Resonance and Modes in Bars

Natural frequencies and modal patterns of longitudinal, transverse and torsional vibration in bars with various boundary conditions.

Conceptual context

An elastic bar confined by its boundary conditions can only vibrate stably at certain discrete frequencies. These are the natural frequencies or resonance frequencies. Each natural frequency corresponds to a spatial vibration pattern called a normal mode.

The fundamental phenomenon is that traveling waves propagating through the bar reflect at the ends and, upon constructive superposition, form standing waves. Only when the bar length is an appropriate multiple of the half-wavelength is the superposition constructively sustained: this defines the allowed frequencies.

This leaf covers longitudinal modes (compression-extension along the axis), torsional modes (rotation about the axis) and mentions the connection with flexural modes (treated in detail in the bending waves in beams leaf). The fundamental boundary conditions are: free end (zero stress, maximum displacement) and fixed end (zero displacement, maximum stress).

🟢 Essential level

When you strike a metal bar, it vibrates at certain specific frequencies: it emits a sound with a definite pitch. Those frequencies depend on the length, the material and how the ends are supported.

The lowest frequency is called the fundamental or first harmonic. The following ones are called higher harmonics. If the bar is free at both ends, the harmonics are integer multiples of the fundamental (2f, 3f, 4f...). If one end is fixed and the other free, only odd harmonics appear (f, 3f, 5f...).

The longer the bar, the lower the frequency: a long bar sounds lower than a short one of the same material. The stiffer the material, the higher the frequency: steel sounds higher-pitched than a soft polymer.

The key idea is that the bar does not choose just any motion: only the patterns that fit its ends survive. This is why the sound simultaneously reveals the material, the geometry, and the way the bar is held.

🔵 Formal level

Basic formulas

Longitudinal standing waves

A longitudinal wave travels through the bar at velocity $v_L$. Upon reaching an end it reflects. The superposition of incident and reflected waves creates a standing wave if the frequency is appropriate.

For a free-free bar, both ends are displacement antinodes (stress nodes). The resonance condition requires that the length contains an integer number of half-wavelengths $\lambda_n$:

Natural longitudinal frequency of a free-free bar

where $n$ is the mode number (1, 2, 3...), $v_L$ is the longitudinal velocity and $L$ is the length.

The longitudinal velocity in a slender bar is calculated as:

Longitudinal wave velocity in a slender bar

where $E$ is the Young modulus and $\rho$ is the material density.

For a fixed-free bar, the clamped end is a displacement node and the free end is an antinode. Only odd harmonics fit:

Natural longitudinal frequency of a fixed-free bar

The fundamental of the fixed-free bar has half the frequency of the free-free fundamental of the same length: the effective wavelength is four times the bar length versus two times in the free-free case.

Real student questions

Why does a fixed-free bar only have odd harmonics? Because the fixed end forces a displacement node and the free end an antinode. For both conditions to be satisfied simultaneously, only odd quarter-wavelengths fit within the length: one quarter-wavelength (1st mode), three quarters (2nd mode), five quarters (3rd mode), etc.

Can I measure Young modulus by measuring the resonance frequency? Yes. The fundamental frequency of a bar of known length and density is measured, the free-free relation is used to obtain $v_L$, and then the longitudinal velocity relation gives $E$. This is the basis of the impulse excitation method (ASTM E1876 standard).

Why are torsional modes lower in frequency than longitudinal ones? Because torsional velocity depends on $G$ and longitudinal velocity on $E$. In common materials, shear stiffness is lower than longitudinal stiffness, so $v_T$ is lower than $v_L$. Lower velocity with the same length gives lower frequency.

Final synthesis

Natural frequencies of a bar are proportional to wave velocity and inversely proportional to length. Boundary conditions select which subset of harmonics can exist. Longitudinal modes use $v_L$, determined by $E$ and $\rho$, and torsional modes use $v_T$, determined by $G$ and $\rho$. The spectrum structure (equally spaced or odd-only) is a direct consequence of the symmetry or asymmetry of boundary conditions. Knowing resonance frequencies allows determining the elastic moduli of the material without destroying the sample.