Abstract
In Chaps. 1–5, we assumed a model and then used that model to determine the system response in the time or frequency domain (or both). More often, however, we have an actual dynamic system and would like to build a model that we can use to represent its vibratory behavior in response to some external excitation. For example, in milling operations, the flexibility of the cutting toolholderspindlemachine structure (and sometimes the workpiece) determines the limiting axial depth of cut to avoid chatter, a selfexcited vibration [1]. In this case, the dynamic response at the free end of the tool (and/or at the cutting location on the workpiece) is measured. Using this measured response, a model in the form of modal parameters can be developed for use in a timedomain simulation of the milling process. How can we work this “backward problem” of starting with a measurement and developing a model? To begin, we need to determine the modal mass, stiffness, and damping values from the measured frequency response function (FRF).
Keywords
 Peak picking
 Model development
 Multiple degrees of freedom
 Rigid body mode
 Node
 Mass matrix
 Stiffness matrix
 Damping matrix
 Linearized pendulum
Il n'est pas certain que tout soit incertain.
(It is not certain that everything is uncertain.)
—Blaise Pascal
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Notes
 1.
In timedomain, or timemarching, simulation, the equations of motion that describe the process behavior are solved at small increments in time using numerical integration [1].
 2.
Let’s begin!
 3.
Shell mills are typically used to machine large flat surfaces.
 4.
One structural modification technique we’ve already discussed is the addition of a dynamic absorber (Sect. 5.4).
 5.
The BEP was introduced in Sect. 2.6.
 6.
There are also other modes of vibration, but we’ll discuss these in Chap. 8.
 7.
The number of nodes is equal to i + 1 for a freefree beam.
 8.
Because the foam base is much more flexible than the beam, freefree conditions are approximated. Alternately, we could support the beam using flexible bungee cords. Both techniques are applied in practice.
 9.
The number of nodes is equal to i − 1 for a fixedfree beam.
 10.
In a square matrix the number of rows and columns is the same.
References
Schmitz, T. and Smith, K.S., 2019, Machining dynamics: frequency response to improved productivity, 2nd ed, Springer, New York
Blevins RD (2001) Formulas for natural frequency and mode shape. Krieger Publishing Co, Malabar, FL, Table 81
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Exercises
Exercises

1.
For a single degree of freedom springmassdamper system subject to forced harmonic vibration, the measured FRF is displayed in Figs. P6.1a and P6.1b. Using the peak picking method, determine m (in kg), k (in N/m), and c (in N s/m).

2.
The direct and cross FRFs for the two degree of freedom system shown in Fig. P6.2a are provided in Figs. P6.2b and P6.2c.

(a)
If the modal damping ratios are ζ_{q1} = 0.01 and ζ_{q2} = 0.016, determine the modal stiffness values k_{q1} and k_{q2} (N/m) by peak picking.

(b)
Determine the mode shapes by peak picking.

(a)

3.
A FRF measurement was completed to give the two degree of freedom response shown in Fig. P6.3. Use the peak picking approach to identify the modal mass, stiffness, and damping parameters for the two modes. Arrange your results in the 2 × 2 modal matrices m_{q}, c_{q}, and k_{q}.

4.
A FRF measurement was completed to give the two degree of freedom response shown in Fig. P6.4 (a limited frequency range is displayed to aid in the peak picking activity).

(a)
Use the peak picking approach to identify the modal mass, stiffness, and damping parameters for the two modes. Arrange your results in the 2 × 2 modal matrices m_{q}, c_{q}, and k_{q}.

(b)
The FRF “measurement” in part a) was defined using the following Matlab^{®} code.
kq1 = 6e6; % N/m kq2 = 6e6; % N/m omega_n1 = 475*2*pi; % rad/s omega_n2 = 525*2*pi; zetaq1 = 0.04; zetaq2 = 0.04; omega = 0:1000*2*pi; % rad/s r1 = omega/omega_n1; r2 = omega/omega_n2; realQ1_R1 = 1/kq1*(1r1.^2)./((1r1.^2).^2 + (2*zetaq1*r1).^2); imagQ1_R1 = 1/kq1*(2*zetaq1*r1)./((1r1.^2).^2 + (2*zetaq1*r1).^2); realQ2_R2 = 1/kq2*(1r2.^2)./((1r2.^2).^2 + (2*zetaq2*r2).^2); imagQ2_R2 = 1/kq2*(2*zetaq2*r2)./((1r2.^2).^2 + (2*zetaq2*r2).^2); realX1_F1 = realQ1_R1 + realQ2_R2; imagX1_F1 = imagQ1_R1 + imagQ2_R2; freq = omega/2/pi; figure(1) subplot(211) plot(freq, realX1_F1, 'k') set(gca,'FontSize', 14) axis([200 800 2e6 2e6]) ylabel('Real({X_1}/{F_1}) (m/N)') grid subplot(212) plot(freq, imagX1_F1, 'k') set(gca,'FontSize', 14) axis([200 800 3e6 5e7]) xlabel('Frequency (Hz)') ylabel('Imag({X_1}/{F_1}) (m/N)') grid
Plot your modal fit together with the measured FRF and comment on their agreement.

(a)

5.
Figures P6.5a, P6.5b, P6.5c, P6.5d, and P6.5e show direct, \( \frac{X_1}{F_1} \), and cross FRFs, \( \frac{X_2}{F_1} \) through \( \frac{X_5}{F_1} \), measured on a fixedfree beam. They were measured at the beam’s free end and in 20 mm increments towards its base; see Fig. P6.5f. Determine the mode shape associated with the 200 Hz natural frequency.
When plotting the mode shape, normalize the free end response at coordinate x_{1} to 1 (this is normalizing the mode shape to x_{1}) and show the relative amplitudes at the other coordinates x_{2} through x_{5}. See Fig. P6.5f

6.
For the same fixedfree beam as Problem 5, the measurement bandwidth was increased so that the first three modes were captured. Again, the direct, \( \frac{X_1}{F_1} \), and cross FRFs, \( \frac{X_2}{F_1} \) through \( \frac{X_5}{F_1} \), were measured. The imaginary part of the direct FRF for the entire bandwidth is shown in Fig. P6.6a. The three natural frequencies are 200, 550, and 1250 Hz.
Use Figs. P6.6b, P6.6c, P6.6d, P6.6e, and P6.6f to identify the mode shape that corresponds to the 1250 Hz natural frequency. Plot your results using the same approach described in Problem 5.

7.
Find the mass matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.7 using the shortcut method described in Sect. 6.6 if m_{1} = 10 kg and m_{2} = 12 kg.

8.
Find the stiffness matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.7 using the shortcut method described in Sect. 6.6 if k_{1} = 2 × 10^{5} N/m, k_{2} = 2 × 10^{5} N/m, and k_{3} = 1 × 10^{5} N/m.

9.
Find the mass matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.9 using the shortcut method described in Sect. 6.6 if m_{1} = 10 kg and m_{2} = 12 kg.

10.
Find the stiffness matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.9 using the shortcut method described in Sect. 6.6 if k_{1} = 2 × 10^{5} N/m, k_{2} = 2 × 10^{5} N/m.
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Schmitz, T.L., Smith, K.S. (2021). Model Development by Modal Analysis. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/9783030523442_6
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DOI: https://doi.org/10.1007/9783030523442_6
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