Simple Analytical Model Research Articles

Key Factors That Influence the Frequency Range of Measured Leak Noise in Buried Plastic Water Pipes: Theory and Experiment

The frequency range of the leak noise in buried water pipes, measured using acoustic correlators, depends significantly on the type of pipe and its location as well as the type of sensors used. Having a rough idea of this frequency range can be beneficial for operators prior to conducting tests; however, there is currently no method of predicting it except through practical experience, and no model-based approach yet exists. This issue is addressed in the present paper by using a concise and relatively simple analytical model of the water-pipe–soil system combined with the sensors’ frequency response. The influence of the various physical parameters of the system, such as the pipe and soil properties and the sensor type, on the cross-power spectral density (CPSD) of leak noise signals and, furthermore, the frequency range are investigated. The main factors that affect the bandwidth are the distance between the sensors, wave speed of the predominantly fluid-borne wave in the pipe and the attenuation of this wave. It is shown that the external medium has a profound effect on the propagation and, in turn, on the bandwidth. The approach to predicting this bandwidth is validated using experimental data from three different test sites.

A proactive decision support tool for road safety audit of new highway projects based on crash modification factors and analytical analysis: Algeria as a case study

The most cost-effective way to improve the performance of a new road is to integrate road safety into the early phase of the project design. Therefore, the information obtained from the design phase is simply used to get an overview of the project in place. This article proposes a simplified analytical tool to target road safety problems proactively, even before inspection visits. The study area contains 110 segments of 100 m long (inspection intervals) of a highway under construction, sited in Algeria in the locality of Ghazaouet, Wilaya of Tlemcen. The methodology adopted consists of combining the International Road Assessment Program (iRAP) and the multiple linear regression method to obtain a simplified analytical model, which allows the prediction of road risk for each section of 100 m. The results obtained showed a 98% correlation between the model values and the real values obtained by the iRAP approach. As a complement to the iRAP method, this approach facilitates the anticipatory assessment of road risks by road safety auditors. Eventually, this tool will support and guide auditors to be acquainted with the recent trends in the domain of road safety.

Development of a bidirectional rocking isolation bearing system (Bi‐RIBS) to control excessive seismic response of bridge structures

AbstractA bidirectional rocking isolation bearing system (Bi‐RIBS) is proposed to provide seismic protection for bridge structures. By using the 3‐D rocking motion of the Bi‐RIBS, this system acts as a mechanical fuse to limit the maximum force transmitted to the bridge piers and as a restoring component to control an excessive girder response. Possible applications in bridge structures were discussed. A simple analytical model was established to characterize the dynamics of an example bridge featuring such a Bi‐RIBS. A series of dynamic analyses were performed by using the proposed model to investigate the effects of several factors on controlling the seismic responses of the bridge, for example, the design parameters of Bi‐RIBS including inclined angle and size, the damping property at the support interface, and the mass ratio. The peak ground accelerations of the bidirectional ground motion record were scaled to various levels to evaluate the maximum performance indices of the bridge structure and the response control effectiveness of the Bi‐RIBS compared to the uncontrolled counterparts. The simulation results demonstrated that the proposed Bi‐RIBS could effectively control the maximum pier displacement while keeping the bearing from overturning if suitable parameters were selected. In particular, the control effectiveness on the maximum pier response becomes more significant as the seismic intensity increases, due to its distinctive negative stiffness property.

Debonding sawtooth analytical model and FE implementation with in-house experimental validation for SRG-strengthened joints subjected to direct shear

A novel and simple analytical debonding model for Steel Reinforce Grout (SRG)-strengthened specimens subjected to direct shear tests is proposed. All the nonlinearity is lumped at the interface between reinforcement chords and mortar, the substrate is assumed perfectly glued to mortar, infinitely stiff and resistant and the steel grid is considered linear elastic. The non-linear interface tangential stress-slip relationship is assumed discontinuous and multi-linear and constituted by four phases, three elastic with progressively reduced stiffness and the last perfectly plastic. The assumption of a non-vanishing residual tangential strength is paramount to reproduce the typical pseudo-linear hardening observed experimentally in the global force-displacement curves when the debonding frontier is triggered starting from the loaded edge. Under such hypotheses, the second order differential field equation describing the slippage of the reinforcement on the support admits closed form solutions, which are suitably derived. An implementation into a standard Finite Element (FE) commercial code is also proposed, which relies in a discretization of the reinforcement through three cutoff bars (one ductile and two brittle) disposed in-parallel, elastic elements for the steel grid and rigid beams used as transversal connections. The reliability of the sawtooth and the FE models is assessed by means of a thorough comparison with experimental data obtained by the authors for this purpose testing five replicates of a concrete specimen strengthen with a SRG and subjected to a standard shear test.

Analysis and experiment of a novel compact magnetic spring with high linear negative stiffness

Improving the isolation performance of vibration isolators by introducing magnetic negative stiffness mechanisms (NSMs) has attracted extensive attention in recent years. Nevertheless, common magnetic NSMs are always accompanied by nonlinear stiffness, resulting in degradation of isolation performance and system instability when subjected to excitations with large amplitude. To improve the stiffness linearity and working range of magnetic NSMs in a limited space, a novel compact magnetic spring with high linear negative stiffness (CMS-HLNS) is proposed in this paper. The CMS-HLNS consists of two sets of permanent magnets (PMs) arranged in an alternate manner, which has both high linearity and high negative stiffness properties. A simplified analytical stiffness model of the CMS-HLNS is deduced based on the magnetic charge method. The effects of design parameters on the stiffness characteristics of the CMS-HLNS are investigated. Finally, experimental prototypes are constructed to verify the feasibility of the analytical model and the superior vibration isolation performance when introducing the CMS-HLNS. The experimental results demonstrate that the CMS-HLNS can achieve linear negative stiffness in a large working range under the constraint of space and is suitable for (ultra-) low-frequency isolation under excitations with large amplitude. It’s believed that the magnetic spring proposed in this paper can be applied as an innovative technology in engineering practice, and its good linearity and compact structure make it extremely adaptable.

Incorporation of Pavement Dynamic Loading in Mechanistic–Empirical Design Frameworks

A framework for incorporating dynamic loading in pavement design is introduced. The framework is composed of five main steps. Firstly, input parameters, such as traffic characteristics, material properties, and the pavement configuration, are determined. Both primary and secondary data were obtained from the literature in this step. Secondly, a database of critical pavement responses, used by AASHTOWare software’s transfer functions to calculate pavement damage, was built using advanced pavement finite element models after the loading input was defined. The database includes various pavement configurations, axle configurations (single and tandem), pavement material properties, temperatures, loading conditions, and tire types (single steering, dual, and wide-base). AASHTOWare’s transfer function predicts pavement damage from pavement responses and is used to calculate the international roughness index (IRI) in the third and fourth steps, respectively. Finally, the load spectrum is adjusted to incorporate roughness-induced dynamic loading once the IRI exceeds 95 in./mi. A simple analytical dynamic-loading model was developed, based on mechanistic truck–trailer results. The model is a function of the tire configuration, speed, and IRI. Multiple case studies were performed, considering various pavement configurations, material properties, average annual daily traffic values, and dynamic-loading percentiles. Results showed that dynamic loading had the most significant impact on fatigue life, followed by rutting potential. The AASHTOWare’s transfer functions were found to be insensitive to an increase in loading when considering thick-pavement structures. The outcome of this effort was validated using the measured and predicted IRI from the Specific Pavement Study sections, which are part of the Long-Term Pavement Performance program.

XPD Enhancement of Dipole Antenna Arrays by Inducing Coherent Current From Feeding Line

Cross-polarization discrimination (XPD) is a critical parameter in designing dual-polarized antennas, particularly for MIMO (Multiple-Input Multiple-Output) base station antenna arrays. A high XPD level leads to a high isolation between two polarized waves and a high input signal-to-noise ratio of the system. In this paper, a new scheme for enhancing XPD of a ±45° dual- polarized dipole antenna pair, the most prevalent antenna element form for MIMO base station antenna arrays, is proposed and is applied to a dipole antenna array. The new scheme induces the required coherent currents on a pair of vertical monopoles from the feeding line. Specifically, a 3-U structure that is printed on the circuit board of the feeding line is proposed. The structure couples the signal from the feeding line and excites two opposite currents on the vertical monopoles, whose radiated fields effectively compensate the θ-component of electric field in a wide scanning angle. The working principle of the scheme is elaborated in detail with a simplified analytic model. The scheme is applied to a dipole array, in which each 3-U structure is personally designed to accommodate the complex array environment. Simulation and experimental results are presented to demonstrate the effectiveness of the proposed scheme.

A Glimpse of the Stellar Populations and Elemental Abundances of Gravitationally Lensed, Quiescent Galaxies at z ≳ 1 with Keck Deep Spectroscopy

Gravitational lenses can magnify distant galaxies, allowing us to discover and characterize the stellar populations of intrinsically faint, quiescent galaxies that are otherwise extremely difficult to directly observe at high redshift from ground-based telescopes. Here, we present the spectral analysis of two lensed, quiescent galaxies at z ≳ 1 discovered by the ASTRO 3D Galaxy Evolution with Lenses survey: AGEL1323 (M * ∼ 1011.1 M ⊙, z = 1.016, μ ∼ 14.6) and AGEL0014 (M * ∼ 1011.5 M ⊙, z = 1.374, μ ∼ 4.3). We measured the age, [Fe/H], and [Mg/Fe] of the two lensed galaxies using deep, rest-frame-optical spectra (S/N ≳40 Å−1) obtained on the Keck I telescope. The ages of AGEL1323 and AGEL0014 are Gyr and Gyr, respectively, indicating that most of the stars in the galaxies were formed less than 2 Gyr after the Big Bang. Compared to nearby quiescent galaxies of similar masses, the lensed galaxies have lower [Fe/H] and [Mg/H]. Surprisingly, the two galaxies have comparable [Mg/Fe] to similar-mass galaxies at lower redshifts, despite their old ages. Using a simple analytic chemical evolution model connecting the instantaneously recycled element Mg with the mass-loading factors of outflows averaged over the entire star formation history, we found that the lensed galaxies may have experienced enhanced outflows during their star formation compared to lower-redshift galaxies, which may explain why they quenched early.

Optimization of powder catchment efficiency for industrial laser cladding processes

Laser clad overlays provide enhanced corrosion resistance and wear performance to components across a wide range of industries. Powder fed laser processes provide flexibility in material selection, however have traditionally yielded lower material usage efficiency (60–80%) compared to wire fed processes. A general model was developed to predict powder catchment efficiency based on the overlap between the molten pool and powder cloud distribution. Mathematical techniques of asymptotic analysis and blending were used to obtain closed-form expressions relating tabulated material properties and the key process inputs with the critical melt pool dimensions of leading length (xf) and maximum width (y m). An improved experimental technique was implemented to measure the powder cloud distribution, a key process input which previously was not well understood. For baseline cladding conditions, melt pool dimensions were smaller than the powder cloud leading to catchment inefficiency. A custom laser optics module was designed to enable adjustment of the relative position of the melt pool and powder cloud independently from other process parameter changes. Combining optimized parameters with a shift of 1mm, catchment efficiency >90% has been achieved and sustained in production cladding operations. This work is part of a broader program at Apollo-Clad, using simple, accurate, and fast analytical modelling techniques to generate engineering design rules for laser-based processes.

Detailed Simulations of a Three-Stage Supercritical Carbon Dioxide Axial Compressor with a Focus on the Shrouded Stator Cavities Flow

This paper describes the findings of detailed simulations performed to investigate the impact of seal teeth cavity leakage flow on the aerodynamic and thermal performance of a three-stage supercritical CO2 axial compressor. The study compares a shrouded stator configuration (with cavities) to a cantilevered stator configuration (without cavities) to highlight their differences. High-fidelity computational fluid dynamics simulations were performed using non-linear harmonic (NLH) and mixing plane assumptions, considering various possible rotor/stator interface configurations for mixing plane calculations. The key performance parameters for each case were compared, and the best-performing configuration selected for further analysis. The individual stage performance parameters are also examined and compared between the cantilevered and shrouded configurations. It was observed that in the shrouded case, the leakage flow enters the cavity downstream of the stator trailing edge and gets entrained into the primary flow upstream of the stator, leading to boundary layer changes at the hub and degradation of stator and downstream rotor performance. Vortical flow structures were also observed in the stator wells, which tended to change the flow angles around the region, thereby affecting mixing and velocity distribution, resulting in a slight deterioration of compressor performance. Additionally, the study examines windage heating due to shear work from rotating walls, including the seal teeth surface. The amount of shear work done on the leakage flow and the corresponding rise in fluid temperature were quantified, tabulated, and further compared with a simple analytical model, showing good agreement between them and, hence, validating the numerical approach used.

Model for Combustion Instability in Hybrid Rocket Engines Burning Liquefying Fuels

Starting from the data measured in several static firings of two laboratory-scale hybrid rocket engines of different sizes burning gaseous oxygen with either a liquefying fuel or a pyrolyzing polymer, a mechanism for the combustion instability observed only with liquefying fuel is suggested. With the latter type of fuels, regression rate is increased for the formation of an unstable liquid layer on the grain surface, for which liquid fuel structures are entrained into the gas stream and burn, upon vaporization and mixing, far from the surface. Based on this circumstance, the hypothesis of a time delay to burn, with the resulting effect of triggering longitudinal acoustic modes of the system, is formulated. A simplified analytical acoustic model of a series of interconnected ducts with different cross sections and gas properties is preliminarily addressed, which allowed demonstrating the acoustic nature of the main pressure oscillation peaks measured in three firing tests along with the influence of the oxygen-feeding line on the system acoustics. Finally, by leveraging an existing acoustic model of the hybrid rocket combustion chamber including the effects of cross-sectional discontinuities and distributed fuel mass addition and combustion, the linear stability of the system is analyzed, whose results are apt to reproduce the experimental findings.

Mechanistically Weighted Metric to Predict In Vivo Antibody-Receptor Occupancy: An Analytical Approach.

In situ clinical measurement of receptor occupancy (RO) is challenging, particularly for solid tumors, necessitating the use of mathematical models that predict tumor receptor occupancy to guide dose decisions. A potency metric, average free tissue target to initial target ratio (AFTIR), was previously described based on a mechanistic compartmental model and is informative for near-saturating dose regimens. However, the metric fails at clinically relevant subsaturating antibody doses, as compartmental models cannot capture the spatial heterogeneity of distribution faced by some antibodies in solid tumors. Here we employ a partial differential equation (PDE) Krogh cylinder model to simulate spatiotemporal receptor occupancy and derive an analytical solution, a mechanistically weighted global AFTIR, that can better predict receptor occupancy regardless of dosing regimen. In addition to the four key parameters previously identified, a fifth key parameter, the absolute receptor density (targets/cell), is incorporated into the mechanistic AFTIR metric. Receptor density can influence equilibrium intratumoral drug concentration relative to whether the dose is saturating or not, thereby influencing the tumor penetration depth of the antibody. We derive mechanistic RO predictions based on distinct patterns of antibody tumor penetration, presented as a global AFTIR metric guided by a Thiele Modulus and a local saturation potential (drug equivalent of binding potential for positron emissions tomography imaging) and validate the results using rigorous global and local sensitivity analysis. This generalized AFTIR serves as a more accurate analytical metric to aid clinical dose decisions and rational design of antibody-based therapeutics without the need for extensive PDE simulations. SIGNIFICANCE STATEMENT: Determining antibody-receptor occupancy (RO) is critical for dosing decisions in pharmaceutical development, but direct clinical measurement of RO is often challenging and invasive, particularly for solid tumors. Significant efforts have been made to develop mathematical models and simplified analytical metrics of RO, but these often require complex computer simulations. Here we present a mathematically rigorous but simplified analytical model to accurately predict RO across a range of affinities, doses, drug, and tumor properties.

Bond behavior between high-strength steel wire meshes and ECC: Experimental study and analytical modelling

In this study, a novel methodology was proposed to investigate interfacial bonding performance between high-strength steel wire mesh (HSSWM) and engineering cementitious composites (ECC), and the inhomogeneous distribution of bond stresses was taken into account. A simple bond-slip model was constructed and compared with the existing average bond-slip model to estimate the bond behavior of the HSSWM-ECC matrix joints. The results showed that the proposed model was more appropriate to estimate the interfacial bond property between HSSWM and ECC. With the bond-slip model, the local bonding performance of HSSWM-ECC interface was analyzed using the numerical simulation method. An analytical model was proposed for predicting the critical embedded length between HSSWM and ECC, while the effects of the matrix properties, high-strength steel wire (HSSW) diameter, transverse HSSW spacing, and HSSW mechanical properties were comprehensively studied. Finally, a simple analytical model was proposed to calculate the interfacial bearing property between HSSWM and ECC and verified by the experiment results.

A Simplified Thermal Analysis Model for Regeneratively Cooled Rocket Engine Thrust Chambers and Its Calibration with Experimental Data

An essential part of the design of a liquid rocket engine is the thermal analysis of the thrust chamber, which is a component whose operative life is limited by the maximum allowable wall temperature and heat flux. A simplified steady-state thermal analysis model for regeneratively cooled rocket engine thrust chambers is presented. The model is based on semi-empirical correlations for the hot-gas and coolant convective heat transfer and on an original multi-zone approach for the wall conduction. The hot-gas heat transfer is calibrated with experimental data taken from an additively manufactured water-cooled nozzle that is connected to a combustion chamber either fed with decomposed hydrogen peroxide or decomposed hydrogen peroxide and automotive diesel. The thrust chamber (i.e., combustion chamber and nozzle) is designed to produce about 450 N of thrust when operating with a chamber pressure of 11 bar. For this application, the calibrated model predicts the total wall heat transfer rate very accurately and the temperature distribution within the wall structure with an uncertainty of a few tens of kelvins. This level of accuracy can be considered more than adequate for the design, and generally for engineering-type thermal analysis, of similar thrust chambers.

A methodology for data-driven modeling and prediction of the drag losses of wet clutches

AbstractIn wet clutches, load-independent drag losses occur in the disengaged state and under differential speed due to fluid shearing. The drag torque of a wet clutch can be determined accurately and reliably by means of costly and time-consuming measurements. As an alternative, the drag losses can already be precisely calculated in the early development phase using computing-intensive CFD models. In contrast, simple analytical calculation models allow a rough but non-time-consuming estimation. Therefore, the aim of this study was to develop a methodology that can be used to build a data-driven model for the prediction of the drag losses of wet clutches with low computational effort and, at the same time, sufficient accuracy under consideration of a high number of influencing parameters. For building the model, we use supervised machine learning algorithms. The methodology covers all relevant steps, from data generation to the validated prediction model as well as its usage. The methodology comprises six main steps. In Step 1, the data is generated on a suitable test rig. In Step 2, characteristic values of each measurement are evaluated to quantify the drag loss behavior. The characteristic values serve as target values to train the model. In Step 3, the structure and quality of the dataset are analyzed and, subsequently, the model input parameters are defined. In Step 4, the relationships between the investigated influencing parameters (model input) and the characteristic values (model output) are determined. Symbolic regression and Gaussian process regression have both been proven to be suitable for this task. Lastly, the model is used in Step 5 to predict the characteristic values. Based on the predictions, the drag torque can be predicted as a function of differential speed in Step 6, using an approximation function. The model allows a user-oriented prediction of the drag torque even for a high number of parameters with low computational effort and sufficient accuracy at the same time.

Sensorless control of switched reluctance motor based on a simple flux linkage model

Introduction. The operation of switched reluctance motor requires prior knowledge of the rotor position, obtaining from either low resolution photocoupler based position sensor or high resolution shaft encoder, to control the on/off states of the power switches. Problem. However, using physical position sensor in harsh environment will inevitably reduce the reliability of the motor drive, in which sensorless control comes into play. Novelty. In this paper, a sensorless control scheme of switched reluctance motor is proposed. Methodology. The method is based on a simple analytical model of the flux-linkage curves rather than the conventional approach that normally uses a look-up table to store all the data points of the flux-linkage curves. By measuring the phase current, rotor position can be deduced from the analytical model. Practical value. Simulation results are given and the proposed sensorless scheme is verified to provide a moderate position estimation accuracy in a wide speed range in both unsaturated and saturated conditions.

Non-linear analytical model for FRCM coupons in tension

Externally bonded FRCM (Fiber Reinforced Cementitious Matrix) materials are nowadays quite diffused to reinforce existing structures: taking as reference FRP (Fiber Reinforced Polymer), the replacement of an organic matrix with mortar is more compatible with masonry substrates, and the reversibility of the reinforcement is more appealing for the architectural heritage conservation. At the same time, due to the low strength of the mortar layers, complex failure mechanisms are observed for FRCMs, which still need comprehensive theoretical investigations. Indeed, not only the mortar-fiber interface may behave non-linearly, but also the mortar may crack and affect the overall behavior. The tensile test on coupons is commonly used as a direct approach to characterize the properties of the reinforcing package, and there is a wide consensus on approximating the global behavior with a trilinear law; however, the different test set-ups and the inhomogeneity of the material itself bring about large dispersion of the experimental results. This paper aims at proposing a simplified analytical model to consider the damage mechanisms in FRCM coupons in tension, including interface slippage, mortar cracking, and their simultaneous occurrence. The accuracy of the model is then verified by comparing the results with experimental data. The conditions under which such failure mechanisms occur and the effects of material properties are also investigated with a view to complement the understanding of experimental observations from a theoretical perspective.

МОДЕЛЮВАННЯ НАПРУЖЕНО-ДЕФОРМОВАНОГО СТАНУ ТОНКОСТІННИХ КУПОЛІВ З ПОЛІКАРБОНАТУ ДЛЯ РАЦІОНАЛЬНОГО ПРОЕКТУВАННЯ

The paper contains the further developed of method for calculating thin-walled dome systemswithout a stationary foundation. Have been carried out the detailed analysis of the fundamental design solutions for frameless collapsible spherical polycarbonate domes, which are used by modern world manufacturers of these structures. Have been done a brief description of the momentless theory of the operation of spherical shells, which is adapted for polycarbonate domes. Have been considered a simplified analytical model of the stress-strain state of a spherical shell with an equatorial diameter of up to5 m under the influence of climatic influences for the subsequent verification of detailed models. Have been developed highly detailed finite element models of domes of different sizes, taking into account technological openings and structural stiffeners (support ring and door frame) under the wind, snow, ice loads and under other climatic influences. Have been identified the fragments with the highest stresses from various loads and forms of the deformation of the structure.Have been considered separately the issues related to the loss of shape stability, position and balance of a thin-walled spherical shell, as a light temporary structure. Have been proven that the worst influence on the dome structures is the windinfluence, based on the stability criterion. Have been determined the estimated value of the aerodynamic lifting force from wind effects on the dome. Have been proven that the lifting force far exceeds the stabilizing force of the weight of a thin-walled dome. Have been revealed with the help of the performed calculations, it was that a frameless spherical polycarbonate dome inevitably БУДІВЕЛЬНІ КОНСТРУКЦІЇНАУКОВО-ТЕХНІЧНИЙ ЖУРНАЛ “СУЧАСНІ ТЕХНОЛОГІЇ, МАТЕРІАЛИ І КОНСТРУКЦІЇ В БУДІВНИЦТВІ”84loses its balance stability due to the action of wind loads and requires unfastening with anchors. Have been proposed a rational method for anchoring dome structures at temporary earthen construction sites using geo-screws or metal screw piles. Have been revealed the addiction between the radius of curvature of a spherical dome and the rational thickness of polycarbonate based on the criteria of stiffness and strength. Have been formulated the constructive recommendations regarding the rational design of polycarbonate dome systems. Have been developed the technological regulations for the further safe operation of domes, and have been outlined the directions for further scientific research on this topic

Effects of electron heating and surface rippling on Rayleigh–Taylor instability in radiation pressure acceleration

The acceleration of ultrathin targets driven by intense laser pulses induces Rayleigh–Taylor-like instability. Apart from laser and target configurations, we find that electron heating and surface rippling, effects inherent to the interaction process, have an important role in instability evolution and growth. By employing a simple analytical model and two-dimensional particle-in-cell simulations, we show that the onset of electron heating in the early stage of the acceleration suppresses the growth of small-scale modes, but it has little influence on the growth of large-scale modes, which thus become dominant. With the growth of surface ripples, a mechanism that can significantly influence the growth of these large-scale modes is found. The laser field modulation caused by surface rippling generates an oscillatory ponderomotive force, directly modulating transverse electron density at a faster growth rate than that of ions and eventually enhancing instability growth. Our results show that when surface deformation becomes obvious, electron surface oscillation at 2ω0 (where ω0 is the laser frequency) is excited simultaneously, which can be seen as a signature of this mechanism.

Two-color polarization control of angularly resolved attosecond time delays

Measured photoionization time delays may exhibit large variations as a function of the emission angles, even for spherically symmetric targets, as shown in recent reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) experiments. The contributions from different pathways to the two-photon quantum channels can explain the observed phase jumps that shape those angular distributions. Here we propose a simple analytical model to describe angularly resolved RABBITT spectra as a function of the relative polarization angle between the ionizing attosecond pulse train and the assisting IR field. We demonstrate that the angular dependences of the measured delays can be analytically predicted and the position of the phase jumps reduced to the analysis of a few relevant parameters.