Upper-Stage Divergence — theory

When a long upper stage bends under its own nose load. Based on C. P. Hoult, Upper Stage Bending Influence on Static Stability (rev. 3), 2012.

Open the calculatorOriginal memo (PDF)

Overview: the Nike-Cajun problem

The Nike-Cajun entered service in 1956 as a small, inexpensive sounding rocket. Flown with a long, heavy payload, it would often break up just before first-stage burnout: camera records showed the second stage bending more and more until the payload snapped off. Beefing up the payload joint helped only a little. The cause, later established by analysis, was static aeroelasticity.

The nose normal force bends the flexible upper stage. The bending tilts the nose further into the wind, which raises its normal force, which bends the stage further. Below a critical dynamic pressure this loop converges to a larger, but finite, load; at the divergence dynamic pressure q_D it runs away.

Flexible upper stage

Take the booster as rigid, so the upper stage is a cantilever from it, loaded at the nose centre of pressure, a distance l from the root. With the bending slope added to the angle of attack, the nose force is

N_N=qSC_{NN\alpha}\left(\alpha+\frac{dy}{dx}\right)

Neglecting inertia loads, the beam equation is N_N(l-x)=EI\,d^2y/dx^2. Integrating once, the slope at the nose is dy/dx=N_Nl^2/2EI, and solving for it,

\frac{dy}{dx}=\frac{qSC_{NN\alpha}\,\alpha\,l^2}{2EI-qSC_{NN\alpha}l^2}=\frac{q\,\alpha}{q_D-q},\qquad q_D=\frac{2EI}{SC_{NN\alpha}l^2}

For a thin-walled tube of radius R and wall t, EI\approx E\pi R^3t. The nose force becomes

N_N=qSC_{NN\alpha}\,\alpha\,\frac{q_D}{q_D-q}

so all nose and bending loads are amplified by q_D/(q_D-q): by 1.25 at q=0.2q_D, by 2 at half of q_D, by 10 at 90 %. The longest stage that stays below divergence at a flight dynamic pressure q is

l_{max}=\sqrt{\frac{2EI}{SC_{NN\alpha}\,q}}

Because q_D\propto1/l^2, a 20 % longer payload lowers the divergence dynamic pressure by 30 %. Stiffness helps linearly: double the wall thickness or use a material twice as stiff and q_D doubles.

Flexible joint

A variation is a vehicle that is rigid except for a single joint — for example where the payload attaches to the body. The joint rotates by \varphi=CM under bending moment M, where C is its compliance. With the joint a distance x_j behind the nose centre of pressure, N_N=qSC_{NN\alpha}(\alpha+CN_Nx_j), so

N_N=\frac{qSC_{NN\alpha}\,\alpha}{1-qSC_{NN\alpha}Cx_j}=qSC_{NN\alpha}\,\alpha\,\frac{q_D}{q_D-q},\qquad q_D=\frac{1}{SC_{NN\alpha}Cx_j}

The amplification law is the same as for the flexible tube. The compliance of a joint is best measured: load the assembled joint as a cantilever and record its rotation.

Effect on the trimmed rocket

Suppose a thrust misalignment \delta trims the rocket at a steady angle of attack. With the first-stage tail, the second-stage tail and interstage, and the nose at arms x_1, x_2 and x_3 from the centre of gravity, the moment balance is

qS\alpha\left[x_1C_{N1\alpha}-x_2C_{N2\alpha}-x_3C_{NN\alpha}\frac{q_D}{q_D-q}\right]=x_4T\delta

Compared with a rigid rocket, the angle of attack of every element grows by (memo eq. 1)

\frac{\alpha}{\alpha_{rigid}}=\frac{x_1C_{N1\alpha}-x_2C_{N2\alpha}-x_3C_{NN\alpha}}{x_1C_{N1\alpha}-x_2C_{N2\alpha}-x_3C_{NN\alpha}\,q_D/(q_D-q)}

and the nose, which also carries its bending slope, by (eq. 2)

\frac{\alpha_N}{\alpha_{rigid}}=\frac{q_D\left(x_1C_{N1\alpha}-x_2C_{N2\alpha}-x_3C_{NN\alpha}\right)}{\left(x_1C_{N1\alpha}-x_2C_{N2\alpha}\right)(q_D-q)-x_3C_{NN\alpha}q_D}

The denominator of eq. (1) can reach zero before q_D: the flexibility of the nose erodes the static margin until the rocket is no longer stable. The calculator reports this case as “unstable”. To apply these factors in a loads analysis, increase the angle of attack of every element except the nose by eq. (1), and that of the nose by eq. (2).

Verdict and fixes

The memo gives the divergence dynamic pressure but no required margin. The calculator asks for yours, n_D (default 2, this site’s choice, not a published limit), and compares the margin q_D/q with it: safe at or above n_D, marginal between 1.25 and n_D, unsafe below 1.25, where the loads are more than five times those of a rigid stage, or when the bending makes the trimmed rocket statically unstable.

The fixes use the scalings of the formulas above, so they are exact within the model. Aiming at q_D=n_Dq:

t_{ok}=t\,\frac{n_Dq}{q_D},\qquad l_{ok}=l\sqrt{\frac{q_D}{n_Dq}},\qquad C_{ok}=C\,\frac{q_D}{n_Dq},\qquad q_{ok}=\frac{q_D}{n_D}

When the trim check finds the rocket unstable, the smallest first-stage tail slope that keeps the denominator of eq. (1) positive is C_{N1\alpha}>(x_2C_{N2\alpha}+x_3C_{NN\alpha}q_D/(q_D-q))/x_1.

Verification

The tests reproduce the field tool’s divergence dynamic pressures for upper-stage lengths from 140 to 220 in, and its load-amplification column. The two trim amplifications are checked against a direct solution of the coupled moment balance and beam equations. The tests also confirm q_D(l_{max})=q, the 1/l^2 and t scalings, and the flexible-joint formula.

Notes on the method

  • Rigid booster. The booster is assumed far stiffer than the upper stage. If it is not, its bending lowers q_D further.
  • Thin wall. EI\approx E\pi R^3t holds for t\ll R. Joints, cut-outs and doors reduce the effective stiffness; the calculator warns when the wall is thick.
  • Static only. Inertia loads and the dynamics of bending (flutter-like interactions with the pitch motion) are not included; see the bending-dynamics and loads topics.

References

  1. C. P. Hoult, “Upper Stage Bending Influence on Static Stability (rev. 3),” RS&T, 18 April 2012. View in library
  2. C. P. Hoult, “Design Bending Loads for a Multiple Longeron Structure,” RS&T, 2013. View in library