Tools › Mechanical Design › Vibration › Multi-DOF Vibration and Critical Speed › Example
Three equal masses on equal springs (the closed-form uniform chain), a three-disc torsional train on 2 in and 1.5 in steel segments, and a 2 in steel shaft with two discs on a 30 in span
| Tool | Multi-DOF Vibration and Critical Speed |
| Source | Trivisonno, R. M. (1973). FORTRAN IV Computer Program for Calculating Critical Speeds of Rotating Shafts. NASA Technical Note D-7385, eqs. (2)–(5), (29), (30) (transfer-matrix conventions); Houbolt, J. C., and Anderson, R. A. (1948). NACA Technical Note 1522, Tables 2 and 7 (lumped torsion check); closed forms: uniform fixed-free chain ω_r = 2√(k/m) sin((2r−1)π/(2(2n+1))), two-disc torsion √(k_t(J₁+J₂)/(J₁J₂)), uniform shaft (nπ/L)²√(EI/ρA), disc on a massless shaft 48EI/(mL³). |
| Level | Undergraduate |
Problem
Chain: three 10 lb masses on 100 lb/in springs, left end fixed, a 10 lb force at 5 Hz on mass 3 with 2 % modal damping, and the free vibration from a 0.5 in initial displacement of mass 3. Torsion: discs of 400, 100 and 200 lb·in² joined by a 2 in × 30 in and a 1.5 in × 20 in steel segment (G = 11.5 Msi), free-free, orders 0.5 to 4 at 1,800 rpm. Shaft: a 2 in steel shaft (E = 30 Msi, 0.284 lb/in³) on rigid bearings at 0 and 30 in carrying 20 lb at 10 in and 15 lb at 20 in, running at 3,600 rpm. Find the natural frequencies and modes, the Holzer table, the Rayleigh and Dunkerley bounds, the steady-state amplitudes, the torsional criticals of every order, and the first three lateral critical speeds by the lumped eigen-solve and the transfer-matrix pass.
Inputs to Enter
- Chain: m 10 / 10 / 10 lbm, k 100 / 100 / 100 lb/in, c 0, left end fixed, k_R 0, ζ 0.02, force 10 lb at row 3 at 5 Hz, transient x₀ 0.5 in
- Torsion: J 400 / 100 / 200 lb·in², segments d 2 / 1.5 in with L 30 / 20 in (rows 2 and 3), G 11.5 Msi, left end free, orders 0.5, 1, 1.5, 2, 3, 4, operating speed 1,800 rpm
- Shaft: section 0 to 30 in, d 2 in, bore 0; discs 20 lbm at 10 in and 15 lbm at 20 in; rigid bearings at 0 and 30 in; E 30 Msi, ρ 0.284 lb/in³, 3,600 rpm, 60 elements
Answers
- Chain: f = 4.401, 12.33, 17.82 Hz = (1/2π) 2√(k/m) sin(π/14), sin(3π/14), sin(5π/14) exactly; modes (0.4450, 0.8019, 1), (1, 0.4450, −0.8019), (−0.8019, 1, −0.4450); effective masses 91.4 / 7.5 / 1.1 % of 30 lb
- Chain: Holzer residual 1e-16 at f₁; Rayleigh 4.423 Hz ≥ exact 4.401 Hz ≥ Dunkerley 4.037 Hz; |X₃| = 0.9021 in at 170.8° lag (direct solve = mode superposition), static 0.300 in
- Torsion: k_t = 602,139 and 285,781 lb·in/rad; f = 0 (rigid), 119.2, 318.3 Hz; mode 2 node 0.337 in from disc 2 in segment 2; critical speeds 7,153 rpm (order 1, mode 2) down to 1,788 rpm (order 4), 0.65 % below the operating speed
- Shaft: critical speeds 6,140 / 24,320 / 95,170 rpm by the eigen-solve and 6,140 / 24,320 / 95,170 rpm by the transfer matrix; Rayleigh 6,140 rpm, Dunkerley 5,931 rpm (by parts 6,009 rpm; bare shaft 10,574 rpm = π²√(EI/ρAL⁴)); δ₁₁ = 1.886 × 10⁻⁵ in/lb; third critical = the uniform shaft's third mode because the discs sit at its nodes
Notes
The tool's default inputs are these three cases (choose the system in the first select); press Submit. Illustrative round-number systems, not textbook examples: every value is a closed form or is reproduced by the independent numpy / scipy script in References/MDOF-Vibration/checks/ (test_mdof_critical_speed.php, 308 checks). The lumped torsion example of NACA TN 1522 (ω² = 4985 for the ten-station model against the report's 5050) is checked in the test, not on the page.