Consider a simple structure such as a beam, plate, ... Assume that you know how to determine the different modes of vibration of the structure. You are given an external input, such as a force or moment at a certain location. How can you then determine which modes will be excited how much? (Just a general question, I am not considering a specific case.)


3 Answers 3


They way to answer this question is transform the dynamics model into modal coordinates, and see what happens to the force term.

Suppose we can describe the stiffness and mass properties of the structure as matrices $\mathbf K $ and $\mathbf M$, and its displacement as a vector $\mathbf x$, in a physical coordinate system, and we apply a vector of forces that vary sinusoidally in time, $\mathbf F e^{i\omega t}$ to the structure.

The equation of motion of the system is then $$(\mathbf K - \omega^2 \mathbf M) \mathbf x e^{i\omega t} = \mathbf Fe^{i\omega t}$$

We can cancel the $e^{i\omega t}$ terms and reduce this to $$(\mathbf K - \omega^2 \mathbf M) \mathbf x = \mathbf F$$ (Note, I'm ignoring damping, for the sake of making things a bit simpler - doing that doesn't affect the final conclusion).

We can find the normal modes of the system and write the eigenvectors as a matrix $\mathbf \Phi$.

We can write the displacements $\mathbf X$ as a linear combination of the eigenvalues, i.e. $\mathbf X = \mathbf \Phi \xi$ where $\xi$ is a vector.

Substitute that in the equation of motion, and pre-multiply both sides by $\mathbf \Phi^T$ and we get $$\mathbf \Phi^T(\mathbf K - \omega^2 \mathbf M) \mathbf \Phi \xi = \mathbf \Phi^T \mathbf F$$ or $$(\mathbf \Phi^T\mathbf K \mathbf \Phi - \omega^2 \mathbf \Phi^T\mathbf M \mathbf \Phi) \xi = \mathbf \Phi^T \mathbf F$$

Now, if we use mass-normalized eigenvectors, we know that $\mathbf \Phi^T\mathbf M \mathbf \Phi$ is a unit matrix, and $\mathbf \Phi^T\mathbf K \mathbf \Phi$ is a diagonal matrix of the eigenvalues squared. So the matrix equation becomes a set of scalar equations, and for the $i$th mode we have $$(\omega_i^2 - \omega^2)\mathbf \xi_i = \mathbf \Phi_i^T \mathbf F$$

Translating that equation back into words answers the OP's question: For each mode, you take the scalar product of each eigenvalue with the applied forces to find the "modal component of the force" (i.e. right hand side term of the final equation), and from that you can find the relative amplitude of that mode (i.e. $\xi_i$).

In many cases we only apply a force to a single degree of freedom of the structure. Then, the result is fairly intuitive - there are two effects which are relevant, when they are taken together:

  1. Looking at the right hand side of the final equation, The modes that will be excited most are those with the biggest displacements at the point where the force is applied.

  2. Looking at the left hand side of the same equation, The modes that will be excited most are those whose natural frequencies are close to the forcing frequency.


Firstly, applying a constant load will not excite any modes. You need a time-varying load to induce vibration (e.g. wind, seismic, blast). Which modes are excited will depend on the frequency content of the applied loading.


For a static problem, you can determine the displacements of a structure using only forces and stiffness of the structure using static equations for text books:

$F_{normal} = K \Delta \\ M_{bending} = EI \phi \\ T_{torsion} = GJ \psi $

Note that none of these equations depend on time. For the purpose of most problems in civil engineering, load application is slow enough that inertia forces are very small compared to other forces, so they are neglected (appart from wind and earthquakes loads which often will require dynamic analysis).

The modes of vibration of a structure require dynamic analysis, so they do not depend only on the stiffness of the structure, they also depend on its weight distribution and to a lesser extent to it's damping. To determine the modes of vibration of a structure, you need to know at least its stiffness and its mass (and how it is distributed). The simplest example is a mass M on wheels tied to a horizontal spring of stiffness K. The natural frequency of this system would be :

$$f = 2 \pi\sqrt\frac{K}M $$

Note here that to obtain the natural frequency of the system, no external loads had to be applied. Since our case in unidirectional, it's quite simple to figure out which way an external load would have to go to excite our structure (at the calculated frequency of course).

For more complex case the mass is usually distributed (rather than punctual) and the stiffness is often in bending, shear and torsion in addition to axial. For these more complex cases, we will usually also be working with more than one degree of freedom, so you will have multiple equations with multiple variables to solve (usually using finite element analysis). In using finite element analysis, your mass will be represented as a matrix M of n by n elements and your stiffness by a matrix K of n by n elements, where n is your number of degrees of freedom. Using eigenvalues, we can solve an equation very similar to the one above to obtain n modes of vibration and n natural frequencies.

Each mode of vibration coupled with it's natural frequency gives you information on how to load each degree of freedom if you want to achieve resonance. Any other loading with any other frequency will not excite one mode alone, so you can only know the behaviour of any other system with more complex calculations.


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