Modal Estimates: Method

Modal Estimates

What the tool is for

Before a stick model or a finite-element model exists, a structural engineer places the first natural frequencies of a part by hand: a bracket as a cantilever with a mass at its end, a tube as a shaft, a panel as a plate, an equipment rack as a chain of masses and springs. This tool carries those estimates out with the exact theory of the simple structures and prints beside them the two classical hand approximations, Rayleigh's quotient and Dunkerley's formula, with their errors, so that the analyst sees both the frequency and how much a quick estimate can be trusted. The results are natural frequencies in Hz and rad/s and the first mode shapes; no load, stress or damping enters.

Uniform beams

The Bernoulli–Euler beam of bending stiffness EI and mass per unit length m vibrating harmonically obeys EI w″″ = m ω² w. Its solution w(x) = A cosh βx + B sinh βx + C cos βx + D sin βx with β⁴ = m ω²/(EI) satisfies two conditions at each end: a clamped end has w = w′ = 0, a pinned end w = w″ = 0 and a free end w″ = w‴ = 0. The four conditions give a frequency equation in βL whose roots are the eigenvalues:

Ends	Frequency equation	β₁L, β₂L, β₃L	Rigid-body modes
pinned–pinned	sin βL = 0	π, 2π, 3π	none
clamped–free	cos βL cosh βL + 1 = 0	1.8751, 4.6941, 7.8548	none
clamped–clamped	cos βL cosh βL − 1 = 0	4.7300, 7.8532, 10.9956	none
clamped–pinned	tan βL = tanh βL	3.9266, 7.0686, 10.2102	none
free–free	cos βL cosh βL − 1 = 0	4.7300, 7.8532, 10.9956	translation and rotation
pinned–free	tan βL = tanh βL	3.9266, 7.0686, 10.2102	rotation

The natural frequencies follow as ω_n = (β_nL)² √(EI/m) / L² and f_n = ω_n/(2π). The tool finds the roots by bisection of the characteristic function written in a form that stays bounded for large arguments (cos βL + 1/cosh βL for the cantilever) and prints beside them the eigenvalues tabulated in NASA SP-160 (Leissa 1969, Tables 4.23 and 4.46, the beam functions used for plate vibration), which agree to every printed digit. The higher roots approach (2n − 1)π/2 for the cantilever, (2n + 1)π/2 for the clamped–clamped and free–free beams and (4n + 1)π/4 for the clamped–pinned and pinned–free beams. The mode shapes are the classical combinations cosh βx − cos βx − σ (sinh βx − sin βx) with σ fixed by the end conditions; the tool draws the first three scaled to a largest deflection of one.

Beams with masses and stepped beams: the transfer matrix

A point mass, a change of section or a mass at mid-span breaks the single closed form, but the exact solution is still elementary by the transfer-matrix method (Pestel and Leckie 1963). The state of a section, z = [w, w′, M = EI w″, Q = EI w‴], is carried across a uniform segment of length l by a 4 × 4 field matrix built from the Krylov–Duncan functions S, T, U, V of βl, the even and odd combinations of cosh, cos, sinh and sin whose derivative chain is S′ = βV, T′ = βS, U′ = βT, V′ = βU. A point mass M_p adds a jump in the shear, Q⁺ = Q⁻ + M_p ω² w, because the beam equation integrated across the mass gives [EI w‴] = M_p ω² w. Two admissible unit states at the left end (for a clamped end the unknown moment and shear) are propagated to the right end, where two conditions must hold; the frequencies are the values of ω at which the 2 × 2 determinant of those conditions vanishes. The tool scans the determinant in the dimensionless variable β_ref L with the first segment's EI and m, bisects every sign change to machine precision and recovers each mode shape from the null vector. For a uniform beam without masses the roots are the classical β_nL above to twelve digits.

For the cantilever with a tip mass the determinant reduces to the frequency equation of NACA Memorandum Report L4K30 (Stowell, Schwartz, Houbolt and Schmieder 1944), 1 + cos θ cosh θ + r θ (sinh θ cos θ − cosh θ sin θ) = 0 with r the tip mass over the beam mass and θ = βL, whose table of roots for r between 0 and 6 the tool reproduces; Storch's NASA conference paper of 1994 gives the same equation (his eq. A.9 with no tip inertia) and the fundamental for r up to 20. As r grows the first root falls towards (3/r)^¼, the single-degree-of-freedom mass on a massless beam, and the second root tends to 3.9266: the tip stops moving and the beam behaves as clamped–pinned. A mass at the mid-span of a simply supported beam lowers the symmetric modes only; the antisymmetric modes have a node at the mass and keep their bare-beam values (βL = 2π, 4π, ...).

Rayleigh's quotient

For any admissible shape w(x), the ratio of the maximum strain energy to the maximum kinetic energy per ω² is an upper bound of the first natural frequency: ω₁² ≤ ∫EI w″² dx / (∫m w² dx + Σ M_p w_p²). The closer the shape to the true mode, the smaller the excess. The classical choice is the static deflection under the structure's own weight, which the tool uses in two forms.

For the uniform beams the static shape is a polynomial in ξ = x/L: ξ⁴ − 4ξ³ + 6ξ² for the cantilever, ξ⁴ − 2ξ³ + ξ for the pinned–pinned beam, ξ⁴ − 2ξ³ + ξ² for the clamped–clamped beam and 2ξ⁴ − 5ξ³ + 3ξ² for the clamped–pinned beam. The integrals are rational: the cantilever gives ∫w″² = 144/5 and ∫w² = 104/45 on [0, 1], hence C² = 162/13 and ω₁ ≈ 3.5301 √(EI/m)/L², 0.40 % above the exact 3.5160; the pinned–pinned beam gives C² = 3024/31 (9.8767 against π² = 9.8696, 0.07 % high), the clamped–clamped beam C² = 504 (22.450 against 22.373, 0.34 % high) and the clamped–pinned beam C² = 238.74 (15.451 against 15.418, 0.21 % high). A beam without supports has no static deflection, so for the free–free and pinned–free beams the tool integrates a bending-moment distribution that vanishes with zero slope at the free ends, ξ²(1 − ξ)² or ξ(1 − ξ)², twice, and removes the rigid-body components so that the shape is orthogonal to the rigid motions; the free–free shape is within 0.011 % of the exact value, the pinned–free one 1.4 % high.

For the beams with masses and the stepped beams the tool computes the static deflection under gravity exactly, segment by segment with the static (cubic) transfer matrices, the distributed weight of each segment and the weight of each mass, and evaluates the quotient on it. It also prints the textbook form with the deflection under a load at the mass alone: ω₁² ≈ k_s/(M_p + m_eff), where k_s is the static stiffness at the mass and m_eff = m ∫w² dx / w_p² the part of the beam mass that moves with it. The classical fractions reappear as computed values: 33/140 of the beam mass at a cantilever tip, so that ω₁² ≈ 3EI/(L³ (M_p + 0.2357 mL)); 17/35 at the centre of a simply supported beam with 48EI/L³; 13/35 at the centre of a clamped–clamped beam with 192EI/L³.

Dunkerley's lower bound

Dunkerley (1894) observed that the first frequency of a shaft carrying several masses is estimated from below by adding the inverse squared frequencies of the parts: 1/ω₁² ≈ 1/ω_beam² + Σ M_i/k_i, with ω_beam the first frequency of the bare beam and k_i the static stiffness at each mass acting alone on the massless beam. The relation is exact for the sum of all 1/ω_n² (the trace of the flexibility–mass product), so dropping the higher modes makes it a lower bound. The tool forms it with the exact bare-beam frequency and the static stiffness at the mass, and states the bracket Dunkerley ≤ exact ≤ Rayleigh. For the default case, a 30 in aluminium cantilever 2 × 0.25 in with a 1 lbm tip mass, the exact first frequency is 4.568 Hz, Rayleigh's point-load shape gives 4.572 Hz (+0.10 %), the gravity shape 4.573 Hz (+0.12 %) and Dunkerley 4.555 Hz (−0.28 %); the bare beam vibrates at 8.817 Hz.

Shafts in torsion and axial rods

Torsion of a uniform shaft (GJ θ″ = ρJ θ̈) and longitudinal vibration of a uniform rod (EA u″ = ρA ü) are wave equations with the speeds c = √(G/ρ) and c = √(E/ρ); the frequencies are ω_n = β_nL · c/L with βL = (2n − 1)π/2 for fixed–free ends and nπ for fixed–fixed or free–free ends. A disk of polar inertia I_e on the free end of a shaft, or a mass M_p on the end of a rod, changes the equation to βL tan βL = r with r the inertia (or mass) of the shaft (rod) over that of the end body; the roots lie in ((n − 1)π, (n − ½)π), tend to √r for a small r (the body on a massless shaft, ω² = GJ/(L I_e)) and to the bare values for a large r. For the shaft the tool takes J both as the torsion constant and as the polar moment of the mass inertia per unit length ρJ, which is exact for circular sections; for other shapes enter the torsion constant and read the frequencies as approximate. Pinned ends have no meaning for a shaft or rod and are treated as fixed.

Rectangular plates

The thin plate obeys D ∇⁴w = ρh ω² w with D = Eh³/(12(1 − ν²)), and every frequency is ω = λ √(D/(ρh)) / a² with a frequency parameter λ. For the plate simply supported on all four edges the parameter is exact, λ_mn = π²[m² + n²(a/b)²] (SP-160 eq. 4.20). For the clamped plate the tool reads Leissa's tables: 35.99 for the square (Young's Ritz value; Tomotika's 35.9866) with the next five modes of Table 4.22, and for rectangles the fundamental ωb²√(ρh/D) of Table 4.28 (27.00, 24.56, 23.76, 23.19 at a/b = 1.5, 2, 2.5, 3), interpolated linearly in b/a between the tabulated points and the clamped-strip limit 22.37, which is the clamped–clamped beam value 4.7300². For the cantilever plate clamped along the edge of length b, the first bending parameter is 3.494 for the square (Table 4.45) and 3.508, 3.472, 3.450 for Barton's a/b = ½, 2, 5 (Table 4.48), interpolated between the narrow-strip limit 3.516 √(1 − ν²) (the cantilever beam, since D carries 1 − ν²) and the wide-plate limit 3.516; at the tabulated ratios the torsion and chordwise families of the tables are listed too. These parameters are for ν = 0.3. The plate cases are a cross-check of the Plate Vibration tool, which solves any aspect ratio, Poisson's ratio and edge combination by the Rayleigh–Ritz method.

Spring–mass chains

A chain of masses m_i joined by springs k_i (k_i between mass i and mass i − 1, k₁ to the ground) has the tridiagonal stiffness matrix with k_i + k_(i+1) on the diagonal and −k_(i+1) beside it. The eigenproblem K x = ω² M x is made symmetric with the diagonal mass matrix, M^(−½) K M^(−½) y = ω² y, and solved by cyclic Jacobi rotations; the mode vectors are x = M^(−½) y scaled to a largest component of one. For n equal masses and springs the exact frequencies are ω_j = 2 √(k/m) sin[(2j − 1)π/(2(2n + 1))], which the tool reproduces to machine precision. Rayleigh's quotient is formed with the static deflection under gravity, x_s = K⁻¹(mg), and Dunkerley's bound with the trace of K⁻¹M; for a chain whose modes lie close together Dunkerley's bound is loose (8 % low for three equal masses) while Rayleigh's static shape is within 0.5 %.

Reading the results

The frequency table lists each mode's β_nL (or λ), ω in rad/s, f in Hz, cycles per minute and the period. The Rayleigh and Dunkerley sections print the assumed shape and the estimate with its signed error; the exact value always lies between the two. The mode-shape figure draws the first three modes against x/L with the mass marked; a node that coincides with the mass explains a mode the mass does not shift. For the stepped beam the roots are written as β_ref L_tot with the first segment's EI and m, so they are not the classical constants.

Limits

The theory is the Bernoulli–Euler beam: shear deformation and rotary inertia (Timoshenko's corrections, about 1 % at a depth of 1/20 of the half-wavelength and growing with the mode number) are not included, nor are axial loads, damping, elastic or intermediate supports, rotary inertia of the attached masses, tapered segments beyond a two-segment step, non-rectangular or loaded plates and forced response. The plate parameters for the clamped and cantilever plates are table values at ν = 0.3 interpolated in the aspect ratio, good to about 1 %; the Plate Vibration tool gives the converged values. The estimates serve a first placement of a design's frequencies; a stick model (Stick Model Lumping tool) or a finite-element model carries the detail.