Vibration · Beam modes

Beam boundaries, bending modes and βₙ

How do end constraints change the way a beam bends?

Compare the first five nonzero bending modes under six combinations of free, clamped and simply supported ends, with the corresponding dimensionless roots βₙ.

Explore the bending modes

Start with something familiar

Imagine a ruler held over a table edge. Clamp one end, rest both ends on supports, or hang it softly: it is still the same ruler, but it is allowed to move in different ways. A boundary condition is simply the rule for what an end may do.

Free end

Translation and rotation allowed

Simple support

Translation restricted; rotation allowed

Clamped end

Translation and rotation restricted

Diagram · End constraints illustrated separately: free, simply supported and clamped. This is a schematic of allowed motion, not three measured mode shapes.

Notice what changes
A free end may move up and down and change its angle. A clamp stops both actions. A simple support stops vertical motion but still permits the end to turn.
Understand why
A mode shape is the outline of one pattern of bending. Its different parts move together at one natural frequency, but not by the same amount. Nodes are locations that remain still in that pattern.
Connect it to wood and making
Wood tapping assumes a particular support and bending pattern. Changing the grip can change that pattern or add loss. Choose the matching boundary model before translating a peak into a material property.

Put names to the ideas

The ends set the constraints

A clamped end prevents displacement and rotation. A simple support prevents displacement but allows rotation. A free end has neither bending moment nor shear force.

βₙ is a root, not a frequency

The dimensionless root βₙ depends on the boundary conditions and bending-mode number. Material properties and geometry then convert it into a natural frequency.

Count the bending modes

The first mode here means the first nonzero bending mode. Rigid-body translation or rotation at zero frequency is excluded from the numbering.

f_n=\frac{\beta_n^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}

Read the frequency equation as a recipe: β chooses the end constraints and mode; E and I describe resistance to bending; ρA describes mass per length; L describes length. β is a dimensionless multiplier, not a frequency in hertz.

A worked example

Free–free and clamped–clamped beams share the same nonzero roots in the ideal model, but their shapes and end motion differ. Equal roots do not imply identical mode shapes or identical experimental support conditions.

Explore it yourself

Choose an end-condition pair to compare all five bending modes. Pause the motion to inspect the shapes, nodal positions and βₙ values.

Change the supports and explore five bending modes

Choose the end conditions to compare mode shapes, nodes and βₙ. The supports determine whether each end can move or rotate, and set the characteristic roots.

Playback speedInternal node

Mode shapes use the same slow playback rate; animation speed does not represent frequency ratios. Each fₙ/f₁ label gives the actual modal frequency ratio for the selected supports.

Mode 1

β₁ = 4.73004f₁ / f₁ = 1.00

Mode 2

β₂ = 7.85320f₂ / f₁ = 2.76

Mode 3

β₃ = 10.99561f₃ / f₁ = 5.40

Mode 4

β₄ = 14.13717f₄ / f₁ = 8.93

Mode 5

β₅ = 17.27876f₅ / f₁ = 13.34

βₙ = kₙL. For a uniform, constant-section Euler–Bernoulli slender beam, frequency is proportional to βₙ squared:

f_n=\frac{\beta_n^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}},\qquad \frac{f_n}{f_1}=\left(\frac{\beta_n}{\beta_1}\right)^2

Numerical reference: β₁–β₅ for six boundary conditions

Constant-section Euler–Bernoulli beam · first five nonzero bending modes
End conditions	First $\beta_1$	Second $\beta_2$	Third $\beta_3$	Fourth $\beta_4$	Fifth $\beta_5$
Free–free	4.73004	7.85320	10.99561	14.13717	17.27876
Clamped–clamped	4.73004	7.85320	10.99561	14.13717	17.27876
Clamped–free (cantilever)	1.87510	4.69409	7.85476	10.99554	14.13717
Simply supported–simply supported	3.14159	6.28319	9.42478	12.56637	15.70796
Clamped–simply supported	3.92660	7.06858	10.21018	13.35177	16.49336
Simply supported–free	3.92660	7.06858	10.21018	13.35177	16.49336

Numbering begins with nonzero bending modes and excludes rigid-body zero-frequency modes. In one bending plane, a free–free beam has two such modes, translation and rotation; a simply supported–free beam has one rigid rotation about the support. Higher bending modes continue beyond the five listed here.

Free–free and clamped–clamped beams share the same nonzero roots but have different end constraints and mode shapes. For a simply supported beam at both ends, βₙ = nπ.

What do the three end constraints mean?

w is transverse displacement; primes indicate derivatives along the beam.

Clamped end: $w=0,\quad w'=0$
Displacement and slope are zero.
Simply supported end: $w=0,\quad w''=0$
Displacement and bending moment are zero; rotation is allowed.
Free end: $w''=0,\quad w'''=0$
Bending moment and shear force are zero.

Further demonstration: Dan Russell · Bending modes of beams ↗Return to the report: infer modulus from bending resonance

Check your understanding

Which end may rotate but cannot move vertically?A simple supportA clamp