Tools › Structures › Loads and Dynamics › Stick Model Lumping › Example
Tapered 180 in wing with an engine at station 3: lumped stick model, harmonic-mean segment stiffness, and the Myklestad and Holzer frequency passes
| Tool | Stick Model Lumping |
| Source | Houbolt, J. C., and Anderson, R. A. (1948). Calculation of Uncoupled Modes and Frequencies in Bending or Torsion of Nonuniform Beams. NACA Technical Note 1522, National Advisory Committee for Aeronautics, Washington, DC. Tables 1, 2, 6 and 7 (the nonuniform-cantilever bending and torsion examples the method is checked against); Myklestad, N. O. (1944). "A New Method of Calculating Natural Modes of Uncoupled Bending Vibration of Airplane Wings and Other Types of Beams." Journal of the Aeronautical Sciences, Vol. 11, No. 2, pp. 153–162. |
| Level | Graduate |
Problem
A wing half of 180 in span is described at ten rows 20 in apart: EI falling linearly from 10⁹ to 10⁸ lb-in², GJ from 6 × 10⁸ to 6 × 10⁷ lb-in², running weight from 4.0 to 1.3 lb/in with its centroid 3.0 to 1.2 in aft of the elastic axis and a radius of gyration of 10 to 5.5 in, and a 300 lb engine at station 3 (y = 60 in) 12 in ahead of the axis with an 8 in radius of gyration. Lump the mass to one station per row by the tributary-length rule, give every segment its harmonic-mean EI and GJ, check the lumps against the input, and find the first three uncoupled bending frequencies by Myklestad's transfer pass and the torsion frequencies by Holzer's.
Inputs to Enter
- Rows y = 0, 20, …, 180 in; EI 1,000 → 100 × 10⁶ lb-in²; GJ 600 → 60 × 10⁶ lb-in²; w 4.0 → 1.3 lb/in; e 3.0 → 1.2 in; r 10 → 5.5 in
- Concentrated weight 300 lb at row 4 (y = 60 in), e_c = −12 in, r_c = 8 in; one lumped station per row, tributary rule, clamped root, 3 modes
Answers
- Input totals: W = 477 + 300 = 777 lb, c.g. at 69.03 in, chordwise c.g. 3.25 in ahead of the axis, I about the elastic axis 251.8 lb-s²-in; lumps: 39.25, 74, 68, 362, 56, 50, 44, 38, 32, 13.75 lb (777 lb, c.g. 68.98 in)
- Segment 1 harmonic-mean EI = (0.9 − 1.0) × 10⁹ / ln 0.9 = 9.491 × 10⁸ lb-in² (arithmetic mean 9.5 × 10⁸); last segment 1.443 × 10⁸
- Myklestad bending: f = 6.790, 25.39, 66.74 Hz (ω = 42.66, 159.6, 419.3 rad/s), identical to the flexibility-matrix eigenvalues and to the numpy transfer-matrix and finite-element pass; continuous-beam reference 6.852 / 25.87 / 69.86 Hz
- Holzer torsion: f = 31.14, 83.93, 178.2 Hz; lever-rule lumping gives 6.780 / 25.36 / 66.58 Hz with the c.g. exact; a root spring of 2 × 10⁸ lb-in/rad lowers the bending modes to 6.450 / 24.12 / 65.49 Hz
Notes
The tool's default inputs are this wing; press Submit. The case is the tool's own worked example, reproduced to ten digits by References/Stick-Model/checks/stick_check.py (numpy 4 × 4 transfer matrices and a Hermite beam FE model); the method is checked in test_stick_model_lumping.php against NACA TN 1522 Example 1 (ten-station lumped ω² = 5.374 against the report's 5.45, converging to 5.461 with 160 stations) and Example 2 (torsion ω² = 4985 against 5050), the uniform cantilever's exact (βL)² = 3.5160 / 22.034 / 61.697 (10 stations −0.5 / −1.6 / −2.6 %, 80 stations within 0.04 %), and the single- and two-mass closed forms.