To measure blood flow in a painless and noninvasive way for patients, medical professionals can use magnetic flow meters, which rely on electromotive force (EMF). In a flow meter, external coils generate a magnetic field and noncontact electrodes measure the induced EMF. When a patient moves during the measurement process (even by merely taking a breath), blood vessels can move, which affects the flow meter’s sensitivity. This phenomenon is an important point of analysis in cardiac and thoracic medicine.
Magnetic flow meters, which rely on coils and electrodes, are a noninvasive way to measure flow in a patient’s blood vessels.
Researchers from ABB Corporate Research in India built a multiphysics model of this process that includes the effects of fluid-structure interaction (FSI) and electromagnetics. Their aim was to understand how blood vessel movement influences flow meter sensitivity by comparing the meter’s performance when blood vessels are displaced versus in their normal positions.
The research team modeled a blood vessel as a pipe, but with the appropriate biological material properties. They coupled multiple physical effects via built-in physics interfaces in the COMSOL Multiphysics® software. The Laminar Flow interface was used to model the blood flow through the vessel. To account for the magnetic field generated by the coils, as well as the EMF induced by the blood flow and magnetic field, they used the Magnetic and Electric Fields interface.
A schematic of the magnetic flow meter model. Image by S. Dasgupta, K. Ravikumar, P. Nenninger, and F. Gotthardt and taken from their COMSOL Conference 2016 Bangalore paper.
The researchers also used the Structural Mechanics Module, an add-on product to COMSOL Multiphysics, to model the vessel displacement during patient movement. They coupled this analysis with the CFD Module to account for FSI, including how the vessel displacement affects blood flow and how the fluid pressure affects the blood vessel.
This model was used to analyze the sensitivity of the magnetic flow meter when blood vessels are displaced and when they are in a normal position.
The researchers compared the results for a blood vessel in a normal position and displaced by 5 cm toward the upper coil. The simulation results show contour plots of the velocity and magnetic flux density across the pipe (i.e., blood vessel) cross section. Other results show the induced electric potential from the flow and magnetic field interface and the potential distribution across the pipe diameter.
The velocity (top left) and magnetic flux density (bottom left) across the pipe as well as the induced electric potential of the cross section (top right) and length (bottom right) of the pipe. Images by S. Dasgupta et al. and taken from their COMSOL Conference 2016 Bangalore paper.
The plots indicate an increase in the sensitivity of the flow meter between the nondisplaced and displaced blood vessels. The team determined that the increase is not due to velocity, since the velocity profiles are the same for both scenarios. Instead, the reason for the increase is that the displaced vessel shifted to a higher magnetic field zone, and the magnetic flux density in the vessel increased as it moved toward the coil.
The researchers from ABB concluded through their simulation results that blood vessel displacement is a potential issue for medical uses of magnetic flow meters. To address the concerns, they theorized that a patient’s body movement can be restricted during these procedures or a breath-synchronous magnetic field can be generated to compensate for the sensitivity changes.
While these simulations proved useful for the medical field, further studies can account for more real-world conditions, such as pulsatile blood flow as well as variations in vessel properties and body locations.
For the team, this research confirmed that COMSOL Multiphysics can be used to analyze how different phenomena — including fluid flow, structural mechanics, and electromagnetics — interact in a complex application.
Check out the full paper from the COMSOL Conference 2016 Bangalore (it won a Best Paper award!): “Measurement of Blood Flowrate in Large Blood Vessels Using Magnetic Flowmeter“
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Canadian Nuclear Laboratories aims to improve nuclear fuel because it limits the efficiency of power generation in nuclear reactors. As Andrew Prudil said in his keynote talk: “If we can increase the power rating of the reactors, that’s worth millions of dollars per day.” Optimized nuclear fuel also enables more green energy on a power grid and reduces the risk of nuclear accidents. Plus, the improved fuel can be used in existing reactors to enhance their performance.
Before engineers can develop improved nuclear fuel, they have to understand its behavior. This is no simple matter, as nuclear fuel experiences multiple physical phenomena during nuclear reaction fission, including high temperatures, radiation, mechanical loading, thermal expansion, the creation of fission products such as xenon and krypton, and more.
To learn more about nuclear fuel behavior during a reaction, in which “everything depends on everything else,” CNL turned to the COMSOL Multiphysics® software.
First, Prudil discussed a multiphysics model — created for his PhD thesis — that studies the behavior of nuclear fuel (or pellets, in this case). The Fuel and Sheath Modeling Tool (FAST) simulates a long row of pellets separated by small gaps inside a metal sheath. Each part of the model involves multiple types of physics. For instance, sheaths in nuclear reactors typically use zirconium-based alloys, which consist of anisotropic crystal structures. For accurate results, the model must account for how the crystals behave when pulled in different directions.
From the video: Results for the FAST simulations.
The simulations show how the ends of the pellets push outward to make room for the hot material at the center. The “hourglassing” phenomenon causes the ends of the pellets to create a wavy pattern in the cladding (exaggerated in the image above). FAST can also plot the radial displacement and various stress and strain fields, such as the hydrostatic pressure, von Mises stress, and axial creep. Prudil noted that the results show “very interesting, very rich spatial fields.”
With FAST, it’s possible to look at how nuclear fuel behaves in a continuum — both in terms of a temperature gradient and mechanical loading.
Prudil then discussed a model created at Canadian Nuclear Laboratories that simulates how fission gas forms bubbles on the boundary of a single grain of uranium oxide, a process that involves fission gas products such as xenon and krypton. At the grain boundary, these insoluble gases try to diffuse pressure by forming bubbles. The bubbles grow larger and larger and let gases escape.
The CNL model simulates this process for individual bubbles. Instead of using the traditional phase field method, which can be computationally expensive, they created the included phase technique to model the phase interface.
Simulation results for the included phase technique. Animation courtesy Andrew Prudil and can be found in the paper: “A novel model of third phase inclusions on two phase boundaries“.
Initially, the simulations show a random distribution of bubbles on the grain boundary. As time progresses, the bubbles combine to minimize the surface energy before collecting at the edges and vertices. CNL validated their approach, determining that they could control the contact angle of a single bubble on an infinite plane.
Wrapping up, Prudil mentioned that COMSOL Multiphysics could also be used to investigate other interesting multiphysics phenomena (e.g., columnar grain growth). With these capabilities, engineers can learn more about nuclear fuel and continue to advance the field.
To learn more about how CNL uses multiphysics modeling to understand the behavior of nuclear fuel, watch the keynote video at the top of this post.
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The Nonlinear Structural Materials Module, an add-on product to COMSOL Multiphysics®, provides a plethora of material models, including models for hyperelasticity, isotropic/kinematic hardening plasticity, viscoelasticity, creep, porous plasticity, soils, and more. These material models cover a vast majority of engineering problems within structural analysis.
However, in some situations, the mechanical behavior of a material is not readily expressed in terms of an existing material model. For instance, suppose you developed a specialized material model for a certain alloy and want to use it to solve a large structural mechanics boundary value problem in COMSOL Multiphysics. What do you do?
As a matter of fact, there are three different ways in which you can define your own material:
The implementation of a material model as an external DLL can seem like a complex endeavor, but this blog post demonstrates how to implement an elastoplastic material model in COMSOL Multiphysics using hands-on steps that you can follow.
As a starting point, we need to decide on a material model to implement. We choose an isotropic linear-elastic material with isotropic hardening. This is a simple plasticity model that already exists in COMSOL Multiphysics, but it serves nicely to convey some key points.
First, let’s go over some assumptions, definitions, and nomenclature:
The example material model: Uniaxial stress-strain curve and yield surface in principal stress space.
Now, let’s discuss approaches for implementing a material model as an external material. There are several different ways of calling user-coded routines for external materials, which we refer to as sockets.
We can use the General stress-strain relation socket of the external material to define a complete material model that includes (possibly) both elastic and inelastic strain contributions. This is the most general of the two modeling approaches discussed here. When we use the General stress-strain relation socket, we are faced with two tasks:
We can also use the Inelastic residual strain socket to define a description of an inelastic strain contribution to the overall material model. An example of this would be if we wanted to add our own creep strain term to the built-in linear elastic material. The Inelastic residual strain socket assumes an additive decomposition of the total (Green-Lagrange) strain into elastic and inelastic parts. Thus, this is an adequate assumption when strains are of the order < 10%. When we use the Inelastic residual strain socket of the external material model, we are faced with two tasks:
Two related External Material sockets are the General stress-deformation relation and the Inelastic residual deformation. These are more general versions of those discussed above. Instead of defining the deformation in terms of the Green-Lagrange strain tensor, the deformation gradient is provided. Many large-strain elastoplastic material models use a multiplicative decomposition of the deformation gradient into elastic and plastic parts. In these situations, you would likely want to use one of these sockets instead.
Tip: We link to the source file and model file at the bottom of this blog post.
The complexity of computing the stress tensor varies significantly between material models. In practice, the computation of the stress tensor often needs to be formulated as an algorithm. This is often called a stress update algorithm in literature. In essence, the objective of a stress update algorithm for a material model is to compute the stresses, knowing:
These quantities are provided to the external material as input.
The term “material state” represents any solution-dependent internal variables that are required to describe the material. Examples of such variables are plastic strain tensor components, current yield stress, back stress tensor components, damage parameters, effective plastic strain, etc. The choice of such state variables will depend on the material model. We must ensure that the material state is properly initialized at the start of the analysis, and that it is updated at the end of the increment.
We first need to investigate if there is plastic flow occurring during the increment. We do this by assuming that the elastic strain is equal to the total strain of the current increment, less the (deviatoric) plastic strain of the previously converged increment, . This assumption would hold true if there was, indeed, no plastic flow during the increment. The deviatoric stress tensor that is computed this way is aptly called a trial stress deviator and is given by
with an effective (von Mises) value .
The effective trial stress is compared to the yield stress of the material by assuming that there is no plastic flow during the increment. The yield stress corresponding to the previously converged increment is given by
Notice that the left-superscripted and in Steps 1 and 2 represent the material state of the previously converged increment, as we discussed earlier.
Now we check if the trial stress causes plastic flow. For another way of expressing this, if the trial stress is inside the yield surface, the response will be purely elastic during the increment. If not, plastic flow will result. The check is performed using the yield condition:
Check of the yield condition to determine elastic or elastoplastic computation.
The stress update algorithm now necessarily branches off into either a purely elastic computation or an elastoplastic computation. We will follow each of these branches, starting with the purely elastic branch.
Because we determined that there is no plastic flow during the increment, the trial stress deviator is, in fact, identical to the stress deviator, and the update of the plastic strain tensor and the effective plastic strain is trivial.
We can directly return the pure elastic stress-strain relation as the Jacobian.
The objective of the elastoplastic branch of the stress update algorithm is to compute the stress deviator and update the plastic strains. We begin by again expressing the stress deviator, now knowing that plastic flow takes place during the increment:
or
In the above equation, we used a discrete form for the flow rule that states that an increment in plastic strain is proportional to the stress deviator through a so-called plastic multiplier . Let’s stop for a moment and consider a graphical representation of this equation for stress:
Graphical representation of the correction of the trial stress deviator.
If we compute a trial stress deviator that lies outside the yield surface, we need to make a correction so that the stress deviator is returned to the to-be-determined yield surface. The plastic multiplier determines the exact amount by which the trial stress deviator should be scaled back to give the correct stress deviator. If we compute the plastic multiplier, it is straightforward to then compute the stress deviator and the plastic strain increment.
The key steps are to:
We can relate the plastic multiplier to the effective plastic strain increment using the flow rule and then transform the equation for stress into a governing scalar equation:
This is in general a nonlinear equation, and we need to solve it using a suitable iterative scheme. We are now ready to compute the stress tensor, the plastic strain tensor, and the effective plastic strain.
The updated plastic strain tensor and effective plastic strain are stored as state variables.
We computed the stresses and updated the material state (the state variables) for our material model. Now, we turn our attention to the Jacobian computation. The Jacobian has other names in literature, such as tangent stiffness, tangent modulus, or tangent operator. In the stress update algorithm, we express the deviatoric and hydrostatic parts of the stress tensor as:
The Jacobian that we want to compute is the derivative of the Second Piola-Kirchhoff stress tensor with respect to the Green-Lagrange strain tensor. For our example material, we assume that strains are small. This means that we do not need to distinguish between various measures of stresses and strains, because they are indistinguishable in the small-strain limit. The derivative of stress with respect to strain is written as
If we use the equations for the deviatoric and hydrostatic stress and the definition of the trial stress, we can express the Jacobian in the following way:
Note that we replaced the increment of the plastic strain tensor by the total plastic strain tensor in the expression above. Their derivatives with respect to strain are the same, by virtue of the additive update of the plastic strains. Recall that our two modeling approaches require differently defined Jacobians. We see immediately how they are related. In the General stress-strain relation, the Jacobian is given by the full expression above. In the Inelastic residual strain, the Jacobian is given by one term in the expression, namely:
The term is the elastic Jacobian. For a purely elastic computation, the total Jacobian of the General stress-strain relation equals this quantity, while the Jacobian of the Inelastic residual strain in this case is zero.
If we use the flow rule and the chain rule for differentiation, we arrive at the following expression:
Notice that this expression depends on the plastic multiplier. This suggests that for the current material model, there is little benefit in choosing the Inelastic residual strain over the General stress-strain relation, because both approaches require a full stress update algorithm to compute the plastic multiplier. For other material models, such as creep models, the benefit would be greater. Using the governing scalar equation and the flow rule, we can compute the last derivative in the expression above.
In order to ensure rapid convergence of the global equation solver and ultimately reduce the simulation time, the computed Jacobian should be accurate. Well, what does accurate mean? Simply put, it means that the computed derivative must be consistent with the stress update algorithm that was used to compute stresses. That is, any assumptions or simplifications used in the stress update algorithm should be reflected in the computation of the Jacobian. A derivative based on the stress update algorithm is often called algorithmic or consistent.
In some situations, the Jacobian computation can be cumbersome. It is often possible in these situations to use an approximate Jacobian. Keep in mind that the accuracy of the solution is determined by the stress update algorithm. As long as the Jacobian is not too far off, the global equation solver will still converge to the correct solution, albeit at a lower rate of convergence.
In the sections above, we developed a stress update algorithm and outlined how to compute a Jacobian. Now, we will consider a special case for the hardening curve. We assume that the yield stress is a linear function of effective plastic strain. This is usually called linear hardening, and it is defined by a constant “plastic modulus” , which is the constant slope of the hardening curve. As it turns out, linear hardening means that the plastic strain increment can be solved on closed form:
In the example’s source code file, we have made use of this specialization into linear hardening.
Let’s consider an example problem of pulling a plate with a hole.
Dimensions, boundary conditions, and loads for a plate with a hole.
The problem has two symmetry planes and only one-quarter of the plate is modeled. We use the following material parameters:
The problem assumes plane stress. We can compare predictions of the implementation of our material model with the built-in counterpart. We expect differences only within the order of numerical round-off as long as the tolerance when solving the nonlinear equations is tight enough.
Computations of the effective von Mises stress in MPa of the external material implementation (left) and built-in material model (right).
Computations of the effective plastic strain for the external material implementation (left) and built-in material model (right).
There are actually more scenarios where you can employ the possibility to add external materials.
Consider a situation where you have a source code for a material model that has been verified in another context. You may have created it yourself or found it in a textbook or journal paper. In this case, it may be more efficient to use the external material functionality than casting it into a new form and enter it as set of ODEs. Even when the code is written in, for example, Fortran or C++, it is usually rather straightforward to wrap it into the C interface used by the external material.
A coded implementation may be computationally more efficient than using the User Defined or extra PDE options. The reason is that the detailed knowledge about the material law makes it possible to devise efficient stress updates, using, for example, local time stepping.
You may want to distribute your material model in a compiled form so that the end user cannot access the source code. As a matter of fact, the third-party product PolyUMod is implemented this way.
PolyUMod software is developed by Veryst Engineering LLC. COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by Veryst Engineering LLC.
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A trebuchet is a long-range weapon that uses a swinging arm to send a projectile toward a target. The machine is generally associated with hurling boulders at a castle wall to bring it down, but trebuchets have also been used to throw Greek fire and wreak all kinds of havoc. Trebuchets have appeared in several films and TV shows, such as The Return of the King (2003); Marco Polo (2014–2016); and even in Monty Python and the Holy Grail (1975), where a cow was catapulted from inside the castle walls toward an unsuspecting King Arthur!
One nonfictional and historically notable trebuchet is War Wolf (known to the English soldiers at the time as “Ludgar”). In 1304, on one of his campaigns to defeat Scotland, King Edward I besieged Stirling Castle and ordered his engineers to build a giant trebuchet. War Wolf was the largest trebuchet ever made and was rumored to send boulders of about 150 kilograms across a distance of over 200 meters.
A small-scale replica of War Wolf, a counterweight trebuchet that uses a boulder-holding sling at the end of a swinging arm. Image by Ron L. Toms. Licensed under CC BY 3.0, via Wikimedia Commons.
Large trebuchets of this type would typically feature a counterweight roughly ten times the weight of the projectile, which would put War Wolf’s counterweight in the neighborhood of 1.5 tons! The poor prospects of surviving an assault from War Wolf prompted the Scottish garrison inside the castle to offer their surrender. However, the king would not have it, as he was eager to try out his new trebuchet. He forced the Scots to remain inside the castle and restarted his siege. War Wolf proved its worth, and the rest is, as they say, history.
The working principle of a trebuchet is simple. The counterweight is raised and the trebuchet is cocked. When the trebuchet is fired, the counterweight drops, and the potential energy of the system is converted into a combination of kinetic and potential energy. The projectile undergoes a swinging motion and is released at some suitable position along its trajectory. This happens when one end of the sling slips off the tip of the swinging arm.
Here, we build a computational model of a basic trebuchet with the Multibody Dynamics Module and version 5.3 of the COMSOL Multiphysics® software.
Our model uses the following assumptions and physical dimensions:
A schematic of the counterweight trebuchet model.
As the projectile is swung around by the swinging arm, it describes a nontrivial motion of varying velocity. If the trebuchet is to be designed for maximum throwing distance, a question arises: At what point during its trajectory should the projectile be released? Elementary mechanics tells us that if we neglect air resistance and the height from the ground at which the projectile is released, the throwing distance s of the projectile (measured in the positive x direction) can be expressed as
where v_{0} and α are the velocity and angle at the time of projectile release, respectively, and g is the gravitational acceleration.
Thus, finding the maximum throwing distance is equivalent to finding the combination of v_{0} and α that maximizes s. Intuitively, you might think that the angle of release should be α = 45°. Let’s see if this holds true for the trebuchet model.
The animation below shows the motion of the trebuchet as it is fired. The quantity s is shown along the projectile trajectory, and it represents the throwing distance that would follow from releasing the projectile at a certain point on this trajectory.
In the results below, the throwing distance is plotted as a function of the release angle α. The maximum throwing distance is obtained if the projectile is released at α ≈ 38°. The plot reveals that deviations of the order of 5° from this optimum only affect the throwing distance by a few meters. In other words, as long as the release angle is roughly correct, the trebuchet will function as intended.
Now, let’s examine what happens if we modify the length of the sling by ±10% using a parametric sweep. The plot below shows that the maximum throwing distance that can be obtained is greatly affected by the length of the sling. So, if you are in the business of designing trebuchets for medieval kings, you should pay attention to this design parameter.
Using a parametric sweep, you could easily examine the effect of changing other physical lengths in the model (while keeping the counterweight at fixed height for consistency). Try for yourself by downloading the model file from our Application Gallery.
In this blog post, we demonstrated that the Multibody Dynamics Module can be used to build a simple model of a counterweight trebuchet. If you are interested in learning more about multibody dynamics modeling, check out these additional blog posts:
Wondering why War Wolf was also called Ludgar? Apparently, the French name Loup de Guerre (“wolf of war”) proved more than a mouthful for the English soldiers, so it was condensed into “Ludgar”.
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Two topics frequently pop up on the COMSOL Blog: 3D printing and metamaterials. Their potential applications, such as generating customized medical implants, printing houses, and being used for cloaking technology, could transform the world around us.
A 3D printer. Image by Jonathan Juursema — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
By combining these two technologies, we might be able to use direct-laser-writing (DLW) printing to fabricate complicated metamaterials, a process that could be challenging or impossible with other manufacturing techniques. Using this idea as inspiration, a team from the Karlsruhe Institute of Technology (Germany) and Université de Bourgogne Franche-Comté (France) investigated a metamaterial that displays the unique mechanical property of negative effective compressibility under stable and static conditions.
The researchers’ poroelastic metamaterial is a 3D manmade composite that experiences isotropic expansion when the hydrostatic pressure generated by the surrounding environment is increased. Most natural elastic materials react in the opposite way, reducing their volume when exposed to an increased hydrostatic pressure at a fixed temperature.
A sponge is a material that is affected by poroelastic phenomena.
So why does this metamaterial expand? To answer this, let’s take a look inside the metamaterial, which consists of a single ordinary constituent solid. Within the material are hollow 3D crosses with concealed internal volumes that contain air at a constant pressure. Each cross also has circular membranes attached to its ends.
These membranes warp inward or outward when the surrounding pressure is different than the pressure within the crosses. The bars asymmetrically connected to the membranes translate this warping into a cross rotation. If the hydrostatic pressure outside is greater than the pressure inside, then the individual rotations translate into an isotropic expansion of the structure, causing a negative effective compressibility.
A unit cell at zero pressure (left) and an elevated pressure (right), depicting the principle of negative compressibility. Image courtesy Jingyuan Qu and Muamer Kadic.
While this negative compressibility may appear to violate the laws of physics, the effective volume increase corresponds with an unseen volume decrease within the material. This ensures that the structure is stable.
To examine the detailed structure of the innovative metamaterial, the researchers turned to the COMSOL Multiphysics® software. When asked about the benefits of this approach, Jingyuan Qu — a member of the research team — noted how easy it was to implement.
The model of the metamaterial is a single unit cell. To see what happens when there is a difference between the pressure inside and outside the material, a pressure increase is implemented as a normal force on all of the outer surfaces of the model. Further, the model is simulated under periodic boundary conditions, enabling the researchers to successfully find the effective material parameters.
Note that built-in periodic boundary conditions are available in the Structural Mechanics and MEMS modules.
For their research, the team performed two main numerical experiments:
In their experiments, the team used the equations for standard linear elasticity:
Now, let’s take a look at the second numerical experiment.
To mimic an infinite material case, periodic conditions are implemented so that each side of the unit cell must contract or expand isotropically. First, selections for each side of the structure are created and named according to the directions x+, x-, y+, y-, z+, and z-. Then, probe variables were created, giving the average displacement on the “minus” sides (dispx, dispy, dispz), shown in the second screenshot below.
Examples for the x direction case, displaying the process of selecting the boundaries that connect to the next unit cell, shown for one of the six planes (top) and the boundary probe settings (bottom). Images courtesy Jingyuan Qu and Muamer Kadic.
Next, the probe variables are used as boundary conditions on both sides (prescribed displacement). That is, on the ‘x-’ boundaries, the x direction displacement is set to dispx, while on the ‘x+’ boundaries, it is set to -dispx. Similar boundary conditions are then set on the other periodic cuts. The idea is that the displacement dispx, still unknown, becomes part of the solution. Since the conjugate reaction force to the prescribed displacements must be zero, the structure will expand or contract in such a way that there is no net force.
Prescribing the probed displacement. Image courtesy Jingyuan Qu and Muamer Kadic.
Moving on, outer pressure is also applied. After selecting the outer boundaries of the geometry and using a high angular tolerance, the model shows that the inner boundaries in the concealed volumes are not selected, as seen below.
The outer boundary settings. Image courtesy Jingyuan Qu and Muamer Kadic.
The hydrostatic load is then applied as a boundary load with pressure (P).
The hydrostatic pressure is implemented as a normal force acting on all outer boundaries. Image courtesy Jingyuan Qu and Muamer Kadic.
The resulting structure of the poroelastic metamaterial at different angles. Images courtesy Jingyuan Qu.
As a point of comparison, the researchers also examined an ordinary porous structure and a cube of a continuous isotropic material. When exposed to an increased hydrostatic pressure, both of these structures shrink in volume. Under the same conditions, the porous metamaterial expands — highlighting its negative effective compressibility.
Thanks to their extensive research, the team was able to capture the metamaterial’s behavior, improve its design, and use this information to move on to the fabrication stage. While fabricating this material may not be possible with conventional machining techniques, 3D printing serves as an alternative option for creating negative compressibility metamaterials. 3D printers can form this metamaterial by using ordinary materials that shrink under hydrostatic pressure.
Qu notes that after it is realized, the metamaterial may find uses in high-pressure applications due to its ability to maintain a constant effective volume, even when subjected to high-pressure environments.
Under plane strain conditions, no expansion in the out-of-plane direction is allowed. There are usually stresses in that direction caused by the coupling to the in-plane strain through a nonzero Poisson’s ratio. On the other hand, when a thin sheet is studied, the plane stress assumption is more useful. In this case, the material is free to contract or expand in the out-of-plane direction and the transverse stress is zero.
If the structure is long in the transverse direction when compared with its in-plane size, but is still not restrained in the transverse direction, then neither of these assumptions is good. This is where a generalized plane strain condition becomes useful.
A possible generalization of the plane strain formulation is to assume that the strains are independent of the out-of-plane coordinate. In the COMSOL Multiphysics® software, this generalization can be implemented using a 2D geometry of the cross section and the Solid Mechanics interface, in which the plane strain formulation is a default option.
The strain tensor components are assumed to be functions of only the in-plane coordinates x and y (and possibly time):
(1)
Under the small-strain assumption, the strain tensor components are related to the displacement field as:
(2)
The above equations have the following 3D solution:
(3)
where a, b, and c are constant coefficients.
The corresponding out-of-plane strains are:
(4)
This strain state differs from the standard plane strain assumption only by the fact that the normal out-of-plane strain is nonzero and can vary linearly over the cross section. At the cross section z = 0, the deformation is in-plane and fully characterized by the in-plane displacement components u(x,y) and v(x,y).
The coefficients a, b, and c in the expression for the normal out-of-plane strain can be introduced as extra degrees of freedom (DOF) that are constant throughout the model (global variables). The extra strain contribution can be incorporated using the External Strain feature available in the Solid Mechanics interface.
A generalized strain formulation is important is when analyzing stress-optical effects, such as birefringence in waveguides composed of several layers of different materials (e.g., silicon-on-insulator waveguides). This stress-optical effects tutorial model shows such a case.
To illustrate the efficiency of this approach, let’s consider a simple beam-like structure composed of two layers with square cross sections of 1 cm. The layers are made of materials with significantly different elastic and thermal properties: aluminum and nylon. The data is taken from the built-in Material Library in COMSOL Multiphysics. The length in the out-of-plane z direction is L = 20 cm. The structure is assumed to be manufactured at an elevated temperature. Due to the mismatch in the thermal expansion properties of the materials, a residual thermal stress builds up in the structure when it has cooled down to the operating temperature. This makes the structure bend slightly in the out-of-plane direction.
The following figure shows plots of the total displacement together with the deformation for a full 3D model and a 2D generalized plane strain condition:
The 2D solution computed for u(x,y) and v(x,y) within the cross section has been extruded in the out-of-plane z direction using the analytical solution for the corresponding 3D displacement field given above.
The 3D solution requires around 32,000 DOF, while the 2D solution only needs around 250 DOF.
The following plots show the variation of the out-of-plane strain and stress along one of the edges.
Strain (left) and stress (right) along the z-axis.
Around 80% of the true 3D structure has stress and strain fields similar to those predicted by the generalized plane strain theory. Only near the free ends, where the stress tends toward zero, does the strain field start to deviate from the linear distribution within the cross section.
One way to incorporate the changes needed for the generalized plane strain approximation is to start with a 2D component and the Solid Mechanics interface and then add the following nodes in the Model Builder tree:
The Model Builder tree, showing the nodes needed to implement a generalized plane strain condition.
In addition to the standard settings for a 2D problem, you must perform the following steps. First, in the Global Equations node, add the a, b, and c coefficients as DOF. Note that you do not set up any equations for those variables here. Thus, all input fields other than the variable names are kept at their default values.
The Global Equations node, showing the a, b, and c coefficients.
In the Variables node, define the out-of-plane normal strain component eZ in terms of a, b, and c.
The Variables node, showing the expression for the variable eZ.
Next, incorporate the extra strain component into the stress-strain relation in the External Strain node. Note that any expression you enter in this node is subtracted from the total strains before the elastic stresses are computed from strains using Hooke’s law. Usually, this node can be used to incorporate inelastic effects; for example, strains caused by various electromechanical multiphysics effects. Here, we use it simply as a mechanism to inject an extra strain component that is zero by default in the plane strain formulation.
The External Strain node, showing the extra strain component.
Lastly, in the Weak Contribution node, include the extra virtual work done by the out-of-plane stress. This sets up equations (in the weak form) to determine a, b, and c. Here, solid.d
is the thickness in the z direction, as defined in the Solid Mechanics interface.
The Weak Contribution node, showing the weak expression.
You can also skip the third and fourth steps above and insert the strain variable eZ directly into the equations of the Linear Elastic Material node. To do this, make sure Equation View is enabled.
The settings for Equation View.
This way, the new strain in the z direction is directly part of the material model and goes into the weak expression that is already generated in the Linear Elastic Material node.
We have shown how to use the functionality available in COMSOL Multiphysics to model elongated structures that are free to extend in the out-of-plane direction. The use of the 2D generalized plane strain approximation allows us to reduce the computation effort significantly while reproducing the possible out-of-plane bending of the structure — a 3D effect that can be important in applications such as piezoelectric devices and optical waveguides. It is also possible to incorporate out-of-plane shearing of the structure, which can be important in some piezoelectric applications.
The rotor system in this example is a simple rotor with a uniform cross section throughout its length. It is supported at both ends by bearings and there are three mounted components called disks at different locations of the rotor.
You can model this rotor using the Beam Rotor interface in the COMSOL Multiphysics® software. The inertial properties and offset of the rotor components are modeled with the Disk node. The bearing support is modeled by an equivalent stiffness-based approach via the Journal Bearing node provided in the Beam Rotor interface.
For more information about the geometric properties and model setup, check out the references in the model documentation.
Geometry of the beam rotor example.
Two types of analyses are commonly used to study rotor vibration characteristics: eigenfrequency and time-domain analyses. As mentioned in a previous blog post, critical speeds of the rotor strongly depend on the rotor’s angular speed. Therefore, while performing the eigenfrequency analysis, you need to consider the variation in the rotor speed to get the correct critical speeds. A time-domain analysis is performed when you want to look at the system response under time-varying excitation.
Now, let’s look at what type of information each analysis provides as well as the steps involved to perform these analyses.
Eigenfrequency analysis is used to determine the natural frequencies of a system. In a rotordynamics scenario, this analysis can be used in two different ways.
First, for the operating speed of a system that is not fixed, you can perform an eigenfrequency analysis of the system for the range of operating speeds and choose the one that is furthest from the critical speed of the system and meets other design considerations. If you cannot find a suitable operating speed for the current system, you might need to make certain design modifications in the system to get a stable operating speed that meets all of the requirements.
In the second type of analysis, the operating speed of the system is fixed. In such a case, you need to perform an eigenfrequency analysis at the given operating speed to check that any of the natural frequencies of the system are not close to the operating speed. If any of the natural frequencies fall closer to the operating speed, design modifications are a must.
The design modifications in the rotor system require an understanding of what kind of modifications will produce the desired effect and at what cost. This is where simulating simple systems to understand the effect of design modifications is very helpful. Simulation can provide guidelines for design modifications, thus reducing the number of iterations in the design process.
Consider the first case, in which the operating speed of the system is not fixed, to understand the analysis steps. In this case, you need to perform a parametric eigenfrequency analysis for the angular speed of the rotor. This requires two steps in the Study node: Parametric Sweep and Step 1: Eigenfrequency, shown below on the left. Settings for the Parametric Sweep node for a sweep over a parameter Ow representing the angular speed of the rotor are shown below in the center. This parameter is used as an input in the Rotor Speed section of the Beam Rotor node settings, shown below on the right.
Steps in the Study node (left), settings for the parametric sweep (middle), and rotor speed input (right).
After performing the analysis, you get a whirl plot of the rotor as the default, shown below. The whirl plot shows the whirling orbit and the deformed shape of the rotor for the given rpm and natural frequency combinations.
Whirl plot of the rotor.
The deformed shape of the rotor also gives you an idea of how strongly the natural frequency will depend on the angular speed of the rotor. If the disks move away from the rotation axis without significant tilting, then the split in the frequency in the backward whirl (opposite to the spin) and forward whirl (same direction as the spin) is not significant. Alternatively, if the disks do not move significantly far from the rotation axis and rather have significant tilting, then the split in the frequency of the backward and forward whirl is noticeable.
To understand this concept in depth, you can plot the variation of the natural frequency for different modes against the angular speed of the rotor, which is often called a Campbell diagram. The Campbell plot for the simply supported rotor example is shown below. You can see the strong divergence of the eigenfrequencies with rotor speed for certain modes; whereas for others, particularly the modes with low natural frequencies, the divergence is not significant. If you look at the mode shapes corresponding to these frequencies, they confirm the behavior previously discussed. Critical speeds of the rotor can be obtained from the Campbell plot by looking at the intersection of the natural frequency vs. angular speed curve with ω = Ω curve. These are the speeds near which a rotor should not be operated, unless sufficiently damped.
Campbell plot of the simply supported rotor system.
The damping in the respective modes can be accessed by plotting the logarithmic decrement with the angular speed of the rotor. The logarithmic decrement is defined as
where A(t) is a time-varying response and ω is the complex eigenfrequency of the system. T is the time period given by .
Logarithmic decrement for different bending modes in the simply supported rotor system.
In the plot above, you can see a logarithmic decrement variation for the different bending modes with the angular speed for the simply supported rotor. The notation ‘b’ and ‘f’ is used for the backward and forward whirl modes, respectively. A logarithmic decrement of zero means that the system is undamped, a negative value indicates an unstable system, and a positive value indicates a stable system.
You can also note the pattern change for some of the curves. The reason is that the modal data is arranged in increasing order of the natural frequency. But we know that the rotor’s natural frequencies decrease in the backward whirl modes and increase in the forward whirl modes. Due to this, there is crossover of the natural frequencies between the higher backward whirl and lower forward whirl modes beyond a certain angular speed. This upsets the initial order of the modes, resulting in switching of the patterns across the crossover points.
Eigenfrequency analysis gives the characteristics of the rotor system operating at steady state. However, before and after reaching the steady state, during the run-up and run-down, the angular speed of the rotor varies with time. In certain cases, the operating speed might be above the first few natural frequencies of the rotor. Therefore, during the run-up and run-down, the rotor will cross over the corresponding critical speeds. Also, there could be nonharmonic time-varying external excitation acting on the rotor. In such cases, the rotor response cannot be completely determined by an eigenfrequency or frequency-domain analysis. Rather, you need a time-dependent simulation to study the response of the system.
You can also perform a time-dependent analysis of the rotor at different angular speeds by performing a parametric sweep to see how the angular speed governs the response. An obvious extension of such an analysis is to evaluate the frequency spectrum of the time-dependent response of the rotor for all of the angular speeds and analyze what combinations of the angular speed and frequency result in a high amplitude response. A waterfall plot shows the response amplitude vs. angular speed and frequency and gives the distribution of the modal participation in the response at different speeds. Such an analysis can be set up using the three steps in the study node, as shown below.
Steps for a waterfall plot analysis. The Parametric Sweep study step is used to sweep the angular speed, a Time Dependent study step is used to perform a time-dependent analysis corresponding to each angular speed in the parametric sweep, and a Time to Frequency FFT study step takes the fast Fourier transform of the time-dependent data to convert into the frequency spectrum.
In the eigenfrequency analysis, bearings are modeled using constant stiffness and damping coefficients. However, in reality, these coefficients are strongly dependent on the journal motion. To highlight the effect of the nonlinearity for the time-domain analysis, a plain journal bearing model is used instead of the constant bearing coefficients. A plain journal bearing model is based on the analytical solution of the Reynolds equation for a short bearing approximation. The system in this case is self-excited due to the eccentric mounting modeled as a disk. To simplify the system, only the second disk is considered with small eccentricity in the local y direction.
The waterfall plot of the z-component of the displacement is shown in the figure below. You can observe three peaks clearly in the spectrum. The third peak, which falls along the ω = Ω curve, corresponds to a 1X synchronous whirl. This is in response to the centrifugal force due to eccentricity changing its direction with the rotation of the shaft. Other peaks correspond to the orbiting of the rotor due to the complex rotor bearing interaction. The reason is that the forces from the pressure distribution around the journal in the bearing have a cross-coupling effect with the journal motion. In other words, the motion of the journal in one of the lateral directions induces a component of the force in the lateral direction perpendicular to it. The effect of this phenomenon is a net force acting on the rotor in the direction of the forward whirl. This causes the subsynchronous orbiting of the rotor.
A waterfall plot shows the response amplitude vs. the angular speed and frequency of the rotor.
The orbit of the different locations along the length of the rotor at 30,000 rpm is shown below. The orbit curve changes its color from green to red with time. You can see that after the initial transient phase, the rotor undergoes a forward circular whirl in the steady state. Also, the second bending mode has the highest participation in the response.
Orbit of the rotor at different locations. The plot changes from green to red with time.
The time variation of the z-direction displacement of a point on the rotor at 30,000 rpm is shown below. Apart from the high-frequency variation, there is also a low-frequency component that envelops the response, but gets damped out with time.
Time variation of the z-displacement.
With this tutorial model, we have demonstrated the approach to set up different analyses in a rotor system, as well as how to plot and analyze the simulation results. Ready to give this tutorial a try? Simply click on the button below to access the MPH-file via the Application Gallery or open it via the Application Library in the COMSOL® software.
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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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 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.