B1. Analysis based on a modal search

B1. Analysis based on a modal search

A reminder of the concept of simple oscillator – Concept of the elastic response spectrum

In the following section, a lot of applied methods return the response of damped oscillators with a degree of freedom often named simple oscillator. 

Its general behavior is briefly reminded here. 

Its behavior is explained starting from the dynamic equation applied to an oscillator with a degree of freedom submitted to a time-dependent harmonic loading and building its transfer function with the following parameters: the ratio between its natural frequency and the natural frequency of the harmonic loading, and its critical damping ratio. 

The dynamic equilibrium equation applied to the mass-spring-damper with only one degree of freedom is reminded below:

It can also be written in its canonical form by dividing all the terms by the mass m such that:

If the harmonic loading is applied  x_s=X.e^jΩt, one can find the transfer function as an absolute response by establishing the ratio between the input loading and the absolute output response below:

The norm of this question is the amplitude of the transfer function that allows highlighting the dynamic amplification phenomenon illustrated for different values of the critical damping in figure 1. This transfer function’s independent variable is the phase.

Transfer function

A response spectrum is different from a transfer function. The curve giving the acceleration (named spectral acceleration) as a function of the period (or frequency) is called the spectrum of elastic response. The latter corresponds to the maximum absolute acceleration seen by a simple oscillator throughout time as a function of its natural period (or natural frequency) and its critical damping ratio. It dimensions the seismic motion. It is possible to construct an approximate relation between the Fourier spectrum of acceleration and the spectrum of spectral velocities for a damping ratio equal to zero. 

Starting from the dynamic equation of the simple oscillator subjected to an arbitrary acceleration: 

The maximum responses are deduced from its resolution according to time using the method of your choice (Duhamel integration, direct calculation…) to then calculate the relative spectral values of displacement, relative velocity, and absolute acceleration:

The concept of pseudo-displacement, pseudo-velocity, or pseudo-acceleration is often used, which means approximating these quantities, starting from one of them, based on the fact that the damping ratio is low (less than 20%).

A spectrum can be presented as shown in figure 2.

Spectrum

Concept of vibration modes

The structure’s response is based on a sum of harmonic responses, which are part of the domain of solutions q(t) of the equation:

They can therefore be written as follows:

If the Basile hypothesis is used (C is neglected), the algebraic solution becomes:

If the Basile hypothesis is not considered, one is now interested in finding complex modes such that:

and the eigenproblem can be resolved:

To get there without neglecting C, one had to postulate that:

With the matrices A and B being:

  and    

The vector ϒ is created such that: 

The solutions presented as pairs of combined complex harmonics are calculated as follows:

with:

• ψ_i: a complex modal deformation vector,

• λ_i: a complex natural frequency.

In any case, real or complex, one must solve an equation in λ_i² of the same degree N (2N for complex cases) than the degree of the matrix N (2N for complex cases). The order of this matrix is equal to the discretized system’s number of degrees of freedom (twice as much for complex cases). To solve the mentioned equation one must find the cases in which the determinant of this matrix and its characteristic polynomials are equal to zero.

To simplify the problem, the case of real vibration modes assuming the Basile hypothesis is considered.

For each natural frequency ω_i a modal shape vector is associated. The latter will be calculated by fixing one of the components to 1, then solving a system with n-1 parameters. 

Be careful! The modes are defined with a precision level of more or less a multiplying constant. Moreover, they are later normalized using various procedures. The most common for FE software being the normalization according to the mass matrix. This procedure is detailed in the following section. The modes can also be normalized using their greatest modal displacement.   

The vibration modes are orthogonal to the mass matrix, which implies that: 

When there are no loadings, the modes have no physical meaning. Different methods can then be used to normalize them to make their visualization more comprehensible. Thus, it is common in FE software to:

• normalize the vibration modes according to the mass matrix, such that:

• normalize the vibration modes according to a particular vibration mode.

For the cases in which the software does not normalize this value, it can be denoted by the generalized mass term. It can then be written that for the mode i:

The objective of this document is to explain case studies for which FE methods were applied, so it is important to remember that the vibration modes are normalized according to the mass matrix, and thus the generalized mass is always equal to 1.

The physical value called modal mass is also created, which will allow identifying the structure’s mass quantity that is dragged by a mode in a given direction Δ_k:

Please note: modal mass is a very different quantity from generalized mass, as the above equation shows.

Use of the modal basis

Time-dependent response by projection onto the modal basis:

A particular solution to the global equation: 

can be broken down on the bases of orthogonal modal shape vectors into N independent problems that are describing the response r_i(t) of a simple oscillator:

In the case of earthquakes, the loading f(t) is an inertial loading applied to the entire structure: 

Harmonic Analysis: 

For this type of analysis, the goal is to construct a function describing the response of the structure according to the frequencies at different nodes of the model. This response can be a quantity of arbitrary interest (displacement, velocity, acceleration – absolute or relative -, efforts…) as a function of a variety of input harmonic loadings that can also be described in various ways similar to the output quantity of interest. 

In this case, the imposed loading on an arbitrary set of nodes are of the harmonic type: 

The result of the analysis at all nodes of the structure is a complex transfer function whose norm (amplitude) and argument (phase) are the most generally used. They enable us to identify the resonant frequencies of a structure of equipment and to know the amplitude of its response according to different harmonic loadings. 

Modal basis truncation – General case

Usually, because it is not possible to extract all the modes using calculations (too computationally costly, even numerically impossible in some cases), one must focus on those that are most likely to respond to the loading of the structure. 

If the representative loading frequency saved is f_c (Hz), one must extract the modes up to 2f_c.

It is also important to make sure the modes inclined to contribute locally are well represented. 

A search for vibration modes must not be accompanied by interpreting the frequency, but by analyzing the modal shapes.