When the input and response are measured at the same point, the transfer function is called a driving point transfer function. When the input and response are measured at different points, a spatial transfer function is obtained.

In modal analysis the transfer functions are usually obtained using a digital test system. The input and response measurements are digitized and then processed using fast Fourier transform techniques. The transfer function is calculated and stored as a finite number of data points. If any one of the six transfer functions listed above is obtained in this manner, any of the other transfer functions can be obtained through mathematical manipulation of the existing transfer function. Once the transfer functions are obtained, a modal analysis can be performed.

There are basically two levels of sophistication available in modal analysis. In the first level, only the natural frequencies and mode shapes are obtained. From this information, "pictures" of how the system is vibrating at its different natural frequencies can be determined and animated for better visualization.

For the second level of sophistication, in addition to the natural frequencies and mode shapes, the modal damping ratios and modal masses are also obtained. When all of these parameters have been obtained, a complete mathematical model of the system can be defined. The mathematical model may then be used in computer simulations to predict the response of the system when it is modified. A designer can now evaluate different design modifications without building prototypes.

In order to understand how mode shapes and modal parameters can be obtained from transfer functions, it is necessary to consider a specific transfer function. We'll begin with a single degree of freedom system.

The system differential equation is obtained using Newton's law: Sum of the forces acting on the mass is equal to the mass times its acceleration.

	(Figure 1)
or

Because the transfer function is defined in the frequency domain, a sinusoidal forcing function of the form f(t) = Fe^{jw t} and a solution of the form x(t) = Xe^{iw t} will be assumed where i = . The velocity and acceleration are obtained through differentiation.

By substituting in the assumed values and dividing out the time varying term (e^{iw t}), the following equation is obtained.

The equation can be solved for (transfer function).

where:

This equation can also be represented as a magnitude and a phase angle.

If the magnitude and phase are plotted as a function of frequency, a bode plot is obtained. The bode plot for this system is shown below for a small nonzero damping value.

(Figure 2)

The transfer function can also be separated into real and imaginary parts.

If the imaginary part of the transfer function is plotted as a function of the real part, a polar plot (argand plot) is obtained. This plot has the form of a circle.

(Figure 3)

If the real and imaginary parts are plotted as functions of frequency the coincident and quadrature responses are obtained, respectively.

(Figure 4)

All of these plots are important in modal analysis and will be used to demonstrate various methods for calculating natural frequencies, damping ratios, masses, and mode shapes. The methods for extracting these parameters will be discussed with reference to the two levels of sophistication mentioned earlier.

For the first level of sophistication, the parameters of interest are the natural frequencies and the mode shapes. For a single degree of freedom system there is only one natural frequency and the mode shape for this natural frequency can be viewed as the amplitude of the transfer function at the natural frequency.

The natural frequency can be approximated using either the coincident response or the bode plot. For the coincident response, the natural frequency occurs at the frequency where the real part is zero. (See Figure 3). Using the bode plot, the natural frequency can be obtained by determining the peak frequency and recalling that for a compliance transfer function:

If the damping ratio is small (), the peak frequency and the natural frequency can be approximated as equal. The damping ratio can be obtained, if desired, by using three different methods.

1. 
2. 

(For methods 1 and 2 see Figure 2)

3. 

(For method 3 see Figure 4)

The amplitude at the natural frequency is obtained from the quadrature response by measuring the height of the peak. (See Figure 4)

For the second level of sophistication, all of the modal parameters must be obtained. Because these parameters are used in the formulation of a mathematical model of the system, the accuracy of the estimations is very important. For this type of analysis, the designer must use a modal analysis computer program. The modal analysis program uses more sophisticated mathematical techniques for parameter estimation so that, when used properly, more accurate results are obtained. For obtaining natural frequencies and damping ratios, most computer programs use a least squares curve fitting routine to curve fit the transfer function. A solution of the form

is assumed and the parameters are varied using an iterative approach until a good fit is obtained. The values of A, w _n, and x are then known. For a single degree of freedom system A=1/k. The mass can then be obtained using the definition of the natural frequency.

As was mentioned earlier, the amplitude can be obtained from the quadrature response, but it can also be obtained from the polar plot. In the real and imaginary coordinate system, as the frequency increases (from zero to infinity for a single degree of freedom system), the transfer function maps out a circle. The diameter of this circle is the same as the peak amplitude of the quadrature response. With a modal analysis program the amplitude can be determined with greater accuracy by using a circular least squares curve fitting routine to curve fit the circle in the polar plot. A solution of the form

(x + a)² + (y + b)² = r²

is assumed. The parameters are varied until a good fit is obtained. The diameter can then be calculated.

The analysis of a single degree of freedom system can be extended to the analysis of a multi-degree of freedom system. For a linear system, the multi-degree of freedom transfer function can be represented as the sum of single degree of freedom transfer functions. (See Figure 5) The multi-degree of freedom transfer function can be represented in equation form as

where:

X_j = response at any point j

F_k = force at any point k

r = mode shape number

n = number of modes

mode shape coefficient (amplitude)

mode shape coefficient

measured at point K for the r^th mode

This equation shows that the transfer function can be represented mathematically if the mode shapes, modal masses, natural frequencies, and damping ratios are known.

If light damping and widely spaced peaks occur in the transfer function (se Figure 6), the simple parameter estimation techniques can be used for the level one modal analysis. If the peaks are closely spaced or if heavy damping exist (see Figure 7), the more sophisticated techniques for parameter estimation should be used. If a complete mathematical model is desired, the more sophisticated techniques must be used.

(Figure 5)

(Figure 6)

(Figure 7)

To perform a modal analysis of a multi-degree of freedom system, the designer must first obtain a set of system transfer functions. The number and location of the test points depends on the particular system being analyzed.

Once the transfer functions have been obtained, the modal parameters can be estimated using the appropriate methods. Because the natural frequencies and damping ratios are global properties, they will be the same for all of the transfer functions. Therefore, these parameters could be estimated from any one of the system transfer functions.

Having obtained the parameters of interest, the mode shapes can now be calculated. Figure 8 shows how the mode shapes, of a simple beam, could be obtained using the quadrature response. The mode shapes for the simple beam are shown in Figure 9. The same mode shapes would have been obtained using the circle fitting technique.

The points where the mode shapes cross the horizontal axis are called nodes. These are points where there is no motion. If one of the test points is at a node for one of the modes, the peak for that mode will not show up in the transfer function. Therefore, when estimating natural frequencies and damping ratios, a transfer function should be selected in which all the modes are present.

The points where the displacement is the largest are called antinodes. Once the modal parameters and the mode shapes have been determined, the dynamic characteristics of the system can be completely defined. This information could then be used to improve the system or to simply correct an existing problem. Either way, modal analysis provides the designer with a powerful tool for the dynamic analysis and design of mechanical systems.

(Figure 8)

(Figure 9)

A simply supported beam driven by a line force f(t) located at x = x_s.

System Equation from Newton's Law.

To get the transfer function, assume that f(t) is harmonic with frequency w such that f(t) = Fe^{jw t} where j = . The response v(x,t) will have the following form:

If we multiply the equation of motion by where m = 1, ... ¥ , integrate over x from 0 to L, and replace m with n we get the following.

for n = 1, ... ¥

Thus, if the beam transverse displacement is measured at x_r, then its response will be:

We define the above ratio as the frequency response function H_rs(w ) = V(x_r,w )/ F(x_s,w ).

Property of reciprocity: H_rs(w ) = H_sr(w ).

Fixed response approach. A sensor is kept fixed at x_r and the excitation point is moved from points x₁, ..., x_s, ..., x_N. By reciprocity we have values for H_rs(w ) for r = 1, ..., N and s = 1, ..., N.