“Vibration sensors are key components in assisting industrial equipment in diagnosing faults and providing predictive protection. However, you may not know that it is the housing of the vibration sensor that affects the extraction of high-quality CbM vibration data. Because the housing used to encapsulate the MEMS accelerometer needs to have a better frequency response than the integrated MEMS. In this regard, ADI uses modal analysis, through theory and ANSYS modal simulation examples, to solve the problem of vibration sensor housing design, and then provides a reliable guarantee for obtaining more effective vibration data.

“

By: Richard Anslow, Systems Applications Engineer, Analog Devices

Vibration sensors are key components in assisting industrial equipment in diagnosing faults and providing predictive protection. However, you may not know that it is the housing of the vibration sensor that affects the extraction of high-quality CbM vibration data. Because the housing used to encapsulate the MEMS accelerometer needs to have a better frequency response than the integrated MEMS. In this regard, ADI uses modal analysis, through theory and ANSYS modal simulation examples, to solve the problem of vibration sensor housing design, and then provides a reliable guarantee for obtaining more effective vibration data.

**What is modal analysis and why is it so important?**

Modal analysis is a basic method to study the vibration characteristics of mechanical equipment structures and a necessary condition for designing high-quality mechanical casings. Typically, MEMS vibration sensors are packaged in steel or aluminum housings that can be securely attached to the asset being monitored and provide water and dust resistance (IP67). A good metal enclosure design is an important factor in ensuring high quality vibration data is measured from an asset.

When using modal analysis, the main concern is to avoid resonance, where the natural frequencies of the structural design are very close to the natural frequencies of the applied vibrational load. For vibration sensors, the natural frequency of the housing must be greater than the natural frequency of the applied vibration load measured by the MEMS sensor. With modal analysis, the natural frequencies and normal modes (relative deformations) of the design can be provided.

The frequency response plot of the ADXL1002 MEMS accelerometer is shown in Figure 1. The ADXL1002 has a 3dB bandwidth of 11kHz and provides a 21kHz resonant frequency. The protective case used to enclose the ADXL1002 needs to have a primary natural frequency of 21kHz or higher.

Figure 1. Frequency Response of ADXL1002 MEMS Accelerometer

**Vibration Sensor Housing Model**

In modal analysis and design, a vibration sensor can be thought of as a stubby cantilever beam. In addition, simulations will be performed using the Timoshenko vibration equations, which will be detailed later. A stubby cantilever cylinder is similar to a vibration sensor mounted on industrial equipment, as shown in Figure 2. The vibration sensor is bolted to the industrial equipment. Both bolt mounting and housing design need to be carefully characterized so that mechanical resonances do not affect the relevant MEMS vibration frequencies. The Finite Element Method (FEM) using ANSYS or a similar program can be used as an efficient method for solving the vibration equations of stubby cylinders.

Figure 2. Vibration sensor housing modeling

**Simulation tools**

In modal analysis, ANSYS and other simulation tools assume harmonic motion at every point in the design. Displacements and accelerations at all points in the design are solved for eigenvalues and eigenvectors, in this case, natural frequencies and mode shapes, respectively.

**• Natural frequencies and mode shapes**

Equation 1 is the relationship between the mass matrix M, stiffness matrix K, angular frequency ωi and mode shape {Φi}, used in FEM programs such as ANSYS. The natural frequency fi can be calculated by dividing 1ωi by 2π, and the mode shape {Φi} provides the relative deformation mode of the material at a specific natural frequency.

For a single degree of freedom system, the frequency can be simply expressed as:

Equation 2 provides a simple, intuitive approach to design evaluation. If the height of the sensor housing is reduced, the stiffness increases, the mass decreases, and therefore, the natural frequency increases. In addition, if the height of the enclosure is increased, the stiffness decreases, the mass increases, and the natural frequency decreases.

Most designs have multiple degrees of freedom. Some designs have hundreds of degrees of freedom. Using FEM, the calculation result of Equation 1 can be obtained quickly, which is very time-consuming if it is calculated manually.

**• Pattern participation factor**

The Mode Participation Factor (MPF) is used to determine which modes and natural frequencies are most important to your design. Equation 3 is the relationship between the mode shape {Φi}, the mass matrix M and the excitation direction vector D for solving the MPF. The square of the participation factor is the effective mass.

MPF and effective mass measure the number of masses moved in each direction in each mode. A higher value in one direction means that in that direction, the mode will be stimulated by a force (such as vibration).

Using MPF in combination with natural frequencies can help designers identify potential design problems. For example, the lowest natural frequency derived from a modal analysis may not be the biggest design problem because the participation factor in the relevant direction (x, y, or z plane) may not be large enough relative to all other modes.

The example shown in Table 1 shows that when the natural frequency of the x-axis is predicted to be 500Hz in the simulation, the mode is weak and unlikely to be a problem. When the housing x-axis is in 800Hz strong mode, it will be a problem if the orientation of the MEMS sensitive axis and the housing x-axis are aligned. However, if the designer orients the MEMS sensor PCB to measure on the z-axis of the housing, the 800Hz strong mode for this x-axis may not matter.

**• Analytical modal analysis results**

It was learned above that the natural frequencies of the relevant axes can be calculated using modal analysis. In fact, designers can use MPF to determine whether a frequency can be ignored in a design. In order to complete a modal analysis analysis, it is necessary to understand that all points on the structure vibrate at the same frequency (global variable), but the vibration amplitude (or mode shape) of each point is different. For example, the 18kHz frequency affects the top of the mechanical enclosure more than the bottom. The mode shapes (local variables) have stronger amplitudes at the top of the enclosure than at the bottom, as shown in Figure 3. This means that while the top of the housing structure is strongly stimulated by the 18kHz frequency, the MEMS sensor located at the bottom of the housing is also affected by this frequency, albeit to a lesser extent.

Figure 3. Natural frequencies of the vibration sensor housing, mode shapes on the relevant axes, and relative amplitudes at the top and bottom of the housing

**Timoshenko differential vibration equation**

The Timoshenko equations are suitable for modeling stubby beams or beams subject to vibrations of several kilohertz. The vibration sensor shown in Figure 2 resembles a stubby cylindrical section and can be modeled using the Timoshenko equation. The equation is a fourth-order difference equation with an analytical solution for the restricted case. As shown in Equation 1 through Equation 3, FEM provides an easy way to solve Timoshenko’s equations using multidimensional matrices that scale with the design degrees of freedom.

**• Governing equations**

While FEM has advantages in efficiently solving Timoshenko’s vibration equations, understanding the trade-offs faced when designing vibration sensor housings requires a more in-depth study of the equation’s 42 parameters.

Using different materials or geometries can affect the natural frequency (ω) of the designed structure.

**• Material and geometry dependencies**

The parameters of Timoshenko’s equation can be classified as geometry-dependent or material-dependent.

**Material dependencies include:**

Ø Young’s modulus (E): This is a measure of the elasticity of a material, that is, the tensile force required to deform it. The deformation pull is at right angles to the surface.

Ø Shear Modulus (G): This is a measure of the shear stiffness of a material, which is the ability of an object to withstand shear deformation stress applied parallel to the surface.

Ø Material density (ρ): mass per unit volume.

**Geometry dependencies include:**

Ø Shear coefficient (k): Shear is a material property, and shear coefficient refers to the change of shear stress in the cross section. The rectangular section is generally 5/6, and the circular section is generally 9/10.

Ø Area Moment of Inertia (I): The geometrical property of an area that reflects how the geometry is distributed around an axis. This property helps to understand the resistance of the structure to applied bending moments. In modal analysis, this can be seen as resistance to deformation.

Ø Cross-sectional area (A): The cross-sectional area of the defined shape (eg cylinder).

The Timoshenko equation predicts the critical frequency fC given by Equation 5. Because Equation 4 is a 4th order equation, under fC, there are 4 independent solutions. For analysis, fC of Equation 5 can be used to compare different housing geometries and materials.

Various methods and solutions can be used to determine all frequencies at fC. Some methods are mentioned in “Free and Forced Vibration of Timoshenko Beams Described by Single Difference Equations” and “Bending Vibrations of Drive Shafts Using Distributed Lumped Modeling Techniques”. These methods use multidimensional matrices, such as FEM.

**What material should be used for the enclosure design?**

Table 2 details information on some commonly used industrial metal materials, such as stainless steel and aluminum.

Of the 4 materials listed, copper is the heaviest and offers no advantages over stainless steel, which is lighter, stronger and less expensive.

Aluminum is a good choice for weight-sensitive applications. Its density is 66% lower than steel. The downside is that aluminium is 20 times more expensive per kilogram than steel. For cost-conscious applications, steel is the obvious choice.

Although titanium is two-thirds heavier than aluminum, its own strength means less is needed. However, except for professional applications where weight reduction is required, the cost of titanium is prohibitive.

Table 2. Young’s Modulus (E), Shear Modulus (G), Density (ρ) and Cost per Kg of Common Industrial Materials

**Simulation example**

The rectangular metal vibration sensor housing shown in Figure 4 is designed to be 40mm high, 43mm long and 37mm wide. For modal analysis, the bottom surface (z,x) is a fixed constraint.

Figure 4. Rectangular enclosure, simulated by changing material type

Figure 5. Rectangular Housing Including Material Type and First Stage Effective Natural Frequency (Hz)

Figure 5 shows FEM modal analysis results for various housing materials. The figure shows the relationship between the natural frequency of the first stage, the effective MPF (the ratio of the effective mass of the system to the total mass is greater than 0.1) and the material type. Clearly, aluminum and stainless steel have the highest effective first-order natural frequencies. They are also a good material choice for low cost or low weight applications.

**Should a rectangular housing or a cylindrical housing be designed?**

Figure 6 shows a hollow rectangular and cylindrical stainless steel extrusion with a wall thickness of 2mm and a height of 40mm. The outer diameter of the cylinder is 43mm, and the dimensions of the rectangular model are also 43mm in both the x and y axes.

Figure 6. Similar rectangles and cylinders for model design studies

Figure 7. Similar Rectangular and Cylindrical First Stage Effective Natural Frequency (Hz)

In a modal analysis, the entire 2mm wall thickness (or x, y cross-sectional area) is a fixed constraint. Figure 7 shows the results of the FEM modal analysis. The figure shows the relationship between the natural frequency of the first stage, the effective MPF (the ratio of the effective mass of the system to the total mass is greater than 0.1) and the material shape. Cylinders have the highest first-order effective natural frequencies in the x and y axes, with similar performance in the z direction.

**Geometry – Area and Inertia**

Both material and geometry dependencies are included in Equation 4. Since both the rectangular and cylindrical models were simulated with stainless steel parameters, the only reason for the better cylindrical performance is its geometry. Figure 8 shows the cylindrical and rectangular cross-sections used to calculate the areal moment of inertia and cross-sectional area of the model.

Figure 8. Area moment of inertia (IYY) and cross-sectional area

The area moment of inertia IYY of the rectangle is almost 50% larger than that of the cylinder, as shown in Table 3. Rectangles are more resistant to deformation. However, the cross-sectional area A of a cylinder is three times that of a rectangle. A larger value of the A parameter means that in both simulation and reality, the fixed constraints are larger, so a cylindrical design helps to increase stiffness or stiffness.

Using the values in Table 3 and Equation 5, the critical frequency is 60.74 kHz for the cylinder and 26.56 kHz for the rectangle. Equation 5 is a useful tool to show the relative performance of different geometries. Equations 4 and 5 predict 4 independent solutions at the critical frequency. Table 4 summarizes the FEM results and identifies 4 first-level valid modes.

Table 3. Area moment of inertia (IYY), shear modulus (G), density (ρ), and cross-sectional area (A) for cylindrical and rectangular models

**bold **= mode participation factor > 0.1

unbold = 0.01

**What is the maximum recommended height for the sensor?**

Equations 4 and 5 are useful, but they do not provide analytical guidance on the trade-off between vertical enclosure height and available first-stage effective natural frequencies. It can be intuitively seen from Equation 2 that the higher the sensor housing, the lower the primary natural frequency.

**Limitations of Analytical Models**

Equations 4 and 5 assume that the section width of the beam is at least 15% of the length of the beam. Other methods for slender beams, such as the Bernoulli equation, assume that the beam’s section width is less than 1% of the beam’s length. For slender beams, Equation 6 can be used, which includes the length (L) or sensor height. Equation 6 does not account for shear forces, but they are important for stubby beams. Equations 4, 5, and 6 remain generally consistent for the first-order effective natural frequencies when applied to solid cylinders. For hollow shapes, Equation 6 underestimates the effective first-order natural frequency by 50%.

Table 5. First-order effective modes for hollow and solid cylinders compared to Bernoulli equations

Parameters used in Equation 6 include stiffness for Young’s modulus (E), diameter (d), length (or height), material density (ρ) used, and Kn constant for a given configuration.

Since the analytical model cannot provide guidance on the height constraints of the hollow shell, height studies are generally performed with the aid of FEM.

**Highly researched**

To provide guidance on performance degradation due to increased enclosure height, ADI simulated the model shown in Figure 9.

Figure 9. Case height study with 5mm base

This stainless steel extrusion features a 5mm base that can be used to mount screws between the housing and monitored equipment such as motors. Increasing the height of the cylinder from 40mm to 100mm resulted in a reduction in the effective first-order natural frequencies of the x- and y-axes from 12.5kHz to 3.3kHz, as shown in Figure 10. The value of the z-axis is also reduced from 31.2kHz to 12.7kHz. To achieve a high-performance sensor, it is clear that the housing height needs to be kept as low as possible.

Figure 10. First Stage Effective Natural Frequency (Hz) with 5mm Base, Increased Height Enclosure

**What is the effect of reducing the shell wall thickness or diameter?**

Reduce shell wall thickness

Table 6 shows how the geometry and material properties of the cylinder shown in Figure 6 would be if the wall thickness were reduced from 2mm to 1mm, but the 40mm height and 43mm outer diameter were retained.

Table 6. Area moment of inertia (IYY), shear modulus (G), density (ρ) and cross-sectional area (A) for cylinders with a height of 40mm and wall thicknesses of 1mm and 2mm, respectively

Using the values in Table 6 and Equation 5, the critical frequency for a cylinder with a wall thickness of 2 mm is 60.74 kHz and for a wall thickness of 1 mm is 61.48 kHz. With both IYY and A parameters reduced by about 50%, the numerator and denominator of Equation 5 are equally affected for a cylinder with a wall thickness of 1 mm. Based on this calculation, it is assumed that in the FEM modal analysis, the two cylinders will behave similarly.

Figure 11 shows the relationship between the natural frequency of the first stage, the effective MPF (the ratio of the effective mass of the system to the total mass greater than 0.1) and the cylindrical wall thickness. The effect of reducing the cylindrical wall thickness is very small compared to the natural frequency.

Figure 11. Effective Natural Frequency (Hz) of the First Stage for Cylindrical 1mm or 2mm Wall Thickness

**Reduce housing diameter**

All examples given so far are dominated by cylindrical housings with an outer diameter of 43mm. Some designs may only require a 30mm or 26mm outer diameter. Figure 12 shows the simulation model and Figure 13 shows the effect of changing the outer diameter of the housing.

When reducing the cylindrical outer diameter from 43mm to 26mm, the first-order natural frequency in the x and y-axes decreases by about 1.5kHz, and the first-order natural frequency in the z-axis increases by 1.9kHz. Both the area moment of inertia (IYY) and the cross-sectional area (A) decrease when changing the outer diameter of the cylinder. The decrease of the IYY parameter was higher than that of the A parameter.

Figure 12. Shell Diameter Study

When reducing the diameter from 43mm to 30mm, the IYY decreases by 2/3 and the A parameter decreases by 1/3. Referring again to Equation 5, the final effect is a gradual decrease in the natural frequency of the first stage. Intuitively, reducing the cylindrical diameter reduces the structural stiffness, so the natural frequency also decreases. However, through the simulation, it is obvious that the reduction of the natural frequency of the first stage is not large. After changing the diameter, the natural frequency of the first stage remains at several tens of kHz.

Figure 13. First Stage Effective Natural Frequency vs. Cylindrical Outer Diameter

**Can changing the orientation of the sensor housing improve performance?**

The previous sections of this article have shown that increasing the housing height reduces the first-stage natural frequency. It is also indicated that a cylindrical case is recommended rather than a rectangular case. However, there are situations in which rectangles can be useful.

Suppose there is a scenario where a sensor and circuit need to be packaged in a housing with a specified height of 60mm, length and width of 43mm × 37mm. If a rectangular enclosure is used, changing the orientation of the fixed constraints (device connections) can help improve performance. The rectangular housing shown in Figure 14 contains multiple attachment holes so the housing can be mounted to the device from multiple orientations. If the enclosure is mounted in the x, z plane, the effective height of the enclosure is 60mm. However, if the housing is mounted in the y, z plane, the effective height is only 37mm. This method works for rectangular shells, but not for cylindrical surfaces.

Figure 14. A rectangular enclosure can be constrained in x and z, or y and z, to reduce height

Figure 15 shows that by changing the orientation of the housing, the first-order resonant frequency is increased on the x-axis and better on the y-axis than cylindrical. Compared with the fixed directions of the x-axis and the z-axis, the first-order resonance frequency of the z-axis in the fixed directions of the y-axis and the z-axis is higher, and the frequency mode is almost doubled. However, cylindrical is currently the best performer in terms of z-axis natural frequencies. Rectangles are a good way to get similar performance in all three axes compared to cylinders.

Figure 15. First Stage Effective Natural Frequency vs. Cylindrical or Rectangular Orientation

**Single-axis 11kHz MEMS sensor with 21kHz resonant frequency**

According to the simulation and analysis results presented in this article, the cylindrical case performs best when encapsulating a single-axis ADXL1002 MEMS sensor with a 21kHz resonant frequency. The orientation of the sensitivity axis of the MEMS sensor should be such that it can take advantage of the first order natural frequency performance of the cylindrical housing in the z-axis.

**Housing Prototype and Assembly Concept**

None of the simulation models shown so far take into account connector selection and its effect on the natural frequencies of the housing design. Figure 16 shows the M12 4-wire connector, TE’s part number is T4171010004-001. This connector is IP67 rated for water and dust resistance and includes a .STEP file from TE for easy integration into housing design files. This interface can be used with M12-to-M12 cables such as TAA545B1411-002 from TE.

Good mechanical installation is critical to ensure excellent vibration transmission and to avoid resonances that can affect performance. A good installation is generally achieved by simply screwing the bolts into the sensor housing and the monitored device. The stainless steel model shown in Figure 16 has a solid 7mm base that provides industry standard ¼”-28 tapped holes for mounting bolted connections to monitored equipment.

Figure 16. Housing prototype

The housing is 24mm in diameter and has a 25mm hexagonal base that can be used to attach the sensor to the monitored device. The overall height of a housing with an M12 connector may be between 48mm and 57mm, depending on manufacturing tolerances and internal wiring assembly or soldering options between the connector and the MEMS PCB. For example, if a straight line connection is used between the M12 bolt cap and the MEMS PCB, the height needs to be at least 5mm.

Figure 17. One possible assembly concept for a MEMS sensor PCB, M12 connector and housing

Figure 17 shows an exploded view of one possible assembly option for the housing, M12 connector and MEMS PCB. The MEMS PCB can be assembled to the housing wall using M3 screws, then connected to the M12 connector, and finally the two housing halves are laser welded together. As shown, the PCB is mounted vertically and the sensitivity axis of the ADXL1002 MEMS is vertically aligned with the z-axis of the housing. Vertical mounting is also important from a system measurement point of view, since measurements from this orientation are generally required when measuring bearing faults on motors (eg, radiated vibration measurements).

**Model simulation**

Before simulating the model, a solid body should be created using the components shown in Figure 17. The resulting simulation model closely matches the assembled and welded sensor. For accurate FEM numerical simulations, especially for connector geometry, fine meshes should be selected. Select the Fine Span Angle Center ANSYS Mesh option for excellent performance. Figure 18 shows the FEM mesh, and the relative deformation of the shell after simulation.

Figure 18. FEM mesh details and relative deformation of the shell

In Figure 18, the gradient from blue to orange to red shows that the relative structural deformation of the housing and the top of the connector is greater.

Figures 19 and 20 show the FEM results for the first stage natural frequency, effective MPF (ratio of effective mass to total mass of the system greater than 0.1) versus total sensor height on the z-axis. Z-axis performance is critical, with a first-stage effective natural frequency of 19.38kHz at a housing height of 52mm. With an overall height of 48mm, performance increases to 22.44kHz. With a case height of 50mm, the performance is about 21kHz.

Figure 19. First Stage Effective Natural Frequency (z-axis) vs. Enclosure Height

Figure 20. First Stage Effective Natural Frequency vs. Enclosure Height (x, y, and z Axes)

**Triaxial 10kHz MEMS Sensor with 21kHz Resonant Frequency**

Controlling the natural frequency of a housing design across three axes is more difficult than with a single-axis sensor, especially when 21kHz performance is required.

**ADcmXL3021**

Fortunately, Analog Devices has developed the ADcmXL3021 ±50g, 10kHz three-axis digital output MEMS vibration detection Module, shown in Figure 21. The ADcmXL3021 is housed in a 23.7mm × 27.0mm × 12.4mm aluminum package with four mounting flanges that can be mounted using standard M2.5 machine screws. The aluminum profile and geometry of the ADcmXL3021 package supports resonant frequencies above 21kHz in the x, y, and z axes.

Figure 21. ADcmXL3021 Triaxial Digital Output MEMS Sensor in Aluminum Package and Flexible Connector

**Attach ADcmXL3021 to IP67 rated enclosure **

Deploying the ADcmXL3021 in an industrial environment requires an IP67 rated (water and dust) enclosure and connectors. Also, the SPI output of the ADcmXL3021 is not suitable for use with long cables. An Industrial Ethernet or RS-485 transceiver circuit is required to convert the SPI output for long cable drives.

Based on the research in this paper, it is not possible to deploy the ADcmXL3021, RS-485 or Ethernet PCB and one connector in the same housing and achieve a 21kHz resonant frequency in all three axes (x, y, and z). Through the combination of components, the housing size can be reduced as much as possible, as shown in Figure 2 above (40mm × 43mm × 37mm). Figure 2 shows that the effective natural frequencies of the first order are between about 10kHz and 11kHz on all three axes. Furthermore, the simulation in Figure 2 does not use connectors, which would increase the actual height and further reduce the natural frequency.

If a simple rectangular aluminum enclosure with dimensions of 23.7mm × 27mm × 12.4mm (such as ADcmXL3021) and a wall thickness of 2mm were simulated using FEM, the effective first-order natural frequency would exceed 21kHz on all axes.

Figure 22. Increase the height of a shape such as ADcmXL3021

If the 12.4mm height is doubled and tripled to provide room for additional circuitry, the natural frequency drops significantly, as shown in Figure 22. Even with only 12.4mm of space left to accommodate additional circuitry, the effective natural frequency of the first stage drops below 15kHz.

**Distributed Systems**

ADI recommends using the distributed system shown in Figure 23 rather than trying to integrate all components into one rectangular enclosure. According to this concept, the ADcmXL3021 is packaged into an IP67 rated enclosure and SPI data is routed over short distances (less than 10cm) into a separate IP67 rated enclosure with integrated cable interface PCB, Ethernet or RS -485 transceivers, and associated power ICs and other circuits.

Using this method, the geometry is greatly reduced and the problem of matching the natural frequency of the housing to the natural frequency of the ADcmXL3021 is also significantly simplified.

Figure 23. ADcmXL3021 and Interface Circuits Packaged in Separate Enclosures

**Design and Modal Analysis**

As shown earlier, a rectangle is a good way to achieve similar natural frequency performance in all three axes compared to a cylinder. In Figure 23, the ADcmXL3021 is packaged in a small form factor hollow rectangular housing with a miniature PCB for the connection between the ADcmXL3021 flex cable and the industrial connector. This model can use a small M8 interface such as TE 7-1437719-5. The rectangular housing has 4 M2.5 mounting holes for fixed mounting to the device. The overall dimensions of the housing are 40.8mm × 33.1mm × 18.5mm. Importantly, the z-axis height is 18.5mm, which helps enable higher frequency modes.

In Figure 24, the y, x planes and 4 M2.5 holes are constrained for modal simulation. The z-direction is the weakest link in the entire design, even under 20mm in height. Figure 25 shows one of the dominant modes of the FEM modal simulation, which shows a greater degree of relative structural deformation at the top of the enclosure.

Figure 24. Hollow case used to encapsulate ADcmXL3021

Figure 25. Dominant Mode for Hollow Case Simulation for Package ADcmXL3021

Figure 26. First-order effective natural frequency vs. wall thickness in z-axis

The stiffness in the z-direction can be increased by increasing the wall thickness. For example, if a 2mm wall thickness is used, the effective first-order natural frequency in the z-direction is 14.76kHz. With a 3mm wall thickness, this frequency will increase to 19.83kHz. As shown in Figure 26, with a 3.5mm wall thickness, the natural frequency in the z direction exceeds 21kHz.

**Add epoxy to the shell**

Epoxy can be added to the vibration sensor housing to keep the hardware PCB in place and prevent the connectors and internal wiring from moving.

To study the effect of epoxy on the natural frequencies of the enclosure, a simple FEM model was created using a 40mm × 40mm hollow stainless steel cube with a fixed wall thickness of 2mm. The cube is filled with 36mm x 36mm epoxy resin. Increase the shell height from 40mm to 80mm to 100mm, alternating with and without epoxy filling. When performing FEM simulation, the x and y planes are used as fixed constraints.

Table 7 shows the simulation results, some of which are very interesting:

► When the sensor height is low, and the height is equal to the length/width, the epoxy resin increases the effective fixed frequency of the first stage of the cantilever axis (z) by 75%.

► When the sensor height is 80mm, which is 2 times the length/width, the effective first-order natural frequency of the cantilever axis (z) is increased by 16% if epoxy is filled. However, the frequencies of the x and y radial axes are reduced by 10%.

► Epoxy reduces the effective natural frequency of the first stage when the height is increased to 3 times the length/width.

Table 7. Height (mm), 40mm (length) × 40mm (width) of stainless steel cube with wall thickness of 2mm,

Epoxy fill (yes/no) and first stage effective natural frequency

As height increases, mass increases and stiffness decreases. At some point, the increase in mass has more of an effect than the increased stiffness of the epoxy. In the given simulation example, this inflection point corresponds to a height greater than 80mm. However, most sensors are generally less than 80mm in height. From this it can be concluded that in most cases the addition of epoxy can help improve the natural frequency performance of the vibration sensor housing solution.

**External cable emulation**

After installing the vibration sensor on the surface of the machine, the cable should be fixed to reduce the stress of the cable terminal and prevent false signals caused by cable vibration. When securing the cable, leave enough slack to move the accelerometer freely.

This section simulates the effect of a vibrating cable on the system response and provides guidance on where (at what cable length) the cable should be clamped.

Create a simulation model with the material properties shown in Figure 27. TE offers connector and cable models, such as the TAA545B1411-002, which can be used as a benchmark. Cable connectors are made of nylon (nylon 6/6) with copper core wires and PVC insulation. The attached sensor is designed in stainless steel and filled with epoxy. The simulation model supports a fixed constraint at the sensor connection, and the entire length of the 0.15m cable is free to vibrate. For simulation, the 0.15m cable length can be increased to 1m.

**Table 8 shows the simulation results with some key findings:**

► If the cable is clamped at lengths shorter than 0.15m, the cable has minimal effect on the frequency response of the vibration sensor. The frequency response of the sensor housing is above 11kHz with or without the 0.15m cable.

► If a 1m cable is attached to the sensor and the cable is allowed to move and vibrate freely along its entire length, the added weight of the cable will determine the frequency response of the system. The 500Hz cable frequency response will be the dominant mode.

In fact, it is impossible for the entire 1m cable to vibrate, as the vibration diminishes as the length of the cable increases. However, this simulation example shows that fixing at about 0.15m helps to achieve precise system response.

Table 8. Cable Length (m) and Primary Effective Natural Frequency (Hz) with and without Vibration Sensor Housing Attached

Figure 27. Cable and sensor model with material properties and 0.15m cable length

**Vibration sensor installation**

Figure 28 shows the effect on mounting resonance and the typical usable frequency range for the bolt, adhesive, adhesive mounting pad, and flat magnet tips shown in Figure 29. Bolt and adhesive mounting keeps the sensor as close to the machine as possible, enabling excellent vibration signal coupling between the machine and the MEMS sensor. When using a clamp with adhesive mounting pads, additional metal material is added between the machine and the sensor. These extra materials reduce the frequency response in the sensor solution. Flat magnet mounting also reduces frequency response and is not as reliable as other methods when securely attached to equipment.

Figure 28 provides typical guidelines only, and each sensor should be characterized by laboratory measurements or simulations.

Figure 28. Effect of Mounting Tips on Sensor Resonance

The bolted installation was simulated by ANSYS modal analysis using default bonded contact constraints.At this point, the bottom of the vibration sensor, specifically the ¼”-28” mounting holes are specified as fixed constraints when using ANSYS. The constraint type is the default bonded or bolted connection

Simulating bonded contact is an advanced topic that requires the use of ANSYS Cohesion Modeling (CZM) and an understanding of contact mechanics. To ensure accuracy, ANSYS CZM requires input parameters based on laboratory test data. For example, the article “Direct Measurement of Adhesive Cohesion Relationships Using the Rigid Dual Cantilever Technique” can be used as an input to ANSYS. If you do not find published experimental data for the adhesive of your choice, you will need to do some lab measurements. Additionally, proper contact formulations need to be set up in ANSYS, with instruction in short courses such as basic topics in contact mechanics. Finally, a combination of CZM and modeling techniques is required in the ANSYS workbench.

Magnetic fields can be simulated using ANSYS Maxwell. However, since magnetic forces are non-contact forces (they push or pull objects, but have no “real” contact), corresponding contact constraints cannot be generated for numerical modal analysis. Modal analysis can be performed with bonded, frictionless, frictional, and non-separating contact. As mentioned earlier, CZM contacts may be able to be achieved.

Figure 29. Mounting techniques for vibration sensors

**Summarize**

Designing a good mechanical enclosure for the MEMS accelerometer ensures that high-quality CbM vibration data is extracted from the monitored asset.

Understanding modal analysis is necessary to design a good mechanical enclosure for a MEMS accelerometer. Modal analysis provides the natural frequencies of the vibration sensor housing in the relevant axis. Additionally, designers can utilize the Mode Participation Factor (MPF) to determine whether a frequency can be ignored in the design.

Material properties and geometry need to be considered when designing a vibration sensor housing to meet natural frequency targets. The housing height needs to be kept as low as possible to achieve higher natural frequencies. Reducing the wall thickness or housing diameter will have a secondary effect on the natural frequency of the housing.

Compared to rectangles, the cylindrical shape has a larger cross-sectional area and its design is more conducive to achieving higher stiffness and natural frequencies in all axes. Rectangular provides more sensor mounting orientation and device connection options than cylindrical. The rectangle helps maintain similar natural frequency performance in all three axes.

In most cases, the addition of epoxy can help improve the natural frequency performance of the vibration sensor housing solution. Mounting with bolts or adhesive gives the vibration sensor an excellent usable frequency range, while the use of magnets or adhesive pads reduces sensor performance.

**About the author**

Richard Anslow is a systems applications engineer in the Interconnected Motion and Robotics team in the Automation and Energy business unit at Analog Devices. His areas of expertise are condition-based monitoring and industrial communication design. He holds a Bachelor of Engineering and a Master of Engineering from the University of Limerick, Ireland.

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