Vibration Isolation for Compact Electronic Subsystem

Abstract

Compact electronic assemblies may experience degraded operational stability under broadband vibration excitation. Mechanical vibration can introduce positioning instability, structural resonance, and sensitivity of precision components to dynamic loading.

This study investigates vibration isolation concepts for a miniature suspended assembly subjected to broadband vibration excitation.

Several isolation approaches are comparatively analyzed, including:

• metallic and polymer helical springs
• elastomer suspensions
• wire rope isolators
• quasi-zero stiffness (QZS) systems

The analysis includes:

• stiffness estimation
• tolerance sensitivity
• geometric scalability
• damping behavior
• transmissibility characteristics

The results show that elastomer-based suspensions provide the most favorable balance between compactness, damping, manufacturability, and robustness for compact low-mass suspended assemblies.

In particular, diagonal elastomer arrangements that utilize combined axial and shear deformation exhibit reduced stiffness sensitivity while maintaining practical dimensions and stable dynamic behavior.

1. Introduction

Compact electronic assemblies may be exposed to broadband vibration generated by cooling systems, airflow excitation, and structural coupling between mounted components.

For miniature suspended structures, even relatively small vibration amplitudes may produce:

• unwanted dynamic motion
• increased structural loading
• reduced positional stability [1–4]

The vibration environment considered in this study covers the frequency range of 10–400 Hz with acceleration amplitudes up to 1.5 g.

The considered miniature assembly has approximate dimensions:

100 × 30 mm

and a mass of:

0.1 kg

Effective vibration isolation for compact, low-mass systems is challenging because low natural-frequency requirements must be met within limited geometric dimensions and allowable displacement constraints.

In addition, the isolation system must remain:

• mechanically robust
• manufacturable
• tolerant to assembly and material variations

Several compact isolation approaches are considered in this study, including:

• metallic and polymer helical springs
• elastomer suspensions
• wire rope isolators
• quasi-zero stiffness (QZS) systems

The analysis focuses on:

• stiffness estimation
• geometric scalability
• damping behavior
• tolerance sensitivity
• transmissibility characteristics

The objective of the study is to identify practical vibration-isolation concepts suitable for miniature suspended assemblies subjected to broadband vibration excitation.

2. Mechanical Layout

The development of the isolation scheme is driven by installation constraints and required dynamic performance.

The miniature suspended assembly may be installed in either a horizontal or vertical orientation, and the final configuration is not predefined.

Therefore, the isolation system must provide approximately identical stiffness along both principal directions.

In addition, the allowable displacement is limited to approximately 5 mm along each axis due to geometric constraints.

A natural starting point is a symmetric configuration with isolators aligned along the principal axes.

Such an arrangement ensures independent control of stiffness in each direction and is well suited for elements operating primarily in axial deformation.

However, this approach has limitations when applied to compact systems.

Independent orthogonal elements tend to:

• utilize the available displacement inefficiently
• not provide coupling between directions
• limit opportunities for additional energy dissipation

When considering elastomer elements that exhibit significant compliance in both axial and shear deformations, an alternative configuration becomes possible.

By orienting the elements at 45°, each isolator participates in both directions of motion simultaneously.

This leads to:

• more efficient use of the available deformation range
• combined tension–shear behavior

Thus, the diagonal arrangement is not a general requirement, but a consequence of using elastomer elements and exploiting their material properties.

The number of supports is chosen as four, which is the minimum required for:

• static stability
• symmetric load distribution

Increasing the number of supports would increase overall stiffness, while fewer supports would compromise stability.

Figure 1: Comparison of isolation schemes: (a) orthogonal arrangement, (b) diagonal arrangement.

Although many geometric suspension arrangements are theoretically possible, the present comparison focuses on two representative configurations.

For spring-based systems, the isolator stiffness is primarily generated through axial deformation. Therefore, orthogonal arrangements naturally provide independent stiffness control and represent the most direct implementation.

Elastomer elements differ because they exhibit substantial compliance not only in axial loading but also in shear deformation. As a result, inclined arrangements allow simultaneous participation of both deformation modes.

Alternative layouts with additional supports or more complex geometries would primarily affect the number of elements and the load distribution, while the underlying deformation mechanisms would remain similar.

Therefore, the comparison focuses on orthogonal and diagonal layouts as representative configurations that illustrate the dominant mechanical behavior.

3. Problem Definition

The miniature suspended assembly must be isolated from vibration generated by the cooling system fans.

• The operating conditions are defined as:
• Frequency range: 10–400 Hz
• Acceleration amplitude: 0.5–1.5 g
• Allowable displacement along one axis: ≈ 5 mm

These constraints define the required balance between low stiffness, compact dimensions, and controlled system motion.

4. Reference Requirement

The key design parameter of the isolation system is its natural frequency (fₙ) [1,2].

To achieve effective vibration isolation within the operating range of 10–400 Hz, the system's natural frequency should ideally be placed below the lower excitation boundary.

However, achieving such low stiffness in a compact miniature suspension introduces significant geometric and manufacturing challenges.

In this study, the initial target natural frequency is set to:

To achieve a required natural frequency, the system stiffness must be adjusted accordingly.

For a single-degree-of-freedom system, the natural frequency is defined as [1,2]:

where:

• m — isolated mass
• ksys— total stiffness of the isolation system

Rearranging this expression, the required stiffness can be written as:

Substituting the system parameters: m = 0.1 kg, fₙ = 10 Hz

the required system stiffness becomes:

Assuming four identical supports, the required stiffness per support is:

5. Displacement Analysis

In addition to stiffness, the allowable displacement of the system is a key design constraint.

The total displacement of the isolated mass consists of two components:

• static deflection caused by gravity
• dynamic deflection caused by external vibration

The static deflection results from the weight of the compact subsystem acting on the isolation system.

It can be expressed as [1]:

where:

• Fg = mg — gravitational force
• ksys — total stiffness of the system

Substituting the system parameters: m = 0.1 kg, g = 9.81 m/s²

gives:

Thus, the static deflection becomes:

The dynamic deflection is caused by external acceleration.

In the worst-case scenario, the system is subjected to vibration with an amplitude of up to 1.5 g.

The corresponding dynamic force is:

where: a — acceleration amplitude

For a = 1.5g :

The dynamic displacement is then:

The total displacement depends on the relative direction of static and dynamic loads.

In general, these components may not coincide in phase or direction.

However, for design purposes, the worst-case scenario is considered, where static and dynamic displacements are aligned.

In this case, the maximum displacement is:

This value exceeds the allowable displacement limit of 5 mm.

The result indicates that although a target natural frequency of 10 Hz is theoretically attractive for vibration isolation, achieving such low stiffness may lead to excessive displacement under worst-case loading conditions.

Therefore, practical implementation may require a compromise between isolation performance and allowable motion constraints.

6. Isolation Concepts

The following sections compare several candidate vibration-isolation approaches for compact, low-mass suspended assemblies.

The analysis focuses on:

• achievable stiffness
• geometric scalability
• manufacturability
• tolerance sensitivity
• damping behavior
• practical implementation constraints

6.1 Metallic Helical Spring

A classical helical spring is considered the reference spring-based isolation concept.

For compact vibration isolation systems, metallic springs primarily operate in axial deformation, with stiffness determined by the torsion of the spring wire.

The stiffness of a helical spring is given by [1,2]:

where:

• G — shear modulus of the material
• d — wire diameter
• D — mean coil diameter
• n — number of active coils

For miniature isolation systems, the spring dimensions are strongly constrained by compact subsystem size and available installation space.

A representative compact geometry is considered:

d = 0.4 mm, D = 4.0 mm, n = 12

For spring steel (G ≈ 77 GPa [2]) the resulting stiffness is:

which corresponds to the required stiffness level for the considered isolation system.

However, the stiffness of a helical spring is highly sensitive to geometric tolerances.

Since:

even small variations in wire diameter produce large stiffness deviations.

For a realistic miniature spring tolerance: d = 0.4 ± 0.1 mm

the resulting stiffness range becomes:

corresponding to nearly one order of magnitude variation.

Since the system's natural frequency depends on stiffness as:

This leads to substantial variation of dynamic behavior and resonance position.

In miniature systems, this problem becomes especially severe because the required wire diameter is extremely small, making the spring highly sensitive to manufacturing repeatability and assembly variations.

As a result, metallic helical springs offer good theoretical control over stiffness but exhibit poor robustness and reproducibility in compact, low-mass isolation systems.

6.2 Polymer Helical Spring

Based on the limitations discussed above, a logical approach is to reduce the sensitivity of spring stiffness to wire-diameter tolerances by increasing the wire diameter.

According to the spring stiffness relation:

This can be achieved in several ways:

• reducing the shear modulus G
• increasing the coil diameter D
• increasing the number of active coils n

However, for compact vibration isolation systems, both the coil diameter and the number of coils are strongly constrained by available installation space.

A significant increase in D or n would result in excessive overall dimensions for the isolator.

Therefore, the most practical approach is to reduce the shear modulus by replacing steel with a polymer material.

Typical shear modulus values are:

• spring steel: G ≈ 77 GPa
• PEEK: G ≈ 1.3–1.5 GPa
• TPC-ET: G ≈ 0.05–0.3 GPa [2,3]

The substantially lower shear modulus allows the use of significantly thicker wire while maintaining the same stiffness level.

As a result, polymer springs can offer improved manufacturability and lower geometric sensitivity than ultrathin metallic springs.

In addition, manufacturing tolerances for miniature polymer springs are typically smaller relative to the wire diameter than for ultrathin steel wire.

Representative achievable tolerances are:

• precision micro-fabrication: ±0.015 mm
• realistic production: ±0.03 mm
• poor reproducibility: ±0.05 mm

For a representative polymer spring geometry: d = 0.75 mm

a realistic tolerance of: Δd = ±0.03 mm

results in a stiffness variation of approximately:

which is significantly lower than the variation observed for the metallic miniature spring.

However, although the geometric sensitivity is reduced, polymer springs introduce additional uncertainty associated with viscoelastic material behavior.

Unlike metallic springs, polymer springs may exhibit:

• residual stresses after molding
• thermal shrinkage
• material relaxation
• frequency-dependent stiffness variation

In addition, the effective number of active coils may differ from the nominal value:

due to end constraints, partial coil contact, and post-processing deformation.

As a result, both the effective shear modulus and the effective number of active coils may vary even for geometrically identical parts.

6.3 Elastomer Suspension

Following the transition from metallic to polymer springs, a natural next step is to consider the direct use of soft polymer materials themselves, i.e. elastomer-based isolators.

Unlike helical springs, elastomer elements do not rely on torsion of a thin wire and therefore avoid the strong geometric sensitivity associated with spring mechanics.

At the initial stage, the elastomer element is considered within the same orthogonal suspension layout previously used for spring-based concepts.

In this configuration, the element primarily operates in axial deformation, allowing direct comparison with metallic and polymer helical springs while preserving the same overall system geometry.

For an elastomer element operating in axial deformation, stiffness is determined by [1,2]:

where:

• E — elastic modulus
• A — cross-sectional area
• L — element length

A cylindrical geometry is selected as the reference elastomer element due to its axisymmetric shape and uniform deformation behavior.

For a cylindrical element:

Substituting the circular cross-section into the stiffness expression gives:

Thus, unlike helical springs:

instead of:

Using representative parameters:

E = 0.24 MPa, d = 2.8 mm, L = 4.7 mm

gives:

Assuming a representative manufacturing tolerance comparable to miniature polymer elements: d = 2.8 ± 0.03 mm

The resulting stiffness variation becomes:

corresponding to approximately ±2%.

This represents a substantial improvement compared to both metallic and polymer helical springs.

However, purely axial deformation does not fully utilize the mechanical properties of elastomer materials.

For the diagonal elastomer configuration, the objective is not necessarily to reduce the system stiffness below the target value, but to achieve the same target stiffness with a more robust element geometry.

For an element installed at 45°, the effective stiffness along a principal direction can be estimated as:

The squared sine and cosine terms result from the double projection of displacement and restoring force between the principal direction and the inclined isolator axis.

The presented relation is intended as a simplified engineering approximation and does not account for nonlinear elastomer behavior, shape factor effects, or large-deformation coupling.

For elastomers [3]:

Therefore:

To obtain the required effective stiffness:

The axial stiffness of the elastomer element can be increased to:

This means that the element itself may be made geometrically stiffer while preserving the required effective stiffness of the suspension.

Since:

The required diameter increase is:

For the reference axial element:

This gives:

Thus, the diagonal arrangement allows the use of a larger elastomer diameter while maintaining the same effective system stiffness.

For the same manufacturing tolerance:

The relative stiffness variation decreases from:

to:

The corresponding natural frequency variation is approximately half of this value:

Therefore, the diagonal elastomer layout not only utilizes shear deformation, but also improves robustness by allowing a larger and less tolerance-sensitive element geometry without increasing the effective stiffness of the suspension.

To better illustrate the practical impact of manufacturing tolerances, Figure 2 compares the estimated natural frequency variation of the considered concepts relative to the target frequency of 10 Hz.

Figure 2: Comparison of Natural Frequency Variation

The comparison shows that the diagonal elastomer configuration produces substantially lower frequency variation, indicating improved robustness and predictability.

6.4 Wire Rope Isolator

Wire rope isolators are widely used in industrial and electromechanical vibration isolation applications.

Their popularity is primarily determined by:

• relatively high damping
• mechanical durability
• fatigue resistance
• stable operation over a wide temperature range

Unlike conventional springs, wire rope elements dissipate energy through combined bending of the cable and friction between individual strands.

For a simplified first-order approximation based on Euler–Bernoulli beam scaling, the stiffness of a wire rope element may be estimated as [1,2,7]:

where:

• E — elastic modulus
• I — area moment of inertia
• L — characteristic loop length

For a circular wire cross-section:

Therefore:

This indicates strong sensitivity not only to wire diameter, but also to geometric scaling.

Using representative parameters for a miniature wire rope element:

d = 0.8 mm, L = 20 mm with E = 200 GPa

gives approximately:

which is within the required stiffness range for the considered suspension system.

However, achieving lower stiffness requires either:

• reducing the wire diameter
• increasing the loop size

For compact low-mass systems, both approaches become problematic.

Reducing the wire diameter imposes manufacturing and durability limitations, while rapidly increasing the loop size makes the isolator disproportionately large relative to the miniature suspended assembly dimensions.

In addition, wire rope isolators are inherently three-dimensional structures, making their integration into a compact assembly more difficult than that of simple elastomer elements.

Thus, although wire rope isolators provide good damping, durability, and stable long-term behavior, their geometric scaling becomes inefficient for miniature vibration isolation systems.

Compared with the elastomer suspension, the wire rope concept offers no substantial advantage in vibration-isolation performance while requiring significantly larger geometric dimensions.

6.5 Quasi‑Zero Stiffness (QZS)

Quasi-zero-stiffness systems represent an alternative approach to achieving very low natural frequencies without requiring extremely soft structural elements [6].

Unlike conventional isolators, a QZS system combines positive and negative stiffness components to achieve a small effective stiffness around the operating point.

The effective stiffness can be expressed as:

where:

• kpos — positive stiffness component
• kneg — negative stiffness component

The key feature of QZS systems is that the resulting effective stiffness is obtained as the difference between two substantially larger stiffness values.

For the considered compact subsystem, the target stiffness per support is approximately:

One possible implementation could use:

and

giving:

Such an approach is theoretically attractive because it allows very low effective stiffness while preserving relatively stiff structural elements.

However, the main limitation of QZS systems is the sensitivity of the resulting stiffness to small variations in parameters.

Both stiffness components are affected by:

• preload accuracy
• assembly tolerances
• alignment errors
• thermal expansion
• material relaxation

If each component changes by only:

The resulting effective stiffness changes significantly.

For the upper-bound case:

giving:

For the lower-bound case:

giving:

Thus, a relatively small variation of the individual stiffness components results in:

corresponding to approximately:

Since the system's natural frequency depends on stiffness as:

a nominal system frequency of:

may shift to:

This demonstrates the primary challenge of QZS systems: although the target stiffness can theoretically be achieved, maintaining stable and reproducible dynamic behavior requires precise tuning and stable preload conditions.

In principle, a QZS mechanism could be implemented for the compact subsystem suspension.

However, unlike platform-type systems with a well-defined vertical preload and a dominant loading direction, the compact subsystem suspension must operate in different installation orientations and within a very limited displacement envelope.

Generating negative stiffness in such a compact system would require additional preloaded springs, flexures, magnetic elements, or bistable mechanisms.

This would increase:

• overall size
• assembly complexity
• sensitivity to preload and alignment

Due to the low mass of the miniature suspended assembly:

The available static preload is also very small:

As a result, manufacturing tolerances, friction, thermal drift, and material relaxation may become comparable to the forces required for tuning the QZS mechanism itself.

In addition, the allowable displacement of the miniature suspended assembly is limited to approximately 5 mm, while QZS systems typically exhibit low stiffness only within a relatively narrow displacement range around the equilibrium point.

Therefore, although QZS systems are theoretically attractive for achieving low effective stiffness, they are not considered a robust practical solution for this miniature assembly suspension.

Compared with the diagonal elastomer suspension, the QZS approach offers no clear advantage in compactness while introducing substantially greater mechanical complexity and tuning sensitivity.

7. Transmissibility Analysis

After evaluating stiffness, manufacturability, and scalability, the next step is to compare the dynamic behavior of the isolation concepts under consideration.

Transmissibility describes how much vibration is transferred from the vibrating base to the isolated mass [1,2]:

where:

• T — transmissibility
• xin — input vibration displacement
• xout — transmitted displacement of the isolated mass

If:

The system amplifies vibration.

If:

The system provides isolation.

For a damped single-degree-of-freedom system, transmissibility can be estimated as [1,2]:

where:

• ζ — damping ratio
• r = f/fn — frequency ratio
• f — excitation frequency
• fn — natural frequency of the isolated system

The damping ratio represents the amount of energy dissipated during vibration.

In this comparison, three representative damping levels are used:

• low damping: metallic helical spring
• moderate damping: polymer spring or spring–damper system
• high damping: elastomer or wire rope isolator

Figure 3: Transmissibility versus frequency for representative damping levels.

The graph illustrates the fundamental trade-off in vibration isolation.

Low-damping systems, such as metallic helical springs, provide the lowest transmissibility at high frequencies.

However, they also produce a strong resonance peak near the natural frequency.

Increasing damping reduces this resonance amplification, which is beneficial for broadband excitation and systems with uncertain operating conditions [1,2].

This behavior is typical for:

• polymer springs
• spring–damper systems
• elastomers
• wire rope isolators

However, higher damping also increases transmissibility in the isolation region.

Therefore, a highly damped system may transmit more vibration at high frequencies than a lightly damped spring system, even when T remains below unity.

For the considered compact subsystem application, resonance control is especially important because the excitation frequency range starts near the target natural frequency.

In this condition, excessive resonance amplification is more critical than achieving the lowest possible transmissibility at very high frequencies.

Thus, elastomer and wire rope concepts provide a more stable broadband response than metallic springs, despite having higher high-frequency transmissibility.

The transmissibility behavior of QZS systems is not included in this simplified linear comparison.

Their response strongly depends on:

• preload tuning
• nonlinear stiffness characteristics
• The exact mechanism used to generate negative stiffness

Therefore, QZS behavior cannot be represented reliably by a single damping ratio, as with conventional isolators.

Overall, the transmissibility comparison shows that damping is not universally beneficial or harmful. It reduces resonance amplification but degrades high-frequency isolation. For the compact subsystem, this trade-off favors solutions with controlled internal damping and stable reproducibility rather than purely low-damping spring-based systems.

8. Practical Design Considerations

In addition to stiffness and transmissibility characteristics, practical implementation considerations play an important role in selecting a vibration isolation concept for a compact subsystem.

For compact low-mass suspensions, real-world performance is influenced not only by theoretical dynamic behavior, but also by:

• manufacturability
• assembly sensitivity
• environmental stability
• long-term reproducibility

Elastomer materials are known to exhibit viscoelastic effects such as:

• creep
• stress relaxation
• compression set
• long-term modulus variation [3,5]

These effects may alter suspension stiffness and natural frequency over time.

However, the considered compact subsystem application operates inside a temperature-controlled enclosure with relatively stable environmental conditions.

Unlike outdoor or automotive systems, the suspension is not exposed to:

• ultraviolet radiation
• large thermal cycles
• moisture
• aggressive contamination

As a result, environmental aging effects are expected to be substantially less severe, improving long-term stability and predictability of the elastomer suspension.

For miniature vibration isolation systems, assembly-related effects may also become comparable to the isolator's stiffness.

In particular:

• adhesive layer stiffness
• mounting frame rigidity
• local structural compliance
• preload variations

may noticeably influence the effective system behavior.

This becomes especially important for very soft isolation systems, where parasitic structural stiffness may significantly alter the resulting natural frequency.

Compared with spring-based and QZS systems, elastomer suspensions remain less sensitive to small geometric and assembly variations due to their distributed deformation behavior and lower sensitivity of stiffness to dimensional tolerances.

Another important practical aspect is geometric integration.

Wire rope and QZS systems typically require relatively large three-dimensional mechanisms or additional structural elements, making integration into miniature suspended assemblies more difficult.

In contrast, cylindrical elastomer elements may be integrated directly into compact frame structures while preserving relatively simple assembly and manufacturing processes.

Thus, for the considered compact subsystem application, practical implementation constraints favor mechanically simple and robust isolation concepts over theoretically optimal but highly sensitive solutions.

The overall analysis indicates that effective vibration isolation for miniature suspended assemblies is determined not only by achievable stiffness or transmissibility, but also by:

• long-term stability
• manufacturability
• assembly robustness
• compatibility with miniature system geometry

9. Conclusion

This study investigated several vibration-isolation concepts for a miniature suspended module operating under broadband vibration conditions typical of compact electronic assemblies.

Metallic helical springs demonstrated high sensitivity to geometric tolerances due to the strong dependence of stiffness on wire diameter.

Polymer springs reduced certain manufacturing limitations by allowing larger wire diameters but still retained significant stiffness sensitivity, along with additional viscoelastic uncertainty.

Wire rope isolators provided good damping and durability, but their geometric scaling became inefficient for compact low-mass systems.

Quasi-zero stiffness (QZS) concepts showed attractive theoretical capability for achieving very low effective stiffness, but required precise tuning and introduced substantial mechanical complexity and sensitivity to preload conditions.

Among the considered concepts, elastomer suspensions demonstrated the most favorable balance between:

• compactness
• manufacturability
• damping behavior
• robustness

The analysis further showed that diagonal elastomer arrangements that utilize combined axial and shear deformation achieve a lower effective stiffness while preserving practical dimensions and maintaining low sensitivity to manufacturing tolerances.

For the considered compact subsystem application, the results indicate that practical vibration isolation performance is determined not only by achievable stiffness, but also by:

• reproducibility
• environmental stability
• assembly robustness
• compatibility with miniature system geometry

Overall, elastomer-based diagonal suspension appears to be the most practical candidate within the scope of the analytical comparison presented.

References

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