Ultrasonic Nonlinearity Research Articles

Nonlinear ultrasonic test applied to discern factors controlling cracking of cement mortar affected by reinforcement corrosion

Intermodulation ultrasonic testing technique was used to detect corrosion- induced cracking in prismatic mortar samples. Three sets of five specimens were prepared and subjected to accelerated corrosion conditions. The different times of exposure allowed to attain unlike damage conditions in every sample of each set. The onset of damage was monitored through the evolution of the accumulated Intensity Modulation Ratio (R) and crack width measurements. The accumulated values of the Intensity Modulation Ratio were used to study the effect of different testing factors (cover depth, corrosion rate or amplitude of the input pump signal) and increased monotonically clearly discriminating the different damage levels. These observations point to the possibility of stablishing damage thresholds to detect cracking using nonlinear ultrasonic (NLU) techniques.

Sideband Peak Count – a new nonlinear ultrasonic technique for monitoring damage progression in engineering materials

Linear ultrasonic (LU) techniques used by majority of the researchers working in material damage monitoring, are reliable for detecting relatively large defects. If the defect dimensions are in the range of the ultrasonic wavelength or larger, then those defects can be detected by analyzing the scattered ultrasonic fields. Material damage affects LU parameters such as ultrasonic wave speed and attenuation and their changes are detectable for relatively large defects. However, for small defects (when defect dimensions are significantly smaller than the wavelength of the propagating signal) the changes in the LU parameters are too small to detect or measure reliably. For detecting small defects engineers often use high frequency ultrasonic signals to make the defect dimensions greater than the wavelength. However, high frequency signals attenuate quickly and therefore, only very small regions near the ultrasonic probe can be inspected in this manner. Inspecting large structures by high frequency ultrasonic signals requires moving the probe mechanically from one point to the next and therefore, can be very time consuming. Nonlinear ultrasonic (NLU) technique on the other hand does not need to satisfy this restricting condition that the wavelength of the signal must be smaller than the defect size. NLU works well when the signal wavelength is much larger than the defect size. Therefore, relatively low frequency signals that can propagate a long distance and monitor a large area of a structure can be used for NLU measurements. Different NLU techniques can be used for detection and monitoring of small damages in a specimen. A relatively new NLU technique called the Sideband Peak Count-Index or SPC-I technique has been developed by the author and his collaborators. SPC-I technique for monitoring different types of materials – composites, metals, concrete, and other cement-based materials has been found to be effective. Along with those success stories of SPC-I the effect of topography and the advantage of topological sensing is also discussed in this presentation.

Precursor Damage Quantification in Composite Structures Using Coda Wave Interferometry and Nonlinear Ultrasonics

Abstract Methods to quantification of precursor damage in carbon fiber reinforced polymer (CFRP) composite structures are reported herein. These techniques include coda wave interferometry (CWI) and nonlinear ultrasonics (NLU). Since low-frequency Lamb wave propagation is insensitive to the early-stage material degradation, it is shown that decoding the information in coda wave can overcome this well-known limitation. To conclude this possibility, CWI technique is cross verified with a traditional high-frequency ultrasound method. To achieve this goal, a tensile–tensile fatigue experiment was designed for CFRP composite specimens. By inducing controlled fatigue damage in these structures, material states are assessed using low-frequency (<500 kHz) ultrasonic guided wave and high-frequency (>10 MHz) P-wave. Stretching guided coda wave is utilized to quantify the precursor damage as a unique approach in this article. However, such method could be illuded by the changes in the signals due to bonds and contacts. To verify if the CWI is successful, and to evaluate the precursor damage in composite structures, additional nonlinear analysis of ultrasonic signals from both guided waves and P-waves is performed. Higher order nonlinearities in both low-frequency guided wave and high-frequency P-wave propagation demonstrate the growth of precursor damage in CFRP composite structures. So does the CWI of low-frequency guided wave data. Accuracy of these ultrasonic techniques is validated with experimentally obtained remaining strengths of the fatigue specimens. With this verification it is envisioned that both CWI and NLU together could quantify the precursor damage in composite structures.

Sideband Peak Count – a new nonlinear ultrasonic technique for monitoring damage progression in engineering materials

Linear ultrasonic (LU) techniques used by majority of the researchers working in material damage monitoring, are reliable for detecting relatively large defects. If the defect dimensions are in the range of the ultrasonic wavelength or larger, then those defects can be detected by analyzing the scattered ultrasonic fields. Material damage affects LU parameters such as ultrasonic wave speed and attenuation and their changes are detectable for relatively large defects. However, for small defects (when defect dimensions are significantly smaller than the wavelength of the propagating signal) the changes in the LU parameters are too small to detect or measure reliably. For detecting small defects engineers often use high frequency ultrasonic signals to make the defect dimensions greater than the wavelength. However, high frequency signals attenuate quickly and therefore, only very small regions near the ultrasonic probe can be inspected in this manner. Inspecting large structures by high frequency ultrasonic signals requires moving the probe mechanically from one point to the next and therefore, can be very time consuming. Nonlinear ultrasonic (NLU) technique on the other hand does not need to satisfy this restricting condition that the wavelength of the signal must be smaller than the defect size. NLU works well when the signal wavelength is much larger than the defect size. Therefore, relatively low frequency signals that can propagate a long distance and monitor a large area of a structure can be used for NLU measurements. Different NLU techniques can be used for detection and monitoring of small damages in a specimen. A relatively new NLU technique called the Sideband Peak Count-Index or SPC-I technique has been developed by the author and his collaborators. SPC-I technique for monitoring different types of materials – composites, metals, concrete, and other cement-based materials has been found to be effective. Along with those success stories of SPC-I the effect of topography and the advantage of topological sensing is also discussed in this presentation.

Monitoring damage growth and topographical changes in plate structures using sideband peak count-index and topological acoustic sensing techniques

Some topographies in plate structures can hide cracks and make it difficult to monitor damage growth. This is because topographical features convert homogeneous structures to heterogeneous one and complicate the wave propagation through such structures. At certain points destructive interference between incident, reflected and transmitted elastic waves can make those points insensitive to the damage growth when adopting acoustics based structural health monitoring (SHM) techniques. A newly developed nonlinear ultrasonic (NLU) technique called sideband peak count – index (or SPC-I) has shown its effectiveness and superiority compared to other techniques for nondestructive testing (NDT) and SHM applications and is adopted in this work for monitoring damage growth in plate structures with topographical features. The performance of SPC-I technique in heterogeneous specimens having different topographies is investigated using nonlocal peridynamics based peri-ultrasound modeling. Three types of topographies – “X” topography, “Y” topography and “XY” topography are investigated. It is observed that “X” and “XY” topographies can help to hide the crack growth, thus making cracks undetectable when the SPC-I based monitoring technique is adopted. In addition to the SPC-I technique, we also investigate the effectiveness of an emerging sensing technique based on topological acoustic sensing. This method monitors the changes in the geometric phase; a measure of the changes in the acoustic wave’s spatial behavior. The computed results show that changes in the geometric phase can be exploited to monitor the damage growth in plate structures for all three topographies considered here. The significant changes in geometric phase can be related to the crack growth even when these cracks remain hidden for some topographies during the SPC-I based single point inspection. Sensitivities of both the SPC-I and the topological acoustic sensing techniques are also investigated for sensing the topographical changes in the plate structures.

Peri-Ultrasound Modeling to Investigate the Performance of Different Nonlinear Ultrasonic Techniques for Damage Monitoring in Plate Structures

Abstract Peri-ultrasound modeling which is based on nonlocal peridynamics is found and proven to be effective for modeling nonlinear waves propagating and interacting with damages in structures. This work presents the peri-ultrasound modeling to investigate the performance of three commonly used nonlinear ultrasonic (NLU) techniques—wave mixing, higher harmonic generation (HHG), and sideband peak count-index (or SPC-I) for monitoring damages (or cracks) in three-dimensional (3D) plate structures. Cracks can be defined as “thin cracks” and “thick cracks” according to the horizon size mentioned in peridynamics. Peri-ultrasound modeling results reveal that the SPC-I results are consistent with other reported numerical modeling and experimental results available in the literature. However, the modulation indicator (MI) from the wave mixing model only shows consistent trends for thin cracks but not for thick cracks and its reliability is affected by the initial excitation bandwidth. The relative acoustic nonlinearity factor β from the HHG technique shows consistent trends for thick cracks but not for thin cracks. It can be concluded from the obtained parametric analysis results that the SPC-I technique is more robust and reliable for monitoring damages in engineering structures.

Nonlinear acoustic response in nanoparticle-dielectric systems and nondestructive assessment of particle agglomeration

Nanoparticle agglomeration reduces the modification effect of nanocomposites, and there is a need to investigate a technique for evaluating the overall particle dispersion within the material, which in turn ensures the modification effect of the material. In this paper, a non-destructive testing technique for nanoparticle dispersion based on nonlinear ultrasonic (NLUS) is proposed to rapidly assess the overall nanoparticle dispersion within a material. An interfacial mismatch model describing different degrees of nanoparticle agglomeration was firstly developed, and then the nonlinear acoustic response of the nanoparticle-dielectric system was quantitatively solved, and the theoretical model was validated in conjunction with NLUS tests. The experimental results show that agglomerated interfaces are the source of the nonlinear response of nanodielectrics and that the microscopically generated nonlinear effects can be equivalently accumulated to the mesoscopic domain, with the error in the theoretically calculated values of the two being less than 5%; The NLUS technique eliminates the need for cumbersome steps such as particle size counting, and its small number of rapid detection results are close to the statistical convergence of the particle size of the 250 particles accumulated in the imaging image of the scanning electron microscope, with an error of between 3% ∼ and 13%. With a single inspection area of 339 mm3, the NLUS technology significantly reduces evaluation time and cost while expanding the inspection area, offering the combined advantages of speed, wide field of view, and non-destructiveness.

Creep strain prediction in power plant material via ANN modelling of nonlinear ultrasonic test results

ABSTRACT Reliable and accurate prediction of the creep life of power plant components is crucial for both economic and safety reasons. Existing prediction models, based on creep test data, can be complex and time-consuming. Nonlinear ultrasonic (NLU) is a widely-accepted non-destructive testing (NDT) technique for evaluating damage progression in crept specimens. The information from NLU measurements alone is insufficient to forecast the life of any component. In real-time applications, intelligent NDT protocols are needed to enable fast and accurate life prediction of such components. A methodology for creep life prediction using artificial neural networks (ANN) has been introduced based on NLU test results of crept P92 steel specimens. The technique involved creep tests of P92 specimens exposed to a temperature of 625⁰C with applied stress ranging from 120MPa to 160MPa, NLU measurements at each step load, and prediction of creep life of the material with a ANN trained with creep strain and NLU test data. The technique involves prediction from previously generated historical data, thus saving both cost and time of conducting continuous experiments. This approach for ANN modeling of NLU data can be considered a reliable, time-saving, and effective technique for assessing creep damage progression in power plant components.

An application of machine learning for material crack diagnosis using nonlinear ultrasonics

Crack diagnosis in non-destructive testing often requires reference data from the structure before damage or a considerable amount of response data. Also, detecting compression cracks is challenging. In this study, a machine learning-based method is proposed for diagnosing cracks in structures under compression. The method consists of convolutional neural networks (CNN) and fully connected networks (FCN). The CNN extracts features from nonlinear ultrasonic signal data, and the features determine the occurrence of fatigue cracks in a target specimen. Four types of input data are defined in accordance with the number of input frequency combinations. The performance of the proposed method is investigated using each data type to secure efficiency and accuracy in diagnosing aluminum specimens under various compression conditions. As a result, the F1 score, a measure of accuracy, of the proposed method depends on the number of input frequency combinations. The method detects high-compression cracks with high accuracy compared to the present technology specialized for compression cracks in a certain data type. A high accuracy of more than 96% is achieved with less computation time. The proposed method will provide an accurate crack diagnosis for compression cracks with reduced time and effort.

Nonlinear effects of dual-frequency focused ultrasound on the on-demand regulation of acoustic droplet vaporization

Dual-frequency ultrasound has been widely employed to enhance and regulate acoustic droplet vaporization (ADV) but the role of ultrasonic nonlinear effects on it remains unclear. The main objective of this study is to investigate the influence of nonlinear effects on the control of ADV nucleation under different dual-frequency focused ultrasound conditions. ADV nucleation of PFC nanodroplets activated by nonlinear dual-frequency ultrasound was modeled and parametric studies were conducted to investigate the influence of dual-frequency ultrasound frequency and acoustic power on the degree of nonlinearity (DoN), nucleation rates and dimensions of the nucleation region in a wide parameter range. The results showed that the ultrasonic nonlinearity caused a significant decrease in peak negative pressure due to waveform distortion, which leads to a lower nucleation rate in the nonlinear model compared to that in the linear model. Furthermore, the distributions of nucleation regions were also affected by the interaction between waves of different frequencies and cloud-like spatial distributions were produced, which could be modulated by the dual-frequency ultrasound parameters and have great potentials in the spatial regulation of the ADV and customized treatment protocols in clinical applications. In addition, represented by 1.5 MHz + 3 MHz, such a dual-frequency combination of fundamental and second harmonic could effectively enhance ultrasonic nonlinear effects with relatively lower peak negative pressure and higher DoN. Therefore, nonlinear effect of the dual-frequency ultrasound plays an important role in the ADV regulation, which should be considered in the numerical model and practical applications.

Frequency sweep study on the generation of dual-mode second harmonics (DMSH) on an isotropic nonlinear elastic cylindrical rod by T(0,1) mode

Nonlinear ultrasonic (NLU) guided wave (GW) measurements are more sensitive to microscale defects than linear measurements. This nonlinear phenomenon is helpful for early-stage damage detection in material for nondestructive evaluation (NDE) and structural health monitoring (SHM) applications. The amplitude of the higher harmonics generated in the material serves as the basis for the NLU measurement. Recent studies have reported the presence of dual-mode second harmonic (DMSH) on an isotropic nonlinear elastic plate and cylindrical structures. The fundamental non-dispersive mode shear horizontal SH0 and torsional T(0,1) modes can generate simultaneously propagating dual-mode second harmonic(DMSH) on plate and cylindrical guided media respectively. For the case of cylinders, at phase matching condition two dominant second harmonic fundamental longitudinal (l(0,1)) and orthogonal torsional (t(0,1)┴) mode are generated with significant amplitude. The particle vibration of the t(0,1)┴ mode was present in both orthogonal directions compared to conventional T(0,1) mode and hence called orthogonal torsional mode. The present work aims to study the behaviour of DMSH at different frequencies through a frequency sweep study on a weakly nonlinear elastic cylinder of circular cross-section via numerical simulations and validated experimentally for some selected cases. The wave propagation of the T(0,1) mode at several selected frequencies from 300 kHz to 2.2 MHz is chosen to explore the second harmonic mode for better understanding. At frequencies below 1.25 MHz, l(0,1) mode is faster than t(0,1)┴ mode; above 1.25 MHz, it was the other way around. The behaviour of harmonics was observed to be in accordance with the group and phase velocities at the corresponding frequencies in the dispersion curves. The evidence for the presence of DMSH was validated experimentally for some selected frequency cases at different propagation distances. The significance of DMSH is explored via numerical simulations and found that t(0,1)┴ mode is sensitive towards material degradation, while all other generated wave modes were not affected. Research shows that enough energy is available in the generated DMSH to be noticed experimentally and in numerical simulations. This improved understanding of the generation of DMSH across different frequencies would be helpful in new advanced NLU-based NDE and SHM applications.

Characterization of multiple debris cloud impacts-induced pitting damage evolution in spacecraft structures using temperature-dependent ultrasonic nonlinearity

Facing the harsh cryogenic-elevated temperature condition, evaluation of the debris cloud-induced pervasive but highly complex pitting damage in manned spacecraft structures, featuring multitudinous small-scale craters and cracks disorderedly scattered over a wide region, accompanied by a diversity of micro-damages, is hitherto still a challenging task. In line with the fact that the use of ultrasonic nonlinearity is restricted by its intrinsic vulnerability to measurement contamination, in particular temperature fluctuations. With this motivation, an insight into the modulation mechanism of typical elastic parameters of material at macroscopic under varying temperatures, as well as microscopic interatomic potential, is achieved via theoretical analysis. Based on this, temperature sensitivity coefficients (TSCs) of these elastic parameters are obtained, and a nonlinear temperature sensitivity index (TSI) is established. On this basis, an active TSI-based evaluation method is proposed to improve the accuracy and reliability of pitting damage evaluation by controlling detection temperature (i.e., exploitation of the benign aspects of the temperature sensitivity of ultrasonic nonlinearity), rather than eliminating or compensating for its adverse effect, when other nonlinear sources interference. This method is experimentally corroborated, in which the multiple debris cloud impacts-induced accumulated pitting damage are accurately and intuitively characterized through temperature control, in terms of its severity and distribution characteristics, which is consistent with the actual results. These findings provide an active structural health monitoring (SHM) strategy for promoting the practical application of this proposed approach for characterizing pitting damage evolution in manned spacecraft in qualitative and semi-quantitative manners.

Depth profiling of residual stress distribution in surface treated metallic structures using nonlinear ultrasonics

Robotic hammer peening (RHP) is a cold-working technique to improve the fatigue life of metallic structures by inducing compressive residual stress in the near-surface region. Measuring the depth profiling of residual stress distribution plays an important role in process design towards the advanced manufacturing of metallic structures. This study investigates the use of the nonlinear ultrasonic method to measure the residual stress profiles through the frequency-dependent penetration depth approach. Stainless steel specimens were treated with varying RHP intensities, and their morphological evolutions were characterized with a step-profilometer. The amplitudes of the second and third harmonic components of the longitudinal critically refracted (LCR) and Rayleigh waves were measured and analyzed. The effect of the surface roughness on the acoustic nonlinearity parameters before and after polishing at various peening intensities was briefly discussed. The results show a large variation in the acoustic nonlinearity parameter at the surface layer, indicating the potential of the proposed nonlinear ultrasonics for the measurement of depth profiling of residual stress.

Non-destructive evaluation of corrosion in steel liner plates embedded in concrete using nonlinear ultrasonics

Nuclear containment structures employ a leak-tight steel liner to avoid any unwanted particle release into the exterior environment. The steel liner can be embedded in concrete which creates large continuous areas where the steel plate is in direct contact with surrounding concrete. The impact of the aging of concrete structures due to the presence of structural discontinuities, such as the presence of foreign matter and a delamination gap at the embedded steel–concrete interface, is investigated for small laboratory-scale specimens manufactured with a novel inlay technique by nonlinear ultrasonic evaluation. A Total Damage Index (TDI) which combines several parameters related to acoustic wave distortion into a single damage index to rank the specimens is introduced. The findings in this work indicate that severe corrosion, the presence of foreign matter, and delamination gap in the steel–concrete interface can be detected using nonlinear ultrasonic techniques based on higher-harmonic analysis and modulation intensity using continuous ultrasonic probe excitation and an impactor to modulate the signal. The results obtained using the TDI are consistent with attenuation, a conventional parameter used in corrosion detection.

Nonlinear ultrasonic techniques to quantify oxidation damage in carbon/carbon composite material

This research applies linear and nonlinear ultrasonic (NLU) techniques to quantify oxidation damage in a carbon/carbon (C/C) composite material cut from an automotive friction disc ring. The oxidation as a result of gasification of carbon at high temperatures (T >∼ 500 °C) is a major degradation mechanism in C/C composites. This is the first research to successfully use a dry-contact mediator wedge setup for nonlinear ultrasonic applications, enabling repeatable in-situ Rayleigh wave measurements on slow wave velocity materials where submersion is impractical. The samples are subjected to thermal damage by exposure to a high temperature atmosphere over multiple holding times, and the oxidation damage is visualized using scanning electron microscopy (SEM). The sample mass is recorded and both linear (velocity and attenuation) and nonlinear (β′) ultrasonic measurements are performed after each holding time. Superior sensitivity to early stage oxidation affirms the suitability of nonlinear Rayleigh wave measurements for the quantification of thermally induced oxidation damage in C/C composites, while the double mediator wedge setup is well suited for high attenuation, slow wave velocity materials, demonstrating potential applicability for in-situ condition monitoring. The results show that a systematic oxidation starts after 3 min of exposure to the temperature in this C/C composite. The oxidation induced fiber–matrix debonding, matrix defects (appearing to be microcracking), and void growth are consistently detected by the Rayleigh wave velocity and acoustic nonlinearity. In addition to the higher sensitivity, the nonlinearity parameter has the capability of selectively measuring the strength related damage such as the observed fiber–matrix debonding, while the wave velocity can detect the oxidation voids. A combined use of these two parameters can provide useful information on the microstructural evolution in C/C composites subjected to high temperatures.

Recent progress on nonlinear ultrasonic phased array for closed-crack imaging

Closed cracks are challenging defects for ultrasonic testing since they are transparent to conventional ultrasound techniques. This can cause the underestimation or overlook of closed cracks, resulting in potential catastrophic accidents. To solve this problem, several types of nonlinear ultrasonic phased array has been developed by combining nonlinear ultrasonics with phased array (PA). In this study, we introduce the recent progress on nonlinear ultrasonic PA. The first one is called fundamental wave amplitude difference (FAD), which is based on the nonlinear incident-wave-amplitude dependence of fundamental responses. The key to the success of FAD is how high incident wave amplitude can be generated in samples to cause the contact vibration of crack faces. However, increasing the incident wave amplitude at MHz frequencies is not easy. To enhance the applicability of nonlinear ultrasonic PA, we have also developed ultrafast imaging (MHz range) with pump excitation (kHz range). We have successfully captured high-speed crack dynamics using plane wave imaging (PWI) during the pump excitation that can generate a large displacement of more than 1000 nm. Furthermore, we will talk about ongoing work toward 3D crack imaging. [Work partially supported by JSPS KAKENHI (19K21910, 21H04592, 22K18745) and JST FOREST program (JPMJFR2023).]

Bolt Axial Stress Detection and Prediction Based on the Nonlinear Ultrasonic Technique

Bolt Axial Stress Detection and Prediction Based on the Nonlinear Ultrasonic Technique

Evaluation of high-temperature degradation of platen superheater tube in thermoelectric power plant using nonlinear surface ultrasonic waves

The operating conditions of Korean standard coal-fired thermal power plants involve high temperatures, with steam reaching approximately 540 °C. To support the life management of platen superheater (PSH) tubes, which operate under the highest temperature and pressure conditions among boiler tubes, various methods have been used. These methods include visual inspection, thickness measurement, and hardness measurement, which are typically used to conduct lifespan diagnosis on facilities that have been operating for more than 100,000 h or 20 y. This study examined the applicability of a nonlinear ultrasonic technique to evaluate the degradation damage of a PSH tube composed of X20CrMoV12.1 steel in a thermal power plant. The specimens of a PSH tube made of X20CrMoV12.1 steel that has been operating for approximately 20 y (160,000 h) in a Korean standard thermal power plant was obtained from three points, and the nonlinear surface ultrasonic wave was measured. Then, the results of the nonlinear ultrasound, microstructure analysis, and tensile test were compared. Based on where the boiler tube specimen was located, a strong correlation was observed between the ultrasonic nonlinearity parameter and changes in the tensile and yield strengths. These results confirm that the nonlinear ultrasonic method can be used to evaluate the changes in the mechanical properties affected by boiler tube degradation.

Fatigue damage assessment using nonlinear critically refracted longitudinal (LCR) wave in pure iron: Experiments and FEM

To achieve early-stage microstructure evolution assessment and damage monitoring of fatigue, a new thickness-independent and nondestructive method based on nonlinear critically refracted longitudinal (LCR) wave is proposed. The nonlinear ultrasonic (NLU) parameter is sensitive to the accumulated fatigue damage, showing two peaks around the 5th and 1000th cycles (0.03% and 6% of fatigue life), respectively. This phenomenon is comprehensively discussed in terms of dislocation structure evolution based on the dislocation string model. A correction considering the weakening of nonlinearity induced by the rough surface is applied to the experiment measurements. The corresponding results are closer to the finite element modeling (FEM) quantitation based on the dislocation-dependent nonlinear constitutive relation. Compared to linear LCR wave, nonlinear LCR wave has weaker sensitivity to roughness effect and higher sensitivity to damage such as dislocation, which offers a new method for monitoring fatigue in engineering components.

Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock’s acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.