## Are My Hammer Impact Results Valid? (3) Time Invariance

As mentioned in our previous two articles, when performing experimental modal testing of a structure or component using impact (or ping) testing, it is important to consider and verify the validity of any results. A modal analysis is valid if it satisfies several assumptions.

This article considers time invariance. A test is said to be time-invariant if an impact test is performed today and then repeated sometime later and the results are identical.

Generally, the basic mathematical model of a single degree of freedom system is expressed as the following Equation:

$M\ddot{x}(t) + C\acute{x}(t) + Kx(t) = F(t)$

Where, M = Mass of the system; C = Damping of the system; K = Stiffness of the system; F(t) = Force; x(t) = displacement excited.

For an impact test, M, C, K are treated as constants that do not change within the short time that the transit excitation force is applied to the system. However, any of these parameters could change due to a variety of reasons. These could result from a change of energy stored in the test object, a change in damping or stiffness or a change in test conditions. Therefore, the transfer function of the system as the ratio of output energy and input energy over the whole frequency band could be changed.

To ensure repeatable results for an impact test, a time–invariance check is necessary. Even for a time-dependent case, it is helpful to have a better understanding of the test results.

Time-invariance check: For an impact test, dynamic characteristics of the test system shall ideally be time-independent. However, due to the test setup issue or test conditions changing, the dynamic characteristics of the structure may be affected and lead to non-repeatable results.

Typical considerations should be given to check if the system is time-invariant before the impact test:

• Mass load/distribution:  In an impact test, roving the accelerometer over the structure to obtain the frequency response function potentially changes mass distribution of the system. It may affect the dynamic characteristics of the structure, especially for lightweight structures. The weight of the accelerometer and position to place the accelerometer could have significant influence on the response measured. Therefore, when the accelerometer needs to be moved in batches for measurement, certain techniques may be considered to reduce this effect: choosing the lightweight accelerometer with acceptable sensitivity; distributing the weight of the sensor to the entire structure instead of a small local area.

The graph in Figure 1 below briefly shows the effect of the accelerometer weight on the frequency response function. The dominant resonant frequency shifts towards low frequency as the accelerometer’s weight increases.

Theoretically, the natural frequency of the basic SDOF system as defined in the equation above is expressed as

$f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{K}{M}}$

However, with an accelerometer (or accelerometers) added, the natural frequency of the system may be altered if the mass of the accelerometer ( or accelerometers) accounts for a significant portion of the total mass of the system, as stated below,

$f^{*} = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{K}{(M+M_{accel})}}$

the frequency decreased as the mass added increased.

• Stiffness of supporting fixture: dynamic characteristics of the structure will also be affected if the stiffness of the supporting fixture changes during the measurement. For example, performing impact test on an air bearing spindle when it is pressurized with different loads, the FRFs are not the same. Therefore, it is necessary to check if the supporting system’s stiffness varies during the measurement process.
• Temperature change: Some properties of the structure, such as material properties, may be affected by temperature, which will influence the dynamic characteristics of the test object. For example, when the modal test of a bridge, sometimes, it may have significant different results when it is performed in winter or in summer, in the morning or in the afternoon caused by temperature effects.

There are still other factors that result in the test being time-dependent.  Only by being aware of these effects can the modal test be carried out to ensure accurate and repeatable results.

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#### Dr Cindy (Xin) Wang

Applications Engineer at Prosig
Cindy has a BSc from the Dept of Jet Propulsion at Beijing University of Aeronautics and Astronautics, a Masters from the University of Sheffield in Mechanical Engineering and an MPhil/PhD from the University of Southampton for her thesis “Computational aeroacoustics of slat track system”. Cindy has extensive experience working in Automotive and joined Prosig in 2019 as an Applications Engineer.

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