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The FRF and its Many Forms!

Siemens Phenom Siemens Phenom
Siemens Phenom

A Frequency Response Function (FRF) gives insight into a structure’s natural frequencies, damping, and mode shapes.

 

Often, the FRF is displayed with units of acceleration (g) over units of force (N) resulting in an accelerance unit of g/N.

 

However, there are alternative formats with which the FRF can be displayed (Figure 1).  These alternative formats are created by performing math operations on the FRF:

  • Integration and differentiation*: Acceleration can be changed to velocity or displacement
  • Inversion: The FRF can be inverted, so it is input over output, rather than output over input

 

(*) In frequency domain, integration and  differentiation simply boils down to respectively division and  multiplication by jω, where ω is the frequency in radians per second.

 

FRF_table.pngFigure 1: Six common forms the FRF can take: Compliance, Mobility, Accelerance, Dynamic Stiffness, Mechanical Impedance, Dynamic Mass. For sake of simplicity, only the magnitude is shown, but be aware that FRFs consists of complex numbers and also have a phase.

There are benefits to displaying the FRF in these different formats, which are discussed in the article below. All of these terms are mathematically related, so by calculating one FRF, any of the other representations can be determined.

 

In Simcenter Testlab, it is easy to move between these different functions by integrating or differentiating in the FRF plot display (see last section of article).

 

Theory and Background

 

A single degree of freedom (SDOF) system, consisting of a mass-spring-damper, is often used to understand dynamic systems and their properties as shown in Figure 2.

 

mass-spring-damper.pngFigure 2: Mass-spring-damper single degree of freedom system.

In the mass-spring-damper system:

  • m – mass of the system [kg]
  • c – damping of system [Ns/m]
  • k – stiffness [N/m]
  • f – force applied to system [N]
  • x – displacement [m]

As a function of frequency, this mass-spring-damper system has a peak response (x) due to the force (f) at the natural frequency (ωn). The natural frequency of the mass spring system is equal to the square root of the stiffness over the mass as given in Equation 1.

 

natural_freq.pngEquation 1: Natural frequency of a mass-spring-damper system is the square root of the stiffness divided by the mass.The FRF of such a system is shown in Figure 3.  The behavior of the system can be broken into three regions:

  • Stiffness – Below the natural frequency, the stiffness governs the response of the SDOF system
  • Damping – At the natural frequency, the damping governs the response of the SDOF system
  • Mass – Above the natural frequency, the mass governs the response of the SDOF system

 

FRF_regions.pngFigure 3: Responses below the natural frequency are governed by stiffness, at the natural frequency damping governs the response, above the natural frequency mass governs the response.

The equation of motion (Equation 2) as a function of frequency (ω, in radian per second) for the SDOF system is:

 

Equation_SDOF_Motion.pngEquation 2: Equation of motion for a single degree of freedom system in the frequency domain.

Rearranging the equation to an output divided by input, and taking the frequency toward zero, the equation at low frequencies becomes one over the stiffness (1/k) as shown in Equation 3.

 

Equation_Stiffness.pngEquation 3: Taking the frequency to zero, the stiffness term dominates.

Expressing the same equation in acceleration over force (by double differentiating the displacement), and taking the frequency to infinity, the equation simplifies to 1/m (Equation 4).

 

Equation_Mass.pngEquation 4: Taking the frequency to infinity, the acceleration over force transfer function becomes 1/m.

No matter what form the FRF is displayed in, the frequencies below the natural frequency in FRF are dominated by stiffness, while the frequencies above the natural frequency are dominated by mass. 

 

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Accelerance:

 

A common form to view the FRF in is accelerance, with units of acceleration over force as shown in Figure 4.

 

accelerance.pngFigure 4: Plot of acceleration over force for a mass-spring-damper system (also called accelerance). The mass line is approximately 1/m (the asymptotic value following a natural frequency).

Below is a plot of accelerance between two points measured on a test structure using an impact hammer and an accelerometer (Figure 5).

 

accelerance_frf.pngFigure 5: The accelerance (g/N) FRF between two points on a brake caliper.

When physically measuring an FRF, an accelerometer and a force sensor are commonly used. It is therefore easy to understand this FRF form in the terms of the equipment used to make the measurement.

 

Dynamic Mass:

 

The dynamic mass FRF (units of acceleration / force) can be calculated by inverting the accelerance FRF, as shown in Figure 6.

 

dynamic_mass.pngFigure 6: Plot of force over acceleration for a mass-spring-damper system (also called dynamic mass).

In a dynamic mass FRF, the data asymptote succeeding the natural frequency is equal to the mass of the SDOF system.

 

The inverse of the FRF from Figure 5 is displayed in Figure 7 below.  This is a dynamic mass FRF.

 

dynamic_mass_frf.pngFigure 7: Dynamic mass FRF of the same FRF as in Figure 5. The FRF was inverted.

Using the mass lines of FRFs collected from a softly suspended test object, the moments of inertia of the object can be calculated (Figure 8). Moments of inertia can be calculated from FRFs using the Simcenter Testlab Rigid Body Property calculator software. 

 

rigid_body_process.pngFigure 8: Rigid body property calculator of Simcenter Testlab (right) can calculate the moments of inertia of a test object (upper left) from FRF data (lower left).

The Simcenter Testlab Rigid Body Property calculator takes 37 tokens to run.  This method utilizes the rigid body modes of a test object on a soft suspension. The rigid body modes of a freely suspended object should be close to zero Hertz. 

 

There is a frequency range between the rigid body modes and first flexible modes of the object from which the mass properties are calculated as shown in Figure 9.

 

mass_line_frf.pngFigure 9: With well separated rigid and flexible modes, the mass line can be used to calculate the moments of inertia of a test object.

For calculating mass for the moments of inertia, the rigid body modes and flexible modes need to be sufficiently spaced apart.  In the FRF, there needs to be a constant amplitude mass line originating from the rigid body modes before the influence of the stiffness line of the flexible modes is significant.

 

Flexible modes of the structure (bending, torsion, local modes, etc) cannot be used to calculate the moments of inertia.  For example, a local panel flexible mode mass line does not contain the mass information of the entire object's inertia. The rigid body modes are required because the entire test object rotates and translates.

 

Mobility:

 

Mobility is another popular form in which to display the FRF. Mobility uses units of velocity over force as shown in Figure 10.

 

mobility.pngFigure 10: Plot of velocity over force for a mass-spring-damper system (also called mobility). The asymptotes preceding and succeeding the natural frequency give insights to the mass and stiffness characteristics of the system.

For SDOF systems, and systems with lightly damped, well-spaced modes, the asymptotes preceding and succeeding the natural frequency are again significant.

 

The asymptote preceding the natural frequency linearly increases with frequency and is inversely proportional to the stiffness value. The slope of the asymptote succeeding the natural frequency decreases with frequency and is inversely proportional to the mass. 

 

In Figure 11 below is the same FRF as in Figure 5, but now displayed in terms of mobility. This was done by doing a single integration (i.e. division by jω).

 

mobility_frf.pngFigure 11: Mobility FRF of the same FRF as in Figure 5. The FRF was integrated once.

Mobility can be a popular form due to various physical phenomenon being proportional to velocity. For example, the sound radiated by a surface is directly proportional to the velocity of vibration on said surface.

 

Using Computer Aided Engineering (CAE) modeling (Figure 12), the mobility between the surface velocity and internal forces would be evaluated as part of the modelling process.

 

simcenter3D_acoustics.jpgFigure 12: In Simcenter3D, surface velocity FRFs can be used in the prediction of radiated sound.

To ensure that a product has the lowest sound level possible, the surface velocities need to be minimized.  Product designs with the lowest amplitude mobility FRFs between internal forces and sound radiating surfaces are preferred.

 

Mechanical Impedance:

 

Mechanical impedance is a measure of how much a structure resists motion when subjected to a unit force. Impedance is defined as the force over velocity FRF (f/v) as shown in Figure 13.

 

mechanical_impedance.pngFigure 13: Plot of force over velocity for a mass-spring-damper system (also called mechanical impedance).

A mechanical impedance FRF is shown in Figure 14.  This is the same FRF as shown in Figure 5, but it has been integrated once and inverted.

 

mechanical_impedance_frf.pngFigure 14: Mechanical impedance FRF of the same FRF as in Figure 5. The FRF was integrated once and inverted.

One use for mechanical impedance is determining the operational forces on a structure. For example, if the mechanical impedance of a structure is known; that structure is placed on a vibrating support; and the operational velocity of the structure can be measured (often with a laser), the operational forces can be calculated as:

 

  • f_operational = v_operational * (f/v)

 

Another way in which mechanical impedance is used in practice is to determine how two different components will interact with one another in an assembly. The coupling point FRFs of the two components can be overlaid as shown in Figure 15.

 

mechanical_impedance_overlay.pngFigure 15: Mechanical impedance FRF overlay of two different components.

If the FRFs are well separated in amplitude, then the two components will not interact dynamically.  If they are close together, then dynamic interaction occurs between the two components.

 

Compliance:

 

Compliance is a measure of how much a structure moves (displacement, x) for a unit input of force (f) as shown in Figure 16.

 

compliance.pngFigure 16: Plot of displacement over force for a mass-spring-damper system (also called compliance). The asymptote preceding the natural frequency is commonly called the “stiffness line”.

For SDOF system and systems with lightly damped, well-spaced modes, the asymptotes preceding and succeeding the natural frequency are again significant. The line preceding the natural frequency is known as the “stiffness line”.

 

When displayed in compliance terms, the stiffness line is approximately flat and equal to the inverse stiffness value of the system.

 

Compliance can be calculated from accelerance by double integrating (Figure 17).

 

compliance_frf.pngFigure 17: The same FRF as in Figure 5, but now displayed in compliance terms.

Compliance is often used, as it is an intuitive quantity to understand. It is more intuitive to think about deflections/displacements rather than acceleration or velocity. It can be difficult to physically measure displacement per unit of input force. However, compliance is often used in CAE modeling.

 

Dynamic Stiffness:

 

Stiffness, or the resistance to deformation from an input force (Figure 18), is very important in structural dynamics and noise and vibration related topics. Vibration can be thought of as a ratio of the forces acting on a structure to its stiffness.

 

dynamic_stiffness.pngFigure 18: Plot of force over displacement for a mass-spring-damper system (also called dynamic stiffness).

Stiffness is especially important when considering elastomeric mounts and mounting locations. Mount manufacturers often work in dynamic stiffness, rather than accelerance or mobility.

 

A FRF in dynamic stiffness format is shown in Figure 19.

 

dynamic_stiffness_frf.pngFigure 19: The same FRF as in Figure 5, but now displayed in dynamic stiffness terms.

Oftentimes, noise and vibration engineers will use dynamic stiffness to determine where to place an isolator. The attachment location should be at least ten times stiffer than the isolator (for example, a rubber mount bushing) as shown in Figure 20.

 

mount_attach.pngFigure 20: Left – Mount between structure and base, Right – Attachment FRF versus Mount FRF in Dynamic Stiffness.

If the attachment location and mount isolator stiffness are similar, then the isolator is not effective.  It does not isolate the vibration at the connection. Think of pushing on two springs that are in series.  If one spring is soft and the other stiff, the soft spring will deflect while the stiff spring does not move.  If both springs are of the same stiffness, they both will move and deflect.

 

Alternatively, manufacturers may come up with a “stiffness target” for the mount attachment location. A single number can be used for the target because the stiffness values are flat versus frequency when using dynamic stiffness.

 

For example, a common target for attachment location stiffness is around 1X10^7 N/m for general mechanical attachments (spring/mount brackets, etc). The manufacturer may create a reference curve for stiffness and overlay the FRF onto this reference curve (see Figure 21, below).

 

attachment_stiffness_target1.pngFigure 21: Two attachment point FRFs for mounts against stiffness target of 1e7 N/m.

The attachment locations can be evaluated for their stiffness.  The higher the amplitude of a dynamic stiffness FRF, the more stiff the attachment location.  In Figure 21, the FRF with black linestyle is well below the desired stiffness, and signficantly less stiff than the FRF with blue linestyle.

 

By modal curvefitting a dynamic stiffness FRF, the intersection of the stiffness line with zero Hertz can be plotted.  This can be used to determine the static stiffness of the test object.

 

Moving between FRF forms in Simcenter Testlab:

 

It is simple to move between forms of the FRF in Simcenter Testlab.

 

Testlab Displays: Processing

 

Click on the y-axis of the FRF, and chose “Processing”. Then choose to integrate or differentiate to move between the different forms as shown in Figure 22.

 

testlab_integrate_differentiate.pngFigure 22: Integrate and differentiate within the Testlab displays to move between different forms of the FRF.

The units are obviously automatically changed, but can be adjusted within the same quantity case (e.g. going from g/N to (m/s2)/N for accelerance FRFs; or from m/N to mm/N for compliance FRFs).

 

Testlab Impact Testing

 

Additionally, it is possible to directly calculate dynamic stiffness in the Simcenter Testlab Impact Testing workbook as shown in Figure 23.

 

impact_testing.pngFigure 23: In addition to accelerance, it is possible to directly calculate dynamic stiffness in the Simcenter Testlab Impact Testing workbook. Under "All Settings" in the Measure worksheet, turn on "Dynamic Stiffness" in the Data Storage settings.

In the “Measure” workbook of Simcenter Testlab:

  • In the upper right of the workbook, select “All Settings”.
  • In the pop-up window, select to calculate dynamic stiffness in the Data Storage section. \

The “All Settings” dialog is available in both the Measure and Impact Setup worksheets.

 

Testlab Data Calculator for Block Math

 

It is also possible to use the Data Calculator to invert and integrate FRFs.  In the Testlab Desktop Navigator, click on the Data Calculator as shown in Figure 24.

 

data_calculator.pngFigure 24: Press Data Calculator in Testlab Desktop Navigator worksheet.

Enter either the INVERSE or INTEGRATE formula as needed (Figure 25).

 

formula_set.pngFigure 25: Enter INVERSE or INTEGRATE as desired in the Formula Set.

A detailed example of how to use the Data Calculator can be found in this Testing Forum post.

 

Questions?  Email peter.schaldenbrand@siemens.com

 

Modal Data Acquisition:

Modal Analysis and Operational Deflection Shapes: