain: [-1, 1]
  },
  marks: [
    Plot.line(data, {
      x: "t",
      y: "y",
      stroke: "blue"
    })
  ]
})

--- title: "Second Order System Response" format: html: code-fold: true code-tools: true html-math-method: mathjax embed-resources: true link-external-newwindow: true # Add this line document-tools: - text: "Slides" href: "second_order-revealjs.html" - text: "PDF" href: "second_order.pdf" other-links: false pdf: papersize: letter geometry: - margin=1in fig-format: pdf revealjs: output-file: second_order-revealjs.html theme: simple slide-number: true width: 1600 height: 900 code-fold: true preview-links: true --- {{< include common/_content.qmd >}} ::: {.content-visible when-format="html"} ## Interactive Plot ```{ojs} viewof dampingRatio = Inputs.range([0, 2], { value: 1.2, step: 0.1, label: "Damping ratio (ζ)" }) viewof naturalFreq = Inputs.range([0.1, 5], { value: 3.4, step: 0.1, label: "Natural frequency (ωn)" }) data = { const t = Array.from({length: 1000}, (_, i) => i * 0.01); return t.map(t => ({ t, y: Math.exp(-dampingRatio * naturalFreq * t) * Math.cos(naturalFreq * Math.sqrt(Math.max(0, 1 - dampingRatio**2)) * t) })); } Plot.plot({ width: 800, height: 500, grid: true, x: { label: "Time (s)", domain: [0, 10] }, y: { label: "Amplitude", domain: [-1, 1] }, marks: [ Plot.line(data, { x: "t", y: "y", stroke: "blue" }) ] }) ``` ::: ::: {.content-visible when-format="pdf"} ## System Response Examples ```{python} #| echo: false #| fig-cap: #| - "Underdamped Response (ζ = 0.3, ωn = 2.0)" #| - "Critically Damped Response (ζ = 1.0, ωn = 2.0)" #| - "Overdamped Response (ζ = 1.5, ωn = 2.0)" #| layout-ncol: 2 import numpy as np import matplotlib.pyplot as plt def create_response_plot(zeta, wn, static=True): t = np.linspace(0, 10, 1000) response = np.exp(-zeta * wn * t) * np.cos(wn * np.sqrt(np.maximum(0, 1 - zeta**2)) * t) plt.figure(figsize=(8, 4) if static else (6, 4)) plt.plot(t, response, 'b-') plt.grid(True) plt.xlabel('Time (s)') plt.ylabel('Amplitude') plt.title(f'Response (ζ = {zeta}, ωn = {wn})') plt.ylim(-1, 1) return plt cases = [(0.3, 2.0), (1.0, 2.0), (1.5, 2.0)] for zeta, wn in cases: create_response_plot(zeta, wn) plt.show() ``` ::: ::: {.content-visible when-format="revealjs"} ## Second Order Systems {background-color="#40666e"} ::: {.incremental} - Model many physical and control systems - Characterized by two parameters: - Damping ratio (ζ) - Natural frequency (ωn) ::: ## Transfer Function {background-color="#40666e"} :::: {.columns} ::: {.column width="40%"} The system is described by: $$H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$ ::: ::: {.column width="60%"} Key parameters: ::: {.incremental} - $\zeta$ controls oscillation behavior - $\omega_n$ determines response speed - Together they shape system dynamics ::: ::: :::: ## Response Types {auto-animate=true} Three characteristic behaviors based on damping ratio: ::: {.fragment .fade-up} 1. **Underdamped** ($\zeta < 1$) - Oscillates with decreasing amplitude ::: ::: {.fragment .fade-up} 2. **Critically Damped** ($\zeta = 1$) - Fastest non-oscillatory response ::: ::: {.fragment .fade-up} 3. **Overdamped** ($\zeta > 1$) - Slow return to steady state ::: ## Underdamped Response {background-color="#2c4a52"} :::: {.columns} ::: {.column width="40%"} Characteristics: - $\zeta < 1$ - Oscillatory behavior - Common in practice ::: ::: {.column width="60%"} ```{python} #| echo: false #| fig-align: center create_response_plot(0.3, 2.0) plt.show() ``` ::: :::: ## Critically Damped Response {background-color="#2c4a52"} :::: {.columns} ::: {.column width="40%"} Characteristics: - $\zeta = 1$ - No oscillation - Fastest settling ::: ::: {.column width="60%"} ```{python} #| echo: false #| fig-align: center create_response_plot(1.0, 2.0) plt.show() ``` ::: :::: ## Overdamped Response {background-color="#2c4a52"} :::: {.columns} ::: {.column width="40%"} Characteristics: - $\zeta > 1$ - No oscillation - Slower response ::: ::: {.column width="60%"} ```{python} #| echo: false #| fig-align: center create_response_plot(1.5, 2.0) plt.show() ``` ::: :::: ## Applications {background-color="#40666e"} ::: {.incremental} - Mechanical Systems - Suspension systems - Vibration dampers - Control Systems - Motor position control - Temperature regulation - Electronic Systems - RLC circuits - Filters ::: ## Summary Key takeaways: ::: {.incremental} - Second order systems are fundamental - Damping ratio determines response type - Natural frequency affects response speed - Design involves balancing speed vs. stability ::: :::

Other Formats

System Transfer Function

Response Types

Interactive Plot