Abstract
In Chap. 6 we solved the “backward problem” of starting with frequency response function (FRF) measurements and developing a model. However, we did not describe the measurement procedure. The basic hardware required to measure FRFs is:
Keywords
 Bandwidth
 Dynamic signal analyzer
 Sine sweep test
 Coherence
 Shaker
 Impact testing
 Accelerometer
 Capacitive sensor
 Laser vibrometer
 Euler integration
 NyquistShannon sampling theorem
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Notes
 1.
You may recognize this Doppler frequency shift as the increase in the pitch (frequency) of an approaching automobile’s horn and subsequent drop in pitch after the automobile passes you.
 2.
The prefix piezo is derived from the Greek word piezein, which translates “to squeeze”.
 3.
This follows from Newton’s second law, F = ma.
 4.
The discrete Fourier transform is applied because our inputs are sampled; they are not continuous in time.
 5.
This \( \frac{1}{k} \) term can be referred to as the DC compliance.
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Exercises
Exercises

1.
Complete the following statements.

(a)
Receptance is the frequencydomain ratio of _____________ to ____________.

(b)
Mobility is the frequencydomain ratio of _____________ to ____________.

(c)
Accelerance is the frequencydomain ratio of _____________ to ____________.

(a)

2.
Find three commercial suppliers of impact hammers for modal testing.

3.
Find three commercial suppliers of dynamic signal analyzers for modal testing.

4.
Digital data acquisition is to be used to record vibration signals for a particular system. If the highest anticipated frequency in the measurements is 5000 Hz, select the minimum sampling frequency.

5.
An impact test was completed using an instrumented hammer to excite a structure and an accelerometer to measure the vibration response.

(a)
Show how to convert from the accelerationtoforce frequency response function (i.e., accelerance) that was obtained to a displacementtoforce frequency response function (i.e., receptance).

(b)
What information is lost in this conversion?

(a)

6.
As described in Sects. 7.2 and 7.4, FRFs are often measured using impact testing. In this approach an instrumented hammer is used to excite the structure and a transducer is used to record the resulting vibration.
Use Euler integration to determine the displacement due to the triangular impulsive force profile shown in Fig. P7.6. The force excites a single degree of freedom springmassdamper system with m = 2 kg, k = 1.1 × 10^{6} N/m, and c = 83 N s/m. For the Euler integration, use a time step of 1 × 10^{−5} s and carry out your simulation for 0.2 s (20,000 points).

(a)
Plot both the force (N) versus time (s) and displacement (μm) versus time.

(b)
Determine the maximum displacement (in μm) and the time at which this displacement occurs.

(a)

7.
For a particular measurement application, an accelerometer must be selected with a bandwidth, or useful frequency range, of 5000 Hz. If the allowable deviation in the scaling coefficient \( {C}_A=\frac{1}{\sqrt{{\left(1{r}^2\right)}^2+{\left(2\zeta r\right)}^2}} \) is ±1% and the damping ratio is known to be 0.65, determine the minimum required natural frequency of the accelerometer.

8.
A single degree of freedom springmassdamper system which is initially at rest at its equilibrium position is excited by an impulsive force with a magnitude of 250 N over a time interval of 0.5 ms; see Fig. P7.8. If the mass is 3 kg, the stiffness is 3 × 10^{6} N/m, and the viscous damping coefficient is 120 N s/m, complete the following.

(a)
Determine x(t) using Eq. 3.44. Plot the response (in μm) over a time period of 0.3 s with a step size of 1 × 10^{−4} s in the time vector.

(b)
Determine x(t) using Euler integration. Use a time step of 1 × 10^{−4} s and carry out your simulation for 0.3 s (30,000 points). Plot x(t) (in μm) versus time.

(a)

9.
Determine the FRF for the system described in Problem 8 using Euler integration to calculate the timedomain displacement due to the impulsive input force. To increase the FRF frequency resolution, use a total simulation time of 1 s. Given the timedomain displacement and force vectors, use the Matlab^{®} function fft to calculate the complexvalued force transform, F, and displacement transform, X. Plot the real and imaginary parts (in m/N) of their ratio, X/F, versus frequency (in Hz). Use axis limits of axis([0 500 5e6 5e6]) for the real plot and axis limits of axis([0 500 1e5 1e6]) for the imaginary plot.

10.
The existence of modes with frequencies higher than the measurement bandwidth leads to an effect referred to as _______________ when performing a modal fit to the measured FRF.
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Schmitz, T.L., Smith, K.S. (2021). Measurement Techniques. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/9783030523442_7
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DOI: https://doi.org/10.1007/9783030523442_7
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