Vibration analysis (Overview)

Anne Schröder, winter semester 2015/16

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The vibration analysis combines measurement and analysis of vibrations in structures. Its purpose is to evaluate the system’s dynamic response at a particular operation point, predict vibration characteristics under different loading conditions and provide quality assurance (structural health monitoring) for the structure. This research method applies e.g. on the aerospace industry, telecommunications and civil engineering.

Motivation

The vibration analysis is employed in many sectors of industry. Undesired vibrations and structural flaws might decrease the product's quality during operation significantly. Vibration analysis helps the engineer to detect critical conditions and hence change the construction of the structure to overcome possible failure, misfunction or unwanted vibrations. Vibration analysis is also used during operation of a structure to provide a method of health monitoring.

Tab. 1: Pros and Cons of vibration analysis ^[1]

Fundamentals

• Equation of motion: describes the dynamic behavior of a mathematical representation of a real system

$//<![CDATA[ \begin{array}{l}M \ddot{x} (t) + D \dot{x} (t) + K x (t) = f (t)\end{array} //]]>$ (eq. 1)

$//<![CDATA[ \begin{array}{l}x(t)\end{array} //]]>$: position vector of discretized system

$//<![CDATA[ \begin{array}{l}M\end{array} //]]>$: mass matrix

$//<![CDATA[ \begin{array}{l}D\end{array} //]]>$: velocity dependent (sum of damping and gyroscopic) matrix

$//<![CDATA[ \begin{array}{l}K\end{array} //]]>$: position dependent (sum of stiffness and zirculatoric) matrix

$//<![CDATA[ \begin{array}{l}f(t)\end{array} //]]>$: vector of external forces

• Mode shapes / deflection shapes: Modes characterize the energy distribution in vibrating structures. Generally, mode shapes represent the possible, generic deflection of the structure under consideration, such as bending, longitudinal and torsional modes of amplitudes varying orders.
• Wave: spacial and temporal description of propagation of vibrations ^[2]
• White noise: signal that has the expectancy value zero; includes all frequencies and features a plain frequency spectrum, that is the power spectral density of the white noise is a constant function over all frequencies
• Young's modulus (also: modulus of elasticity) $//<![CDATA[ \begin{array}{l}E\end{array} //]]>$ in [$//<![CDATA[ \begin{array}{l}N/m^2\end{array} //]]>$]: is the specific material property that relates the stress distribution to a given strain distribution in a structure in the linear regime, i.e. for fully reversible deformations

$//<![CDATA[ \begin{array}{l}E = \frac{\sigma}{\epsilon}\end{array} //]]>$ (eq. 2)

$//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$: stress distribution in the specimen

$//<![CDATA[ \begin{array}{l}\epsilon\end{array} //]]>$: strain distribution in the specimen

$//<![CDATA[ \begin{array}{l}E = \frac{9KG}{3K + G}\end{array} //]]>$ (eq. 3)

• Shear modulus (also: modulus of rigidity) $//<![CDATA[ \begin{array}{l}G\end{array} //]]>$ in [$//<![CDATA[ \begin{array}{l}N/m^2\end{array} //]]>$]: is a specific constant for each material, which makes statements about linear-elastic deformations in cause of shear stress
  Bulk modulus $//<![CDATA[ \begin{array}{l}K\end{array} //]]>$ in [$//<![CDATA[ \begin{array}{l}N/m^2\end{array} //]]>$]: describes the elastic deformation respectively volume changes due to pressure loading
• Poisson’s ratio $//<![CDATA[ \begin{array}{l}v\end{array} //]]>$: is a material constant which characterizes the deformation perpendicular to the direction of a prescribed strain $//<![CDATA[ \begin{array}{l}σ\end{array} //]]>$. Poissons's ratio of 0.5 represents an incompressible material

Excitation of the structure

Vibrations of structures emerge from energy put into the system. Excitation mechanisms can be an impact, frequency sweeps, stochastic or periodic excitation. The dynamic response of the structure depending on the excitation type can be classified in:

Free oscillation

An external force acts on the structure for a short period of time. This excitation contains a large frequency spectrum and thus excites a large number of the system's Eigenfrequencies which can then be detected out of the structure response. Also, the regular operation deflection shapes can be measured. Examples for this kind of excitation are the Impact Echo Method and the modal hammer analysis.

Forced vibration

An excitation vibration is affection the structure, so it is adopting the vibration over a longer time period. The dynamical response is related to the force in terms of frequency and amplitude. Examples in practice can be found through excitation by a Shaker or a turning lathe (imbalance leads to vibrations).

Sensors

Sensors record vibrations of structures. After receiving these they immediately transform the vibrations into electric signals for a later treatment and discussion. For the recording the following sensors can be used:

• Laser Doppler Vibrometer (LDV): is an optical gauge for quantification of displacements at high frequencies and large range of orders of magnitude
• Accelerometer: measures proper acceleration in various axial directions
• Microphone: converts acoustic noise into electronic signals
• Seismometer: an instrument to measure motions of the ground

Before each measurement the sensors need to be calibrated for correct signals. The main purpose of calibrating a sensor is to gain the actual signal, the real output response of the structure without interaction of the sensor.

For the later handling of the raw data the investigater needs to make sure if, whatever measurement system is used, the signal needs further treatment. So the raw signal of the sensor might need a filtration or correction before the data gets conveyed and used in analysing algorithms.

An example therefor is the random noise contained in recorded signals. This noise is a disturbance variable with wide unspecific frequency spectrum which can be described approximately by statistical parameters. For the vibration analysis it is important to cancel out this disturbance value to gain the actual signal measured.

Evaluation of the signal

There are two different kinds of signal analysis: the evaluation in the time domain and the one in the frequency domain. By regarding these different perspectives the origin of the structural vibrations can be understood better.

Time analysis

Is a signal (e.g. displacement) evaluation as a function of time. ^[2]

Frequency analysis

With the help of the Fast Fourier Transformation (FFT) the frequency spectrum can be calculated from the time signal by fitting multiple harmonics to the time series.

Modal analysis – Identification of elastic parameters (modulus of elasticity)

The modal analysis i.a. suits to the derivation of elastic parameters of tested materials. Using the velocities of elastic waves transmitted by the sensors the identification of mode shapes dynamic properties of elastic components can be identified. Thus, the material class as well as the material consistence can be determined. ^[2]

Two principles to experimentally identify the modal parameters through modal analysis are:

Ambient Vibration Testing (AVT)

A natural, operational valid excitation of the construction is taken, e.g. cars driving on a bridge. This excitation is not measurable, therefore the excitation is only reconstructable through stochastic account. Hence, no transfer function can be generated (input and output is required) but unscaled mode shapes. System accounts are: the system being linear, in steady-state and observable. The excitation level is very low, so sensible sensors are required. The application range of the AVT is e.g. the defect detection of wind power plants. Accounts to the excitation of the structural vibrations:

• Gauß-Process: Overlapping of stochastic processes
• Stationary Process: time independencies of the mean value and the variance
• Ergodic Process: the expected value is identic to the mean value, the process is stationary

Further accounts:

• The mean value results in zero because of its unreliable measured value in the operation vibration analysis

Knowing the equation of motion every vibration status can be described by linking the modal natural oscillation form (deflection shape, represented by the right-hand Eigenvectors). For calculation of the Eigenvectors the eigenvalue problem needs to be solved.

For the derivation of the following formulas an equation of motion of first order is taken. For systems with more than one degree of freedom the equation of motion needs to be transferred in a differential equation of first order.

The following step describes the transformation of the differential equation to an ordinary algebraic equation using the Laplace- or Fourier transformation. The Fourier transformation of the equation of motion reads in the frequency domain:

$//<![CDATA[ \begin{array}{l}E = (-\omega^2[M] + i\omega[C]+[K])Y(\omega) = F(\omega)\end{array} //]]>$ (eq. 4)

Hence it can be rewritten into a frequency response function (FRF) which relates (excitation) input and (vibration) output:

$//<![CDATA[ \begin{array}{l}[H(\omega)] = \frac{Y(\omega)}{F(\omega)} = \frac{adj(-{\omega}^2[M] + i\omega[C] + [K])}{det(-{\omega}^2[M] + i\omega[C] + [K])}\end{array} //]]>$ (eq. 5)

With the help of partial fraction decomposition the frequency response function can be transformed into a modal form and be calculated by an experimental modal analysis:

$//<![CDATA[ \begin{array}{l}[H(\omega)] = \sum_{r=1}^{N_m} \frac{[R_r]}{i\omega - \lambda_r} + \frac{[R_r]^*}{i\omega - \lambda_r^*} = \sum_{r=1}^{N_m} \frac{Q_r \Phi_r \Phi_r^T}{i\omega - \lambda_r} + \frac{Q_r^* \Phi_r^* \Phi_r^{*T}}{i\omega - \lambda_r^*}\end{array} //]]>$ (eq. 6)

$//<![CDATA[ \begin{array}{l}N_m\end{array} //]]>$: amount of mode shapes of the system

$//<![CDATA[ \begin{array}{l}\lambda_r\end{array} //]]>$: pol that belongs to the r-mode

$//<![CDATA[ \begin{array}{l}\Phi_r\end{array} //]]>$: mode shape of r-eigenmodes

$//<![CDATA[ \begin{array}{l}Q_r\end{array} //]]>$: modal scale factor

By gaining the frequency response function we now can also calculate the modal parameters. In the operational vibration analysis this remains unsolved due to the unknown stochastic excitation.

Gaining these modal parameters for the operational vibration analysis the frequency response function is also connected to the power spectral density of the signal answer and excitation:

$//<![CDATA[ \begin{array}{l}[S_{yy}(\omega)] = [H(\omega)]^* \cdot [S_{FF}(\omega)] \cdot [H(\omega)]^T\end{array} //]]>$ (eq. 7)

With the assumption of the white noise the matrix of the power spectral density of the signal excitation becomes a diagonal matrix with constant entries and values R_uu. Hence, the frequency response function remains dependent sole by the power spectral density and a scaling factor.

$//<![CDATA[ \begin{array}{l}[S_{yy}(\omega)] = [H(\omega)] \cdot R_{uu} \cdot [H(\omega)]^T\end{array} //]]>$ (eq. 8)

On this account the informal content of both functions is the same and can be described by partial fraction decomposition:

$//<![CDATA[ \begin{array}{l}[S_{YY}(\omega)] = \sum_{r=1}^{N_m} \frac{\Phi_r \gamma_r^T}{i\omega - \lambda_r} + \frac{\Phi_r^* \gamma_r^H}{i\omega - \lambda_r^*} + \frac{\lambda_r \Phi_r^T}{-i\omega - \lambda_r} + \frac{\lambda_r^* \Phi_r^H}{-i\omega - \lambda_r^*}\end{array} //]]>$ (eq. 9)

Out of this equation one can also extract the Eigenfrequencies, the modal damping and the mode shapes from the frequency response as well as the power spectral density. Yet, the modal forms cannot be scaled due to the missing measurement of the excitation.

As seen in eq. 9 the pole shape of the power spectral density shows various poles for each mode, two of them conjugate-complex. To avoid this circumstance only the positive semi spectrum is considered:

$//<![CDATA[ \begin{array}{l}[S_{YY}(\omega)] = \sum_{r=1}^{N_m} \frac{\Phi_r \gamma_r^T}{i\omega - \lambda_r} + \frac{\Phi_r^* \gamma_r^H}{i\omega - \lambda_r^*}\end{array} //]]>$ (eq. 10)

This semi spectrum is the fundament for most of the algorithms for identification of modal parameters. For the determination of Eigenvibrations, respectively modes, there are two types of methods: either the evaluation of the signal in frequency or in time domain.

• Approach in the frequency domain
  • Peak-Picking: analysis of the power spectral density of the response function; applied if the modes are well seperated and the system is hardly damped
  • Frequency Domain Decomposition Method (FDD): applied if the modes are less good differentiable; identification of modal damping parameters and the amount of dominant modes whilst determining at certain excitation frequencies
• Approach in the time domain:
  • Covariance driven stochastic subspace identification method (SSI-COV): Signal values are assumed as zero, extracting of stochastic state-space models from the correlation matrix of the measured system response, separation of numerical modes and modes with constant frequencies, damping and mode shape

Forced Vibration Testing (FVT)

The structure is excited by an external impact, which is measured simultaneously with the system response. Thus the transfer function and the incidental modal parameters are identified.

Test execution: To measure the structural vibration, firstly the metering points must be scheduled in strategic good positions, i.e. the different modes should be determinable. Thus, sensors must be placed in locations where no node of vibration is assumed.

Afterwards an impact is applied on one of these points. This impact is measured together with the system responses of the other points. This procedure can also be performed with one sensor and many impacts in various metering points.

Signal analysis and evaluation: The obtained signals can now be examined and the Eigenfrequencies and modal parameter respectively as well as the damping of the modes.

To gather the Eigenfrequencies the signals are plotted as frequency response function and the maxima can be read as Eigenfreuqencies. The damping can be determined by the median width of the resonance maximum ^[2]. With the estimated modes and the read Eigenfrequencies we can now calculate the elastic parameters. Therefore each mode has its specific elastic parameter:

• The shear modulus belonging to the torsional mode can be calculated as: ^[3]

^{$//<![CDATA[ \begin{array}{l}G = \frac{f_{tors}^2}{k^2} \cdot \frac{m \cdot l}{A} \cdot 4R\end{array} //]]>$}(eq. 11)

$//<![CDATA[ \begin{array}{l}f_{tors}\end{array} //]]>$: frequency of the torsional mode

$//<![CDATA[ \begin{array}{l}m\end{array} //]]>$: structural mass

$//<![CDATA[ \begin{array}{l}l\end{array} //]]>$: structural length

$//<![CDATA[ \begin{array}{l}k\end{array} //]]>$: harmonic order

$//<![CDATA[ \begin{array}{l}A\end{array} //]]>$: cross-sectional area

$//<![CDATA[ \begin{array}{l}R\end{array} //]]>$: correction value of the geometry (circular cross-section: $//<![CDATA[ \begin{array}{l}R = 1\end{array} //]]>$, else: $//<![CDATA[ \begin{array}{l}R = \frac{\frac{b}{h} + \frac{h}{b}}{4 \cdot \frac{b}{h} - 2,52 (\frac{b}{h})^2 + 0,21(\frac{b}{h})^6}\end{array} //]]>$)

• The Young’s Modulus of the longitudinal mode can be determined as:

$//<![CDATA[ \begin{array}{l}E = \frac{f_{long}^2}{k^2} \cdot \frac{m \cdot l}{A} \cdot 4C\end{array} //]]>$ (eq. 12)

$//<![CDATA[ \begin{array}{l}v\end{array} //]]>$: Poisson’s ratio ($//<![CDATA[ \begin{array}{l}\nu = \frac{E}{2 \cdot G} - 1\end{array} //]]>$)

$//<![CDATA[ \begin{array}{l}I\end{array} //]]>$: moment of inertia

$//<![CDATA[ \begin{array}{l}C\end{array} //]]>$: correction value of the geometry ($//<![CDATA[ \begin{array}{l}C = 1 + \frac{k^2 \cdot \pi^1 \cdot \nu^2 \cdot I}{A \cdot l^2}\end{array} //]]>$)

• Calculation of the Young’s Modulus of the bending mode:

$//<![CDATA[ \begin{array}{l}E = \frac{f_{flex}^2}{(2k + 1)^4} \cdot \frac{m \cdot l^3}{I} \cdot \frac{64}{\pi} \cdot T\end{array} //]]>$ (eq. 13)

$//<![CDATA[ \begin{array}{l}n\end{array} //]]>$: control variable

$//<![CDATA[ \begin{array}{l}T\end{array} //]]>$: correction value of the geometry (calculated by the Taylor series expansion:

$//<![CDATA[ \begin{array}{l}T = \frac{1}{2} + \frac{(2n + 1)^2 \cdot \pi^2 \cdot I}{8 \cdot A \cdot l^2} \cdot [1 + \frac{2(1+\nu)}{k}] + \sqrt{\frac{1}{4} + \frac{(2n + 1)^2 \cdot \pi^2 \cdot I}{8 \cdot A \cdot l^2} \cdot [1 + \frac{2(1+\nu)}{k}] + \frac{(2n + 1)^4 \cdot \pi^4 \cdot I^2}{64 \cdot A^2 \cdot l^4} \cdot [1 - \frac{2(1+\nu)}{k}]^4 }\end{array} //]]>$)

Application ranges

• Aerospace Engineering
• Automotive Industry
• Manufacturing Industry
• Wood and Paper Production
• Power Generation
• Armaments Industry
• Entertainment Electronics
• Telecommunications
• Civil engineering

Literature

• Brincker, Rune: Some Elements of Operational Modal Analysis. In: Shock and Vibration 2014 (4), S. 1-11. DOI: 10.1155/2014/325839. 2014.
• Große, C.U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen. Grundlagen und Anwendungsbeispiele. Vorlesungsunterlagen, Version 01.07.2015, p. 47, 201 - 211.
• Grüningen, D. C.: Digitale Signalverarbeitung. Mit einer Einführung in die kontinuierlichen Signale und Systeme. 5., neu bearb. Aufl. München: Hanser. 2014.
• Magalhaes, Filipe; Cunha, Álvaro: Explaining operational modal analysis with data from an arch bridge. In: Mechanical Systems and Signal Processing 25 (5), S. 1431-1450. DOI:10.1016/j.ymssp2010.08.001. 2011.
• Mitchell, B.S.: An Introduction to Materials Engineering and Science for Chemical and Materials Engineers. John Wiley & Sons, Inc., Hoboken, New Jersey, 2004, p. 381 - 389.
• Rainieri, Carlo; Fabbrocino, Giovanni: Operational modal analysis of civil engineering structures. An introduction and guide for applicants. New York, 2014. Springer Science+Business Media.
• Tcherniak, D.; Chauhan, S.: Dynamic Characterization of Operational Wind Turbines by Output-only Modal Analysis. In: 3. VDI-Fachtagung Schwingungen von Windenergieanlagen 2012 mit Fachausstellung. Bremen, 07. Und 08. Februar 2012. Nichtred. Ms.-Dr. Düsseldorf: VDI-Verl. (VDI-Berichte, 2168), S. 73 - 86. 2012.
• Wagner, J.; Mlejnek, H.P.: Vorlesungsunterlagen – Dynamik II. Institut für Statik und Dynamik der Luft und Raumfahrtkonstruktionen der Universität Stuttgart. Stuttgart, 2012.
• Weiler, Bernd; Große, Christian: Elastic Constants – Their dynamic Measurement and calculation. Abgerufen am 29.01.2016.
• Vibration. Wikipedia.

References

1. Große, C.U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen. Grundlagen und Anwendungsbeispiele. Vorlesungsfolien, Version 01.07.2015
2. Große, C.U.: Einführung in die Zerstörungsfreie Prüfung im Ingenieurwesen. Grundlagen und Anwendungsbeispiele. Vorlesungsskript, Version 01.07.2015
3. Weiler, Bernd; Große, Christian: Elastic Constants – Their dynamic Measurement and calculation. Abgerufen am 29.01.2016.