Mode shape

In the study of vibration in engineering, a mode shape describes the expected curvature (or displacement) of a surface vibrating at a particular mode. To determine the vibration of a system, the mode shape is multiplied by a function that varies with time, thus the mode shape always describes the curvature of vibration at all points in time, but the magnitude of the curvature will change. The mode Shape is dependent on the shape of the surface as well as the boundary conditions of that surface.

Modes

A mode of vibration is characterized by a modal frequency and a mode shape, and is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one valley) it would be vibrating in mode 2.

In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have a radial coordinate and an angular coordinate. If you measured from the center outward along the radial coordinate you would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the antisymmetric (also called skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2-1 or 1-2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).

Each mode is entirely independent of all other modes. Thus all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes (with lower modes having greater amplitude).

Since the lower modes vibrate with greater amplitude, they cause the most displacement and stress in a structure. Thus they are called fundamental modes.

Nodes

In a one dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero. These nodes correspond to points in the mode shape where the mode shape is zero. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times.

When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about 1/3 of the

way from the center to the edge, and the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In a real system these lines would equal zero exactly, as shown to the right.

Vibrations :

Typically, an object will vibrate at several modes at once, thus, assuming linear behavior, the total displacement will be a superposition of the mode shapes of the individual modes. Each mode is multiplied by a different time function, such that all modes vibrate at a different frequency.

For example, a beam might have a mode shape of:

Where n is the mode number, x is the distance from a given end of the beam, and L is the overall length. The n subscript denotes that this is for a single n-th mode.

The time function may look like:

Where t is time and T is the period of vibration.

Thus the vibration for a given mode is given by:

Since the total vibration of the beam is given by the superposition of all modes, the total vibration for our example system is given by:

A. What is modal analysis?

Modal analysis studies the dynamic properties or “structural characteristics” of a mechanical structure under dynamic excitation:

1. resonant frequency 2. mode shapes 3. damping

To explain this in a simple manner, we’ll take a plate as a theoretical example. We’ll apply a force that varies in a sinusoidal fashion on one corner. Then, we’ll change the rate of oscillation (frequency rate) of the sinusoidal force, but the peak force stays the same. And then, we’ll measure the response of the excitation with an accelerometer attached to the other corner of the plate.

The measured amplitude can vary depending on the frequency rate of the input force. The response amplifies as we apply a force with a frequency rate that gets closer and closer to the system’s resonant or natural frequencies.

The resonant frequency is the frequency at which any excitation produces an exaggerated response. This is important to know since excitation close to a structure’s resonant frequency will often produce adverse effects. These generally involve excessive vibration leading to potential fatigue failures, damage to the more delicate parts of the structure or, in extreme cases, complete structural failure.

Example: when spinning, the washing machine’s drum vibrations induce such a powerful resonant frequency that the machine begins to actually move causing the door to spring open.

If we take the time data and transform it to the frequency domain using a Fast Fourier Transform algorithm to compute something called the “frequency response function”, we see the functional peaks that occur at the resonant frequencies of the system.

Left: example of a simple plate frequency response function

Right: example of online FRF curves in LMS Test.Lab during impact testing

Deformation patterns (bending, twisting …) at these resonant frequencies take on a variety of different shapes depending on the excitation force frequency. These deformation patterns are referred to as the structure’s mode shapes.

Example of second mode animation: Noise and vibration engineers appreciate the animated resonance display of LMS Test.Lab

Structural damping provides information about how quickly the structure dissipates vibrational energy and returns to rest when the excitation force is removed.

Example: in the well-known case of the Tacoma bridge, the damping was not high enough to absorb all the excitation energy. 

Modal analysis refers to a complete process including both an acquisition phase and an analysis phase. The structure is excited by external forces such as an impact hammer or shaker. In this case, we talk about experimental modal analysis.

Modal testing systems consist of transducers (typically accelerometers and force cells), an analog to digital converter or front-end to digitize the analog instrumentation signals and a host PC to review and analyze the data.

Impact Hammer Modal Testing

An ideal impact to a structure is a perfect impulse, which has an infinitely small duration, causing a constant amplitude in the frequency domain; this would result in all modes of vibration being excited with equal energy. The impact hammer test is designed to replicate this; however, in reality a hammer strike cannot last for an infinitely small duration, but has a known contact time. The duration of the contact time directly influences the frequency content of the force, with a larger contact time causing a smaller range of bandwidth. A load cell is attached to the end of the hammer to obtain a recording of the force. Impact hammer testing is ideal for small light weight structures; however as the size of the structure increases issues can occur due to a poor signal to noise ratio.

Shaker Modal Testing

A shaker is a device that excites the structure according to its amplified input signal. Several input signals are available for modal testing, but the sine sweep (chirp) and random frequency vibration profiles are by far the most commonly used signals.

Shakers can have an advantage over the impact hammer as they can supply more energy to a structure over a longer period of time.

Eigenvalue, eigenvector and eigenspaceIn mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of applied mathematics as diverse as finance and quantum mechanics.

In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with that eigenvector. An eigenspace is the set of all eigenvectors that have the same eigenvalue.

Mathematical definition

In linear algebra, there are two kinds of objects: scalars, which are just numbers, and vectors, which can be thought of as arrows, and which have both magnitude and direction (though more precisely a vector is a member of a vector space). In place of the ordinary functions of algebra, the most important functions in linear algebra are called "linear transformations", and a linear transformation is usually given by a "matrix", an array of numbers. Thus instead of writing f(x) we write M(v) where M is a matrix and v is a vector. The rules for using a matrix to transform a vector are given in the article linear algebra.

If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then the vector is called an eigenvector of that matrix. A vector which is "flipped" to point in the opposite direction is also considered an eigenvector. Each eigenvector is, in effect, multiplied by a scalar, called the eigenvalue corresponding to that eigenvector. The eigenspace corresponding to one eigenvalue of a given matrix is the set of all eigenvectors of the matrix with that eigenvalue.

Many kinds of mathematical objects can be treated as vectors: ordered pairs, functions, harmonic modes, quantum states, and frequencies are examples. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenstate, and eigenfrequency.

Example

If a matrix is a diagonal matrix, then its eigenvalues are the numbers on the diagonal and its eigenvectors are basis vectors to which those numbers refer. For example, the matrix

stretches every vector to three times its original length in the x-direction and shrinks every vector to half its original length in the y-direction. Eigenvectors corresponding to the eigenvalue 3 are any multiple of the basis vector [1, 0]; together they constitute the eigenspace corresponding to the eigenvalue 3. Eigenvectors corresponding to the eigenvalue 0.5 are any multiple of the basis vector [0, 1]; together they constitute the eigenspace corresponding to the eigenvalue 0.5. In contrast, any other vector, [2, 8] for example, will change direction. The angle [2, 8] makes with the x-axis has tangent 4, but after being transformed, [2, 8] is changed to [6, 4], and the angle that vector makes with the x-axis has tangent 2/3.

Definition

Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation

for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.

A acts to stretch the vector x, not change its direction, so x is an eigenvector of A.

The key equation in this definition is the eigenvalue equation, Ax = λx. That is to say that the vector x has the property that its direction is not changed by the transformation A, but that it is only scaled by a factor of λ. Most vectors x will not satisfy such an equation: a typical vector x changes direction when acted on by A, so that Ax is not a multiple of x. This means that only certain special vectors x are eigenvectors, and only certain special scalars λ are eigenvalues. Of course, if A is a multiple of the identity matrix, then no vector changes direction, and all non-zero vectors are eigenvectors.

Geometrically (Fig. above), the eigenvalue equation means that under the transformation A eigenvectors experience only changes in magnitude and sign—the direction of Ax is the same as that of x. The eigenvalue λ is simply the amount of "stretch" or "shrink" to which a vector is subjected when transformed by A. If λ = 1, the vector remains unchanged (unaffected by the transformation). A transformation I under which a vector x remains unchanged, Ix = x, is defined as identity transformation. If λ = −1, the vector flips to the opposite direction; this is defined as reflection.

If x is an eigenvector of the linear transformation A with eigenvalue λ, then any scalar multiple αx is also an eigenvector of A with the same eigenvalue. Similarly if more than one eigenvector shares the same eigenvalue λ, any linear combination of these eigenvectors will itself be an eigenvector with eigenvalue λ. Together with the zero vector, the eigenvectors of A with the same eigenvalue form a linear subspace of the vector space called an eigenspace.

The eigenvectors corresponding to different eigenvalues are linearly independent meaning, in particular, that in an n-dimensional space the linear transformation A cannot have more than n eigenvectors with different eigenvalues.

Vibration analysis

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies of vibration, and the eigenvectors determine the shapes of these vibrational modes. The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis.

Tensor of inertia

In mechanics, the eigenvectors of the inertia tensor define the principal axes of a rigid body. The tensor of inertia is a key quantity required in order to determine the rotation of a rigid body around its center of mass.

Stress tensor

In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.