Rayleigh and Rayleigh-Ritz energy methods

Rayleigh's quotient, admissible trial shapes, the upper-bound property, effective mass corrections and the Rayleigh–Ritz method for several modes.

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Why it matters

Exact natural frequencies exist only for simple, uniform structures. A tapered wing with engines, fuel and a fuselage attached needs an approximation, and energy methods give one quickly and reliably: Rayleigh's method for a fast estimate of the fundamental frequency, the Rayleigh–Ritz method for several modes with controlled accuracy. The same idea, assumed shapes plus energy, is the basis of the finite-element method and of the assumed-mode models used in flutter analysis.

Key ideas

Energy balance. In undamped free vibration at frequency ω, every point moves as w(x, t) = W(x)·cos(ω·t). Energy is conserved, so the maximum strain energy (at extreme displacement, zero velocity) equals the maximum kinetic energy (at the equilibrium position): U_max = T_max. With T_max = ω²·T*, where T* is the kinetic energy computed with W(x) in place of velocity, this gives Rayleigh's quotient ω² = U_max/T*.

Rayleigh's method. Assume a shape W(x), compute U_max and T*, divide. For a beam, U_max = ½∫EI·(W″)²·dx and T* = ½∫m·W²·dx + ½·Σ M_i·W(x_i)², where m is mass per unit length and M_i are lumped masses. The assumed shape must satisfy the geometric (essential) boundary conditions: zero deflection and zero slope at a clamp, zero deflection at a pin. It need not satisfy the force (natural) conditions, but shapes that also do so give better answers.

Upper bound. Any admissible shape constrains the structure to deform in a way it would not choose, which acts like extra stiffness. Rayleigh's quotient is therefore stationary at the true modes and always gives a frequency greater than or equal to the true fundamental. The lower the estimate, the better it is. Because the quotient is stationary, a shape error of a few per cent produces a much smaller frequency error.

Good trial shapes. The static deflection curve under the structure's own weight (applied in the direction of vibration) is an excellent choice, often within 1 % of the exact fundamental. Polynomials such as (x/L)² for a cantilever, or sin(πx/L) for a simply supported beam (exact in that case), are common.

Effective (equivalent) mass. Rayleigh's method gives the classic corrections: a spring of mass m_s adds m_s/3 to the end mass; a cantilever of mass m_b adds (33/140)·m_b ≈ 0.236·m_b to a tip mass; a simply supported beam adds about 0.49·m_b (17/35) to a central mass.

Rayleigh–Ritz method. Take W(x) = Σ a_j·ψ_j(x), a combination of n admissible functions, and require Rayleigh's quotient to be stationary with respect to each a_j. This gives ([K] − ω²·[M])·{a} = {0}, with K_ij = ∫EI·ψ_i″·ψ_j″·dx and M_ij = ∫m·ψ_i·ψ_j·dx (plus lumped-mass terms). The result is n approximate frequencies, each an upper bound on the corresponding exact one, which converge downward as functions are added. The lowest is accurate; the highest is usually poor.

Links. Rayleigh–Ritz with local piecewise functions is the finite-element method. In aeroelasticity, assumed bending and torsion shapes turn a wing into a few generalised coordinates for divergence and flutter calculations. A complementary lower bound is given by Dunkerley's formula, 1/ω₁² ≈ Σ 1/ω_ii², where ω_ii is the frequency with only mass i present.

Formulas

ω² = U_max/T* — Rayleigh's quotient; U_max maximum strain energy (J), T* = T_max/ω² (kg·m²).

U_max = ½·∫₀ᴸ EI·(d²W/dx²)²·dx — beam bending; EI flexural rigidity (N·m²), W assumed shape (m).

T* = ½·∫₀ᴸ m·W²·dx + ½·Σ M_i·W(x_i)² — m mass per unit length (kg/m), M_i lumped masses (kg).

ω² = Σ m_i·g·y_i / Σ m_i·y_i² — Rayleigh with static deflections y_i of lumped masses m_i.

K_ij = ∫EI·ψ_i″·ψ_j″·dx, M_ij = ∫m·ψ_i·ψ_j·dx, det([K] − ω²·[M]) = 0 — Rayleigh–Ritz.

m_eff = M + m_s/3 (spring), m_eff = M + 0.236·m_b (cantilever with tip mass M).

Exact uniform cantilever: ω₁ = 3.516·√(EI/(m·L⁴)), ω₂ = 22.03·√(EI/(m·L⁴)). Simply supported: ω_n = (nπ)²·√(EI/(m·L⁴)).

Worked examples

Example 1 (standard). A uniform cantilever has L = 2 m, m = 5 kg/m and EI = 2000 N·m². Estimate ω₁ with W = 1 − cos(πx/(2L)) and compare with the exact value.

  1. Admissibility: W(0) = 0 and W′(0) = 0 at the clamp. W″ = (π/(2L))²·cos(πx/(2L)) is zero at the free end, so the moment condition is also met.
  2. U_max = ½·EI·(π/(2L))⁴·∫cos²(πx/(2L))dx = ½·EI·(π/(2L))⁴·(L/2).
  3. T* = ½·m·∫(1 − cos)²dx = ½·m·L·(3/2 − 4/π) = ½·m·L × 0.22676.
  4. ω² = (π⁴/32)/0.22676 × EI/(m·L⁴) = 3.0440/0.22676 × EI/(m·L⁴) = 13.42·EI/(m·L⁴), so ω₁ ≈ 3.664·√(EI/(m·L⁴)).
  5. √(EI/(m·L⁴)) = √(2000/(5 × 16)) = √25 = 5 rad/s, so ω₁ ≈ 18.32 rad/s. Exact: 3.516 × 5 = 17.58 rad/s.

Answer: ω₁ ≈ 18.3 rad/s (2.92 Hz), 4.2 % above the exact 17.6 rad/s, as an upper bound must be.

Example 2 (GATE level). Use Rayleigh–Ritz with ψ₁ = (x/L)² and ψ₂ = (x/L)³ for a uniform cantilever. Find the first two frequency coefficients, then ω₁ for L = 1.5 m, m = 10 kg/m, EI = 2.0 × 10⁴ N·m².

  1. With ξ = x/L: ψ₁″ = 2/L², ψ₂″ = 6ξ/L². K₁₁ = (EI/L³)·∫4dξ = 4·EI/L³, K₁₂ = (EI/L³)·∫12ξdξ = 6·EI/L³, K₂₂ = (EI/L³)·∫36ξ²dξ = 12·EI/L³.
  2. M₁₁ = mL·∫ξ⁴dξ = mL/5, M₁₂ = mL/6, M₂₂ = mL/7.
  3. With λ = ω²·m·L⁴/EI: det([[4 − λ/5, 6 − λ/6], [6 − λ/6, 12 − λ/7]]) = 0 → λ²/1260 − (34/35)·λ + 12 = 0 → λ² − 1224·λ + 15 120 = 0.
  4. λ = (1224 ∓ √1 437 696)/2 = 12.48 and 1211.5. Coefficients √λ: 3.533 and 34.81 (exact 3.516 and 22.03).
  5. √(EI/(m·L⁴)) = √(20 000/(10 × 5.0625)) = 19.88 rad/s, so ω₁ = 3.533 × 19.88 = 70.2 rad/s (11.2 Hz).

Answer: ω₁ ≈ 70.2 rad/s, only 0.5 % above exact (69.9 rad/s); the two-term second frequency (34.8 against 22.0) is poor, as expected for the highest Ritz mode. The one-term (x/L)² estimate would give √20 = 4.47, 27 % high.

Common mistakes

  • Using a trial shape that violates a geometric boundary condition (for example sin(πx/L) for a cantilever); the answer can then fall below the true value and is meaningless.
  • Forgetting lumped masses in T*, or squaring EI instead of W″.
  • Writing T_max = U_max without separating ω²: the quotient divides by T*, not by T_max.
  • Expecting Rayleigh's method to give accurate higher modes; it targets the fundamental.
  • Treating a higher Rayleigh estimate as "safer": the lowest admissible estimate is the most accurate.

For GATE AE

Questions ask for the fundamental frequency of a beam with an assumed polynomial or trigonometric shape, the effective mass of a spring or beam (m_s/3, 0.236·m_b), Rayleigh with static deflections of lumped masses, the upper-bound property and admissibility of trial functions. Practise the integrals of polynomials and of sin² and cos² quickly.

Quick check

  1. Is Rayleigh's estimate above or below the true fundamental frequency?
  2. Which boundary conditions must a trial function satisfy?
  3. A spring of mass 0.3 kg carries a 1 kg mass. What effective mass do you use?
  4. For a simply supported beam with W = sin(πx/L), is Rayleigh's answer exact?
  5. Cantilever, W = (x/L)²: what is the frequency coefficient?

Answers: 1. above (or equal); 2. the geometric (essential) ones; 3. 1.1 kg; 4. yes, it is the exact mode shape; 5. √20 = 4.47.

Interview questions

All Structural Dynamics and Aeroelasticity interview questions

Try answering each one aloud before you open it.

  1. 1.What is the Rayleigh method in structural dynamics?Concept

    The Rayleigh method is an approximate technique used to estimate the natural frequencies of a system. It involves assuming a mode shape for the system and using energy principles to calculate the frequency. The method is particularly useful for systems where an exact solution is difficult to obtain.

  2. 2.Explain the Rayleigh-Ritz method and its application in structural dynamics.Concept

    The deflection is written as a combination of several admissible functions, W(x) = Σ a_j·ψ_j(x), each satisfying the geometric boundary conditions. Requiring Rayleigh's quotient (maximum strain energy over the kinetic-energy function) to be stationary with respect to every a_j gives a matrix eigenvalue problem ([K] − ω²[M]){a} = {0}, with K_ij = ∫EI·ψ_i″·ψ_j″dx and M_ij = ∫m·ψ_i·ψ_j dx. It yields several approximate frequencies and mode shapes, each an upper bound that improves as functions are added, and it is used for tapered wings, plates and assumed-mode aeroelastic models; with piecewise local functions it becomes the finite-element method.

  3. 3.Why is the Rayleigh method used in aeroelasticity?Application

    The Rayleigh method is used in aeroelasticity to estimate the natural frequencies of aircraft structures, which is crucial for predicting flutter and other dynamic instabilities. By understanding these frequencies, engineers can design structures that avoid resonant conditions that could lead to failure.

  4. 4.What are the limitations of the Rayleigh method?Concept

    It gives essentially one frequency, the fundamental, and its accuracy depends on the assumed shape; it is poor for higher modes unless the shape is chosen to resemble them and is orthogonal to the lower ones. The estimate is always an upper bound, so it never shows by how much it is high unless compared with another estimate or a lower bound such as Dunkerley's. The trial shape must satisfy the geometric boundary conditions, otherwise the result is not a bound at all, and the method as normally used assumes a conservative, linear, undamped system.

  5. 5.How does the Rayleigh-Ritz method improve upon the Rayleigh method?Concept

    The Rayleigh-Ritz method improves upon the Rayleigh method by using a set of assumed mode shapes rather than a single one. This allows for a more accurate approximation of the system's behavior, especially for complex structures with multiple degrees of freedom. The method also provides a systematic way to improve the approximation by increasing the number of assumed mode shapes.

  6. 6.What happens if the assumed mode shape in the Rayleigh method is not close to the actual mode shape?Application

    If the assumed mode shape is not close to the actual mode shape, the estimated natural frequency will be inaccurate. The Rayleigh method relies heavily on the accuracy of the assumed mode shape, so a poor assumption can lead to significant errors in the frequency estimation.

  7. 7.In what scenarios is the Rayleigh-Ritz method particularly useful?Application

    The Rayleigh-Ritz method is particularly useful in scenarios where the structure is complex and analytical solutions are not feasible. It is also beneficial when dealing with structures that have multiple degrees of freedom, as it allows for a more comprehensive approximation of the system's behavior.

  8. 8.Calculate the approximate natural frequency of a cantilever beam using the Rayleigh method, given that the beam has a length of 2 m, a mass per unit length of 5 kg/m, and a flexural rigidity of 2000 Nm².Numerical

    Take an admissible shape satisfying W = W′ = 0 at the root, for example W = 1 − cos(πx/2L). Then U_max = ½EI(π/2L)⁴(L/2) and T* = ½mL(3/2 − 4/π), so ω² = (π⁴/32)/(0.2268) × EI/(mL⁴) = 13.42 EI/(mL⁴). With EI/(mL⁴) = 2000/(5 × 16) = 25 s⁻², ω ≈ √(13.42 × 25) = 18.3 rad/s (2.92 Hz). The exact value is 3.516 × 5 = 17.6 rad/s, so the estimate is about 4 % high, as expected of an upper bound.

  9. 9.Explain how energy principles are used in the Rayleigh-Ritz method.Concept

    For harmonic free vibration the maximum strain energy equals the maximum kinetic energy, which gives Rayleigh's quotient ω² = U_max/T*. In the Ritz method the shape is a sum of trial functions with unknown coefficients, and the coefficients are chosen to make this quotient stationary (equivalently, Hamilton's principle applied to the assumed-mode system). The stationarity conditions form the stiffness and mass matrices of a small eigenvalue problem, whose roots are upper bounds on the true natural frequencies.

  10. 10.What is the significance of choosing appropriate trial functions in the Rayleigh-Ritz method?Application

    Choosing appropriate trial functions in the Rayleigh-Ritz method is crucial because the accuracy of the approximation depends on how well these functions represent the actual mode shapes of the system. Poorly chosen trial functions can lead to inaccurate results, while well-chosen functions can provide a close approximation to the true behavior of the system.

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