From a Free Vibration Equation of a structure, we have [M}{ϋ(t)}+[K]{u(t)}={0}, and with a trial function {u}={φ} sin ωt , we will come to an Eigen Equation ( [K]-ω²[M] ) {φ} = 0. Solving this Eigen Equation we will get Eigen vector, {φ}, (mode shapes) and Eigen value, ω², (natural frequency).

So, theoritically the mode shapes {φ} and the natural frequency of a structure, ω, are determined only by the mass [M] and the stiffness [K] of the structure and independent from the Force.

Physically "The mode shape" is the shape of the deformed structure if it is excitated by a dynamic force which has the same frequency as the natural frequency of the structure. The mode shape has no unit.

In modal analysis, we make the deformation vector, {u(t)}, as a superposition of multiplication from mode shapes, {φⁿ}, and a time function, qⁿ(t), i.e. : {u(t)}=Σ{φⁿ}qⁿ(t) , and the deformation vector has unit of length.

I think you can read more detail about this topics in Dynamics of Structures, by Anil K. Chopra or another books about Structural Dynamic, but according to my experience his book is the best.

In practice, if we measure the vibration of a structure, let's say by using accelerometer at certain locations, the displacement, {u(t)}, can be obtained by integrating its acceleration twice. The mode shapes, {φ} can be obtain by transforming their acceleration using Fourier Transform.

From the peaks of the Imaginary part of transformed function in frequency domain, we can get the mode shapes of every frequency included.

For more information and detail about this modal analysis and testing, you can read : "Theoritical and Experimental Modal Analysis" by Maia & Silva, and "Modal Testing, Theory, Practice and Application" by D.J. Ewins.

The amplitude is the same than or a multiple of the "mode shape value" (if my interpretation of how you define this term is correct) when the beam is oscillating exactly in the (eigen-)mode that corresponds to that mode shape.

But in general I would think, that the amplitude at a certain location can be determined from the superposition of (infinitely) many mode shapes.

Indeed like Jochen said, the vibration shape is something, and modal shapes are something else. Some theories suggest that vibration shape can be obtained from a combination of infinite number of modes. For modes, the shape is what matters, the magnitude has no meaning, you can pick any location along the beam give it a value of 1, or 1000, doesn't matter, but what matters is the relation of the modal values of other location with respect to that 1 or 1000 you assumed.

From a Free Vibration Equation of a structure, we have [M}{ϋ(t)}+[K]{u(t)}={0}, and with a trial function {u}={φ} sin ωt , we will come to an Eigen Equation ( [K]-ω²[M] ) {φ} = 0. Solving this Eigen Equation we will get Eigen vector, {φ}, (mode shapes) and Eigen value, ω², (natural frequency).

So, theoritically the mode shapes {φ} and the natural frequency of a structure, ω, are determined only by the mass [M] and the stiffness [K] of the structure and independent from the Force.

Physically "The mode shape" is the shape of the deformed structure if it is excitated by a dynamic force which has the same frequency as the natural frequency of the structure. The mode shape has no unit.

In modal analysis, we make the deformation vector, {u(t)}, as a superposition of multiplication from mode shapes, {φⁿ}, and a time function, qⁿ(t), i.e. : {u(t)}=Σ{φⁿ}qⁿ(t) , and the deformation vector has unit of length.

I think you can read more detail about this topics in Dynamics of Structures, by Anil K. Chopra or another books about Structural Dynamic, but according to my experience his book is the best.

In practice, if we measure the vibration of a structure, let's say by using accelerometer at certain locations, the displacement, {u(t)}, can be obtained by integrating its acceleration twice. The mode shapes, {φ} can be obtain by transforming their acceleration using Fourier Transform.

From the peaks of the Imaginary part of transformed function in frequency domain, we can get the mode shapes of every frequency included.

For more information and detail about this modal analysis and testing, you can read : "Theoritical and Experimental Modal Analysis" by Maia & Silva, and "Modal Testing, Theory, Practice and Application" by D.J. Ewins.

For free vibration of the beam (due to the beam natural frequency) you will get the mode shapes of the vibration in which every mode shape is proportional to an eigenvalue , and for a certain position which is related to a certain mode shape you can get the amplitude in every point using the equation of motion with or without defining a damping ratio to the system, this process is very easy using frequency analysis in ABAQUS , you can get all the values and compare to those you got via numerical solution.

Indeed, in free vibration of a beam, we have to know two things about eigenvalue problem: 1) Eigen values and 2) Eigen vector. Eigen value is related to mode frequency and Eigen vector is related to mode shape. Each mode frequency has a corresponding mode shape. The mode frequency is a value and is measured in Hertz. Mode shape values represent the displacement of the beam during vibration. The mode shape displacements values are relative and the maximum value is often taken as one.

There are two kinds of Eigenproblems. Damped and undamped vibration. In numerical simulation, we often neglect damping.

Dear Mr. Mohamed Abdo, you told that mode shape displacement values are relative.

Ok ! How to calculate the relative values & w.r.t. what value you considered ???

Dr Rupakhety is right to say that at resonance the deflection of a beam under a harmonic force excitation is a mode. However, this applies to a structure with proportioanl damping. If damping is not proportional, severa lexcitation forces must be applied in the right pattern to excite a single mode.

This is an important question that addresses the difference between mode shapes and operational deflection shapes (ODS). A mode shape is a natural characteristic (due interaction between inertia and stiffness) of the structure (eigen-function), and simply represents the structure’s deformation shape at a certain resonant frequency. The coordinates of point on the mode shape represent the position of such a point with respect to other points of the structure, and that is why mode shapes do not have units. Mode shapes can be determined numerically by solving the eigenvalue problem, or measured experimentally using Experimental Modal Analysis (EMA).

On the other hand, the operational deflection shapes (ODS) represent the forced structure deformation shape (due to applied forces), and can be measured at a resonant frequency or any other frequency. Operation deflection shapes have units, and therefore the amplitude represents the point’s deformation. ODS is simply the forced-response curve. It can also be experimentally measured using Operational Modal Analysis (OMA). I hope my answer would help clarify this point.

Dear Mr. Ranjan: Relative displacement is often related to the maximum value. Thus, the maximum value of the mode shape is 1.0, other values of course will be less than one. One may take the maximum value as 100 or 1000, as mentioned above. I hope my answer is satisfactory.

Mode shapes are dimensionless representations of the shape that the structure undergoes when vibrating at a specific frequency (the corresponding natural frequency of the mode shape). You can't get any information on the absolute displacement of the structure for a specific point, you can only get information on the relative position of the points of the structure for that specific natural mode.

I recommend the following websites. Modal analysis is explained in simple words, witch i specially useful when your just staring in this field.

http://sdasl.uml.edu/umlspace/mspace.html

Neglecting the damping term is not physically true because it corresponds to the dissipative energy in which atoms are interacting with each others. This phenomenon is not easier to describe since there is not only two particles or atoms in interaction for an isotropic material. The problem becomes difficult if we have a mixture of materials or if the material is complex like concrete (made of cement, sand, gravels, and others). The free or excited motion of the structure will stop vibrating after a time duration even if it has not any additional external damping. If you touch your structure after vibrating you will find it hot. We can ask a question is it the strain energy resulting from internal forces corresponding to stresses or is there another term of energy corresponding to the calorific one as it is observed in chocs where the kinetic energy is transformed after stopping the movement when using the theorem of energy conservation.

Hi All,

Suggest the originator reads Modal Analysis by David Ewins of Imperial. This text will answer all of his worries.

Regards

The mode shape value is certainly different from the amplitude value at a particular point of a vibrating system. You must understand that a vibrating beam will in general consist of a linear combination of 2 or more modes of the beam or even all the modes depending on how the beam was initially excited.

A structural system usually has an infinite number of natural frequencies but when the lumped-mass approach is used this is reduced to a finite number of natural frequencies. The eigenvalue of the dynamic stiffness matrix, lambda, can be configured to equal omega^2 , where omega is the circular natural frequency.

A mode shape is obtained to be the eigen vector corresponding to an eigen value (natural frequency) of the dynamic stiffness matrix of the beam. It is essential that one dof be given a prescribed value (say one or hundred or one thousand units) in order to define the eigen vector. The decision as to the prescribed dof lies with the analyst. Thus, a mode shape value consists of the vibration amplitude when the beam is undergoing vibration in only one of its natural frequencies. A beam can be constrained to vibrate in a mode shape by simply making its initial deflection configuration to be proportional to the eigen vector when oscillation is initiated.

A beam undergoing vibration in a mode can be described simply by

y(x,t) = A_1*Sin(w_n*t)

A beam undergoing complex vibration will vibrate with frequencies containing a number of the natural frequencies viz:

y(x,t) = A_1*Sin(w_1*t) + A_2*Sin(w_2*t) +A_3*Sin(w_3*t) +........

with A_3 being less than A_2 that is in turn less than A_1.

In conclusion the mode shape value will be different from the amplitude value.

The value would be the same if there is no coupling with torsion or with the lateral component. Otherwise the mode shape will differ based on where you determine location of the mode shape in section or plan.

Dear Mr.Ranjan,

My understanding of modal shapes is that it denotes the relative displacement (if you are measuring displacement of a building due to horizontal loads like earthquake) between the adjacent masses so, the maximum amplitude of a modal shape can be anything from unity to a million or more. But the important thing is the ratio of two amplitudes which tells us that if the first body has moved by unity, then the other one has displaced by the ratio calculated above.

You would need to formulate a differential equation for displacement as a function of time, position and natural circular frequency of the beam if it a Single degree of freedom system. You may need to solve the differential equation of the form ma+cv+kx=P(t) here a, v, x are a function of displacement and time and c and k are the damping and stiffness of the beam.

There can be infinite number of modes for a vibration case but the contribution of each of them to the final result keeps decreasing. One needs to find the eigen values in order to get the different modes. Refer a specialised book like Mario Paz for Structural Dynamics for more details. 

Hope this helped sir. 

Thank you for good information.