Application of virtual sensing to an outdoor air-conditioning compressor unit

Abstract

This study demonstrates an acceleration-based virtual sensing (VS) approach for estimating low-frequency vibrations in a commercial air-conditioning compressor unit. Frequencies corresponding to the fundamental rotational frequency (1f) of the compressor and its higher harmonics up to the sixth harmonic (6f) dominate the spectrum and are tied to reliability issues in piping, yet are difficult to monitor due to the system’s geometric complexity. A linear Kalman filter (KF) was implemented under a zero-input assumption (u = 0) during quasi-steady intervals, explicitly avoiding reliance on high-fidelity finite-element (FE) models or known excitations. A speed-sweep test on a full-scale outdoor unit was conducted; estimated accelerations were benchmarked against independent measurements using a cross signature scale factor (CSF). With the observation noise parameter α fixed at 100 and the process noise parameter β tuned, the method achieved an average CSF of 0.79 (0–200 Hz), while capturing the dominant resonance near 122 Hz and overall trends despite model discrepancies. These results indicate that accurate low-frequency vibration estimation is feasible with limited sensors and imperfect FE models, supporting scalable diagnostics and predictive maintenance in HVAC equipment and other vibration-sensitive products.

Introduction

Low-frequency noise and vibration in commercial air-conditioning compressor units have become a significant concern in both residential and industrial environments, as they are inherently difficult to mitigate using conventional sound-absorbing treatments. Unlike higher-frequency components, low-frequency structural responses propagate efficiently through mechanical paths and radiate with limited attenuation, often bypassing typical enclosure or insulation measures. International noise regulations for HVAC equipment vary across jurisdictions. Nevertheless, there is broad demand to reduce low-frequency emissions to improve acoustic comfort and meet tightening environmental standards. In particular, frequency components corresponding to one to six times the compressor’s rotational frequency (1f–6f) are of interest, as they often dominate the low-frequency spectrum and are closely associated with vibration-induced reliability issues such as fatigue cracking and contact wear. Similar issues have been reported in HVAC compressors, where low-frequency noise due to internal impacts was mitigated using semi-active control strategies. ¹

During product development, accurate evaluation of vibration and stress in refrigerant and water piping systems is essential to ensuring both structural integrity and acoustic performance. However, the complex geometry, numerous connection interfaces, and distributed nature of the piping network make full-field vibration measurements impractical. Incident reports from regulatory and safety agencies in Japan over the past two decades^2–5 have documented multiple cases of vibration-induced pipe failures, highlighting the urgent need for diagnostic and monitoring techniques capable of operating with limited sensor coverage.

Virtual sensing (VS) based on Kalman filtering has been widely applied to estimate unmeasured states from sparse measurements combined with models.^6–8 Its effectiveness has been validated on simple structural elements such as beams, frames, and acoustic enclosures,^9–16 and in rotating machinery and vibroacoustic applications.^17,18 However, most existing VS approaches assume either known excitations or high-fidelity FE models—assumptions rarely satisfied in full-scale industrial products. Recent contributions in JLFNVAC^1,19–21 have addressed low-frequency vibration and compressor diagnostics, yet the challenges of unknown inputs and imperfect FE models remain largely unresolved. In practice, full-system FE models of HVAC equipment inevitably require simplifications—such as idealized joint stiffness, local flexibility approximations, or lumped-mass elements—resulting in frequency-dependent discrepancies. ²² To address these uncertainties, this study applies acceleration-based VS using a linear KF, under a zero-input (u = 0) assumption during quasi-steady intervals. This assumption, previously validated in simplified structures, is here extended to a full-scale outdoor unit as a product-level validation under realistic industrial conditions.

From an industrial perspective, failures such as pipe fatigue or joint loosening lead to costly downtime and warranty claims.^2–5 To address this challenge, the present study applies an acceleration-based linear KF to a fully assembled outdoor air-conditioning compressor unit—a structurally complex product with bolted joints, rubber isolators, and multiple coupled substructures—operating under partially unknown excitations.

The robustness of the proposed method is evaluated by combining experimental sweep tests with a full-system FE model containing realistic structural simplifications. Performance is benchmarked against independent acceleration measurements using the Cross Signature Scale Factor (CSF) metric, ²³ with systematic tuning of noise parameters to optimize accuracy. The findings show that reliable low-frequency vibration estimation is achievable for structurally complex systems under realistic industrial conditions, underscoring the potential of acceleration-based VS as a diagnostic tool for condition monitoring and predictive maintenance in HVAC and other vibration-sensitive equipment.

Theory of virtual sensing

Overview of the state-space model

The mechanical behavior of the system is described using a state-space representation derived from the governing finite-element equation of motion. Assuming a single-input, multiple-output (SIMO) system, the dynamics can be written as

${\mathit{M}}_{\mathit{v}}\ddot{\mathit{d}}\left(\mathrm{t}\right)+{\mathit{C}}_{\mathit{v}}\dot{\mathit{d}}\left(\mathrm{t}\right)+{\mathit{K}}_{\mathit{v}}\mathit{d}\left(\mathrm{t}\right)={\mathit{F}}_{\mathit{v}}\mathrm{u}\left(\mathrm{t}\right)$

(1)

where ${\mathit{M}}_{\mathit{v}}$ , ${\mathit{C}}_{\mathit{v}}$ and ${\mathit{K}}_{\mathit{v}}$ are the mass, damping, and stiffness matrix, respectively; ${\mathit{F}}_{\mathit{v}}$ is the input location selector; $\mathrm{u}\left(\mathrm{t}\right)$ is the excitation; and $\mathit{d}\left(\mathrm{t}\right)$ is the displacement vector. In this study, these matrices are directly extracted from the full-system FE model of the compressor unit, which was developed using Siemens NX Nastran (see Section “Finite element model” for details). Defining the state vector as

$\mathit{x}\left(t\right)=\left\{\begin{array}{l}\mathit{d}\left(\mathrm{t}\right)\\ \dot{\mathit{d}}\left(\mathrm{t}\right)\end{array}\right\},$

(2)

the first-order state-space form becomes

$\dot{\mathit{x}}\left(\mathrm{t}\right)=\mathit{A}\mathit{x}\left(\mathrm{t}\right)+\mathit{B}\mathrm{u}\left(\mathrm{t}\right)+\mathit{w}\left(\mathrm{t}\right)$

(3)

$\mathit{y}\left(\mathrm{t}\right)=\mathit{C}\mathit{x}\left(\mathrm{t}\right)+\mathit{D}\mathrm{u}\left(\mathrm{t}\right)$

(4)

where $\mathit{w}\left(\mathrm{t}\right)$ is the process noise, and $\mathit{y}\left(\mathrm{t}\right)$ contains the measured accelerations. The matrices $\mathit{A}$ , $\mathit{B}$ , $\mathit{C}$ , $\mathit{D}$ are obtained systematically from the FE model.

To reduce computational cost, modal reduction is applied. The displacement vector $\mathit{d}\left(\mathrm{t}\right)$ is projected into a reduced modal space using the mode matrix $\mathit{V}$ , yielding modal coordinates $\mathit{\xi }\left(\mathrm{t}\right)$ . This results in diagonalized mass, damping, and stiffness matrices, facilitating efficient computation.

Kalman filtering for virtual sensing

To estimate unmeasured states such as accelerations or stresses at inaccessible locations, this study employs a discrete-time KF. The continuous-time model introduced in Section “Overview of the state-space model” is discretized using a zero-order hold on the inputs, and the corresponding matrices ( ${\mathit{A}}_{\mathit{d}}$ , ${\mathit{B}}_{\mathit{d}}$ , ${\mathit{C}}_{\mathit{d}}$ , ${\mathit{D}}_{\mathit{d}}$ ) are computed via Padé approximation of the matrix exponential.^24–26

The KF operates recursively through two steps: prediction and correction. In the prediction step, the state vector and its error covariance are propagated using the discrete system matrices^27,28:

${\widehat{\mathit{x}}}^{-}\left[\mathrm{k}\right]={\mathit{A}}_{\mathit{d}}\widehat{\mathit{x}}\left[\mathrm{k}-1\right]+{\mathit{B}}_{\mathit{d}}\mathrm{u}\left[\mathrm{k}-1\right],$

(5)

${\mathit{P}}^{-}\left[\mathrm{k}\right]={\mathit{A}}_{\mathit{d}}\mathit{P}\left[\mathrm{k}-1\right]{{\mathit{A}}_{\mathit{d}}}^{\mathrm{T}}+{\mathbf{\Sigma }}_{\mathit{S}}\hspace{1em}.$

(6)

The Kalman gain is then computed as

$\mathit{g}\left[\mathrm{k}\right]={\mathit{P}}^{-}\left[\mathrm{k}\right]{{\mathit{C}}_{\mathit{d}}}^{\mathrm{T}}{\left({\mathit{C}}_{\mathit{d}}{\mathit{P}}^{-}\left[\mathrm{k}\right]{{\mathit{C}}_{\mathit{d}}}^{\mathrm{T}}+{\mathbf{\Sigma }}_{\mathit{R}}\right)}^{-1},$

(7)

where the process and observation noise covariance matrices are assumed to be isotropic,

${\mathbf{\Sigma }}_{\mathit{R}}=\alpha \mathit{I},$

(8)

${\mathbf{\Sigma }}_{\mathit{S}}=\beta \mathit{I}\hspace{0.5em}.$

(9)

In the correction step, the predicted state is updated with the measurement residual:

$\widehat{\mathit{x}}\left[\mathrm{k}\right]={\widehat{\mathit{x}}}^{-}\left[\mathrm{k}\right]+\mathit{g}\left[\mathrm{k}\right]\left(\mathit{y}\left[\mathrm{k}\right]-{\mathit{C}}_{\mathit{d}}{\widehat{\mathit{x}}}^{-}\left[\mathrm{k}\right]\right)$

(10)

This recursive scheme provides an efficient framework for fusing limited acceleration measurements with reduced-order FE models. Importantly, the process noise parameter $\beta$ is tuned to account for model uncertainties, while the observation noise parameter $\alpha$ reflects sensor accuracy. This balance between model and measurement reliance is particularly critical when applying VS to complex industrial products, as considered in this study.

Zero-input assumption for industrial application

In this study, the input $\mathrm{u}\left[\mathrm{k}-1\right]$ is assumed to be zero, as the objective is to estimate the overall vibration response of the outdoor unit using only acceleration sensors. Previous work on beam structures ¹² has shown that, when steady-state vibrations dominate, the $\mathrm{u}=0$ assumption still yields accurate amplitude estimates.

Here, we extend this assumption to industrial equipment where coupled substructures, bolted joints, and isolators introduce model uncertainties. The aim is to demonstrate that accurate low-frequency vibration estimation remains feasible for complex, large-scale systems under realistic operational conditions without explicit input force information. A successful validation on the outdoor unit would further suggest applicability of the method to a broad range of vibration-sensitive industrial products.

Experimental equipment and analytical model

Test configuration

The test object was the outdoor unit of a commercial air-conditioning compressor, chosen as a representative example of a structurally complex and vibration-sensitive product. Figure 1(a) shows the external view, while Figure 1(b) indicates the location of the internal accelerometer used for validation. The sensor measured vibration at the marked position as the compressor speed was swept from 20–110 rps over a 45 s interval.

Figure 1.

Experimental setup for vibration measurement of the compressor unit.

No external shaker was employed; instead, the experiment relied on self-excitation generated by the compressor operation. This sweep test procedure is widely used in the HVAC industry and in other rotating machinery such as electric motors, as it excites a broad range of structural modes under conditions representative of real-world operation.

In service, compressors usually operate at discrete fixed speeds (e.g., high-cooling or energy-saving modes), producing weaker dynamic excitation than the sweep scenario. The chosen test therefore represents a stringent validation case for the proposed vibration estimation method, while also illustrating its applicability to a broader class of industrial machinery.

Analytical model

Finite-element model

A full-system FE model was developed under the same physical configuration as the tested unit (Figure 2). The compressor and fan motor were represented as lumped masses, the rubber isolators as bushing elements, the heat exchanger as solid elements, and the remaining structural parts as shell elements. The assembled model comprised 267,740 nodes and 332,812 elements.

Figure 2.

FE model of the compressor unit showing different isometric views for clarity.

To capture the dynamic behavior up to 200 Hz, a modal reduction was applied. The first 315 modes were retained, ensuring coverage of the frequency range of interest, and used to construct the reduced-order modal truncation matrix for the state-space formulation.

Assessment of model accuracy

The accuracy of the FE model strongly affects the performance of KF-based virtual sensing. To evaluate this accuracy, frequency response functions (FRFs) obtained from the FE model were compared with experimental measurements at two representative locations: one near the excitation source (Figure 3(a)) and another at the furthest point (Figure 3(b)).

Figure 3.

Measurement and analysis points selected for FRF comparison in the compressor unit.

For the near location, Figure 4(a) shows that the FRF in the X-direction agrees closely with the measurements except for a deviation near 10 Hz. Larger discrepancies appear in the Y- and Z-directions, particularly around 10 Hz and within the 60–100 Hz range, reflecting the effects of structural complexity, unmodeled joint compliance, and model simplifications. At the far location (Figure 4(b)), the overall FRF trend is captured, but notable frequency-dependent errors remain. These discrepancies highlight the inherent limitations of full-system FE models while raising the central question of this study: whether KF-based VS can still provide reliable vibration estimates under such model uncertainties.

Figure 4.

Comparison of FRFs between measurements (black solid line) and FE model calculations (red solid line) at the points indicated in Fig. 3.

Validation of virtual sensing for vibration estimation accuracy

The performance of the proposed VS approach was validated by comparing estimated accelerations with independent measurements at selected verification points. The accelerometer configurations are shown in Figure 5. Eight accelerometers were used as observation sensors (KF inputs), while five additional accelerometers were reserved exclusively for validation.

Figure 5.

Sensor configurations for VS implementation and accuracy assessment.

Parameter tuning

The influence of the KF noise parameters $\alpha$ and $\beta$ in Eqs. (8)–(9) was investigated following the authors’ prior study, ¹² which demonstrated that the ratio $\alpha /\beta$ strongly governs estimation accuracy. Specifically, estimation error decreases below a certain threshold of this ratio. In the present study, the KF parameters ( $\alpha$ fixed at 10° to represent sensor noise scaling, and $\beta$ tuned from 10^‒18 to 10^‒8) were adjusted to evaluate sensitivity.

Accuracy evaluation metric

To quantify estimation accuracy, vibration data spanning 86 s—including transitional phases before and after speed changes—were processed. The peak acceleration spectrum was obtained from 1,000 FFTs using peak-hold averaging.

As the objective is amplitude accuracy rather than phase reconstruction, the CSF ²³ was adopted. While CSF is traditionally applied to FRFs, in this study it is extended to response spectra, enabling a direct comparison between measurements and VS predictions without relying on input-output relationships.

The average CSF was computed across three validation points, corresponding to Acc.1–Acc.3 in Figure 5. The relationship between CSF and $\beta$ is shown in Figure 6, revealing a maximum at $\beta$ = 1.33 × 10^‒12.

Figure 6.

Relationship between CSF and noise parameters ( $\alpha$ , $\beta$ ) with $\alpha$ fixed at 10° and $\beta$ swept from 10^‒18 to 10^‒8.

Estimation results under quasi-steady conditions

During the 45 s sweep, the change in compressor rotational speed was sufficiently gradual to approximate the response as steady state over short intervals. Accordingly, the KF was applied under the zero-input assumption (u = 0).

Figures 7 and 8 show the measured and estimated responses at Acc. 4 and Acc. 5 (see Figure 5) using the optimal parameter $\beta$ . The VS method employs measured accelerations to correct FEM-predicted responses, thereby improving agreement despite model uncertainties.

Figure 7.

Comparison of acceleration spectra (black: measured; red: VS-estimated).

Figure 8.

Comparison of acceleration time histories (black: measured; red dotted: VS-estimated).

The agreement between measured and estimated responses indicates consistent robustness. Across the 0–200 Hz range, the average CSF was 0.79, with values at individual validation signals ranging from 0.68 to 0.87. This indicates consistently meaningful correlation despite model uncertainties. At the dominant resonance near 122 Hz, the estimated amplitude deviated from the measured value by 43%, yet the resonance peak location and overall trend were accurately captured. Time-domain traces overlapped closely, confirming that the KF-based VS not only reproduces global resonance characteristics but also fine-scale temporal dynamics. Such robustness, demonstrated despite significant FE model discrepancies, highlights the practical reliability of the proposed method. These results confirm that acceleration-based VS can provide sufficiently accurate low-frequency vibration estimates for industrial applications, even under substantial model uncertainty.

The KF used in this study operates as shown in Eq. (10), where the state vector is sequentially corrected using acceleration measurements at the observation points to estimate the modal coordinates $\xi \left(\mathrm{t}\right)$ . $\xi \left(\mathrm{t}\right)$ represents how strongly each mode is excited, and the corrected $\xi \left(\mathrm{t}\right)$ is combined with the modal mass, damping, stiffness matrices, and the mode matrix V to reconstruct the full-system acceleration response. Importantly, only $\xi \left(\mathrm{t}\right)$ is corrected, while the modal matrices remain dependent on the FE model. Up to 200 Hz, 315 modes were retained, and the contribution of each mode is adjusted sequentially through $\xi \left(\mathrm{t}\right)$ . However, when the mode shapes differ from the real unit, residual errors appear in reconstructed responses at unmeasured points even if good agreement is achieved at the observation points. Since mode shapes are spatially smooth, agreement is generally better near the sensors, whereas discrepancies become more pronounced in regions strongly influenced by local flexibility. In fact, clear differences were observed in the 40–100 Hz range at Acc. 4. This frequency band is governed by bending and torsional modes of the piping, and the discrepancies are likely due to simplified modeling assumptions. Major factors include the following: (i) welded joints modeled as rigid connections, neglecting local flexibility; (ii) shape deviations introduced during manufacturing, leading to differences from the CAD model; and (iii) The rubber and butyl rubber sheets, as indicated in Figure 5 and wrapped around the pipes, were modeled using the density and Young’s modulus of butyl rubber, but without considering damping. Since Acc. Four is located near the butyl rubber sheets, discrepancies may arise from mismatches in the actual material properties of the sheets in use, as well as from damping effects not included in the model. These factors result in differences in the mode shapes between the FE model and the real unit, which in turn lead to residual errors at unmeasured points. Even so, the KF correctly reproduces the dominant modes and overall trends. This demonstrates that KF-based VS remains practically effective even under unavoidable modeling simplifications and product-to-product variability.

Spatial vibration distribution

To further illustrate the system behavior, Figure 9 shows the spatial distribution of VS-estimated vibration amplitudes at t = 40.9 s, corresponding to the dominant resonance near 122 Hz. Localized resonances are visible in the side panel and along sections of the piping network, consistent with expected structural mode shapes. These distributions provide additional evidence that the VS method can capture not only global vibration levels but also spatial features relevant to fatigue-sensitive regions.

Figure 9.

Spatial distribution of VS-estimated vibration amplitudes at t = 40.9 s (dominant resonance at 122 Hz).

Summary of findings

The validation results confirm that the proposed VS method can reliably estimate dominant resonances and overall vibration trends in a structurally complex industrial product, even under quasi-steady operating conditions and with significant FE model uncertainties. Importantly, the ability to achieve accurate estimates without relying on high-fidelity models highlights the robustness of the approach for practical low-frequency vibration monitoring in HVAC systems and other industrial equipment.

Conclusion

This work demonstrates that acceleration-based virtual sensing using a linear Kalman filter (KF) enables reliable low-frequency vibration estimation in full-scale HVAC compressor units without requiring high-fidelity FE models or known inputs, addressing persistent industrial problems such as pipe failures, joint loosening, and costly downtime. By achieving robust estimates under realistic industrial conditions, the approach directly addresses a key barrier to practical deployment and underscores its relevance for predictive maintenance.

The main findings are as follows:

• Accuracy under model uncertainty: Average CSF 0.79 over 0–200 Hz across three validation points (range 0.68–0.87), with the dominant resonance near 122 Hz and overall trends captured despite FE discrepancies.

• Validated operating regime: The u = 0 assumption holds for quasi-steady intervals of speed sweeps dominated by periodic compressor forces; it is not intended for transients or broadband random excitations.

• Practical calibration: With $\alpha$ fixed (10°) and β tuned, the best agreement occurred at $\beta$ = 1.33 × 10^‒12, providing a practical calibration strategy when input excitations are unknown.

Overall, the findings confirm that acceleration-based VS using a linear KF is a practical and scalable solution for low-frequency vibration monitoring and fatigue diagnostics in structurally complex industrial systems.

In addition to HVAC applications, the demonstrated robustness under model uncertainty suggests broader relevance. Potential extensions include fatigue diagnostics in piping systems, predictive maintenance frameworks for rotating machinery, and low-frequency noise reduction in industrial products. By enabling accurate estimation without high-fidelity models, the approach offers a scalable tool for industrial monitoring and design optimization.

Future work will extend the framework to partially known or stochastic input models, adaptive noise tuning, and long-term integration with predictive maintenance pipelines.