Building physics engineers aim to improve the energy performance and sustainability of building envelopes. Although their practices are based on past experience, new materials and building techniques are constantly being developed that offer a wide set of options in building design and thermal management. Let’s see how to model heat and moisture transport in building materials to help reduce energy costs and preserve buildings.
Building envelopes can be analyzed by modeling heat and moisture transport.
Controlling moisture is necessary to optimize the thermal performance of building envelopes and reduce energy costs. The thermal properties of insulation or isolation materials usually depend on both temperature and moisture content. Therefore, a coupled heat and moisture model helps us fully analyze the thermal performance of a building component. One example is the dependence of a lime silica brick’s thermal conductivity on relative humidity.
The moisture dependence of thermal conductivity for lime silica brick.
The figure above shows that lime silica brick becomes two times less thermally isolating for high relative humidity values.
In addition, we must consider moisture control in the building design process to choose building components that can reduce the risk of condensation. The coupled modeling of heat and moisture transport enables us to analyze different moisture variations and phenomena in building components, such as:
Let’s consider a wood-frame wall between a warm indoor environment and a cold outdoor environment. Vapor diffuses through the wall from the high-moisture environment inside to the low-moisture environment outside. This creates high relative humidity values associated with high temperature values close to the exterior panel, with the risk of condensation as a direct consequence.
The relative humidity distribution in a wood-frame wall.
Condensation leads to mold growth, which directly affects human health and building sustainability. The rate of mold growth is key data for the preservation of historical buildings, for example. To prevent the risk of interstitial condensation, it is common practice to add a vapor barrier between the interior gypsum panel and the cellulose isolation board. This reduces the moisture values where they are at a maximum. The figure below shows the relative humidity distribution across the wood-frame wall through a wood stud (red lines) and a cellulose board (blue lines), with and without the vapor barrier (dashed lines and solid lines, respectively).
Effect of a vapor barrier on relative humidity distribution across the wood-frame wall in a wood stud and cellulose board.
For this model, we consider the building materials to be specific unsaturated porous media in which the moisture exists in both liquid and vapor phases and only some transport processes are relevant. The norm EN 15026 standard addresses the transport moisture phenomena that is taken into account in building materials, following the theory expressed in Ref. 1.
The transport equation established as a standard by the norm accounts for liquid transport by capillary forces, vapor diffusion due to a vapor pressure gradient, and moisture storage.
We model the latent heat effect due to vapor condensation by adding the following flux in the heat transfer equation:
In addition, the moisture dependence of the thermal properties is assessed.
Find details about the moisture transport equation in building materials in the Heat Transfer Module User’s Guide.
When using the Heat Transfer Module, the Heat and Moisture Transport interface adds a:
Finally, the latent heat source due to evaporation is added to the heat transfer equation by the Building Material feature of the Heat Transfer interface.
The model tree and subsequent subnodes when choosing the Heat Transfer in Building Materials interface, along with the Settings window of the Building Material feature.
Modeling heat and moisture transport in an unsaturated porous medium is important for analyzing polymer materials for the pharmaceutical industry, protective layers on electrical cables, and food-drying processes, to name a few examples.
For these applications, phenomenological models, such as the one presented above for building materials, may not be available. However, by considering the conservation of heat and moisture in each phase (solid, liquid, and gas), and volume averaging over the different phases, we can derive a mechanistic model.
To compute the moisture distribution, we solve a two-phase flow problem in the porous medium. Two equations of transport are solved: one for the vapor and one for the liquid water. The coupling between the vapor and liquid water operates through the definition of saturation variables, S_{vapor} + S_{liquid} = 1. The changing water saturation is taken into consideration for the definition of the effective vapor diffusivity and liquid permeability.
For quick processes, with a time scale comparable to the time it takes to reach equilibrium between the liquid and gas phases inside the pores of the medium, a nonequilibrium formulation can be defined through the following evaporation flux:
In this definition, the equilibrium vapor concentration, defined as the product of the saturation concentration c_{sat} and the water activity a_{w}, is used to account for the porous medium structure. Indeed, due to capillary forces, equilibrium is reached for concentrations that are lower than in a free medium.
By letting the evaporation rate K go to infinity, an equilibrium formulation is obtained with the vapor concentration equal to the equilibrium concentration.
Let’s consider a food-drying process. A piece of potato, initially saturated with liquid water, is placed in an airflow to be dried. Inside the potato, the vapor is transported by binary diffusion in air. We use a Brinkman formulation to model the flow induced by the moist air pressure gradient in the pores. As the liquid phase velocity is small compared to the moist air velocity, Darcy’s law is used for the liquid water flow due to the pressure gradient. The capillary flow, due to the difference between the relative attraction of the water molecules for each other and the potato, is also considered in the liquid water transport.
The vapor and liquid water distributions over time for this model are shown in the following two animations. Note that water can leave the potato as vapor only.
The liquid water concentration over time.
The vapor is transported away by the airflow, as shown in this animation:
The water vapor concentration over time.
The evaporation causes a reduction of the temperature in the potato. The temperature distribution over time is shown below.
Temperature distribution over time.
You can implement the equations in the Heat Transfer in Porous Media interface within the Heat Transfer Module and the Transport of Diluted Species interface within the Chemical Reaction Engineering Module. This process requires some steps in order to couple the multiphase flow in a porous medium together with the evaporation process.
Read the article “Engineering Perfect Puffed Snacks” on pages 7–9 of COMSOL News 2017 to see how Cornell University researchers used COMSOL Multiphysics to model rice puffing. In this numerically challenging process, the rapid evaporation of liquid water results in a large gas pressure buildup and phase transformation in the grain.
In this blog post, we discussed COMSOL® software features for modeling heat and moisture transport in porous media. COMSOL Multiphysics (along with the Chemical Reaction Engineering Module and Heat Transfer Module) provides you with tools to define the corresponding phenomenological and mechanistic models for a large range of applications. Depending on the dominant transport processes, you can use predefined interfaces or define your own model.
Künzel, H. 1995. Simultaneous Heat and Moisture Transport in Building Components. One and two-dimensional calculation using simple parameters. PhD Thesis. Fraunhofer Institute of Building Physics.
In powder compaction, a metal powder enters a die and is compacted through applied pressure. This high pressure comes from a punching tool inside the die cavity (often at the bottom surface). The powder is ejected from the cavity once it has been compacted and molded into a certain shape.
Through powder compaction, metal powders are transformed into solid components. Image by Alchemist-hp — Own work. Licensed under CC BY-SA 3.0 DE, via Wikimedia Commons.
With production rates averaging between 15 to 30 parts per minute, the powder compaction process enables manufacturers to quickly design strong components. Another benefit of this process is that it saves costs, as the component doesn’t need much additional work.
From a simulation standpoint, we need to perform a highly nonlinear structural analysis on powder compaction that accounts for:
As we demonstrate here, COMSOL Multiphysics® version 5.3 is ideal for handling such analyses.
For our example, let’s consider the fabrication of a cup-shaped component via powder compaction. The model geometry includes the workpiece (metal powder in this case) and the die. Note that the punch tool is not part of the model setup. We instead apply a prescribed displacement in the normal direction to the upper and lower faces of the powder in order to compact it. Because of the axial symmetry of the model, we can reduce its size to a 2D model, thus reducing the computational time of the simulation.
The model geometry for a powder compaction analysis.
The latest version of COMSOL Multiphysics includes five new porous plasticity models that cover various porosity values.
These models are important for simulating powder compaction, as they allow us to accurately represent the porosity of the workpiece and produce reliable results. In this case, we combine the Fleck-Kuhn-McMeeking and Gurson-Tvergaard-Needleman models to describe an aluminum metal powder. Note that the die’s material properties assume it to be rigid.
In addition to the Prescribed Displacement boundary condition mentioned above, we also set the inner and outer dies as fixed domains.
From our simulation results, we can assess various properties of the metal powder at the end of compaction. To start, let’s look at the volumetric plastic strain. The strain at the center of the fillet appears to be minimal, while the strain near the ends is high. At the corner points of the workpiece, the strain is around 12% — likely the result of friction against the die.
The volumetric plastic strain of the workpiece as the compaction process ends.
During compaction, the porosity of the aluminum powder decreases, while the density and strength of the component increases. Based on the geometry and loading used in this scenario, we can expect that the changes to the porosity will be nonuniform.
The plot below shows the current void volume fraction contours of the powder; i.e., the porosity of the powder. Compared to the middle and top portions of the workpiece, the metal powder in the thin lower portion is more compact. Near the central area of the fillet, the powder is less compact because of material sliding on the rounded corner. The following animation illustrates how the volume fraction evolves over time.
The current void volume fraction as the compaction process ends.
Changes in the volume fraction over time.
Lastly, let’s consider the von Mises stress in the workpiece. The results indicate that the stress is greater in the areas where more compaction occurs.
The von Mises stress within the workpiece.
When simulating powder compaction, it is important to access the appropriate plasticity model that is preferably predefined in your analysis tool and available for direct use. To meet your modeling needs, COMSOL Multiphysics® version 5.3 brings you five new models that cover a wide range of porosity values.
For a helpful introduction to using these porous plasticity models, try out the example from today’s blog post.
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Humidified air not only affects human comfort, but also conditions the sustainability of buildings and the operation of electronic devices. This makes accounting for the presence of moisture critical when modeling heat transfer and phase change in ambient air surrounding equipment and in structures.
A standard variable used to quantify the amount of moisture in air is the relative humidity, φ. It expresses a relative state toward saturation and is the ratio between the vapor’s partial pressure in air, p_{v}, and the saturation pressure at a given (usually standard) temperature, p_{sat}(T):
As a first approximation, we can assume the vapor’s partial pressure p_{v} to be homogeneous. Yet, because of the dependence of saturation pressure on temperature, we should note that the relative humidity is not actually homogeneous, as temperature gradients are present.
Typical ambient moisture conditions can be defined from tabulated data such as weather records. These can be used to define, for example, the air’s thermodynamic properties when solving the heat transfer equation:
The moisture dependence on the density, thermal conductivity, and heat capacity at constant pressure are set through a mixture formula based on dry air and pure steam properties.
In a previous blog post, we detail how to use typical weather data for temperature and relative humidity in COMSOL Multiphysics®.
By exclusively solving the equation above for the temperature while knowing the (homogeneous) vapor’s partial pressure, we can already identify zones where condensation is likely to happen. Indeed, condensation happens at the saturation state, corresponding to φ = 1, where the detection of condensation relies on the relationship between both temperature and moisture.
As an example, let’s consider an electronic device within a box that produces 1 W of heat. Moist air flows through the box via 2 small slits located at the left and right sides of the box. From the computed temperature and relative humidity distributions, the risk of condensation inside the box is evaluated. Note that in this computation, the latent heat associated with the condensation is not accounted for in the heat transfer model. As shown in the figures below, condensation forms on the walls close to the slits after around 3 hours, for about 30 minutes, as well as after 4 hours and 30 minutes. These times correspond to when the ambient temperature is low and the relative humidity is high, at different points in the box.
Temperature distribution after 3 hours (left); relative humidity distribution after 3 hours (center); and the evolution of the condensation indicator variable, ht.condInd
, over time (right).
You can find more information in a previous blog post on modeling convective heat transfer.
When using the Heat Transfer Module, the Moist Air option in the Fluid Settings window from the Heat Transfer in Fluids interface defines the moisture-dependent thermodynamic properties of the modeling domain. This option also provides the ht.condInd
variable to be used when postprocessing results to identify condensation detection.
The model tree and Settings window of the Fluid feature with the Moist air option selected.
In some situations, we need to describe the moisture distribution more precisely. This includes cases where the amount of moisture is locally high due to evaporation and when the diffusion and convection of vapor can’t be neglected.
Compared to the previous approach, we need to compute the moisture distribution by solving an additional transport equation for the convection and diffusion of the vapor concentration c_{v} in air:
Note that in this equation, the temperature dependence is still accounted for through the vapor concentration c_{v} = φc_{sat}(T), with c_{sat}(T) as the saturation concentration of vapor.
Let’s consider a beaker filled with hot water (80°C) and placed in an air flow with a velocity of 2 m/s. Evaporation occurs from the water surface due to the flow of the air. Evaporation creates a state of saturated vapor (dependent on the temperature) at the air-liquid water interface, where this is transported away, and replenished by unsaturated air through convection and diffusion (see the figure below).
Vapor concentration distribution after 20 minutes, with contour lines for the relative humidity.
The energy required to sustain the evaporation is primarily extracted from the internal energy of the liquid water, which cools down as a result, as shown in the animation below. This process is known as evaporative cooling. It is the main process used in evaporative coolers and cooling towers, taking advantage of water’s relatively large heat capacity and latent heat when heating and vaporizing water for air cooling.
Temperature distribution over time and streamlines indicating the flow field.
In the model, evaporation occurs when the vapor concentration stays below the saturation state and just above the liquid surface. The evaporation flux is expressed as:
where K is an evaporation rate depending on the application.
The latent heat variation in the liquid is taken into account by adding the following heat source in the heat transfer equation:
where L_{v} is the latent heat of the evaporation of water.
When using the Heat Transfer Module, the Heat and Moisture Transport interface adds the subnodes shown in the screenshot below, including the:
The model tree and subsequent subnodes when choosing the Heat Transfer in Moist Air interface, along with the Settings window of the Moist Air feature.
When defining a fully coupled simulation of evaporative cooling, the Heat Transfer in Moist Air and Moisture Transport in Air interfaces are included together with the Heat and Moisture multiphysics interface. This also sets up the situation by including the first three subnodes under both interfaces by default. Further subnodes (e.g., the Boundary Heat Source and Wet Surface subnodes) can be included depending on the participating conditions of the process being simulated.
We have now reviewed the COMSOL® software features dedicated to the modeling of heat and moisture transport in moist air. Depending on the application, you may want to solve only for heat transfer and use the temperature prediction to detect condensation, or you may need to go further by computing the temperature and moisture distributions in a coupled way. In addition, you can account for the latent heat effects or disregard them. COMSOL Multiphysics (along with the Heat Transfer Module) provides the tools to define the corresponding models for a large range of applications.
Stay tuned for an upcoming blog post covering how to model heat and moisture transport in building materials and porous media.
Editor’s note: You can read the follow-up post in this blog series here: “How to Model Heat and Moisture Transport in Porous Media with COMSOL®“.
Solar-grade silicon is one of three grades of high-purity silicon. Each grade has different applications and specific purity percentage requirements:
The structure of monocrystalline silicon. Solar-grade silicon is almost pure silicon.
Traditionally, solar-grade silicon is produced using high temperatures (2000°C) to reduce silicon quartz and carbon, resulting in silicon with a 98.5% purity. This isn’t quite pure enough to be considered solar grade, so the silicon must be refined further through a gas phase. With multiple steps and different processes, this method isn’t efficient. It is also energy intensive, expensive, and requires experienced operators.
The method that JPM analyzed starts with raw materials that are highly pure. The silicon is placed into a contaminant-free microwave oven that performs both the heating and gas phase stages of the traditional production process. Since there’s no consecutive refinement processes, this approach is more efficient and cost effective.
The setup for the microwave furnace. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The microwave furnace consists of five parts:
One advantage of an optimized microwave furnace design is that there is reduced heat loss. This is partially due to the selective heating, which heats materials on a volumetric heat input, leading to a temperature drop from the inside out. In addition, there’s less diffusion of the silicon’s impurities because the furnace has a faster warming time and shorter residence time.
To optimize the microwave furnace for solar-grade silicon production, JPM Silicon GmbH studied its internal processes with the COMSOL Multiphysics® software.
The research team set up their model to include the electromagnetic, chemical, and physical phenomena occurring within the microwave furnace. Since some materials have electromagnetic properties that are strongly temperature dependent, the model couples the electromagnetic field distribution and temperature field.
You can learn more about the model setup by reading the full conference paper “Multiphysics Modelling of a Microwave Furnace for Efficient Solar Silicon Production“.
It’s important to use chemically stable structural materials and an inert gas in the microwave furnace to avoid unwanted reactions. Also, the insulation materials must be effective in minimizing heat losses.
The research team used the RF Module to simulate the electromagnetic intensity and distribution in the resonator and silicon sample. They used Maxwell’s equations to determine the propagation of the microwave radiation.
The electric field is higher at the height of the waveguide ports than at any other part of the reaction chamber. The field enhancement in the crucible’s core indicates that this is the optimal location for the crucible to be heated, as shown in the results below.
The distribution of the electric field in the resonator and waveguide. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The researchers also wanted to see how varying the height of the insulation plate affects the operation of the furnace. They tested three different heights for the plate (which the crucible sits on top of) and reexamined the electric field. The different insulation plate heights include:
The distribution of the electric field when the height of the insulation plate is 30 mm (left), 40 mm (middle), and 50 mm (right). Images by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The simulation results show that the 40-mm insulation plate performs best. The electric field is focused at the center of the crucible, thus on the silicon sample.
The CFD Module solves for the Navier-Stokes equations, allowing the researchers to find the gas flow velocity distribution. The gas flows from the inlet over the surface of the silicon sample, rather than having a homogeneous velocity. The wall then deflects the flow toward the outlet. The simulation shows that only a slight gas flow exists near the waveguide ports as well as near the top and bottom walls.
The distribution of gas velocity in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.
To analyze how well the electromagnetic waves heat the silicon sample, the research team examined the heat distribution in the resonator. Their model includes forced heat equations to calculate conduction, convection, and radiation from solids and liquids (Planck’s radiation law) as well as gases (Stefan-Boltzmann law). The dissipated heat, solved with the RF Module, is used as a volumetric heat source. The gas velocity profile, calculated with the CFD Module, helps find the convective thermal losses.
As expected from the electromagnetics study, the hottest point in the resonator is at the crucible’s core. Further, the surrounding insulation layers don’t heat up as much, thanks to their lower thermal conductivity.
The distribution of heat in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.
By gaining insight into the internal processes of a microwave furnace, researchers from JPM Silicon GmbH were able to optimize their design and pave the way for efficient solar-grade silicon production.
Let’s say you just finished creating a CAD assembly of a fitting for the threaded steel pipe referenced above. Now, you want to analyze stress in your assembly in order to better understand how this portion of the pipe system performs. With our LiveLink™ interfacing products, you can perform such analyses by integrating the COMSOL Multiphysics® software into your design workflow.
Threaded pipes are common in fire sprinkler systems. Image in the public domain, via Wikimedia Commons.
Threaded geometries include a large number of details. The complex nature of these CAD assemblies causes additional preprocessing work and takes up more computing resources during analysis. One solution is to assume that the thread is symmetric and compute the solution in a 2D section cut from the 3D object.
In previous versions of the COMSOL® software, selections from the original geometry had to be manually redefined after synchronization — a process that can be time consuming. Thanks to improvements in version 5.3, setting up CAD assembly selections is now a more efficient process. All of the relevant selections are automatically loaded and properly assigned in the COMSOL Multiphysics environment. This makes it possible to run parametric studies as well as improve 3D designs from 2D analyses.
Want to see a firsthand example? Good news: There’s a new tutorial model in the Application Gallery that highlights this functionality.
Note: While today’s example uses LiveLink™ for SOLIDWORKS®, this functionality is also available for LiveLink™ for Inventor®. For more details, see the 5.3 Release Highlights page.
In this example, you can synchronize a full threaded pipe fitting geometry built in SOLIDWORKS® software into the COMSOL Desktop® environment via LiveLink™ for SOLIDWORKS®. To compute a reduced stress analysis, you obtain a 2D section from the 3D geometry via the Cross Section node. The analysis assumes that a torque of 5000 Nm is applied to the male thread part (shown below). This part is made up of the same steel material as the other parts in the design.
Left: Full 3D assembly synchronized in COMSOL Multiphysics. Right: 2D section cut for the stress analysis.
To compute the force transmission between each part of the assembly, the model uses structural contact. In SOLIDWORKS® software, these contact surfaces are defined as face selections. After synchronizing the assembly, all of the selections are automatically transferred over to the 2D axisymmetric model. This simplifies the process of setting up the contact pair, as it is no longer necessary to manually and individually select boundary entities in contact with one another. In particular, when it comes to the thread, you only need to create a selection for two surfaces in SOLIDWORKS® software instead of selecting fifteen edges in the 2D axisymmetric model.
Looking at the results of our stress analysis, we can see the von Mises stress when the maximum torque (5000 Nm) is applied. The plot indicates that the maximum value of stress is less than that generally reported for using a class 10.9 alloy steel, highlighting the potential of using this material in this pipe fitting design.
Simulation plot depicting the von Mises stress with the maximum applied torque.
In version 5.3 of the COMSOL® software, you can combine your complex CAD assemblies and COMSOL Multiphysics analyses for an efficient modeling workflow.
Ready to try this tutorial yourself?
SOLIDWORKS is a registered trademark of Dassault Systèmes SolidWorks Corp.
Autodesk, the Autodesk logo, and Inventor are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries.
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A critical speed is the angular speed of a rotor that matches one of its natural frequencies. Finding the natural frequencies of a stationary rotor, however, is not enough to determine the critical speed. The real challenge comes from the fact that the natural frequency of the rotor depends on the rotor angular speed. Therefore, it is important to compute the natural frequency of a rotating component by considering the effect of the rotation.
This effect can be automatically included in the underlying model of a COMSOL app, which shows only the important design parameters as inputs. Let’s see how we can find the critical speeds of various rotating systems using an example from the Application Gallery: the Rotor Bearing System Simulator.
Video demonstrating the Rotor Bearing System Simulator.
A typical rotor system has three standard components:
A rotor system, which includes a rotor (shaft), disk, and bearing.
In most cases, the shaft is either a solid or hollow cylinder on which various components are mounted. In rotordynamics terminology, these mounted components are often called disks and they are modeled as rigid objects due to their high stiffness when compared with the shaft. Therefore, only inertial properties of the disks are important in the critical speed analysis. Shafts are flexible elements and also have inertia. A complete specification of the shaft requires its geometric dimensions and material properties, such as Young’s modulus, Poisson’s ratio, and density. Bearings are the components on which the shaft is supported. These components are described by their equivalent stiffness and damping coefficients.
Now, let’s see how this information is passed to the app. There are various sections in the app for different uses, including:
The sections to specify the input data are Rotor Properties, Disks, Bearings, and Study Parameters. The Critical Speeds section is used to evaluate the critical speed of the modeled rotor. The Geometry State and Information sections contain the information about the geometry and the solver, respectively. On the right panel of the app, the geometry of the rotor, whirl plot, and Campbell plot can be accessed. Various items on the top ribbon are provided to perform the different actions in the app.
The user interface of the Rotor Bearing System Simulator.
In the Rotor Properties section, you can specify the geometric dimensions of the rotor (shaft) and its material properties. There are two ways to specify the material properties of the rotor:
The Rotor Properties section, with a material from the list (left) and a user-defined material (right).
The Disks section has a combo box to either specify the geometric dimensions of the disks together with the density or to specify the inertial properties directly. The properties of disks can be given as tabular data in which each row in the table represents the disk. You can add as many rows as there are disks on the rotor. If Geometric dimension is chosen, then the location of the disk, outer diameter, thickness, and density can be specified. For Inertial properties, you can specify the location, mass, polar, and diametral moment of inertia.
The Disks section, showing the properties specified through geometric dimensions (left) and mass and moment of inertia (right).
Data entered in the table can be saved to a file for later use. Also, if you have a text file of the disk data in the appropriate format, it can be directly imported into the Disks table, thus simplifying the process of entering data.
You can specify the bearings’ stiffness and damping coefficients in the Bearings section. This section again requires a tabular input, with each row representing a bearing. Cross-coupled stiffness (k_{yz} and k_{zy}) and damping coefficients (c_{yz} and c_{zy}) for the bearing can also be specified (if available). These coefficients play an important role in determining the stability of the bearing. Tabular input for the bearing, like the disk, has the advantage of saving the data for later use and also importing the data from a text file.
The Bearings section.
After you are done setting up the system properties, you can evaluate the rotor system to be analyzed by updating the geometry.
The Study Parameters section provides you with the inputs to specify the maximum angular speed of the rotor and the steps in the parametric sweep of the angular speed from zero to the maximum value. You can also specify the number of natural frequencies to be computed by the app.
The Study Parameters section is used to specify the angular speed and eigenfrequency information.
As discussed above, the critical speed of a rotor is determined by obtaining the variation of the natural frequency with its angular speed. To do so, after setting up the model, you first perform an eigenfrequency analysis by clicking the Compute button. As a result, in the Graphics window, you can see the whirl, orbit, and Campbell plots for the modeled rotor system. In the Campbell plot, critical speeds are the points where the frequency is equal to the angular speed. In other words, critical speeds are the intersection points of the eigenfrequency curves with the ω = Ω curve, as shown below.
In the Campbell plot, critical speeds are marked as points (light blue).
In the underlying model, there is no direct way of calculating the critical speed. This is where you harness the power of the Application Builder. Using the Method Editor (available in the Application Builder), you can easily write your own method to compute the critical speeds. This is what is done in the Rotor Bearing System Simulator. A screenshot of the code for computing the critical speed is shown in the figure below.
The code shows how the critical speed is calculated.
The calculated critical speeds are then displayed as a table in the Critical Speeds section.
The Critical Speeds section.
Simple apps such as this can help designers to quickly come up with a good starting point for their design. Further, the app allows them to test various configurations without spending excessive money on experiments. Apps also make such investigations convenient because they hide the technical details while highlighting the important parameters in the design process. This provides designers with the accessibility and flexibility to control the design parameters and evaluate their findings with only a few clicks and without worrying about the underlying technical details.
Apps are not limited to modeling only simple physics. The underlying model for an app could be as complex as possible, simulating multiple physics simultaneously. The app itself could further extend the model, with the help of the Method Editor, to bring the simulation closer to reality.
A gearbox assembly generally consists of gears, shafts, bearings, and housing. When operated, a gearbox radiates noise in its surroundings for two main reasons:
Out of all of the components in a gearbox, the primary source of vibration or noise is the gear mesh. A typical path followed by the structural vibration, seen as the noise radiation in the surrounding area, can be illustrated like this:
The noise generated due to gear meshing can be classified into two types: gear whine and gear rattle.
Gear whine is one of the most common types of noise in a gearbox, especially when it runs under a loaded condition. Gear whine is caused by the vibration generated in a gear because of the presence of transmission error in the meshing as well as the varying mesh stiffness. This type of noise occurs at the meshing frequency and typically ranges from 50 to 90 dB SPL when measured at a distance of 1 m.
Gear rattle is observed mostly when a gearbox is running under an unloaded condition. Typical examples are diesel engine vehicles such as buses and trucks at idle speed. A gear rattle is an impact-induced noise caused by the unloaded gear pairs of the gearbox. Backlash, required for lubrication purposes, is one of the gear parameters that directly impact the gear rattle noise. If possible, simply adjusting the amount of backlash can reduce gear rattle.
We know that transmission error is the main cause of gear whine, but what exactly is it? When two rigid gears have a perfect involute profile, the rotation of the output gear is a function of the input rotation and the gear ratio. A constant rotation of the input shaft results in a constant rotation of the output shaft. There can be various unintended and intended reasons for modifying the gear tooth profile, such as gear runouts, misalignment, tooth tip, and root relief. These geometrical errors or modifications can introduce an error in the rotation of the output gear, known as the transmission error (TE). Under dynamic loading, the gear tooth deflection also adds to the transmission error. The combined error is known as the dynamic transmission error (DTE).
Reducing gear whine or rattle to an acceptable level is a big challenge, especially for modern complex gearboxes, which consist of many gears meshing simultaneously. By accurately simulating these complex behaviors, we can design a quieter gearbox. COMSOL Multiphysics gives designers the ability to accurately identify problems and propose realistic solutions within the allowable design constraints. With such a tool, we can optimize existing designs to reduce noise problems and gain insight into new designs earlier in the process, well before the production stage.
A gearbox model in the COMSOL Desktop®.
Let’s consider a five-speed synchromesh gearbox of a manual-transmission vehicle in order to study the vibration and radiation of gear whine noise to the surrounding area. The gearbox is in a car and used to transfer power from the engine to the wheels.
Geometry of a five-speed synchromesh gearbox of a manual transmission vehicle.
In order to numerically simulate the entire phenomenon of gearbox vibration and noise, we perform two analyses:
In the multibody analysis, we compute the dynamics of the gears and housing vibrations, performed at the specified engine speed and output torque in the time domain. For the acoustic analysis, we compute the sound pressure levels outside the gearbox for a range of frequencies using the normal acceleration of the housing as a source of noise.
First, we look into the gear arrangement in the synchromesh gearbox. Here, helical gears are used to transfer the power from the input end of the drive shaft to the counter shaft and further from the counter shaft to the output end of the drive shaft.
The gear arrangement in the five-speed synchromesh gearbox, excluding the synchronizing rings that connect the gears with the main shaft.
The gears used in the model have the following properties:
Property | Value |
---|---|
Pressure angle | 25 [deg] |
Helix angle | 30 [deg] |
Gear mesh stiffness | 1e8 [N/m] |
Contact ratio | 1.25 |
All of the gears on the counter shaft are fixed to the shaft, whereas the gears on the drive shaft can rotate freely. Only one gear at a time is fixed on the shaft. In real life, this is achieved with the help of synchronizing rings. In the model, hinge joints with an activation condition are used to conditionally engage or disengage gears with the drive shaft.
Looking at the shafts, they are assumed rigid and rested on the housing through hinge joints, whereas the housing is assumed flexible, further mounted on the ground, and connected to the engine at one of its ends. The driving conditions considered for the simulation in terms of engine speed, load torque, and the engaged gear are as follows:
Input | Value |
---|---|
Engine speed | 5000 [rpm] |
Load torque | 1000 [N-m] |
Engaged gear | 5 |
With these settings, it is possible to run a multibody analysis and compute the housing vibrations as shown in this animation:
The von Mises stress distribution in the housing together with the speed of different gears.
In order to have a better understanding of the variation of normal acceleration as a function of time, we can choose any point on the gearbox housing. The time history of the normal acceleration at that point is shown below. Let’s transform this result to the frequency domain using the FFT solver. In this way, we can find the frequency content of the vibration. It is clear from the frequency response plot that the normal acceleration of the housing contains more than one dominant frequency. The frequency band in which the housing vibration is dominant is 1000–3000 Hz.
Time history and frequency spectrum of the normal acceleration at one of the points on the gearbox housing.
Once we have simulated the vibrations in a gearbox, let’s see how to model the noise radiation in COMSOL Multiphysics. To begin, we create an air domain outside the gearbox to simulate the noise radiation in the surrounding.
In order to couple multibody dynamics and acoustics, we assume a one-way coupling, as the exterior fluid is air. This implies that the vibrations from the gearbox housing affect the surrounding fluid, whereas the feedback from the acoustic waves to the structure is neglected. It is a good assumption that the problem is one-way coupled.
The acoustic analysis is performed for a range of frequencies. As the multibody analysis is solved in the time domain, the FFT solver is used to convert the housing accelerations from the time domain to the frequency domain.
The air domain enclosing the gearbox for acoustic analysis. The two microphones placed to measure noise levels are shown.
As a source of noise, the normal acceleration of the gearbox housing is applied on the interior boundaries of the acoustics domain. In order to avoid any reflections from the exterior boundaries of the surrounding domain, we apply a spherical wave radiation condition. With these settings, we can solve for the acoustic analysis and look at the sound pressure level in the near field as well as on the surface of the gearbox housing at different frequencies. For a better understanding of the directivity of the noise radiation, we can create far-field plots in different planes at different frequencies.
The sound pressure level in the near field (left) and at the surface of the gearbox (right).
The far-field sound pressure level at a distance of 1 m in the xy-plane (left) and xz-plane (right).
After visualizing the sound pressure level in the outside field, it is interesting to find out the variation of sound pressure with frequency at a particular location. For this purpose, two microphones are placed in specific locations.
Microphone | Placement | Position |
---|---|---|
1 | Side of the gearbox | (0, -0.5 m, 0) |
2 | Top of the gearbox | (0, 0, 0.75 m) |
These microphone locations are defined in the Parameters node in the results and can be changed without updating the solution every time.
The frequency spectrum of the pressure magnitude at the two microphone locations.
The pressure response plot at the microphone locations gives a good idea of the frequency content present in the noise. However, wouldn’t it be nice if we could actually listen to the noise recorded at the microphone, just like in a physical experiment? This is possible by writing Java® code in a model method using the magnitude and phase information of the pressure as a function of frequency.
Let’s listen to the sound files corresponding to the noise received at the two microphones…
We have already looked at the acoustics results for various frequencies. It would also be nice to see them in the time domain. Let’s transform the results from the frequency domain to the time domain using the FFT solver so that we can visualize the transient wave propagation in the surrounding area of the gearbox.
Animation showing the transient acoustic pressure wave propagation in the surrounding area of the gearbox.
The above approach describes a technique to couple multibody analysis and acoustics simulation in order to accurately compute the noise radiation from a gearbox. This technique can be used early in the design process to improve the gearbox in such a way that the noise radiation is minimal in the range of operating speeds of the gearbox. Additionally, model methods — new functionality as of version 5.3 of the COMSOL Multiphysics® software — enable us to actually hear the noise generated by the gearbox — making the simulation one step closer to a physical experiment.
Some devices require a very high degree of frequency stability with respect to changes in the environment. The most common parameter is temperature, but the same type of phenomena could, for example, be caused by hygroscopic swelling due to changes in humidity. In very high precision applications, the frequency stability requirements might specify a precision at the ppb (parts-per-billion, 10^{-9}) level. Setting up simulations that accurately capture such small effects can be a challenging task, since several phenomena can interact.
Consider a rectangular beam with the following data:
Property | Symbol | Value |
---|---|---|
Length | L | 10 mm |
Width | a | 1 mm |
Height | b | 0.5 mm |
Young’s modulus | E | 100 GPa |
Poisson’s ratio | ν | 0 |
Mass density | ρ | 1000 kg/m^{3} |
Coefficient of thermal expansion, x direction | α_{x} | 1·10^{-5} 1/K |
Coefficient of thermal expansion, y direction | α_{y} | 2·10^{-5} 1/K |
Coefficient of thermal expansion, z direction | α_{z} | 3·10^{-5} 1/K |
Temperature shift | ΔT | 10 K |
The beam geometry and mesh used in the example.
The material parameters have values that are of the same order of magnitude as those for many other engineering materials. To better separate the various effects, Poisson’s ratio is set to zero, but this assumption does not change the results in any fundamental way. Orthotropic thermal expansion coefficients are used to highlight some properties of the solution.
To analyze the effect of thermal expansion, add a Prestressed Analysis, Eigenfrequency study.
Adding the Prestressed Analysis, Eigenfrequency study.
This study consists of two study steps:
The two study steps shown in the Model Builder tree.
To compute the reference solution, you either add a separate Eigenfrequency study or run the same study sequence, but without thermal expansion.
The eigenfrequencies of the beam have been calculated for two different types of boundary conditions:
The doubly clamped beam results are shown below.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 50713.9 | 50425.1 | 0.9943 |
First bending, y direction | 97659.6 | 97526.2 | 0.9986 |
First twisting | 266902 | 266917 | 1.00006 |
First axial | 500000 | 500025 | 1.00005 |
Mode shapes for the doubly clamped beam.
The following table shows the cantilever beam results.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 8063.79 | 8066.92 | 1.00039 |
First bending, y direction | 16049.1 | 16053.7 | 1.00028 |
First twisting | 132233 | 132265 | 1.00025 |
First axial | 250000 | 250050 | 1.0002 |
Mode shapes for the cantilever beam.
The first thing to note is that the bending eigenmodes for the doubly clamped beam stand out and have a strong temperature dependence. The change is 0.6% in the first mode. For all other modes, the relative shift in frequency is significantly smaller. If you make the beam thinner, this difference would be even more pronounced. The reason for this behavior is discussed in the following sections.
In the case of the doubly clamped beam, the thermal expansion causes a compressive axial stress. With the given data, the stress is -10 MPa (computed as Eα_{x}ΔT). This stress causes a significant reduction in the stiffness of the beam — an effect often called stress stiffening, since it typically occurs in structures with tensile stresses. However, compressive stresses soften the structure.
Another way of looking at this is by performing a linear buckling analysis. You can do so by adding a Linear Buckling study to the model and using the thermal expansion caused by ΔT = 10 K as a unit load. You will then find that the critical load factor is 80.
The first buckling mode.
With a linear assumption, the beam becomes unstable at an 800 K temperature increase. At the buckling load, the stiffness has reached 0. Assuming that the stiffness decreases linearly with the compressive stress, the stiffness at ΔT = 10 K should be reduced by a factor of
Since a natural frequency is proportional to the square root of the stiffness, you can estimate the decrease to , which matches the computed value of 0.9943 well.
Stress softening also affects the twisting and axial modes, but the effect is not as obvious as it is in the bending modes.
In the cantilever beam, no stresses develop when it is heated, as it simply expands. In this case, the frequency shift is due solely to the change in geometry — an effect that is much smaller than the stress-softening effect.
The natural frequencies for the bending, torsional, and axial vibration of a beam have the following dependencies on the physical properties:
Here, the following variables have been introduced:
It is assumed that the initial dimensions of the beam are L_{0} x a_{0} x b_{0}, where a_{0} > b_{0}. After thermal expansion, the size is L x a x b.
The expansions (strains) in the three orthogonal directions are called ε_{x}, ε_{y}, and ε_{z}; respectively. In this case, they are linearly related to the thermal expansion by ε_{x} = α_{x}ΔT, ε_{y} = α_{y}ΔT, and ε_{z} = α_{z}ΔT; but in principle, it could be any type of inelastic strain.
The geometric properties scale as:
The mass density also changes. Since the same mass is now confined in a larger volume,
By introducing these expressions into the formulas for the natural frequencies, you arrive at the following expected eigenfrequency shifts:
Since the thermal expansions are very small, the approximate first-order series expansions can be expected to be accurate.
For the torsional vibrations, the situation is slightly more complicated, since the powers of a and b are mixed in the expression for the polar moment J. But if you make use of the fact that a = 2b for this geometry, then it is possible to derive a similar expression.
Now, compare the computed frequency shifts with the analytical predictions for the cantilever beam. The results are shown in the table below and the agreement is very good.
Mode Type | Computed | Predicted |
---|---|---|
First bending, z direction | 1.00039 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 |
First twisting | 1.00025 | 1.00025 |
First axial | 1.00020 | 1.00020 |
The fixed constraints at the ends of the beam cause local stress concentrations when the temperature is increased, as the transverse displacement is constrained.
The axial stress in the doubly clamped beam caused by a 10 K temperature increase.
This can have two effects:
To determine what effects the constraints should have, you must rely on your engineering judgment. Usually, the component and its surroundings are subject to temperature changes. In this situation, the possibility to add a thermal expansion to constraints in COMSOL Multiphysics comes in handy. Let’s see how the solution is affected.
Thermal expansion added to the fixed constraints for the doubly clamped beam.
For the cantilever beam, the results now change so that they perfectly match the analytical values.
Mode Type | Fixed Constraints | Stress-Free Constraints | Predicted |
---|---|---|---|
First bending, z direction | 1.00039 | 1.00040 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 | 1.00030 |
First twisting | 1.00025 | 1.00026 | 1.00025 |
First axial | 1.00020 | 1.00020 | 1.00020 |
In the analysis above, it is assumed that the material data does not depend on temperature. When looking at constrained structures (dominated by the stress-softening effect), this might be an acceptable approximation. However, with the small frequency shifts caused by geometric changes, the temperature dependence of the material must also be taken into account.
In this guide, you can see the temperature dependence of Young’s modulus for a number of metals. The stiffness decreases with temperature. For many materials, the relative change in stiffness is of an order of 10^{-4} K^{-1}. This means that for a temperature change of 10 K, you can expect a relative change in material stiffness that is of the order of 0.1%. This effect might actually be larger than the geometric effect computed above.
A small note of warning: When measuring the temperature dependence of Young’s modulus, it is important to know whether or not the geometric change caused by thermal expansion has been taken into account. In other words, you must know whether the Young’s modulus is measured with respect to the original dimensions or the heated dimensions.
When it comes to mass density, the situation is easier. When performing structural mechanics analyses in COMSOL Multiphysics, the equations are formed in the material frame. Thus, the mass density should never be given an explicit temperature dependence, since that violates mass conservation.
The coefficient of thermal expansion (CTE) usually increases with temperature. The relative sensitivity is often of the order of 10^{-3} K^{-1}. This sounds large, but it isn’t usually important when looking at the way the CTE enters the equations.
Most materials in the Material Library in COMSOL Multiphysics come with temperature-dependent material properties. In this example, you manually add a linear temperature dependence to the Young’s modulus with the following steps:
Alternatively, you can create a function and call it, with T as the argument.
Adding a linear temperature dependence to the material.
In the settings for the Linear Elastic Material, the Model Input section is now active. You then provide a temperature to be used by the material.
Adding the temperature to the material using Model Input.
After including a reduction of Young’s modulus by 1·10^{-4} K^{-1}, the resulting frequency shift turns out to be negative, rather than the positive shift observed with a constant Young’s modulus (shown in the table below).
Mode Type |
Stress-Free Constraints Constant E |
Stress-Free Constraints Temperature-Dependent E |
Difference |
---|---|---|---|
First bending, z direction | 1.00040 | 0.99990 | -0.00050 |
First bending, y direction | 1.00030 | 0.99980 | -0.00050 |
First twisting | 1.00026 | 0.99976 | -0.00050 |
First axial | 1.00020 | 0.99970 | -0.00050 |
The shift is exactly as expected for all modes — Young’s modulus is reduced by a factor 1·10^{-3} and the natural frequencies are proportional to its square root. Actually, you can include the change in Young’s modulus in the linearized expressions for the frequency shifts as:
Here, it is assumed that . The value of the coefficient β is usually negative; In this case, β = -10^{-4} K^{-1}.
For the common case of isotropic thermal expansion, these expressions simplify to:
We are looking for frequency changes that are at the ppm (parts-per-million) level. This means that it is important to avoid spurious rounding errors. There are some actions that you can take to ensure optimal accuracy.
In the settings for the Eigenfrequency node, set Search for eigenfrequencies around to a value of the correct order of magnitude.
The updated settings in the Eigenfrequency node.
Then, decrease the Relative tolerance in the settings for the Eigenvalue Solver node.
The decreased Relative tolerance in the settings for the Eigenvalue Solver node.
Change only the parameters necessary for capturing the physics. For example, use the same mesh for all studies.
If you have reason to believe that the problem is ill-conditioned, as can be the case for a slender structure, select Iterative refinement in the settings for the Direct solver.
The settings for the Direct solver, showing the option for Iterative refinement.
In version 5.3 of COMSOL Multiphysics®, the method for how inelastic strains are handled under geometric nonlinearity has been changed. A multiplicative decomposition of deformation gradients is the current default, rather than the subtraction of strains that was used in previous versions. This is one key concept to understand why it is now possible to perform this type of analysis with a very high accuracy. Let’s look at a (somewhat artificial) case where the temperature increase is 3·10^{4} K and there are no temperature dependencies in the material properties. This means that the stretches are
resulting in the volume changing by a factor of 3.952.
You can then compare the results from the prestressed eigenfrequency analysis with a standard eigenfrequency analysis on a bigger beam with L = 13 mm; a = 1.6 mm; b = 0.95 mm; and lower density, scaled by a volume factor of 3.952, ρ = 253.036 kg/m^{3}. This of course leads to large increases in the natural frequencies, as the heated object is much larger but with a lower density. The relative changes in frequency for the two approaches are shown in the following table.
Mode Type |
Thermal Expansion and Prestressed Eigenfrequency |
Larger Geometry and Lower Density |
---|---|---|
First bending, z direction | 2.2309 | 2.2308 |
First bending, y direction | 1.8759 | 1.8759 |
First twisting | 1.6702 | 1.6695 |
First axial | 1.5292 | 1.5292 |
As can be seen above, the correspondence is in excellent agreement. There is a slight difference in the twisting mode, but that disappears with a refined mesh. Actually, refining the mesh shows that the best prediction is from the prestressed eigenfrequency analysis.
We have discussed how to accurately determine changes in eigenfrequencies caused by temperature changes with COMSOL Multiphysics, as well as the effects of stress softening, geometric changes, and the temperature dependence of material properties.
Artificial ground freezing is a construction technology that involves running an artificial refrigerant through pipes buried underground. As the refrigerant circulates through the pipe network, heat is removed from the ground and ice begins to form around the pipes. This in turn causes the soil to freeze. In other words, the process converts soil moisture into ice. Once the soil is frozen, it is both stronger (sometimes as hard as concrete) and has a greater resistance to water. This allows the soil to provide effective support to the relative infrastructures, particularly those that are larger and more complex.
Once frozen, soil becomes stronger and more resistant to water.
For the AGF method to be effective, we need to know the temperature distribution inside the system. Of the physical processes that occur in AGF, the most prominent is the phenomenon of transient heat conduction with phase change. Further, it is also important to consider the relationship between this phase change and the groundwater flow — particularly when there is a higher flow velocity. These elements can impact the development of the freezing wall and thus the strength and reliability of the AGF method.
To study the AGF method, a team of researchers from Hohai University turned to the COMSOL Multiphysics® software. Their case study involves using the method to strengthen soil at a metro tunnel entrance in Guangzhou, China.
For this specific example, the refrigerant that circulates throughout the pipe system is -30ºC brine. The subsurface temperature is reduced until the pore water is frozen and the freezing wall forms. The formation within the frozen area is made up of muddy sand, and the direction of the groundwater flow is primarily horizontal and normal in relation to the axial direction of the tunnel.
To simplify modeling heat transport in a saturated aquifer, the researchers used a 2D model based on a coupling of temperature and flow fields. The model, shown below, is 20 m in both length and height. Note that five monitoring points are included. These points are used to verify the accuracy of the model by comparing the calculated temperature results with in situ measurements.
The AGF model’s geometry, with the monitoring points highlighted (left), and the model grid’s mesh (right). Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
In this analysis, the following assumptions are made:
According to previous temperature monitoring data from the frozen area, there is an initial ground temperature of 15°C. The figure below shows the initial temperatures in various holes of thermal observation.
The initial temperatures in different holes for thermal observation. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
The cooling source of the freezing system is the lateral wall of the freezing pipe. Changes in the temperature of the lateral wall have the greatest impact on the temperature distribution within the system. It is possible to use the values from the temperature monitoring of the main pipe as approximations for the estimated temperature of the lateral wall. The plot below shows the fitting function and curve for the lateral wall temperature of the main pipe after a monitoring period of 40 days.
The fitting function and curve for the lateral wall temperature. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
With regards to groundwater flow, a flow velocity of 0.2 m/d is obtained via field tests. Between upstream and downstream, the head difference is calculated as 0.8 m.
Now onto the results. Let’s consider the temperature distribution and permeability coefficient for a range of times. In terms of temperature, when the freezing time increases, the cold temperature from the freezing pipes is primarily led downstream — with less of an influence upstream. The permeability coefficient results, which illustrate the formation of the freezing wall, indicate that the top and bottom walls form at a faster rate than those walls at upstream and downstream. Note that the freezing wall is entirely closed after 35 days.
The temperature distribution (left) and permeability coefficient results (right) at various points in time. Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
When comparing the closure of the freezing wall and flow velocity, the closing time increases nonlinearly as the flow velocity increases. The time of closure dramatically increases when the velocity is greater than 1.5 m/d. As for the average wall thickness in all directions and relative flow velocity, the influence of the flow velocity on the thickness of the upstream wall is most prominent.
The successful validation of this model offers guidance for the metro tunnel project in Guangzhou, China. With plans to further develop this model, the researchers hope to use it as a resource for improving applications of the AGF method.
When performing a structural analysis with plate or shell elements, there is an underlying assumption that the variation of the in-plane stresses through the thickness is linear. In a local coordinate system, where z is oriented along the normal to the shell surface, it is thus possible to write
where d is the thickness. The indices i and j can be either x or y. In this decomposition, is called the membrane stress and (or ) is called the bending stress.
The decomposition of a linear stress distribution into a membrane stress and bending stress.
For the other stress components, shell theory implies that
and
Unless the shell is thick, the transverse shear stresses, σ_{iz}, are significantly smaller than the in-plane stresses.
A membrane stress has the same value all through the section. If the material is assumed to be elastoplastic with no hardening, then all points reach the failure stress at the same time. The load that causes incipient plasticity is thus also the failure load.
The stress-strain curve for an elastoplastic material with no hardening. The variable σ_{y} is the yield stress.
Now, consider pure bending with a uniaxial stress state, as in a beam. As long as the material is elastic, the stress distribution is linear through the section, with the value being zero at the midsurface. As the load increases, the stress in the outermost fibers reaches the yield limit. However, the rest of the section is still elastic. It is thus possible to further increase the load without a complete failure.
The stress distribution at incipient yielding (left), partly through yielding (middle), and collapse (right).
The bending moment at failure is 1.5 times the bending moment at initial yield. Thus, if the allowed stress only takes the maximum stress into account, the risk of collapse is larger for a membrane state than it is for a bending state.
If we consider a state of mixed bending and tension, it is possible to compute the combinations of moment, M, and axial force, N, which cause failure.
The stress state at collapse for combined tension and bending.
The membrane and bending stresses are, for an elastic case, related to the moment and axial force through
and
By writing the moment and axial force in terms of membrane and bending stresses, we arrive at the following interaction formula:
In a full 3D case, the stress distribution differs significantly from linear in the vicinity of geometric discontinuities. This is where the concept of stress linearization becomes important. The sum of the membrane and bending stress provides a linear approximation to the true stress distribution, having the property that the resultant force and moment are preserved.
The linearization of a stress tensor component from a 3D solution.
In the graph above, the maximum computed stress is 305 MPa. If the stress state is uniaxial — and the yield stress of the material is 350 MPa — this means that 87% of the load giving initial yield has been reached. However, the linearized stress predicts only 64% of the yield stress. The membrane stress contributes 32% of the yield stress.
If we want to compute a safety factor against collapse, the actual stress distribution does not matter. At failure, the stress everywhere is equal to the yield stress, either in tension or in compression. The relation between tensile and compressive stresses is uniquely determined by force and moment equilibrium.
In the figure below, we can see an example of how the stress is distributed along a stress linearization line as the load is increased in an elastoplastic analysis. The yield stress is first reached when the load parameter rises slightly above 0.38. When the load parameter reaches 0.76, a collapse ensues.
The stress distribution over a cross section as the external load is increased. The load parameter value is the ratio between the membrane stress and yield stress.
In this example, the values have been chosen so that σ_{m} = σ_{b}. Using the interaction formula above, this means that collapse should occur when
This value matches the final parameter value of 0.76 rather well. The difference can be explained by the fact that a small plastic hardening is used in the model to stabilize the analysis.
The conclusion is that for determining safety conditions within plastic collapse, the linearized stress is the relevant parameter, since it is proportional to the axial force and bending moment. Using the true peak stress gives an overly conservative design. The safety factor, which is implicit in the bending collapse, must also be taken into account.
If the structure is subjected to cyclic loading, the peak stresses are of utmost importance, as they determine the risk of fatigue crack initiation at the surface.
The concept of stress linearization is an important part of the qualification of pressure vessels, as described in ASME Boiler & Pressure Vessel Code, Section III, Division 1, Subsection NB. Here, we are required to classify stresses as either primary or secondary.
A primary stress is a stress that is required for maintaining force and moment equilibrium. Secondary stresses are caused by other effects. Typically, secondary stresses are local effects caused by either geometric discontinuities or displacement-controlled loading. Secondary stresses do not lead to a collapse when they exceed elastic limits, since they are just redistributed.
During the analysis, the stress is studied along a number of lines through the section, referred to as stress classification lines (SCLs). The choice of SCL is not unique, so here we must use our engineering judgment to find the critical locations.
Although not fully correct (but conservative), the linearized stresses are sometimes viewed as equivalent to the primary stresses. Without going into detail, the basic requirements of the code are:
Interestingly enough, this means that if the membrane stress is at the limit allowed by the first criterion, it is still allowed to add a certain amount of bending stress. The discussion above tells us why: The bending stress reduces the stress over part of the section.
As noted above, the detailed stress state is not important when it comes to static failure, as the stress distribution in the collapse state is fully determined by the force and moment equilibrium. In the figure below, the collapse interaction curve is compared with the stress limits imposed by the code.
The fundamental ASME criteria for primary stresses. The stresses are normalized by the yield stress.
It should be noted that because pressure vessels often operate at elevated temperatures, room temperature values of allowed stresses might not be sufficient.
The requirement on the secondary stresses is set to avoid cyclic plastic deformation upon repeated loading–unloading cycles. The purpose is to avoid plastic strains accumulating in each load cycle, which can lead to a fast failure due to low-cycle fatigue.
Some rules for qualifying structural elements are based on the stresses being “hand calculated” or the result of a shell or plate analysis. When we do a full 3D analysis, the effect can be that we get results that are “too good”. The effects of local stress concentrations are already taken into account by providing low allowable nominal stresses. Because of this, we might end up in a situation where using the accurate results of a full 3D analysis leads to a highly conservative design. In this case, stress linearization can provide a useful tool for converting the 3D stress state back into a set of nominal stresses.
For instance, this situation can occur when analyzing welds. Typically, the local geometry at the weld is not even well defined (unless it is a very high-quality weld that has been ground smooth). Thus, the actual local stress is not even meaningful to compute, so we must resort to methods based on nominal stresses.
A weld in a pipe used for district heating. Image by Björn Appel, Benutername Warden. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
A stress linearization does not affect the analysis as such; it is a type of result presentation. The variables to be used are set up in the Solid Mechanics interface. We add a line for stress linearization either under Variables in the context menu for the Solid Mechanics interface or under Global on the Physics tab in the ribbon.
Adding a Stress Linearization node from the context menu.
Adding a Stress Linearization node from the ribbon.
Depending on whether the component is in 3D or not, the definition of the stress linearization line comes in two different flavors. In either case, we select an edge (or set of edges) that forms a straight line through the thickness of the component that we are evaluating. In 3D, we must also define the axis orientation of the local coordinate system in which the stresses along the SCL are represented.
The settings for stress linearization in 3D.
The stress tensor components along an SCL are represented in a local coordinate system, where 1 is the direction along the line. The 2 direction is perpendicular to the line and has the following orientations:
For the last bullet point, note that the Second Axis Orientation section of the Stress Linearization node provides several options for entering the orientation.
If we have defined the SCLs prior to the analysis, then one edge data set is generated for each SCL. At the same time, a default plot called Stress Linearization is added.
The default data sets and graph plot group.
The stress linearization plot contains three graphs along the selected SCL:
An example of a default stress linearization plot.
In the stress linearization plot, we can change to another SCL by selecting the corresponding edge data set. In the default plot, the 22 stress tensor component is displayed. Of course, we can change to other components. Usually, 33 and 23 are the most important.
If we add Stress Linearization nodes after running the analysis, we must click on the Update Solution button to make the newly created variables accessible for result presentation. No default plots or data sets are automatically generated in this case.
Graphing along the SCLs is important for understanding the stress state at different locations, but at the end of the day, it is the stress intensity that is important. The maximum stress intensity for each SCL can be presented by adding a Global Evaluation node. When computing the stress intensity for the bending stress plus the membrane stress, the bending part of the out-of-plane stress components (which are supposedly small) is ignored. This approach is customary in this type of analysis.
The result quantities for stress linearization when selecting data for a Global Evaluation node.
In addition to the stress intensities, the peak stress tensor at the two ends of the SCL is available. We can also directly access the section forces and moments corresponding to the linearized stresses.
As of version 5.3 of COMSOL Multiphysics® and the Structural Mechanics Module, the functionality for stress linearization provides us with a set of built-in tools for converting a 3D stress state to one of pure bending and tension. This makes it much easier to produce results that comply with various design codes.
While both Nikola Tesla and Galileo Ferraris built early versions of AC induction motors in the 19^{th} century, Tesla (a large proponent of AC) is more often credited with the motor’s invention. This device turned out to be a popular machine, with future iterations proving to be durable, reliable, and adaptable.
Left: A Tesla induction motor. Image by Ctac — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons. Right: A modern three-phase induction motor. Image in the public domain, via Wikimedia Commons.
Engineers can continue to improve these motors by accurately analyzing their performance, something that requires accounting for all of the relevant physical effects. To accomplish this, we can couple the Multibody Dynamics Module and AC/DC Module to analyze electromechanical effects in a three-phase induction motor. A new example model, added to the Application Library in COMSOL Multiphysics® version 5.3, demonstrates this functionality. (You can also find it in the online Application Gallery).
We can see all of the parts included in the 3D model of a three-phase induction motor in the schematic below. We physically model each part except for the bearings and foundation, which we model as massless springs.
The geometry of the three-phase induction motor housing assembly.
In this example, the stator and rotor are slightly misaligned, causing the small air gap between them to be asymmetric. As a result of this asymmetry, vibrations occur in the motor, which can be analyzed with simulation. To induce eddy currents into the rotor, we rely on the rotor’s rotation and time-harmonic currents in the stator windings.
Next, we perform two different studies: a 2D electromagnetics simulation and a 3D multibody dynamics simulation. In these studies, we use the Rotating Machinery interface to account for the motor’s electromagnetic fields and the Multibody Dynamics interface to simulate the rotor’s motion and housing vibration.
Let’s first discuss the electromagnetic case. For this analysis, we simplify the model to include only three parts:
This 2D geometry, shown in the cross section below, is a transverse section of the full 3D geometry. We also apply an alternating current of 60 Hz to the stator winding in this geometry via a Homogenized Multi-Turn Coil feature that has 2045 turns.
For more information about the geometrical dimensions and electromagnetic model, check out the references in the model documentation.
A cross section of a three-phase induction motor model. The three different coil regions in the stator (labeled A, B, and C) represent the motor’s three phases.
Switching gears, let’s explore the multibody dynamics case. This time, we use the full 3D geometry and model the stator, rotor, and shaft as rigid, with the rotor rigidly mounted on the shaft. The elastic hinge joints between the rotor and structural steel housing represent the bearings, which support the rotor and transmit its forces to the housing. As for the housing, we assume that it is elastic and use elastic fixed joints to connect it to the foundation. To compute the rotor’s angular speed, we use rotational torque, which is calculated as a function of time.
Using calculations from both of these cases, we run an electromechanical analysis that couples our electromagnetics and multibody dynamics simulations. For instance, we add values calculated with the Rotating Machinery interface — such as the electromagnetic forces caused by the stator and rotor misalignment and the electromagnetic torque — to the rotor and stator in the Multibody Dynamics interface.
We can find the rotor’s speed by combining these interfaces once again, transferring the hinge joint’s angular motion computed in the Multibody Dynamics interface to the Rotating Machinery interface.
Let’s now take a closer look at the magnetic flux density norm over time and the rotor’s electromagnetic forces. When calculating these electromagnetic forces, we observe vibrating forces in the transverse direction that are caused by the misaligned stator and rotor.
The magnetic flux density norm of the rotor and stator over time (left) and the rotor’s electromagnetic forces in both the transverse and axial directions (right).
In regards to electromagnetic torque, when the rotor speed equals the stator electrical frequency, the electromagnetic torque falls to zero if there is no loading torque on the shaft. The time delay for the rotor speed to equal the stator electrical frequency is dependent on the rotor’s inertia. In this case, the rotor takes 0.7 seconds to achieve a steady-state speed.
The rotor’s electromagnetic torque (left) and angular speed (right) as a function of time.
To find areas of high stress in the motor, we combine our analysis of the rotor’s velocity with the housing’s von Mises stress distribution. As indicated in the animation below, the areas near the bearing and where the housing and foundation connect have the highest stress values.
The housing’s von Mises stress distribution and the rotor velocity profile.
The plots below explore the forces acting on Bearing 1, Bearing 2, and Foundation 1 as a function of time. These forces travel through the elastic housing to the motor foundation.
The forces on Bearing 1 (left) and Bearing 2 (middle) in the transverse and axial directions. The forces at the connection between the housing and foundation at the location of Foundation 1 (right).
By analyzing the frequency spectrum of the electromagnetic forces, we can conclude that the frequency is 120 Hz, double the stator electrical frequency. Despite this, the frequency spectrum plot for the housing-foundation connection shows a dominant frequency contribution of around 60 Hz, with a few peaks around 83 Hz — the first natural frequency of the induction motor’s housing assembly.
The frequency spectrum of the rotor’s electromagnetic forces (left) and forces in the housing-foundation connection (right).
Lastly, let’s examine the rotor’s orbital motion, which results from the rotor vibrating in the transverse direction, with respect to the stator. This occurs due to the electromagnetic forces acting on the rotor in the transverse direction and the finite stiffness of the bearings supporting the rotor ends. The orbits seen in the following plot are not concentric due to the rotor’s asymmetric inertia in the axial direction.
Rotor orbital motion, combining its rotation and vibration, at both bearing locations.
Want to take this electromechanical analysis for a spin? Access the tutorial model with the button below.