Elevator Deflection (trim)⚠ unverified

Understanding the Parameters

• What trimming means — an aircraft is trimmed when the net pitching moment is zero and it holds its attitude with no stick force. Because the airframe alone has a nonzero moment at most angles of attack, the elevator must supply the balancing moment. This formula returns exactly the deflection that does it.

• Trim sets the speed — each angle of attack corresponds (in level flight) to one airspeed. So choosing the trim $\alpha$ is really choosing the trim speed, and the pilot's trim wheel effectively resets which $\alpha$ the aircraft holds hands-off. To slow down, trim to a higher $\alpha$ (more nose-up elevator); to speed up, a lower $\alpha$.

• The numerator is the airframe's own moment — $C_{m0} + C_{m\alpha}\alpha$ is the pitching moment the wing and body produce at angle $\alpha$ before the elevator acts (see Longitudinal Trim Moment). For a stable aircraft $C_{m\alpha} < 0$, and a positive $C_{m0}$ (nose-up bias) is needed so the trim angle is positive. The elevator cancels whatever this evaluates to.

• Divide by control power $C_{m\delta}$ — the more effective the elevator (larger $|C_{m\delta}|$), the smaller the deflection needed. A powerful elevator trims with a flick; a weak one needs large travel and may run out of authority at the extremes of the CG range (the forward CG limit is where trim demands full elevator — see Control Power Cm Delta).

• Sign and direction — with a stable airframe and a conventional elevator (both $C_{m\alpha}$ and $C_{m\delta}$ negative), a positive trim $\alpha$ needs a specific elevator sign to balance; the leading minus keeps the accounting straight. The magnitude is what matters operationally — how much elevator, and whether it is within travel limits.

Derivation (Approaching a Proof)

Model the total pitching-moment coefficient as a linear sum of the airframe contribution and the elevator contribution:

$$C_m = C_{m0} + C_{m\alpha}\,\alpha + C_{m\delta}\,\delta_e.$$

The first two terms are the airframe's moment at angle of attack $\alpha$; the third is the extra moment the elevator adds, proportional to its deflection through the control power $C_{m\delta}$. Trim is the condition of zero net pitching moment — rotational equilibrium about the pitch axis (Torque, zero angular acceleration):

$$C_m = 0.$$

Set the sum to zero and solve for the elevator deflection:

$$C_{m0} + C_{m\alpha}\,\alpha + C_{m\delta}\,\delta_e = 0 \quad\Longrightarrow\quad C_{m\delta}\,\delta_e = -\big(C_{m0} + C_{m\alpha}\,\alpha\big),$$

$$\delta_e = -\frac{C_{m0} + C_{m\alpha}\,\alpha}{C_{m\delta}}. \qquad\blacksquare$$

The structure is transparent: the elevator deflection is "minus the airframe moment, divided by how much moment each degree of elevator makes." Setting $\delta_e = 0$ recovers the neutral-elevator trim angle $\alpha_{trim} = -C_{m0}/C_{m\alpha}$ of Longitudinal Trim Moment — the two calculators are the same equation solved for different unknowns.

Dimensional check. $C_{m0}$ is dimensionless, $C_{m\alpha}\alpha$ is (1/rad)(rad) = dimensionless, and dividing by $C_{m\delta}$ (1/rad) gives radians ✓ — an elevator angle.

History and Development

• The trim/stability pairing. The linear moment model $C_m = C_{m0} + C_{m\alpha}\alpha + C_{m\delta}\delta_e$ is the foundation of classical longitudinal stability and control (Perkins & Hage, Etkin, Nelson). Its two readings — solve for $\alpha$ (stability, trim angle) or for $\delta_e$ (control, trim deflection) — are the two questions every empennage design must answer.

• Elevator vs stabilator vs trim tab. Real aircraft achieve this deflection through different mechanisms: a hinged elevator, an all-moving stabilator, or a trim tab that offloads the stick force. The underlying moment balance is identical; the hardware differs in how the deflection is commanded and held (see Control Surface Hinge Moment for the force the mechanism must supply).

• CG and trim scheduling. Because the neutral point and $C_{m0}$ shift with configuration and the CG moves as fuel burns, the trim deflection changes throughout a flight. Autopilots and FMS continuously retrim; some aircraft (Concorde, A340) pump fuel to shift the CG and minimise trim drag — actively managing this very balance.

Related Concepts: Longitudinal Trim Moment, Control Power Cm Delta, Control Power Moment, Static Margin, Control Surface Hinge Moment, Moment Coefficient, Pitching Moment

Notes: Solves the trim condition $C_m=C_{m0}+C_{m\alpha}\alpha+C_{m\delta}\delta_e=0$ for the elevator $\delta_e$ at a given $\alpha$ — the companion of Longitudinal Trim Moment (which solves for the angle at $\delta_e=0$). Trim $\alpha$ = trim speed. Deflection $\propto 1/C_{m\delta}$ (elevator power); running out of travel = the forward CG limit. All coefficients per radian; $\alpha$ in rad; returns 0 if $C_{m\delta}=0$.