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Model Development by Modal Analysis

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Abstract

In Chaps. 15, 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 tool-holder-spindle-machine structure (and sometimes the workpiece) determines the limiting axial depth of cut to avoid chatter, a self-excited 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 time-domain 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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Fig. 6.1
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Notes

  1. 1.

    In time-domain, or time-marching, simulation, the equations of motion that describe the process behavior are solved at small increments in time using numerical integration [1].

  2. 2.

    Let’s begin!

  3. 3.

    Shell mills are typically used to machine large flat surfaces.

  4. 4.

    One structural modification technique we’ve already discussed is the addition of a dynamic absorber (Sect. 5.4).

  5. 5.

    The BEP was introduced in Sect. 2.6.

  6. 6.

    There are also other modes of vibration, but we’ll discuss these in Chap. 8.

  7. 7.

    The number of nodes is equal to i + 1 for a free-free beam.

  8. 8.

    Because the foam base is much more flexible than the beam, free-free conditions are approximated. Alternately, we could support the beam using flexible bungee cords. Both techniques are applied in practice.

  9. 9.

    The number of nodes is equal to i − 1 for a fixed-free beam.

  10. 10.

    In a square matrix the number of rows and columns is the same.

References

  1. Schmitz, T. and Smith, K.S., 2019, Machining dynamics: frequency response to improved productivity, 2nd ed, Springer, New York

    Google Scholar 

  2. Blevins RD (2001) Formulas for natural frequency and mode shape. Krieger Publishing Co, Malabar, FL, Table 8-1

    Google Scholar 

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Exercises

Exercises

  1. 1.

    For a single degree of freedom spring-mass-damper 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. 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.

    1. (a)

      If the modal damping ratios are ζq1 = 0.01 and ζq2 = 0.016, determine the modal stiffness values kq1 and kq2 (N/m) by peak picking.

    2. (b)

      Determine the mode shapes by peak picking.

  3. 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 mq, cq, and kq.

  4. 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).

    1. (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 mq, cq, and kq.

    2. (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*(1-r1.^2)./((1-r1.^2).^2 + (2*zetaq1*r1).^2); imagQ1_R1 = 1/kq1*(-2*zetaq1*r1)./((1-r1.^2).^2 + (2*zetaq1*r1).^2); realQ2_R2 = 1/kq2*(1-r2.^2)./((1-r2.^2).^2 + (2*zetaq2*r2).^2); imagQ2_R2 = 1/kq2*(-2*zetaq2*r2)./((1-r2.^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 -2e-6 2e-6]) ylabel('Real({X_1}/{F_1}) (m/N)') grid subplot(212) plot(freq, imagX1_F1, 'k') set(gca,'FontSize', 14) axis([200 800 -3e-6 5e-7]) 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.

  5. 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 fixed-free 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 x1 to 1 (this is normalizing the mode shape to x1) and show the relative amplitudes at the other coordinates x2 through x5. See Fig. P6.5f

  6. 6.

    For the same fixed-free 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. 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 m1 = 10 kg and m2 = 12 kg.

  8. 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 k1 = 2 × 105 N/m, k2 = 2 × 105 N/m, and k3 = 1 × 105 N/m.

  9. 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 m1 = 10 kg and m2 = 12 kg.

  10. 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 k1 = 2 × 105 N/m, k2 = 2 × 105 N/m.

Fig. P6.1a
figure 43

Measured FRF

Fig. P6.1b
figure 44

Measured FRF (smaller frequency scale)

Fig. P6.2a
figure 45

Two degree of freedom spring-mass-damper system under forced vibration

Fig. P6.2b
figure 46

Direct FRF \( \frac{X_2}{F_2} \)

Fig. P6.2c
figure 47

Cross FRF \( \frac{X_1}{F_2} \)

Fig. P6.3
figure 48

Measured direct FRF for two degree of freedom system

Fig. P6.4
figure 49

FRF measurement for two degree of freedom system

Fig. P6.5a
figure 50

Direct FRF \( \frac{X_1}{F_1} \)

Fig. P6.5b
figure 51

Cross FRF \( \frac{X_2}{F_1} \)

Fig. P6.5c
figure 52

Cross FRF \( \frac{X_3}{F_1} \)

Fig. P6.5d
figure 53

Cross FRF \( \frac{X_4}{F_1} \)

Fig. P6.5e
figure 54

Cross FRF \( \frac{X_5}{F_1} \)

Fig. P6.5f
figure 55

Coordinates for direct and cross FRF measurements

Fig. P6.6a
figure 56

Direct FRF \( \frac{X_1}{F_1} \)

Fig. P6.6b
figure 57

Direct FRF \( \frac{X_1}{F_1} \) (mode 3 only)

Fig. P6.6c
figure 58

Cross FRF \( \frac{X_2}{F_1} \)

Fig. P6.6d
figure 59

Cross FRF \( \frac{X_3}{F_1} \)

Fig. P6.6e
figure 60

Cross FRF \( \frac{X_4}{F_1} \)

Fig. P6.6f
figure 61

Cross FRF \( \frac{X_5}{F_1} \)

Fig. P6.7
figure 62

Two degree of freedom system with a rigid, massless bar connecting the two masses, m1 and m2

Fig. P6.9
figure 63

Two degree of freedom system with a rigid, massless bar connecting the mass, m2, to a fixed pivot

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Schmitz, T.L., Smith, K.S. (2021). Model Development by Modal Analysis. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/978-3-030-52344-2_6

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  • DOI: https://doi.org/10.1007/978-3-030-52344-2_6

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