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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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.
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. An 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 via the button below.
Multibody systems involve multiple rigid or flexible bodies that are connected to each other and may be subjected to large displacements. Such systems have applications in the aerospace industry and robotics and are used to create machines such as helicopter swashplate mechanisms and reciprocating engines.
A helicopter and helicopter swashplate, an example of a multibody system. Left image by Riley Kaminer — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
When designing multibody systems, engineers need to detect and avoid defects, thereby preventing future issues and optimizing their designs. They can achieve this by performing simulation analyses with the Multibody Dynamics Module, an add-on product to COMSOL Multiphysics and the Structural Mechanics Module. To confirm the reliability of these simulations, engineers can benchmark their results against existing data.
Here, we’ll show you a benchmark example of a four-bar mechanism, which is used in bicycles, for example. We analyze the mechanism’s dynamic behavior with the Multibody Dynamics Module and check how well the results compare to existing research. With this benchmark model, we hope to show that the results generated by the Multibody Dynamics Module can be relied on for these types of analyses.
Looking at the geometry of the planar four-bar mechanism example, we see that it consists of three links (labeled Link 1–3). The mechanism has a defect so modeling these links as rigid causes the mechanism to lock. Instead, we model them as flexible parts via the Linear Elastic Material node where Link 1, the mechanism’s left crank, has an angular velocity of 1 rad/s.
Connecting these links are four hinge joints (labeled Hinge joint 1–4), which are located at points A–D in the image below. Hinge joints 2 and 3 (located at points B and C, respectively) connect Links 1 and 3 to Link 2. Meanwhile, Hinge joints 1 and 4 (located at points A and D, respectively) connect Links 1 and 3 to a fixed constraint.
The geometry of the four-bar mechanism model. Here, Links 1 and 3 are 0.12 meters long, while Link 2 is 0.24 meters long. All of the links have a circular cross section with a diameter of 5 millimeters.
Without a defect, the mechanism model would only move within the plane, as the four hinge joints would have an axis of rotation that is perpendicular to the mechanism’s plane. However, in this model there is an assembly defect in Hinge joint 3 (point C). This alters the joint’s angle of rotation, skewing it 5° from the normal to the plane. As a result, the mechanism experiences out-of-plane motion.
Using this model, we can study the displacement caused by the four-bar mechanism’s defect and validate our results against an existing study. First, we plot the mechanism’s displacement and trajectories for Points B (a hinge joint without a defect) and C (the hinge joint with a defect) after 10 seconds. By doing so, we see how the assembly’s defect affects the trajectories in a four-bar mechanism.
The four-bar mechanism model after 10 seconds, showing the trajectories of points B and C during those 10 seconds. The visualization is enhanced by scaling the out-of-plane displacement by a factor of 20.
Next, we compare the simulation results for out-of-plane displacement to an existing study (Ref. 1 in the model documentation). In our comparison, we study the displacement’s y-component at points B and C.
The comparison shows that the computed results are in very good agreement with the results from the reference. Further, both the computed and existing results confirm that the out-of-plane displacement vanishes when there is no defect in the joint.
Comparison of the out-of-plane displacements for points B (left) and C (right) between the COMSOL Multiphysics model and reference.
With this benchmark model, we demonstrate that we can confidently use the Multibody Dynamics Module to generate verifiable results.
Before the invention of gears, people used wheels to transfer the rotation of one shaft to another with the help of friction. The major drawback in using these frictional wheels was the slippage beyond a certain torque value, as the maximum torque that could be transmitted was limited by the frictional torque. To overcome this limitation, people began using toothed wheels, more commonly known nowadays as cogwheels or gears.
Gear pair created using the Parts Library in the Multibody Dynamics Module.
The main purpose behind gears is to avoid slippage. This is why the teeth of one gear are inserted between the teeth of the mating gear, a process referred to as gear meshing. Compared to the gear’s core region, the gear’s mesh region is more flexible. Hence, accounting for the stiffness of the gear mesh is important when trying to accurately capture the dynamics and vibrations in the system.
Gear mesh stiffness depends on several different parameters and, most importantly, it varies with the gear rotation. This makes the problem nonlinear, and the continuously varying gear mesh stiffness gives rise to vibrations in the system. These vibrations in different parts of the transmission system result in noise radiation. Therefore, it is crucial to evaluate gear mesh stiffness and include it in the gear model.
To examine gear mesh stiffness, we assume that the gears are elastic bodies and model the contact between them. We then perform a stationary parametric analysis to determine the mesh stiffness of the gears for different positions in a mesh cycle. A mesh cycle is defined as the amount of gear rotation after which the next tooth takes the position of the first one.
Now, to understand this process, let’s take an example in which two gears, both made of steel, have the following properties:
Properties | Pinion | Wheel | |
---|---|---|---|
Number of teeth | n | 20 | 30 |
Pitch diameter | d_{p} | 50 mm | 75 mm |
Pressure angle | a | 25° | 25° |
Gear width | w_{g} | 10 mm | 10 mm |
In this example, both gears are hinged at their respective centers. Using the penalty contact approach, we model the contact between the teeth of the two gears. The boundaries of the two gears in contact with each other are shown below. For more details about how to set up this model, you can check out the tutorial titled: Vibrations in a Compound Gear Train.
The contact pair boundaries (left) and the finite element mesh (right) in the gear pair.
Because the mesh stiffness changes for the gears’ different positions in the mesh cycle, we rotate both gears parametrically to compute the variation of gear mesh stiffness. The rotation of the pinion (θ_{p}) about the out-of-plane axis is prescribed in such a way that the pinion rotates for two mesh cycles. The rotation of the wheel (θ_{w}) about the out-of-plane axis is defined as the following:
where g_{r} is the gear ratio with a value of 1.5 and θ_{t} is the twist with a value of 0.5°.
The wheel is given a twist, θ_{t}, and the required twisting moment, T, is evaluated on the hinge joint. Hence, the torsional stiffness of the gear pair is prescribed as:
Once we know the torsional stiffness, we can define the stiffness along the line of action as:
where d_{pw} is the pitch diameter of the wheel and α is the pressure angle.
The von Mises stress distribution in the gear pair for different positions in a mesh cycle. This shows high stress levels at the contact points along the line of action.
The figure below shows the variation of computed gear mesh stiffness with the rotation of the pinion for two mesh cycles. We can see that the gear mesh stiffness is periodic in each mesh cycle as well as across multiple mesh cycles, increasing in the beginning and then later decreasing. This happens due to the changing contact ratio. In the beginning of a mesh cycle, the contact ratio increases from 1 to 2, but then drops back down to 1.
The variation of gear mesh stiffness with the pinion rotation.
In the previous section, we saw that gear mesh stiffness varies with the gear’s position in the mesh cycle. It also depends on several other parameters, some of which are listed here:
Let’s focus on investigating the effect of gear tooth parameters on the mesh stiffness. While doing so, we keep the same geometric and material properties that were given in the first table.
To look at the effect of the number of teeth or module on gear mesh stiffness, we consider different values for the number of teeth on the pinion.
We then compute the number of teeth on the wheel by using the gear ratio, which is set to 1.5. The other two gear tooth parameters are fixed to the following values:
Gear meshes for three different values of the number of teeth (n_{p} = 20, 28, 36).
The von Mises stress distribution in the gear pair for different values of n_{p}.
The variation of gear mesh stiffness with pinion rotation for three different values of the number of teeth (n_{p} = 20, 28, 36). The stiffness is comparatively higher and smoother for a greater number of teeth or for a smaller module.
To understand the effect of pressure angle on gear mesh stiffness, we look at three different values of the pressure angle.
The other two gear tooth parameters are fixed to the following values:
Gear meshes for three different values of the pressure angle (α = 20°, 25°, 35°).
The variation of gear mesh stiffness with pinion rotation for three different values of the pressure angle (α = 20°, 25°, 35°). The stiffness increases with a larger pressure angle.
After investigating the effects of module and pressure angle, we now examine the effect of different addendum values on gear mesh stiffness.
The other two gear tooth parameters are fixed to the following values:
Gear meshes for three different values of the addendum-to-pitch-diameter ratio (adr = 0.6, 0.75, 0.9).
The variation of gear mesh stiffness with pinion rotation for three different values of the addendum-to-pitch-diameter ratio (adr = 0.6, 0.75, 0.9). The stiffness is comparatively higher for higher values of addendum, however it also has more fluctuations. This may lead to higher vibration levels in the transmission system.
After evaluating gear mesh stiffness using the static contact analysis, the next step is to include the stiffness in the gear model so that we can perform an NVH analysis of the full transmission system.
The gear mesh stiffness and damping added along the line of action between the two gears.
In the multibody dynamics analysis, we use the evaluated gear mesh stiffness in the Gear Elasticity node under the Gear Pair node. In this analysis, we write gear mesh stiffness as a function of gear rotation. By default, gear mesh stiffness is assumed periodic in a mesh cycle. However, it is also possible to assume that it is periodic in a full revolution.
In order to dampen the vibrations, we can add gear mesh damping in the Gear Elasticity node. This can be entered either as a function of mesh stiffness or explicitly. The latter technique works well when we have the gear-mesh stiffness variation available. If we don’t have the exact gear-mesh stiffness variation, we can use the gear tooth stiffness for the wheel as well as the pinion. The tooth stiffness can simply be evaluated by applying a load on the gear tooth and measuring the deflection. The gear tooth stiffness is also the function of a mesh cycle, although as an approximation, and we can enter it as a constant average value.
Finding the overall gear mesh stiffness also requires determining the contact ratio. In simple words, the contact ratio can be defined as a measure of the average number of teeth in contact during the period in which a tooth comes and goes out of contact with the mating gear. To show how different values of the contact ratio affect the stiffness, let’s examine a few cases.
In the first case, only a single pair of teeth is in contact for all positions in the mesh cycle. The typical variation of the gear tooth stiffness is shown below.
The typical variation of the gear tooth stiffness for the pair of teeth in contact.
In this case, two pairs of teeth are in contact for all positions in the mesh cycle. We can see from the following image that except for a phase difference, the second pair of teeth has the same stiffness as that of the first pair. The total stiffness of the gear mesh is the summation of individual tooth stiffness.
The typical variation of the gear tooth stiffness for the first and second pair of teeth when the contact ratio equals 2.
In the third case, the pairs of teeth that are in contact change for different positions in the mesh cycle. For certain positions, there is only one pair of teeth in contact, whereas in other positions, there are two pairs of teeth in contact. The stiffness of the second pair of teeth goes to zero when it loses contact for certain positions in the mesh cycle. This results in large fluctuations in the overall gear mesh stiffness, which leads to vibrations in the system.
The typical variation of the gear tooth stiffness for the first and second pair of teeth when the contact ratio is between 1 and 2.
To demonstrate the effect of gear mesh stiffness on gear dynamics, let’s use a pair of helical gears as an example. We first perform a transient study to compare a rigid gear mesh, gear mesh with a constant stiffness, and a gear mesh with a varying stiffness. We then analyze the effects of different types of gear mesh on the angular velocity of the driven gear as well as on the contact force. More details about this tutorial model can be found in the Application Gallery.
The figure below shows the variation of the driven gear’s angular velocity for the constant angular velocity of the driver gear. For a rigid gear mesh, the driven gear rotates at a constant speed. When the gear mesh stiffness is constant, the driven gear initially fluctuates before settling down to a constant speed. The gear mesh that has a varying stiffness continues to fluctuate about the mean value, giving rise to the vibrations.
Driven gear angular velocity for different types of gear meshes.
We can observe a similar trend in the contact forces. The rigid and constant-stiffness gear mesh eventually begin to maintain a constant contact force, but the varying-stiffness gear mesh causes the contact force to fluctuate about the mean value. The contact force variation is periodic with respect to the mesh cycle, and the contact force varies from about 150 N to 450 N, with a mean value of 250 N. This large variation in the contact force within a mesh cycle rotation causes vibrations in other parts of the system. This may lead to noise radiation in the surrounding area.
Variation of the contact force with gear rotation for different types of gear meshes.
The variation of gear mesh stiffness, which depends on several geometric and material parameters, plays an important role in the NVH analysis of a transmission system. With COMSOL Multiphysics and the Multibody Dynamics Module, we can calculate its variation by combining a contact analysis with the parameterized gears in the Parts Library. We can then use the computed gear mesh stiffness in the multibody dynamics model to accurately capture the dynamics of gears working together with the other parts of the transmission system.
Stay tuned for the next blog post in our Gear Modeling series, where we’ll show you how to simulate gearbox noise and vibrations generated due to varying gear mesh stiffness. In the meantime, we encourage you to browse the additional resources below.
As a refresher, let’s begin by reviewing some of the key concepts behind modeling gears in COMSOL Multiphysics. A gear is defined in a Gear node as a rigid body with six degrees of freedom in the form of translations and rotations at the center of rotation. It is used in a Gear Pair node in the model tree in order to connect with another gear. Here, you can specify a finite stiffness for the gear mesh or gear tooth, either for individual gears or for the pair. A mathematical formulation is used to describe the connection between two gears, without any need for a defined, realistic gear geometry to detect the contact between the two gears. Therefore, you can represent a gear with either a realistic gear geometry or any similar geometry of a disc.
It is possible to compute the inertial properties of a gear from the geometry using its calculated mass density, or you can directly enter the properties in the form of mass and moment of inertia in the node’s edit fields. You can also apply external forces and moments on the gear as well as constrain certain degrees of freedom of a gear. For instance, when modeling torsional vibrations, all of the degrees of freedom except the axial rotation can be constrained.
COMSOL Multiphysics offers a number of standard gear types, each with its own merit and applications. As mentioned above, the gear is an abstract object, but if you want to add a realistic geometry for visualization, you can access the Part Libraries, where you can find various types of gears and racks.
In the following images, you can see the various types of gears and racks available and the geometrical parameters needed for their mathematical descriptions.
A Spur Gear (left) and Helical Gear (right) with their external gear mesh.
A Spur Gear (left) and Helical Gear (right) with their internal gear mesh.
A Bevel Gear (left) and Worm Gear (right).
A Spur Rack (left) and Helical Rack (right).
The inputs required to model each gear type are shown in the respective figures. They are as follows:
After selecting the appropriate gear type, you can then define the parameters controlling the size and shape of the gear teeth. As an example, these parameters are required to define a helical gear:
A screenshot showing the settings window for a helical gear. Various inputs required to model a helical gear, including gear properties, gear axis, center of rotation, and density are shown.
The next step is to define the position and orientation of the gear. The gear position is defined in terms of the center of rotation. This is the point at which the degrees of freedom are created and the rotation is interpreted. The forces and moments acting on the gear due to meshing with other gears are also interpreted about this point. By default, the center of rotation is set to the center of mass of the gear, but there are other ways to define it explicitly as well.
The gear orientation is specified in terms of the gear axis, which is the axis of rotation passing through the center of rotation. The gear axis is used when creating the gear local coordinate system. Also interpreted about this axis is the gear rotation, a degree of freedom in the Gear Pair node.
You can mount gears in one of two ways: on a flexible or a rigid shaft. These devices can be mounted either rigidly or with a finite stiffness using a fixed joint. Joints are the features used to connect two components by allowing certain relative motion between them.
When there is no clearance between the gear and the shaft in the geometry, the objects can be either in an assembly state or a union state. For a flexible shaft, gears are by default rigidly mounted on the shaft if both the gear and shaft are in a union state.
It is not necessary to model a shaft in order to mount gears, as the devices can be mounted directly to the ‘ground’ either rigidly or with a finite stiffness using a hinge joint. The prescribed displacement/rotation subnode of a gear can also be used for this purpose.
Note that it is also possible to support shafts on:
This can be done using hinge joints, which can be rigid or have a finite stiffness.
Figure showing gears with an actual geometry as well as those modeled through equivalent discs. Different mounting methods for gears and shafts are also depicted.
In order to connect the different types of gears that you have defined in your model, you can use a Gear Pair node. This node can connect spur, helical, and bevel gears. You can also use Worm and Wheel as well as Rack and Pinion nodes for their specific cases. These nodes connect two gears in such a way that there is no relative motion along the line of action at the contact point. The remaining displacements and rotations of the two gears are independent of each other.
Each Gear Pair node adds two degrees of freedom:
The following constraints are added by the Gear Pair node in order to connect two gears:
For a line contact model, one more constraint is added to restrict the relative rotation about a line joining the two gear centers. If friction is included, frictional forces are obtained using the contact force, which is computed as the reaction force of the contact point constraint. These frictional forces are then applied on both gears in a plane perpendicular to the line of action.
In a Gear Pair node, you can select any two gears defined in the model. But in order to achieve proper tooth meshing, a set of gears must fulfill the following compatibility criteria:
All these checks are automatically performed and an error message is issued during equation compilation if the two selected gears are not compatible.
Examples of incompatible gear mesh. In the figure on the left, the gears have different modules. In the figure on the right, the gears have different pressure angles.
A coordinate system for each gear is defined using the gear axis and center of rotation of both gears. The first axis of the coordinate system triad is the gear axis itself. The second axis is the direction pointing from the center of rotation to the contact point. The third axis is normal to the plane containing the first two axes. This coordinate system is attached to the gear and varies with the changes in gear orientation. Note, however, that it does not rotate with the gear rotation about its own axis.
A schematic showing coordinate systems and other parameters for both gears connected by a gear pair.
These quantities are illustrated in the above figure of a gear pair:
The gear tooth coordinate system is defined for both gears by rotating the gear coordinate system with the tooth angle matrix. This matrix is constructed using the helix angle and the cone angle.
The line of action, meanwhile, is defined as the normal direction of the gear tooth surface at the contact point on the pitch circle. This is the direction along which the forces are transferred from one gear to another. It is defined by rotating the third axis of the gear tooth coordinate system (gear tangent) about the first axis of the gear tooth coordinate system with the pressure angle (α). Based on the direction of the driver gear, the gear tangent can be rotated either clockwise or counterclockwise.
Two figures depicting the line of action and the direction of rotation of the driver gear. The line of action is defined due to the fact that the driver gear and tangent rotate in the clockwise direction (left) and counterclockwise direction (right).
The contact between the two gears is modeled through analytically founded equations. These are independent of the finite element mesh and thus much faster and more robust compared to mesh-based contact methods. To compute contact forces and moments, you can choose one of two methods:
The point of contact on each gear is defined via the center of rotation, displacement vector at the center of rotation, contact point offset from the gear center, pitch radius, and cone angle. Based on the orientation of both gears, different gear pairs can be classified into one of two configurations:
For a parallel or intersecting configuration, the contact point offset from the pinion center is the input and the contact point offset from the wheel center is automatically computed. The contact model can be selected as either:
For a configuration that is neither parallel nor intersecting, the contact point offset from the pinion, as well as the wheel center, is automatically computed. The reason for this is that there is always a point contact and the contact point can be uniquely determined.
From left to right: Thin gears (point contact model), thick gears (line contact model), and thick gears with an axial offset.
Now that we’ve explored gears in further detail and how to connect them, let’s look at various examples of gear pairs classified based on their configurations. You can use many gear pairs together in order to model complex parallel and planetary gear trains.
Some examples of the parallel axis configuration are as follows:
Bevel gears, meanwhile, offer an example of an intersecting axis configuration.
Set of spur gears and parallel helical gears with an external gear meshƒ.
Set of spur gears, one with an internal gear mesh and the other with an external gear mesh, as well as a set of bevel gears.
Some examples of a crossed (neither parallel nor intersecting) axis configuration are as follows:
Set of crossed helical gears with an external gear mesh and the worm and wheel.
Rack and pinion with a straight gear mesh.
When it comes to modeling gears, there are many important elements to consider to optimize your simulation results. As we’ve demonstrated here today, the new features and functionality for gear modeling in COMSOL Multiphysics allow you to address such elements, providing you with more useful insight into how to improve your gear design.
In the next blog post in our Gear Modeling series, we’ll discuss how you can use advanced features on your gear pairs (i.e., gear mesh elasticity, backlash, transmission error, and friction) in order to perform simulations requiring greater fidelity. We’ll also show you how these parameters affect the dynamics of your system. Stay tuned!
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In principle, we can analyze mechanical devices with gears by explicitly including the contact interactions between gears as part of the simulation, but this method is computationally time-consuming when performing a multibody dynamics analysis. Instead, we can implement a mathematical formulation to model the contact interactions between the gears.
With this formulation, we can include a realistic gear geometry, which provides accurate inertial properties when used in transient and frequency-domain studies. Realistic gear geometries from the Part Library can also be used to evaluate gear mesh stiffness in a static contact analysis and for multiphysics simulations. Note that the gear mesh stiffness is not analyzed through finite element analysis, but the stiffness of pairs of gear teeth are still in contact. Another benefit of having realistic gear geometries in a multibody dynamics analysis is that this provides better visualization when either setting up the physics or when postprocessing.
Geometry of a helical gear pair built using the Part Library.
We could manually build the geometry, but using built-in parts is both easier and faster. These parts are parametric in nature, which means that we can change their shape by readjusting the geometric parameters, and they come with optional features that can be added, such as shafts and fillets. The parts also have extensive checks to validate the input data as well as selections for the gear, shaft, and contact boundaries, therefore ensuring realistic physical entities and behavior. With the Part Library, it’s easy to specify the position and orientation of the gears as well as to align the gear mesh with their counterpart. These parts also contain robust geometric operations when creating complex gear geometries and the ability to manually change geometric operations.
The gear parts in the Part Library are divided into three categories based on whether they are gears with an external mesh, a gear with an internal mesh, or a rack. To learn more about the gear parts available in the Part Library, please read the previous blog post in our Gear Modeling series.
While the gear geometries in the Part Library are for individual gears or racks, gears are always used in pairs. Due to this, we need to build a gear train using individual gear parts. To illustrate the steps involved, we use a 2D spur gear pair example. The known quantities are as follows:
A spur gear pair showing the center distance of the two gears and the angular position of the second gear.
To place the second gear correctly, the first step is to compute the center distance ():
The position of the second gear () can be defined as:
Once the second gear is placed at the correct location, the next step is to align the teeth, or in this case mesh, of both gears. To accomplish this task, rotate the second gear with a mesh alignment angle () defined as:
where and are the mesh cycle of both gears, and they are defined as:
where and are the number of teeth of the first and second gear, respectively.
After computing the position of the second gear as well as the mesh alignment angle, we enter them as either expressions or numbers in the input parameter fields of the second gear, as shown below:
The input parameters of a 2D spur gear part with the gear center and mesh alignment angle highlighted.
For the gear tooth, we define the profile using an involute curve. The tooth shape and size are specific to the gear’s application, so a different application would require another type of gear tooth. Here is a list of input parameters through which we can control the shape and size of a gear tooth:
In the case that the fillet is not required in these places, we can set the tip or root fillet radius to zero.
An external gear tooth showing various input parameters.
The input parameters are mostly relative quantities for better scalability. We can compute different tooth profile parameters in terms of these input parameters:
Some applications require a specific type of gear tooth. High-pressure angle gears are better for high-speed applications as their wear rate is less than that of a standard tooth profile. Similarly, backlash is needed in high-speed applications because it provides space for a film of lubricating oil between the teeth, which prevents overheating and tooth damage. On the other hand, backlash is not desirable in precision equipment, such as instruments, machine tools, and robots. Backlash in these devices causes lost motion between input and output shafts, making it difficult to achieve accurate positioning.
Gears for different pressure angles and modules. Left: Gear with a standard tooth profile. Middle: High-pressure angle gear. Right: High-module gear.
After exploring the details of a gear tooth, we look at other parameters that influence the shape and size of a gear. The gear geometry is divided into three components: the gear teeth, gear blank, and shaft. For the gear shaft, the parameters are as follows:
Although the shaft is not an integral part of a gear, we can create one at the gear center with built-in gear parts. It is also possible to set the axial position of the gear on the shaft.
By default, a gear is placed at the origin and its axis is set to the z-axis, but it’s possible to control the position and orientation of the gear using the following parameters:
In order to align the gear mesh with the mating gear, we use a mesh alignment angle parameter to rotate the gear around its own axis.
A helical gear geometry showing different input parameters.
These input parameters, like the ones for the gear tooth, are relative quantities that we can use to calculate the gear parameters. They are as follows:
By default, a gear geometry comes with a set of features. Some of these are optional, and we can remove them by setting the appropriate input parameter to zero. It is possible, for example, to build a gear geometry without a shaft, gear blank ring, center hole, and fillets at the root and tip.
Geometry of spur gears where optional features are removed sequentially from (A) to (F). (A) Default geometry; (B) Without shaft; (C) Without gear blank ring; (D) Without center hole; (E) Without tip fillet; (F) Without root fillet.
While the gear blank shape is rather standard in all of the built-in gear parts, we can create a ring by removing the material in the gear blank. To customize the gear blank shape, we need to perform various manual geometric operations on the built-in parts.
Gears with customized gear blanks.
The built-in gear parts provide selections that we can use when setting up the physics or postprocessing. The available selections are for different components of the gear as well as for the gear teeth boundaries. We can use these boundaries to model contact between the two gears.
A spur gear where the geometry of the gear body, excluding the shaft, (left) and the gear teeth boundaries (right) are highlighted.
Since the gear parts are highly parametric, it is important to have an extensive set of checks to validate the input data. These checks ensure that the input parameters are correct independently as well as when combined with other parameters. We perform these checks before proceeding to build the geometry.
In the case that the set of input parameters is invalid, an appropriate error message is displayed. A few examples of nontrivial geometry checks, an external gear for instance, are as follows:
Next, we’ll look at some examples of gear geometries created using built-in parts.
The first example is a differential gear mechanism used in automobiles. This gear allows the left and right axles to rotate at different speeds. A differential gear uses five pairs of bevel gears, six bevel gears in total, to perform its operation.
Geometry of a differential gear mechanism.
The next example is a three-stage wind turbine gearbox. The first stage is a planetary gear train, which has three planet gears, one sun gear, and one ring gear. The second and the third stages are parallel gear trains that each have a pair of gears. This gearbox uses eight pairs of helical gears, nine in total, to perform its operation. The typical gear ratio of this gearbox varies from 50 to 100.
Geometry of a wind turbine gearbox with the top and front view showing.
Designed to transfer rotary motion from one shaft to another, gears are important devices in a variety of machines, from automobiles to wind turbines. New functionality in COMSOL Multiphysics provides you with several possibilities for quickly building gear geometries. With these robust and highly parametric built-in parts, you can change the shape of a gear to create an application-specific gear geometry.
In the next blog post in our Gear Modeling series, we’ll show you how to simulate gearbox noise and vibration. Stay tuned! We encourage you to browse the additional resources below in the meantime.
Let’s begin with a simple definition. A gear is a rotating machine part that is comprised of a set of toothed wheels, with the purpose of transmitting power from one part of a machine to another.
Model of a gear.
Gears can be connected to one another and they can also vary in size. Transferring power from one gear to another gear enables you to do one of the following things:
An animation illustrating the gear configuration that is needed to increase the speed of the second gear.
An animation illustrating the gear configuration that is needed to increase the torque in the second gear.
Because they reduce the torque and create a mechanical advantage through their gear ratio, gears can be considered a simple machine. A gear train or a transmission refers to two or more meshing gears that work together in a sequence, while a rack is the term used to describe a linear toothed part. In the latter case, the gear’s rotary motion is converted into the translational motion of the rack.
Now that we’ve looked at some of the dynamics behind how gears work, let’s explore some of their applications.
Just as the mechanical devices that gears are used in vary, so do the tasks that they are designed to perform. Of these tasks, the most important is gear reduction. Take the example of an electric screwdriver. It needs very high torque while in operation, whereas the electric motor generates very little torque at a high speed. With gears, it is possible to increase the torque at the expense of a reduced speed.
Now consider the example of an automobile. The engine of an automobile generates power at a fairly high speed. This same speed cannot be directly transferred to the wheels of the vehicle. Why? The reason is two-fold: The speed is very high compared to the required vehicle speed and the amount of torque required to move a vehicle, from an idle position, is much higher than the torque generated by the engine. So we need a device that converts high-speed, low-torque power into low-speed, high-torque power. A gearbox, placed between the crankshaft and the driveshaft, is the solution. By reducing its speed, the gearbox increases the torque in the driveshaft. In other words, it changes the form of power, matching the total power of that generated in the engine.
One question that may come to mind is why should you numerically model devices that include gears when you can do analytical calculations with certain assumptions. While analytical calculations serve the purpose at the preliminary design stage of a transmission system, there is a greater emphasis today on optimizing these systems to make them smaller, lighter, quieter, more durable, and more reliable. Numerical modeling provides a path for accomplishing this, as it accounts for all of the realistic situations that create nonlinearity in the system. Such factors include the flexibility of shafts, bearing stiffness, gear mesh stiffness, gear mesh damping, backlash, transmission errors, and friction, among others.
The numerical modeling of gears is designed to address the following elements:
COMSOL Multiphysics version 5.2a provides new functionality to easily model a pair of gears. The functionality, included in the Multibody Dynamics interface, allows you to design a transmission system that consists of a number of gears and shafts. Several types of gears and racks can be modeled, such as the following:
Additionally, you can model spur and helical gears as internal gears.
Schematic of a Helical Gear (left) and Spur Rack (right), depicting various gear parameters.
Gears are always used in pairs, which creates a need for a pair feature in COMSOL Multiphysics that connects two gears that satisfy the compatibility criteria. The following modeling nodes are available for connecting various types of gears:
Schematic of a Gear Pair (left) and Rack and Pinion (right), depicting various coordinate systems and other important parameters.
An ideal gear pair is both rigid and frictionless, without any static transmission error or backlash. To make the gear pair more realistic, you can add the following effects via subnodes:
The series of images below highlight some of the gear pairs you can model with the new functionality.
From left to right: Spur Gears (External), Spur Gears (Internal), and Helical Gears (Cross).
From left to right: Bevel Gears, Worm and Wheel, and Rack and Pinion.
In addition to this functionality, new parameterized gear geometry parts are also available. These gear parts are available for 2D and 3D models, with the option to customize the gear tooth and gear blank shape via input parameters. You can use these parts to build a range of items, from an individual gear to a pair of gears.
A helical gear geometry created with new gear parts included in the Parts Library.
You learn more about these upgrades in the Multibody Dynamics Module on the COMSOL Multiphysics version 5.2a Release Highlights page.
To showcase the new gear modeling capabilities, we’ve introduced several new tutorial models, each highlighting a different application.
Take the vibrations in a compound gear train tutorial model, for instance. In this case, we use spur gears, which are mounted on rigid shafts, to model the gear train. By performing a transient analysis, we can study the dynamics of not only the gears but also the vibrations within the elastic housing. The gear mesh stiffness is also calculated as a function of gear rotation via a parametric analysis.
Normal acceleration in the elastic housing due to vibrations.
Von Mises stress distribution in the gears while analyzing the gear pair’s mesh stiffness.
Our differential gear mechanism example, meanwhile, models a differential gear that is used within automobiles. With a differential gear, the outer drive wheel can rotate faster than the inner drive wheel — a necessary capability for a car to turn. Here, we compute the spider gears’ motion for two cases: when a car moves along a straight and a curved path. In both scenarios, the velocity magnitude of the components and the wheels’ angular velocity are calculated.
Differential gear mechanism that enables two of the vehicle’s axles to rotate at different speeds.
Also included in the mix is a tutorial model that computes the forces and moments that occur on bevel gears, as well as a tutorial that analyzes the dynamics behind helical gears. Both of these examples are highlighted below.
Bevel gear motion as an incremental rotation is prescribed.
Helical pair eigenfrequency analysis.
Modeling gears, a common element in mechanical devices, is now easy with new features and functionality available in the Multibody Dynamics Module. You have the ability to model various types of gears as well as include advanced effects, from the flexibility of shafts and backlash to gear mesh stiffness and damping. You can easily couple these gear dynamics with other physics to further extend the scope of your simulation analyses. The fatigue analysis of a gear tooth or the acoustic analysis of radiated noise from a gearbox are just some relevant examples.
Stay tuned for additional blog posts relating to gear modeling, where we’ll share more details on additions to the Parts Library, implementing features, and case studies. In the meantime, contact us for a software evaluation or browse the resources highlighted below.
A centrifugal governor is a specific type of governor known as a feedback system, which maintains a constant speed in an engine by regulating the amount of fuel that is let into the device. These devices rely on both centrifugal forces and the principle of proportional control, meaning that the output of the device is in direct proportion to the difference in the actual speed of the engine versus the desired speed. You can see proportional control in action when driving your car — you adjust how much pressure you apply to the gas pedal in direct correlation to how fast you aim to drive.
A centrifugal governor in a beam engine. Image by Biswarup Ganguly — Own work. Licensed under CC BY 3.0, via Wikimedia Commons.
The centrifugal governor was developed by James Watt during the late 1700s and is credited with aiding the growth of the Industrial Revolution. Factories and textile mills during this time often ran on engines that were powered by centrifugal governors, and they were incorporated into the design of steam engines for more efficient mass transportation as well. Even today, the design of the centrifugal governor is commonly used in machinery and engines, and its simple nature has not changed much since earlier versions.
In an effort to study the behavior of a centrifugal governor, analyze its parameters for an optimized design, and in turn create more efficient engines, we can model such a device using the multibody dynamics capabilities of COMSOL Multiphysics.
Breaking down the geometry of our spring-loaded centrifugal governor, we find a spindle in the center with a sleeve; two arms that consist of a top, bottom, and extension portion; and of course, a flyball on each arm that contributes to the centrifugal force of the device.
The geometry of a centrifugal governor model.
To perform a rigid body analysis of a centrifugal governor in COMSOL Multiphysics, we use the Multibody Dynamics Module with included transient, stationary, and eigenfrequency studies. The Rigid Domain feature is ideal for modeling the governor’s links, while the link connections can be modeled with the Hinge Joint and Prismatic Joint features. In an effort to simulate the model’s rotation using frame acceleration, the Rotating Frame feature can also be used. The details of this model are further explained in a previous blog post.
But what if you want to test the wide range of parameters for your centrifugal governor design quickly and easily without having to run the whole simulation over and over again? Further, you may want to share this model with others on your team to run their own tests, yet they rely on your simulation expertise at every step of the way. Fortunately, by building a simulation app, you can streamline your design workflow in many ways.
With the Centrifugal Governor Simulator, you can easily perform three different types of physical analyses simultaneously. A transient analysis computes the sleeve motion and trajectory of the governor’s flyballs, while a stationary analysis computes the equilibrium configuration of the device. Lastly, an eigenfrequency analysis computes the mode shape and damping characteristics of the governor design.
The user interface of the Centrifugal Governor Simulator.
As mentioned, there are a multitude of parameters that affect the operation of a centrifugal governor, and all of them can easily be included as input within an app. The range of parameters include:
To use the app, you can first set the geometric parameters listed above in the Geometry Parameters section. The details of the device geometry are shown in the Sketch section. When you do so, the geometry will automatically be built and shown in the Geometry tab. This tab also highlights if there are any failed geometry checks. The Information section will show the simulation status at this stage. Next, input the data for the Model Parameters and Study Parameters sections. Note that by simply selecting the Reset button, these parameters can all be reset.
Clicking on the Transient, Stationary, or Eigenfrequency buttons will showcase specific simulation results for that study type. The data will be displayed in the respective tab for that study. On the average computer system, the transient study will take 3 minutes, while the stationary and eigenfrequency studies will take 15 seconds each.
To see an attractive animation of the centrifugal governor’s trajectory, you can go to the Results section and click on the Animate button. The Report button, meanwhile, will generate a text report of the app for one, two, or all three of the study types. The Information section will further show the memory usage and computation time for the simulation.
For models with many parameters and multiple study types, like the multibody dynamics model of a centrifugal governor discussed here, simulation apps offer valuable help. Whether you’re running your own tests and parameter checks or empowering other team members to harness these simulation capabilities for their own benefit, we encourage you to get started exploring and building apps today. After all, designing efficient products calls for a just-as-efficient workflow.
Pole vaulting is a sport with a storied history. What began as an ancient competition for Greeks, Celts, and Cretans has evolved into a medaled event in the Olympic Games. Several tournaments, including the upcoming IAAF World Indoor Championships, are also hosted throughout the year, giving pole vaulters the opportunity to showcase their skills.
The sport itself, recognized as one of the major jumping events, involves the use of a long, elastic pole to clear a bar. In the past few decades, carbon fiber and fiberglass poles have arrived on the pole vaulting scene. These advancements are helping to bring athletes to new heights and break previous world records. While the pole has an important impact on performance, there are many other elements to consider that can affect the overall jump.
When it comes to clearing a height in pole vaulting, the general approach taken by athletes can be broken down into a series of phases. Each of the phases, listed here, places different constraints on the body:
In each phase, athletes control several of the initial conditions. Such conditions include: speed; grip height (the height at which the pole vaulter grips the pole); stiffness, which differs between different pole categories; the angle of attack (the angle between the pole and the ground at takeoff); and body position while airborne.
Angelica Bengtsson sets the Swedish pole vaulting record in 2015, achieving a 4.68 m clearance. Later that year, Bengtsson increased the national record to 4.70 m and finished in 4^{th} place in the 15^{th} IAAF World Championships.
Here, we’ll provide some more details about the individual phases.
The run up phase refers to when an athlete holds the pole in an upright position and successively tilts it forward while approaching the box, the hole in the runway where the pole is placed. By holding the pole close to the body, the torque created by the weight of the pole decreases. The muscular strength thus becomes less fatigued, with most of the muscular energy retained in the body. While approaching the box, the athlete maximizes his or her speed in order to maximize the kinetic energy, E_{K}, which is transferred to the next phase.
During pole plant and takeoff, the pole is initially placed in the box. The athlete then bends the pole and jumps up. What we have here is a multibody system, a combination of the pole itself and the pole vaulter. To get the pole into a vertical position, the system must rotate forward. Several variables can affect the angular position of the pole, θ, including the jump force, F; the jump velocity, v; and the body mass, m.
The jump force is transferred through the body to the pole at the hand grip. This pole force creates a forward-rotating torque at the takeoff and provides a positive contribution to the forward rotation. The athlete’s velocity affects the angular momentum, which further adds to the forward rotation. The body mass, assisted by the gravity, g, creates a counteracting gravitational torque throughout the entire movement that decelerates the rotation. Additionally, the pole vaulter rotates around the hand grip, φ, and moves his or her body parts. Such motion alters the position of the body mass and the rotational inertia, influencing the pole rotation.
The take-off phase. The double dots denote rotational acceleration.
Let’s now walk through a few pole vaulting scenarios.
At a high angle of attack — when the pole vaulter’s body is straight, with arms stretched and hands held high in the air — the torque leverage, the distance between the ground and the hand grip, is maximized. As a result, the pole rotates forward. If an athlete bends his or her arms, the leverage might not be sufficient enough to produce the amount of torque needed to drive the pole vaulter forward. Because of this, the pole will not reach the vertical position; instead, it will spring the athlete back to the runway. The same situation will occur if the speed of the athlete is not fast enough.
The grip height has a major influence on the take-off phase as well. On one hand, with increasing grip height, the pole vaulter will come higher up along the pole in its straight vertical position. On the other hand, an increased grip height will result in a lower angle of attack, while also increasing the horizontal distance between the pole plant and the body mass, which is the leverage of the counteracting torque from the body mass. However, as an athlete becomes stronger and faster, it is possible to increase the angular momentum, compensating for the additional counteracting torque due to higher grip height.
To maximize the energy transfer to the pole, it is also important that the athlete has a pretensed body. With a looser trunk, as well as shoulders and arms, some of the energy will be dissipated in the body. Body tension has a strong influence on the variables of the pole rotation as well. At takeoff, the athlete pushes backward with his or her leg and generates a forward-acting force. The pole counteracts, rotating the athlete backward. With a loose body, the pole vaulter will come down further on the runway, closer to the pole, and tilt backward. Such a position not only gives the athlete a smaller angle of attack, but it creates a lower jump force and velocity as well — all of which reduce the desired forward rotation of the pole.
At takeoff, the pole vaulter jumps up. This results in a vertical upward and horizontal forward velocity and force. If the angle of the jump is too low, the forces on the pole will bend it substantially. Once the tensile strength of the material has passed, the pole will snap, sending the athlete straight into the landing mat and unfortunately, below the bar. The most common reason for a pole to break is surface damage. When a pole is thrown on the ground or stepped on by track spikes, surface scratches can develop. These small surface marks can be large enough to initiate a pole fracture. Since the materials used in poles (carbon fibers and fiberglass) are brittle, they have a poor tolerance to damage.
Once an athlete has jumped, he or she can no longer utilize the runway that previously helped to increase the kinetic energy and counteract the initial pole bending. In this phase, the athlete rotates around the hand grip on the pole, φ, and generates a centripetal force, F_{C}, which further bends the pole. Since the elastic energy of the pole, E_{S}, depends on the deformation of the pole, δ, a higher elastic potential energy is transferred over to the next phase. Further, with greater bending, a higher spring force is stored in the pole. Note that the amount of stored energy and spring force is limited by the material strength.
As we discussed earlier, too much bending of a pole can cause it to break. An athlete can opt to use a pole with a higher stiffness, k, to increase the energy and force, but a stiffer pole exerts a greater stress on the body during the pole plant and takeoff.
Bending of the pole. The dots indicate rotation velocity.
During the swing, a pole vaulter lifts his or her legs, followed by the torso, to place them above the head when the pole reaches an upright position. The motion reduces the radius between the center of mass and the hand grip, thus increasing the rotation around the hand grip on the pole and sending the athlete higher up into the air. Moreover, the spring force from the pole now comes into play, as it catapults the pole vaulter upward.
With the ability to position the body in a certain shape, an athlete can control the inertia and position of the center of mass. Since both variables affect the angular motion around the hand grip, the athlete can optimize the angular motion of the pole; the elastic energy stored in the pole; and the spring force in the pole (theoretically, the sequence of motion that prompts an increase in jumping height). This involves considering several variables, from the position of multiple body parts to the dynamics of the pole vault. In reality, a pole vaulter’s body must respond to the dynamic changes during the vault, and with perfect timing.
When the pole is in an upright position, muscular energy and the arms are used to pull the body higher up. The velocity of the pull affects the generated power and the work done by the athlete. By increasing the velocity, more work is added to the potential energy at the grip height. This increases the potential energy of the pole vaulter, E_{P}, and therefore enables the clearing of heights above the grip height, h. The timing of such motion is of crucial importance. With an early pull, the athlete will not make it to the bar; with a late pull, the athlete will fly into the bar.
From the point at which the athlete releases the pole, he or she is moving as a free body, with the center of gravity following a parabolic path. The initial velocity is mainly directed upward and the gravitational force is acting downward. The pole vaulter’s legs clear the bar. As they are pulled downward, the legs generate a downward force, F_{L}, which is assisted by Newton’s third law of motion. As this happens, the hips are influenced by a counteracting upward force, F_{H}, and the pole vaulter ends up in an upside-down “U” shape. In this configuration, the athlete’s center of mass can pass below the bar, with the body soaring above it. On the way down, Newton’s third law is reused. The athlete moves his or her hips forward, stretching the arms and legs backward so that the upper body can clear the bar.
In a simple analysis of the pole vault, all of the kinetic energy from the run is transferred to the potential energy at clearance. The kinetic energy is . Here, m is the mass of the athlete and v is the velocity. The potential energy, meanwhile, is , where g is the acceleration of gravity and h is the height of the elevation. A perfect energy conversion results in a maximum achievable height difference for the center of mass: .
An elite male athlete can reach 9.5 m/s during the run up, while an elite female athlete can reach 8.4 m/s. This corresponds to and , respectively. Since the center of mass is initially about 1 m above the ground, it is evident that even a perfect conversion of kinetic energy into potential energy brings the pole vaulters to 5.5 m and 4.5 m, respectively. In reality, the best male athletes clear about 6 m and the best female athletes clear about 5 m. The athlete’s muscles supply additional energy during the jump.
Pole vaulting consists of many phases. By improving the details behind the technique, centimeter by centimeter and inch by inch, an athlete can work their way up to the limitations of the laws of physics and muscular strength. For many elite athletes, however, such success comes after more than 15 years of training.
Typically, there are two approaches to developing a successful jumping technique. Some people believe that a certain jumping sequence is the perfect approach and thus try to mimic it. Others, however, do not believe that one jumping sequence is the best option for everyone. Instead, they set out to develop their own technique. Incremental improvement can help athletes find local maximum in their height clearance, but to reach higher levels, they must make a significant change. Coping with this modification, which introduces a different response on an athlete’s body, requires the pole vaulter to not only be mentally and physically strong, but also to have a feeling for the physics underlying the sport.
The modern bicycle hasn’t really changed much since the safety bicycle, an early bicycle design that came into use in the 1880s. Over a century later, scientists are still trying to work out what makes a bicycle self stable. In other words, how does an uncontrolled bicycle keep balance and stay upright? Numerous papers explain the dynamics of bicycle motion through analytical equations. One of the earliest notable works is a paper by Francis Whipple in which he derived general nonlinear equations for the dynamics of a bicycle with a leaning, hands-free rider.
The long-standing consensus has been that a bicycle is stable because of two factors: the gyroscopic precession of the front wheel and the caster effect, or trail, as a result of the front steer axis meeting the ground ahead of the front contact point. More recently, a team of researchers from Delft and Cornell (see Reference 3) published a comprehensive review on the linearized equations of the motion of the Whipple bicycle model. They used their findings to present on bicycle self-stability. Their study shows that there isn’t one simple reason for this phenomenon. A combination of factors, including gyroscopic and caster effects, bicycle geometry, speed, and mass distribution come into play to keep an uncontrolled bicycle upright.
Inspired by their research, we developed a multibody dynamics model to demonstrate the self-stable riding of a bicycle with a hands-free rider.
The motion of a bicycle at different time instances.
This tutorial model of a bicycle demonstrates the motion of a bicycle with a hands-free rider on a flat surface when perturbed with a lateral force to produce a lean about the forward-riding direction. We extended the analysis to study the self-stability of the bicycle when the forward speed and front steer axis tilt vary.
To set up the bicycle model, we make the following assumptions:
The bicycle comprises four rigid components: the rear wheel; the rear frame, including the hands-free rider; the front frame, including the handlebar; and the front wheel. Although wheels have a finite thickness, the formulation assumes a single point of contact with the ground. The rear frame connects to the rear wheel through a hinge joint. Another hinge joint connects the two frames. The axis of this joint forms the steer axis of the bicycle. A third hinge joint connects the front wheel and the front frame.
The geometry of a bicycle in COMSOL Multiphysics.
The bicycle geometry used in this model is shown above. The inertial properties of the bicycle’s components are defined in terms of mass, moments of inertia, and the center of mass location. The important geometry parameters are the wheel radii, wheelbase (wb), and steer axis tilt (st). The steer axis tilt controls the trail (c).
The schematic of a bicycle.
As mentioned earlier, a pure rolling motion is assumed between the wheels and the ground. In the case of pure rolling, the velocity at the point of contact with the ground is zero. In this model, the wheel rolling is implemented without the use of any contacts to reduce the computational cost. Instead, we use an alternative multibody dynamics approach to model this motion. To describe the yaw, lean, and spin motions of the wheel, three hinges and three corresponding massless links are created. We implement three no-slip constraints to restrict the wheel slip in the forward, in-plane lateral, and out-of-plane directions. Since it is not possible to apply constraints on velocity, we convert them into displacement constraints.
Below, we describe the process for setting up these constraints, which have also been explained in more detail in the Rolling of a Rigid Wheel tutorial model.
To ensure the pure rolling of the wheel, we need three constraints to restrict the wheel slip in three directions.
The wheel model, showing the three constraint directions.
These constraints are as follows:
No slip in the forward direction:
No slip in the in-plane lateral direction:
No slip in the out-of-plane direction:
where , , and are the instantaneous forward direction (lean axis), in-plane lateral direction (spin axis), and out-of plane direction , respectively; is the translational velocity at the CG; is the wheel radius; is the spin angular velocity; and is the lean angular velocity.
As it is not possible to apply constraints on velocity, these constraints are discretized in time and implemented as follows:
where , , and are the displacement vector, spin, and lean angles at the previous time instance, respectively.
These time-discrete, no-slip constraints require the previous configuration of the wheel. The rigid body translation, rotation, and the instantaneous axes of the previous step are stored by using global equations and the Previous Solution node in the time-dependent solver.
We chose a bicycle with a steer tilt of 18° for this analysis. The bicycle is given an initial forward velocity of 4.6 m/s. After 1 second, the bicycle is perturbed through a force of 500 N, applied for a very short period of time. This force produces a lean about the forward axis.
For the first second, the bicycle moves forward along the initial forward direction at a constant speed. Next, the lateral force causes the bicycle to start leaning. The cyclist is hands free and cannot steer to balance the bicycle. What happens now? We observe that as the bicycle starts leaning, the handlebar starts steering in the same direction of the fall. This corrective action of steering towards the fall causes the bicycle to start straightening itself.
As the bicycle continues to move forward in a different forward trajectory, it starts leaning in the opposite direction. The lean is smaller in magnitude, with the steer motion closely following the lean with a slight phase lag. This back-and-forth leaning and steering perturbation continues and eventually damps down. The bicycle moves forward in an upright position and acquires a slightly increased forward velocity. The oscillations in lean and steer angles, as well as angular velocities, progressively damp and die out.
Motion of a bicycle with a hands-free rider on a flat surface when leaned about the forward axis. Here, the arrow indicates the lean of the bicycle.
The results of the lean and steer rotation (left) and relative angular velocity (right) of the bicycle.
So far, we have learned that the bicycle has the ability to be self stable. Research tells us that there isn’t just one parameter that influences the stability of a bicycle. The bicycle’s geometry, mass distribution, and forward velocity are all responsible for its stability. To gain some understanding of this, we conducted a further analysis to study the effect of two parameters: the initial forward velocity and the steer axis tilt. We used the bicycle model mentioned above, which has an 18° steer axis tilt and a forward riding speed of 4.6 m/s, as a reference configuration and conducted a parametric analysis to study the effect of these two factors.
A bicycle has no way of staying upright when stationary. In order to study the effect of bicycle speed, we varied the initial forward velocity of the bicycle from 2.6 m/s to 6.6 m/s in steps of 1 m/s. At 2.6 m/s and 3.6 m/s, the bicycle leans too much and is unstable. At 5.6 m/s, the bicycle’s lean velocity damps to zero, but the lean angle acquires a nonzero value. Although this is a stable configuration, the bicycle will move in circles with a slight lean. At 6.6 m/s, the lean and steer angles increase with time, making the motion of the bicycle unstable.
Unstable | Stable | Unstable | ||
---|---|---|---|---|
2.6 m/s | 3.6 m/s | 4.6 m/s | 5.6 m/s | 6.6 m/s |
The stable case, with an initial forward velocity of 5.6 m/s (left) and the unstable case, with an initial forward velocity of 6.6 m/s (right).
The steering assembly is quite important for bicycle self-stability. If the bicycle is not allowed to steer (for instance, if the handlebar is jammed), then the bicycle has no way of counteracting the lean, so it will fall. In relation to this, the steer axis tilt, which controls the trail, plays a role in bicycle self-stability.
In this analysis, we varied the steer tilt from 15° to 21° in steps of 1°, thereby varying the trail to study its effect on the stability of the bicycle. At a 15° tilt, the lean and steer angles keep increasing with time, making this configuration unstable. The bicycle remains stable in the range of 16° to 19° and unstable for larger angles. For a steer tilt greater than 19°, the lean and steer angles oscillate and these oscillations increase with time, causing the bicycle to be unstable.
Unstable | Stable | Unstable | |||||
---|---|---|---|---|---|---|---|
Tilt | 15° | 16° | 17° | 18° | 19° | 20° | 21° |
Trail | 0.066 m | 0.0706 m | 0.0753 m | 0.08 m | 0.0848 m | 0.0896 m | 0.0945 m |
Two unstable cases, with the steer axis tilt at 15° (left) and 21° (right).
In this blog post, we have described how you can simulate the motion of an uncontrolled, self-stable bicycle through the Multibody Dynamics Module in COMSOL Multiphysics. We have shown how to implement no-slip constraints on a rigid wheel through equations and then couple these constraints with a multibody model of a bicycle. We then analyzed the effect of initial forward velocity and steer axis tilt on the self-stability of the bicycle. By evaluating these parameters, we have seen that a bicycle that is stable in one configuration becomes unstable in another.
Bicycle self-stability is an effect of a combination of factors. Through our analysis, we concur with prior research by demonstrating that a bicycle’s stability is related to its ability to steer in the same direction as the lean.