BMEN90033 · Week 7
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BMEN90033 · WEEK 7 · BIOINSTRUMENTATION

Intro to Transfer Functions.

A transfer function expresses the input–output relationship of a linear time-invariant system in the frequency domain as a compact rational function. The exposition here adopts a canonical second-order system — a gravity pendulum — as its illustrative vehicle.

linear systems Laplace domain poles & zeros
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01the idea

Input → System → Output.

Physical systems encountered in bioinstrumentation uniformly map an input signal to an output through an intervening transformation: a transducer converts acoustic pressure to voltage; an amplifier scales a voltage; a muscle fibre converts neural excitation to mechanical contraction.

Provided the system is linear — the response scaling proportionally with the input — and time-invariant, its dynamics are fully specified by a single complex-valued function of frequency, the transfer function $H(s)$.

$$Y(s) = H(s)\,X(s)$$

All constitutive parameters — inertial, elastic, dissipative, and biological — are encoded within $H(s)$. Once determined, it permits the response to arbitrary inputs to be evaluated algebraically, obviating repeated integration of the governing differential equation.

The variable $s$. The complex variable $s = \sigma + j\omega$ represents a generalised frequency. Steady-state sinusoidal response at a real frequency $\omega$ is recovered by the substitution $s = j\omega$, from which the magnitude (gain) and phase (delay) of the output follow directly.
Input
X(s)
→
System
H(s)
→
Output
Y(s)
02the model

A child on a swing.

Consider a pendulum of rope length $L$ and bob mass $m$, subject to linear viscous damping and gravitational restoration, driven by a periodic external force. The objective is to determine the steady-state amplitude as a function of the forcing frequency.

showfull derivation · 5 steps▶

Step 1 · choose a coordinate

Let $\theta(t)$ denote the angular displacement from vertical, taken positive in the direction of forcing. The corresponding arc-length kinematics are

$$s = L\theta, \qquad \dot{s} = L\dot{\theta}, \qquad \ddot{s} = L\ddot{\theta}.$$

Step 2 · sum the torques about the pivot

Newton's second law for rotation gives $I\ddot{\theta} = \sum \tau$, with $I = mL^2$ the moment of inertia of a point mass at radius $L$. Three torques act about the pivot:

  • Gravity acts vertically with magnitude $mg$; its moment arm about the pivot is $L\sin\theta$, yielding a restoring torque $\tau_g = -mgL\sin\theta$.
  • Viscous dissipation (aerodynamic drag and pivot friction) is, to leading order, linear in angular velocity: $\tau_d = -b\,\dot{\theta}$.
  • External forcing $F(t)$ applied tangentially at radius $L$ contributes $\tau_p = F(t)\,L$.

Combining these contributions:

$$mL^2\,\ddot{\theta} = -mgL\sin\theta - b\,\dot{\theta} + F(t)\,L.$$

Step 3 · small-angle approximation

For modest displacements the Taylor expansion $\sin\theta = \theta - \theta^3/6 + \ldots$ is truncated at first order, an approximation accurate to within $2\%$ for $\theta \le 20^\circ$. The governing equation thereby becomes linear:

$$mL^2\,\ddot{\theta} + b\,\dot{\theta} + mgL\,\theta = F(t)\,L.$$

Each term admits a direct physical interpretation:

$$\underbrace{mL^2\,\ddot{\theta}}_{\text{inertia}} + \underbrace{b\,\dot{\theta}}_{\text{damping}} + \underbrace{mgL\,\theta}_{\text{gravity restoring}} = \underbrace{F(t)\,L}_{\text{forcing}}.$$

Step 4 · normalise to canonical form

Normalising by $mL^2$ so that the coefficient of $\ddot{\theta}$ is unity:

$$\ddot{\theta} + \frac{b}{mL^2}\,\dot{\theta} + \frac{g}{L}\,\theta = \frac{F(t)}{mL}.$$

Comparison with the canonical second-order form $\ddot{\theta} + 2\zeta\omega_0\,\dot{\theta} + \omega_0^2\,\theta = u(t)$ identifies:

$$\omega_0^2 = \frac{g}{L} \;\Rightarrow\; \omega_0 = \sqrt{\tfrac{g}{L}}, \qquad 2\zeta\omega_0 = \frac{b}{mL^2} \;\Rightarrow\; \zeta = \frac{b}{2mL^2\omega_0} = \frac{b}{2mL\sqrt{gL}}.$$

with forcing $u(t) = F(t)/(mL)$, yielding the canonical form:

$$\ddot{\theta} + 2\zeta\omega_0\,\dot{\theta} + \omega_0^2\,\theta = u(t).$$
  • Natural frequency $\omega_0 = \sqrt{g/L}$ — the undamped oscillation frequency, determined by rope length alone. For $L = 2.45\text{ m}$, $\omega_0 \approx 2.0\text{ rad/s}$ (period $\approx 3.1\text{ s}$).
  • Damping ratio $\zeta$ — a dimensionless measure of dissipation. Typical playground swings exhibit $\zeta \approx 0.02$–$0.05$, corresponding to light damping.

Step 5 · Laplace transform

The Laplace transform maps differentiation to multiplication by $s$. Under quiescent initial conditions $\theta(0) = \dot{\theta}(0) = 0$, each term transforms as:

$$\mathcal{L}\{\ddot{\theta}\} = s^2\,\Theta(s), \qquad \mathcal{L}\{\dot{\theta}\} = s\,\Theta(s), \qquad \mathcal{L}\{\theta\} = \Theta(s), \qquad \mathcal{L}\{u(t)\} = U(s).$$

Substitution into the canonical equation gives:

$$s^2\,\Theta(s) + 2\zeta\omega_0\, s\,\Theta(s) + \omega_0^2\,\Theta(s) = U(s).$$

Factoring $\Theta(s)$:

$$(s^2 + 2\zeta\omega_0\, s + \omega_0^2)\,\Theta(s) = U(s).$$

The ratio of output to input defines the transfer function, stated below.

The ratio $\Theta(s)/U(s)$ yields the transfer function of the pendulum:

$$H(s) = \frac{\Theta(s)}{U(s)} = \frac{1}{s^2 + 2\zeta\omega_0\, s + \omega_0^2}.$$

This is the canonical second-order low-pass. RLC circuits, seismometers, and analog ECG filter stages share this structure; only the physical interpretation of $\omega_0$ and $\zeta$ differs.

swing · geometry
03resonance

Push at the right rhythm.

Forcing is in practice impulsive rather than continuously sinusoidal: a brief push delivered once per cycle constitutes an impulse train at rate $\omega$, whose fundamental Fourier component is situated at $\omega$. The transfer function determines the steady-state amplitude of this component:

$$|H(j\omega)| = \frac{1}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}}$$

Maximum amplitude is attained at $\omega = \omega_0$, where the forcing remains in phase with the motion; at other frequencies the phase relationship is incoherent and amplification is suppressed. This phase dependence is precisely what $|H(j\omega)|$ captures.

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ω₀—
ω / ω₀—
|H(jω)|—
peak angle—
Observe. For $\omega \ll \omega_0$ or $\omega \gg \omega_0$ the response is negligible. At $\omega = \omega_0$ the amplitude attains $1/(2\zeta)$, here approximately tenfold the input — the defining feature of resonance.

The peak of the Bode magnitude defines the quality factor $Q \approx 1/(2\zeta)$. Lightly-damped systems exhibit tall, narrow peaks; heavily-damped systems exhibit broad, flat responses. Increasing $\zeta$ progressively flattens the peak.

04the s-plane

Poles. Zeros. A fingerprint.

The transfer function carries information beyond its values on the real-frequency axis. Treating $s$ as a complex variable, one identifies the loci at which $H(s)$ diverges and at which it vanishes. These points, termed poles and zeros respectively, together fully characterise the system.

Equating the denominator of $H(s)$ to zero and solving the resulting quadratic:

$$s = -\zeta\omega_0 \pm j\,\omega_0\sqrt{1-\zeta^2}$$

The pendulum possesses two poles and no finite zeros. Writing $\omega_d = \omega_0\sqrt{1-\zeta^2}$, the poles are located at $-\zeta\omega_0 \pm j\omega_d$.

Interpretation of the s-plane

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× poles left half-plane · stable impulse response
Observe. For vanishingly light damping the poles approach the imaginary axis and the impulse response persists indefinitely. Increasing $\zeta$ migrates the poles inward along the $\omega_0$ arc, accelerating decay. At $\zeta = 1$ the poles coalesce on the real axis and oscillation is superseded by pure exponential decay.

Pole positions encode the principal qualitative features of the system: stability (left half-plane), oscillation frequency (imaginary part), decay rate (real part), and resonance sharpness (proximity to the imaginary axis).

05where zeros come from

A weight tied halfway down.

The simple pendulum's transfer function contains poles alone. Zeros — frequencies at which transmission is suppressed — require a second dynamical mode capable of opposing the first through the same input channel.

The minimal modification introduces an intermediate mass $m_1$ at a joint along the rope, with the bob $m_2$ suspended below. Horizontal forcing is applied to $m_2$. The resulting system constitutes a double pendulum.

Equations of motion

Let $\theta_1,\theta_2$ denote the two segment angles. In the small-angle limit the bob's horizontal position is $x_2 = L_1\theta_1 + L_2\theta_2$. A Lagrangian formulation with gravity as the sole restoring force yields a coupled $2\times 2$ system. The off-diagonal inertia term $m_2 L_1 L_2$ mediates modal coupling — displacement of one angle excites the other. This coupling, in conjunction with the intermediate mass, generates a transmission zero.

The zero

$$\omega_z^2 = \frac{g\bigl[L_1 m_2 + L_2(m_1+m_2)\bigr]}{m_1 L_1 L_2}$$

At $\omega = \omega_z$ the intermediate mass oscillates with precisely the amplitude and phase required to cancel the forcing transmitted to the bob; the bob remains nearly stationary while the intermediate mass absorbs the input. In the time trace — green denoting the bob's horizontal displacement and blue the intermediate mass — the bob response collapses as $\omega \to \omega_z$.

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06walking up the jω axis

The notch, made visible.

A real forcing frequency $\omega$ corresponds to the single point $s = j\omega$ on the positive imaginary axis. Traversing $\omega$ advances a marker along this axis, at each position of which the transfer function evaluates to

$$|H(j\omega)| = \frac{\prod_i |j\omega - z_i|}{\prod_k |j\omega - p_k|}$$

— the ratio of the product of distances from the marker to each zero to the product of distances to each pole. Proximity to a pole produces a resonance peak as the denominator contracts; coincidence with a zero produces a transmission notch as the numerator vanishes. The Bode magnitude is the trace of $|H(j\omega)|$ so obtained.

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× poles ○ zeros jω marker |H(jω)|

Observe. As $\mu \to 0$ the intermediate mass vanishes, the zero recedes to infinity, and the transfer function reduces to that of the simple pendulum. Increasing $\mu$ deepens the notch. Displacing $\alpha$ from $0.5$ shifts $\omega_z$: higher placement of the intermediate mass along the rope raises the notch frequency.

Taipei 101. The 660-tonne tuned mass damper suspended within the upper storeys of Taipei 101 is an instance of this mechanism. The tower serves as the primary mass and the pendulum as the intermediate mass along an effective compliance of stiffness and gravity. Its parameters are tuned such that $\omega_z$ coincides with the wind-driven resonance, notching out the sway that would otherwise be excited.