Glossary term
Mode Shape
Relative deformation pattern associated with a natural frequency of a vibrating structure, used in modal analysis, testing and dynamic validation.
Definition
quantityA mode shape is the relative deformation pattern associated with a natural frequency of a vibrating structure or dynamic system.
A mode shape describes how the coordinates of a structure move relative to one another in a specific vibration mode. It is commonly represented as an eigenvector or modal vector. Its scale and sign are usually arbitrary unless a normalization convention is stated, so a plotted mode shape shows deformation pattern, not physical displacement amplitude by itself.
A mode shape is the relative deformation pattern associated with a vibration mode. In a linear structural model, undamped free vibration is often written as:
where [K] is stiffness matrix, [M] is mass matrix, \omega_n is natural circular frequency and \phi is the mode-shape vector. Each mode has a frequency and a corresponding shape. The shape says how different coordinates move relative to one another.
A mode shape is usually not an absolute displacement field. The same physical shape may be normalized to unit tip displacement, unit modal mass, maximum component equal to 1, or another convention. Multiplying the vector by a nonzero constant, including -1, does not change the physical mode shape.
Engineering Role
Mode shapes tell engineers where a structure moves, bends, twists or strains when a mode is excited. They are essential for sensor placement, shaker placement, finite-element correlation, fatigue screening, flutter analysis, rotating-machinery critical-speed review, vibration isolation and structural modification decisions.
Frequency alone is not enough. A mode at 15\ \text{Hz} may be a local bracket motion, a global bending mode, a torsional mode or a control-surface mode. The engineering consequence depends on the shape. A mode that has little motion at a sensor may be missed; a mode with large motion at a fatigue-sensitive detail may control durability even if its frequency is not the lowest.
In modal superposition, physical response is often written as a sum of mode shapes multiplied by modal coordinates:
The vector \phi_i gives the pattern. The modal coordinate q_i(t) gives the amplitude for that mode in a particular response. Confusing the two is a common source of overinterpreting mode-shape plots.
Normalization and Coordinate Meaning
Mode-shape scale depends on the normalization convention. A maximum-component normalization may set:
Mass normalization may set:
Both can describe the same physical deformation pattern. They are not interchangeable in formulas for modal mass, participation factors, effective modal mass or forced response unless the convention is known.
The coordinates also matter. A measured shape from accelerometers may include only translational degrees of freedom at sensor locations. A finite-element shape may include translations and rotations at many nodes. Comparing them requires coordinate reduction, interpolation, sign convention, unit review and mode pairing.
Worked Example: Normalized Shape and Physical Amplitude
A simplified wing, beam or machine support has four measurement stations along its length. A first bending mode is normalized so that the tip component is 1:
| Station | Normalized mode-shape component |
|---|---|
| Root or fixed end | 0.00 |
| Inboard station | 0.35 |
| Mid-outboard station | 0.80 |
| Tip or free end | 1.00 |
During a response calculation, the modal coordinate for this mode is:
The physical displacement estimate at each station is:
Root or fixed end:
Inboard station:
Mid-outboard station:
Tip or free end:
If another software exports the same mode as:
the sign has changed but the deformation pattern is the same. The engineer should not interpret the sign reversal as a different physical mode unless the coordinate convention or phase reference has also changed.
Engineering comment: the normalized shape identifies where the structure moves relative to itself. The modal coordinate gives the amplitude for a specific response condition. A mode-shape plot without a scale, normalization and load or response context should not be read as a real displacement measurement.
If the same shape is scaled so that the maximum component is 2 instead of 1:
it is still the same physical shape if the modal coordinate is scaled consistently. The ratio between the inboard station and the tip remains:
That ratio, not the plotted scale, is the useful deformation-pattern information.
Distinction from Related Terms
Mode shape is not natural frequency. Natural frequency is the rate of free vibration; mode shape is the deformation pattern associated with that frequency.
Mode shape is not damping ratio. Damping ratio describes how quickly a mode decays; mode shape describes where and how it moves.
Mode shape is not modal assurance criterion. MAC compares two mode-shape vectors; it is not itself a mode shape.
Mode shape is not mesh deformation magnitude. Finite element plots often exaggerate deformation for visibility. The plotted shape may be scaled arbitrarily.
Mode shape is not a stress result, although it can indicate where strain or stress may concentrate when that mode is excited.
Validation and Common Mistakes
A defensible mode-shape record states the coordinate system, degrees of freedom, normalization, boundary conditions, mass state, sensor or node locations, frequency, damping estimate, mode-pairing evidence, and whether the shape is measured, computed or operationally identified.
Useful validation evidence includes ground vibration test data, frequency response functions, MAC comparisons, sensor layout, coherence, calibration, finite-element mesh and boundary-condition review, mass-property records and uncertainty in mode pairing. If the shape supports flutter, fatigue or critical-speed decisions, the record should identify which degrees of freedom and which structural details control the decision.
Common mistakes include:
- comparing mode-shape plots that use different boundary conditions;
- treating a plotted deformation scale as physical displacement;
- ignoring rotational degrees of freedom when they are important;
- assuming the first mode is always the only important mode;
- placing sensors near modal nodes where response is small;
- comparing mode shapes without coordinate alignment or sign convention;
- tuning a model to match frequency while leaving the mode shape physically wrong;
- using a mass-normalized vector as if it were a plotted displacement shape, or the reverse.