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:

- Position of the first gear (x_1,y_1)
- Pitch radius of the first gear (r_1)
- Pitch radius of the second gear (r_2)
- Angular position of the second gear (\theta)

*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 (d):

d = r_1+r_2

The position of the second gear (x_2,y_2) can be defined as:

x_2 = x_1 +d \cos{\theta}

y_2 = y_1 +d \sin{\theta}

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 (\theta_a) defined as:

\theta_a = \frac{\pi}{n_2}+\mod \left(\frac{\theta}{\theta_{m1}}\right)\frac{n_1}{n_2}+\mod \left(\frac{\theta}{\theta_{m2}}\right)

where \theta_{m1} and \theta_{m2} are the mesh cycle of both gears, and they are defined as:

\theta_{m1}=\frac{2\pi}{n1};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\theta_{m2}=\frac{2\pi}{n2}

where n_1 and n_2 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:

- Number of teeth (n)
- Pitch diameter (dp)
- Pressure angle (\alpha)
- Helix angle (\beta)
- Addendum-to-module ratio (adr)
- Tooth-height-to-addendum ratio (htr)
- Backlash-to-pitch-diameter ratio (blr)
- Tip-fillet-radius-to-pitch-diameter ratio (tfr)
- Root-fillet-radius-to-pitch-diameter ratio (rfr)

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:

- Normal module: m = dp/n*cos\beta
- Addendum: ad = adr*m
- Tooth height: ht = htr*m
- Dedendum: dd = ht-ad
- Base diameter: db = dp*\cos\alpha
- Tip fillet radius: tf = tfr*dp
- Root fillet radius: rf = rfr*dp
- Tooth thickness at the pitch circle: t = \pi *m/2 -blr*dp

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:

- Gear-width-to-pitch-diameter ratio (wgr)
- Ring-width-to-gear-width ratio (wrr)
- Ring-outer-diameter-to-root-diameter ratio (dorr)
- Ring-inner-diameter-to-hole-diameter ratio (dirr)

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.

- Shaft-length-to-pitch-diameter ratio (lsr)
- Relative axial position of shaft center (zs)

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:

- Gear center (\{xc, yc, zc\})
- Gear axis (\{egx, egy, egz\})

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.

- Mesh alignment angle (th)

*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:

- Gear width: wg = wgr*dp
- Ring width: wr = wrr*wg
- Ring outer diameter: dor = dorr*dr
- Ring inner diameter: dir = dirr*dh
- Shaft length: ls = lsr*dp

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:

- Addendum check: ad<=(dp-db)/2
- Dedendum check: (2*dd/dp)<=0.9
- Hole diameter check: dh<(dp-2*dd)

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.

- Check out other blog posts in the Gear Modeling series
- Interested in multibody dynamics modeling? Read related posts on the COMSOL Blog
- Learn more about the new gear modeling features and functionality in COMSOL Multiphysics version 5.2a® by visiting the Release Highlights page

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:

*Increase the speed*: Say you are connecting two gears to one another, with the first gear featuring more teeth than the second gear. If this is the case, the second gear needs to turn faster than the first gear. As a result, the torque in the second gear decreases, keeping the power the same in both gears.*An animation illustrating the gear configuration that is needed to increase the speed of the second gear.**Increase the torque*: Say you are connecting two gears to one another, with the first gear featuring fewer teeth than the second gear. In this case, the second gear needs to turn slower than the first gear. As a result, the torque in the second gear increases.*An animation illustrating the gear configuration that is needed to increase the torque in the second gear.**Change the direction of rotation*: Now consider the situation in which two external gears are meshed together. Here, the second gear will always turn in the opposite direction. Therefore, if the first gear turns clockwise, the second gear must then turn counterclockwise. Specially shaped gears can also be used to transfer the power at an angle.

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:

- Transmission efficiency
- Loads on the other parts of the system (e.g., bearings)
- Stresses in the shafts
- Vibratory motion of the system
- Natural frequencies of the system
- Radiated noise
- Stability regions
- Whirling of rotors
- Reliability and life

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:

*Bevel Gear**Helical Gear**Spur Gear**Worm Gear**Helical Rack**Spur Rack*

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:

*Gear Pair**Rack and Pinion**Worm and Wheel*

*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:

*Gear Elasticity*: Defines the elastic properties of the gear mesh (i.e., mesh stiffness)*Transmission Error*: Specifies the static transmission error, which can result from geometrical errors and geometrical modifications*Backlash*: Defines the backlash in a gear pair, which impacts the dynamics of gears that are unloaded or lightly loaded*Friction*: Accounts for frictional forces that occur at the contact point

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, browse the resources highlighted below.

- Visit the COMSOL Multiphysics version 5.2a Release Highlights page to learn more about the new gear modeling features and functionality as well as additional upgrades
- Read another blog post that discusses the modeling of gears: “Modeling Magnetic Gears in COMSOL Multiphysics“

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:

- Geometric parameters
- Spindle height
- Length of arms
- Diameter of spindle, arms, and flyballs
- Radial distance of hinges
- Inclination angle for the top arm

- Model parameters
- Spring constant
- Damping coefficient
- Density of flyballs and linkages

- Study parameters
- RPM of spindles
- Number of spindle revolutions
- Spindle rotation per time step

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.

- Try it yourself: Download the Centrifugal Governor Simulator
- Read a blog post to learn more about centrifugal governors and the underlying COMSOL Multiphysics model
- Browse the COMSOL Blog for further discussions on how apps are enhancing other areas of simulation

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:

- Run up
- Pole plant and takeoff
- Pole bend and swing
- Pull and release
- Clearance

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,

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,

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,

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 E_K= \frac{mv^2}{2}. Here, *m* is the mass of the athlete and *v* is the velocity. The potential energy, meanwhile, is E_P= mgh, 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: \Delta h = \frac{v^2}{2g}.

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 \Delta h = 4.5\, \mathrm m and \Delta h = 3.5 \, \mathrm m, 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.

- Interested in modeling pole vaulting? Read more about the Multibody Dynamics Module and the Structural Mechanics Module, which can be coupled to model such mechanisms in COMSOL Multiphysics.
- You can find several other blog posts pertaining to the physics of sports on the COMSOL Blog. Have a look here.

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:

- All of the components are assumed rigid.
- All of the joints are considered frictionless.
- The wheels of the bicycle are modeled assuming knife-edge contact with the ground.
- Pure rolling constraints are implemented for the wheels.
- The bicycle is assumed to move on a flat surface.
- The bicycle model assumes a hands-free rider, defined through added mass on the rear frame.

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:

{\frac{d\bold{u}}{dt}.\bold{e}_{2}=r\frac{d\bold{\theta}_s}{dt}}

No slip in the in-plane lateral direction:

\frac{d\bold{u}}{dt}.\bold{e}_{3}=r\frac{d\bold{\theta}_{l}}{dt}

No slip in the out-of-plane direction:

\frac{d\bold{u}}{dt}.\bold{e}_{4}=0

where \bold{e}_{2}, \bold{e}_{3}, and \bold{e}_{4} are the instantaneous forward direction (lean axis), in-plane lateral direction (spin axis), and out-of plane direction (\bold{e}_{4}=\bold{e}_{2} \times\bold{e}_{3}), respectively; \frac{d\bold{u}}{dt} is the translational velocity at the CG; r is the wheel radius; \frac{d\bold{\theta}_{s}}{dt} is the spin angular velocity; and \frac{d\bold{\theta}_{l}}{dt} 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:

(\bold{u}-\bold{u}_{p}).\bold{e}_{2}=r(\bold{\theta}_{s}-\bold{\theta}_{sp})

(\bold{u}-\bold{u}_{p}).\bold{e}_{3}=r(\bold{\theta}_{l}-\bold{\theta}_{lp})

(\bold{u}-\bold{u}_{p}).\bold{e}_{4}=0

where \bold{u}_{p}, \bold{\theta}_{sp}, and \bold{\theta}_{lp} 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.

- Tutorial model: Rolling of a Rigid Wheel
- Tutorial model: Simulating the Motion of a Bicycle on a Flat Surface
- Meijaard, Jaap P., Jim M. Papadopoulos, Andy Ruina, and Arend L. Schwab. “Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review.” In proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 463, no. 2084, pp. 1955-1982. The Royal Society, 2007.
- Blog post: Using the Previous Solution Operator in Transient Modeling

The *gyroscope* — a term coined by Léon Foucault during the middle of the 19^{th} century — has been recognized as an instrument in science and engineering for nearly two hundred years. Its predecessor, the *spinning top*, has been known since ancient times, with uses as a toy, for gambling, and as an object with magic properties.

As an instrument, the gyroscope is valued for its precision in measuring and maintaining orientation. Such properties have facilitated its use in airplanes, spacecraft, and submarines, as well as within the sensors of inertial guidance systems.

*A replica of the first gyroscope.*

The classical gyroscope is based on the law of conservation of angular momentum. A spinning body wants to maintain the orientation of its axis in the absence of external moments. The resistance to changing orientation when perturbed depends on the *angular momentum*, that is, the product of the angular velocity and the mass moment of inertia. When a moment that is not parallel to the axis of rotation is applied to the rotor, the effect can be quite surprising.

Note: Several types of devices exist today with the same purpose as the classical gyroscope, but they are based on different physical properties. Recent developments in physics and microscale engineering have made this possible.

As the schematic below illustrates, a gyroscope consists of a disc that is forced to spin with a high angular velocity around an axis. The axis is journaled in an inner ring called a *gimbal*. The inner gimbal is attached to the outer gimbal with another pair of journals. These journals have an axis that is positioned at a right angle to the spinning shaft. A third pair of journals attaches the outer gimbal to the *frame*. As a result, the rotor has three rotational degrees of freedom, one about each axis. Note that the frame is attached to the surroundings (i.e., the vessel).

If the frame is rotated around an arbitrary axis, the rotor axis strives to maintain its direction. While doing so, both gimbals are forced to rotate.

*A sketch of a classical gyroscope.*

Using the Multibody Dynamics Module in COMSOL Multiphysics, we can simulate the mechanical properties of a gyroscope. Our Modeling Gyroscopic Effect tutorial model focuses on such studies. The example, which we will discuss next, is actually comprised of two models: a gyroscope and a spinning top.

Let’s begin with our gyroscope model. The geometry for the model includes four rigid bodies: the rotor, the two gimbals, and the frame. Steel is used as the material in the rotor, while aluminum is used in other parts. Because of these material choices, the rotor’s moment of inertia is large as compared to the supporting structure. The frame is given a prescribed rotation around an axis oriented at 90° from the rotor axis and at 45° from both gimbal journal axes. The frame rotation is harmonic with a magnitude of 2 rad and a frequency of 2 Hz. Each of the journals is modeled as a hinge joint.

Two different situations are addressed in the analysis to illustrate the effect of a rotor spinning on its orientation. In the first case, the rotor is not spinning. In the second case, an angular speed of 350 rad/s (3342 RPM) is prescribed as the initial value for the rotor.

The first animation below shows how the rotor is forced to change its orientation when it is not spinning. There is no gravity in the problem, and kinematically, it is possible for the rotor to maintain its orientation, so the rigid body dynamics of the system cause the change in the orientation of the rotor. In the second animation, we can see that the rotor essentially maintains its orientation since it is spinning.

*Orientation of the rotor under the imposed rotation of the frame when the rotor is not spinning.*

*Orientation of the rotor under the imposed rotation of the frame when the rotor is spinning.*

Plotted in the graph below is the inclination angle of the rotor axis, with the difference in stability shown. An angle error of about 1°, which occurs in the rotating case, may still not be useful in a precision instrument. Design changes, however, can reduce such deviation. In our example, the frame’s rate of rotation is rather high. The frame rotates approximately 115° and then back in the 0.25 seconds that are covered by the simulation study. To improve the stability of the axis orientation under such an external disturbance, a higher rotor speed or a heavier rotor is necessary.

*Comparing the rotor axis inclination with and without spinning.*

Shifting focus, let’s now look at the spinning top model. Here, we use only a single rigid body: the rotor from the previous example. The rotor axis is initially oriented at 20° from the vertical axis, and a gravity load is added. Further, an initial angular velocity is given to the rotor about its own axis. Together with the reaction force at the bottom, the gravity load creates a moment pointing out of the plane that is spanned by the rotor axis and the vertical axis.

*The force couple acting on the spinning top.*

This moment causes an angular acceleration in the direction out of the plane, and the spinning top starts to alter its orientation. This change in orientation of the spinning top, together with the spinning about its own axis, causes a gyroscopic torque on the spinning top. Under the effect of gyroscopic torque, the tip of the spinning top slowly moves along a circular path. Such rotation of the rotor axis orientation is referred to as *precession*. The plot below illustrates the trajectory of the tip of the axis.

*Trajectory of the tip of the axis for the spinning top.*

As we can see, the wide circular path is overlaid by a smaller periodic disturbance — a movement known as *nutation*. The nutation depends on the initial conditions. Since the spinning top study begins with only a rotation around the rotor axis and no precession velocity, the initial conditions are not compatible with a pure precession movement. In a real physical system, damping would decrease the nutation amplitude over time.

When you want to solve problems of this type, it is important to limit the time step used in the analysis. Typically, the time step must be limited to correspond with a rotation angle of the order of a few degrees per time step. In the above examples, a time step of 0.1 ms is used. This corresponds to approximately 2° of rotation of the rotor around its axis during each time step.

You can download the tutorial model presented here from our Application Gallery. If you are interested in learning about another technology for designing MEMS gyroscopes, we encourage you to check out our Piezoelectric Rate Gyroscope tutorial model as well.

]]>Truck-mounted cranes are designed to handle heavy loads. With this in mind, manufacturers and engineers look to optimize the machine’s payload, or *carrying*, capacity. Simulation apps can help expedite the optimization process by extending simulation capabilities into the hands of those who are not experts in simulation through a customized and intuitive interface. Our Truck Mounted Crane Analyzer demo app shows the benefits of this approach.

With their mechanical advantage in providing lifting power as well as mobility, truck-mounted cranes are valued in numerous industries, from construction to electrical line maintenance. When designing these machines, an important consideration is the payload capacity of the crane. In other words, how much weight is the crane able to lift when in operation? Addressing this question involves taking into account the orientation and extension of the crane as well as the capacity of the crane’s hydraulic cylinders, which control the machine’s motions.

The Truck Mounted Crane Analyzer demo app, released with COMSOL Multiphysics® software version 5.1, provides answers — all within an easy-to-use, customized format. Designers who are not experts in setting up simulations need a simplified simulation tool in order to modify the configuration of the crane and the positioning of the payload as well as the capacity of the hydraulic cylinders. By turning your models into apps, you will provide them with the easy-to-use tool that they need to run their own simulations. Let’s take a closer look at this example.

The truck-mounted crane demo app is based on the Truck Mounted Crane tutorial model, which we have previously blogged about. The crane features five hydraulic cylinders: the inner boom cylinder, the outer boom cylinder, and three extension cylinders. The inner boom cylinder is used to rotate the inner boom with respect to the base, while the outer boom cylinder is used to rotate the outer boom with respect to the inner boom. Meanwhile, the extension cylinders are designed to modify the length of the three extensions. The different components of the crane’s geometry are illustrated below.

Parts | Color |
---|---|

Inner boom | Cyan |

Outer boom | Green |

Inner boom cylinder | Red |

Outer boom cylinder | Blue |

Extension cylinders | Magenta |

*The geometry of the crane.*

When building your own app, you can adjust what parameters your app user should be able to modify. In our example case, we are creating an app for someone who wants to improve the payload capacity of the truck-mounted crane. Therefore, the demo app has been customized to allow modifications in the capacity of the hydraulic cylinders. In *your* app, you can include the elements that are important to your particular design scenario.

For the inner boom and outer boom cylinders in the crane, the capacity can range between 0.1 ton and 1000 ton (where 1 ton = 1000 kg); the capacity of the extension cylinders, on the other hand, can range between 0.1 ton and 100 ton. These parameters are listed under the *Capacity of Hydraulic Cylinders* section in the demo app, as shown in the screenshot below.

The intuitive user interface (UI) also includes options for changing the angle to the horizontal of the inner boom, the angle between the booms, and the total extension length. Located in the *Orientation and Extension* section, these parameters control the crane’s configuration as well as the payload’s position. To visualize the crane’s new configuration in the graphics window, the app user can click the *Update* button in the ribbon. Note that the weight of the crane is fixed in this application as 6.48 ton.

*The app’s user interface.*

After modifying the input parameters, the app user can click the *Compute* button in the ribbon to begin the computation of the simulation. On a typical desktop computer, it will take only about 25 seconds to compute the crane’s payload capacity with the simulation app.

Once the computation is complete, the payload capacity and the hydraulic cylinder usage results will be displayed in the *Results* section, with the updated input parameters reflected in the position of the linkages in the graphics window in the *Configuration* section. The *Results* section also includes the percent usage of each hydraulic cylinder, highlighting the limiting cylinder (i.e., the cylinder with a usage of 100%).

For the specified configuration of the crane, it is possible that the specified capacities of some of the hydraulic cylinders will not be sufficient to lift the weight of the crane. If this is the case, a warning message will appear as well as the minimum capacity that is required for each hydraulic cylinder in order to lift the crane’s weight. The minimum required capacity will also be highlighted if it exceeds the cylinder’s specified capacity.

Finally, by clicking the *Create Report* button, a Microsoft® Word document can be generated that compiles a report of the app. This report captures the current state of the application, noting modifications to the input parameters that you have decided to make available in your application and the simulation results.

By making simulation capabilities available to people without simulation expertise, simulation apps offer a more integrated approach to product design and development. The revolutionary Application Builder in COMSOL Multiphysics expands the scope of simulation by making the physics and functionality of your model available in a user-friendly format that can be tailored to design needs. Our truck-mounted crane demo app emphasizes the benefits of making your simulations accessible to a wider audience, offering a more efficient method for developing a high-performance machine.

- Download the demo app: Truck Mounted Crane Analyzer
- Learn how to create a simulation app (Corrugated Circular Horn Antenna Simulator example)
- Explore our other demo apps and additional updates to the Application Builder in COMSOL Multiphysics version 5.1 on our Release Highlights page

*Microsoft is a registered trademark of Microsoft Corporation in the United States and/or other countries.*

Not everyone has access to a laundry room or the ability (or time!) to go to a laundromat. You may not want to spend your change on coin-operated machines or leave the warmth of your apartment to go to a laundromat. Portable washing machines are a solution to these problems. They do have their own setbacks, though. The lightness of these machines coupled with an unbalanced distribution of clothes causes them to become unstable.

When the laundry is in a spin cycle, it generates a centrifugal force — causing the machine to destabilize. This problem could be solved by making the washing machine heavier, but since this machine is meant to be portable, this isn’t an ideal option. To learn more, we turn to simulation.

For this problem, we chose to model a simplified horizontal-axis front-load washing machine (the horizontal model has more severe instability than the vertical version). We used this model to figure out how walking instability affects the washing machine during a spin cycle. In order to try to remove the instability, our model makes use of an active balancing method.

In order to simplify our model, we made a few assumptions.

Assumptions about the drum and washer:

- Both are rigid.
- The rotation around the axis of the drum is the sole relative motion between the drum and the washer.
- The laundry spins at the same speed as the drum. We assume this because the RPM of the drum is high enough to produce substantial centrifugal forces.

Assumptions about the machine’s interaction with the surrounding environment:

- The washing machine cannot tip over. It remains connected to the ground throughout the simulation.
- The Coulomb friction model, with a constant friction coefficient, is used to simulate friction between the washer and the ground.

When setting up the model, we have to pay attention to where everything is placed. The balancing mass should be placed on both sides, the front as well as the back side, of the drum however for the ease of modeling, we place the balancing mass only on the front side of the drum making sure that its center of mass moved to the center of mass of the drum. When calculating mass, we need to remember that the mass of the drum adds to the clothing mass.

The drum is connected to the washer and the slot through hinge joints and a prismatic joint connects the slot and balancing mass. Both the joints, hinge joint connecting the drum to the slot and the prismatic joint, are necessary for active balancing, as we will learn more about in the next section. (Please note that in our model the prismatic joint is always locked since it isn’t used, while the drum/slot joint is only locked when there is no balancing in the system.)

Planar joints function as the washer’s four supports and connect the washer to the ground at four separate points. We’re able to analyze the joint forces independently on all joints because the joints have elasticity.

The instability of our washing machine model could result in multiple kinds of slip. For our needs, though, we are focusing on rotational slip and instability, since it happens at a lower critical speed than translational slip.

We measure our model’s stability by using the slip margin. Slip margin is the difference between the maximum possible friction force and the actual friction force. In the case of our example, it decides the critical operational speed needed to avoid walking instability. Walking instability will happen when the slip margin reaches zero. Our machine slips when three or more of our supports have a zero slip margin.

In order to make our machine more stable, first we have to eliminate what is making our washing machine unstable. In this case, the culprit is the net unbalanced centrifugal force acting on the rotating clothing. To combat this, we can apply an equal and opposite force. This is where active balancing comes into play.

To balance the forces within our machine, we adjust the angular and radial positions of the balancing mass. An angular correction serves to fix the direction of the centrifugal force and can be achieved by rotating the slot-balancing mass in relation to the drum. A radial position fixes the magnitude of the centrifugal force through the translation of balancing mass in the slot. In our simulation only angular correction is required because the radial position of the balancing mass is already set based on the weight of the laundry.

Our model makes it easy to use active balancing. Since we know the angular drum speed and acceleration rate of our model, we can activate our balancing system at a specific time instead of waiting for the slip margin to get close to zero.

After taking all the previously mentioned factors into account, we ran our simulations. At first we looked at our washing machine without the aid of active balancing:

*Left: The washer rotation (magnified by a factor of 100). Right: The drum rotation and friction force at the washer supports.*

Next, we saw what happens when we add active balancing to our simulation by viewing the total imbalance of our washing machine in two ways.

Our first plot shows the imbalance in the rotating frame and displays a clear correlation between active balancing and reduced imbalance. A similar effect is viewed in the fixed frame plot.

*Left: Total imbalance in the rotating frame. Right: Total imbalance in the fixed frame.*

Another area we looked into was the slip margins of our model. First, we analyzed the differences in the individual supports without active balancing. We compared the slip margin of a support in the front of the washing machine (support 1) and a support in the back (support 3). The plot to the left below reveals that the a washer with front support has a lower slip margin and is therefore more prone to slipping than a washer with back support.

Widening the parameters to look at the total slip margin of our model, we saw that the total slip margin does become zero for short periods. This means that the washing machine will experience walking instability during these moments. The right-side plot shows that active balancing increases the total slip margin upon activation.

*Left: The slip margins of supports 1 and 3 in the absence of active balancing. Right: The washing machine’s total slip margin with and without balancing.*

Active balancing is further shown as helpful when looking at the movement of the washer around a vertical axis. Rotational instability is eliminated with the balancing mechanism.

*The Z-axis rotation of washer with and without balancing.*

We modeled other aspects of our portable washing machine as well. When looking into the revolutions per minute (RPM) of the drum and correction motor, we found that the correction motor works efficiently by starting when needed and stopping when the system is stabilized. We also calculated the appropriate correction angle for stabilizing the system.

*Left: RPM of the correction motor and the drum. Right: The necessary correction angle in an active balancing system.*

Creating an active balancing system helped our model avoid the effects of walking instability and rotational slip during its spin cycle. Now it’s your turn to try it out yourself — download the model via the link below to get started.

- Model download: Walking Instability in a Washing Machine
- Related blog post: Simulating Vibration and Noise in a Washing Machine

By design, cranes offer a mechanical advantage, lifting and lowering heavy materials that require strength beyond that of a human. In many applications of this machine — from construction to electric line maintenance — a favorable feature is mobility. Truck-mounted cranes are free to move in various directions as well as travel on highways, which can help avoid the need for additional transport equipment.

*An example of a truck-mounted crane. (“A truck-mounted crane from Palfinger (Austria). The concrete component (build in Germany) is a small sewage treatment plant for a house with up to four residents.” by TM — Own work. Licensed under Creative Commons Attribution-Share Alike 2.0 Germany, via Wikimedia Commons.)*

In these types of cranes, there are several hydraulic cylinders that control the motion of the crane as well many other mechanisms. When handling heavy loads, the components are subjected to large forces. Through simulation, we can explore the impact of these forces during the machine’s operating cycle, determining ways to enhance its performance by building a more efficient design.

Combining the Multibody Dynamics Module with the Structural Mechanics Module, the Truck Mounted Crane model analyzes the forces on the cylinders and hinges of a crane during an operating cycle. The crane geometry, which is imported from a CAD model, is comprised of 14 parts that move in relation to one another.

*Geometry of the truck-mounted crane.*

The figure below provides a more detailed overview of the crane link mechanisms, followed by a table defining the individual components.

Part | Color |
---|---|

Base | Blue |

Inner boom | Green |

Outer boom | Yellow |

Telescopic extensions | Cyan, Magenta, Gray |

Boom lift cylinders | Red, Gray |

Boom lift pistons | Yellow, Magenta |

Inner link mechanism | Magenta, Black |

Outer link mechanism | Cyan, Blue |

In this example, there are two loads applied — self weight in the negative *z*-direction and a payload of 1,000 kg at the crane’s tip. The operating cycle consists of lifting the payload from a position that is far away and placing it below the crane. The load is initially moved upwards and then drawn inwards to a position that is close to the crane. The plot below depicts the crane tip’s trajectory during the operating cycle.

*Crane tip trajectory during operating cycle.*

In reality, the crane is operated by controlling three cylinder lengths — the inner cylinder, the outer cylinder, and the extension cylinders. The inner cylinder raises the inner boom, the outer cylinder regulates the angle between the inner boom and the outer boom, and the extension cylinders determine the reach of the extensions. Here, the angles of the booms are used as parameters rather than the cylinder lengths, as this method is more convenient.

The image below illustrates the 9^{th} position of the operating cycle, which features an inner boom angle to the horizontal of 45°, a -30° angle between the inner and outer boom, and a total extension of 1.5m.

*The crane during the 9 ^{th} position of the operating cycle. The color shows the total displacement of the crane components.*

We can now address the impact of the forces on various parts of the crane. In each of the following graphs, the solution number correlates to the position of the crane. Initially, the crane picks up a load in an extended position and then, at the final solution, releases the load close to its own position.

Let’s begin with the forces in the cylinders controlling the boom. Here, the compressive forces are positive. As can be expected, when the payload is far from the crane base, the cylinder forces are greater. The maximum force during the operating cycle determines the required cylinder capacity.

*Forces in the cylinders controlling the boom.*

The next graph highlights the forces in the extension cylinders. As in the previous case, a compressive force is defined as positive. Because they have to carry the weight of extension segments a further distance, the inner cylinders endure greater forces.

*Forces in the extension cylinders.*

Finally, we can observe the forces acting on the hinges between the main parts of the crane. This same tactic can be used to analyze the forces within the connections between any parts of the crane. The results below are a valuable resource in the structural dimensioning of such details.

*Forces on the hinges.*

We can now take things a step further and use the Optimization Module to enhance the crane link mechanism. This can be accomplished through the Optimization of a Crane Link Mechanism model, a continuation of the Truck Mounted Crane model. In this case, the focus is on reducing the cylinder force that is necessary to haul a particular payload in a worst-case load cycle scenario.

*A detailed look at the link mechanisms.*

The table below identifies each of the parts and their colors considered in this model.

Part | Color |
---|---|

Base | Blue |

Inner boom | Green |

Boom lift cylinder | Red |

Boom lift piston | Yellow |

Link mechanism | Magenta, Black |

Since this example is designed to test for the worst-case scenario, the operating cycle is chosen so that the link mechanism will experience as much force as possible. To ensure this, the inner boom is raised to its highest position, the telescopic extensions are extended as far out as possible, and the angle of the outer boom is selected to ensure that the crane tip is as far out as it can be. The same loads from the original model are applied.

Within this optimization problem, the positions of three axles can be changed. These include the axle that connects the first link arm to the base, the axle that connects the second link arm to the boom, and the axle that connects the two link arms and the hydraulic cylinder’s piston.

Now let’s compare our results. The first graph below shows the variation of the cylinder force during the operating cycle. Here, we compare the maximum cylinder force during the operating cycle, which determines the capacity of the hydraulic cylinder.

In comparison to the original design, the optimized version enables a reduction in the maximum force from 597 kN to 413 kN — that’s a 31% decrease, a sizable improvement! With this enhancement, the allowed payload can become greater; the decreased forces will enable the link mechanism to meet stress criteria more easily.

*Comparing the cylinder forces in the optimized and original designs.*

The second plot illustrates the *y-* and *z*-components and the magnitude of the force acting on the axle that forms the hinge between the base and the boom. As indicated by the results below, the total force of the original design is greater than the total force of the optimized design.

*Forces acting on the axle in the optimized and original designs.*

With COMSOL Multiphysics version 5.0, we introduced two new models designed to analyze the interactions between different components of a truck-mounted crane and evaluate the role of optimization methods in enhancing these mechanisms. These examples demonstrate how simulation can be used to investigate the impact of loads on such complex mechanical systems and how this knowledge can be applied to developing a stronger design.

Download the models featured here:

One last thing… We’re in the process of creating an app based on this model. Stay tuned for that.

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When buying a home, there are several modern conveniences that people look for. One of these is often a washing machine. Having this device within your own home eliminates the time-consuming task of taking your clothes to the laundromat or washing them by hand, giving you the freedom to tackle other tasks around the house — or better yet: relax.

Imagine that it’s laundry day and you are fortunate enough to have a washer in your home. How would you notice that your laundry cycle is complete? One common tell-tale sign is silence.

During the cycle, your washing machine is likely to produce vibration and noise, a result from the uneven distribution of clothes and the structural properties of the machine. We now have a multibody dynamics model that enables you to analyze this common issue via simulations.

The Vibration in a Washing Machine Assembly model depicts a horizontal-axis portable washing machine. The model’s geometry accounts for various parts of the washing machine assembly (including the housing), with varying components defined by different colors.

*The geometry of the washing machine.*

The machine’s parts are represented as follows:

Color | |
---|---|

Clothes | Red |

Drum | White |

Tub | Cyan |

Motor | Yellow |

Pistons | Green |

Cylinders | Magenta |

Mountings | Blue |

Base supports | Black |

Housing | Gray |

Within this multibody dynamics example, we assume that the housing is modeled as a flexible shell and that the other components of the washing machine are modeled using rigid solids. Additionally, we assume that the clothes do not move in relation to the drum.

The diagram below explains the connection details between various parts of the machine through the use of joints and springs.

In an eigenfrequency analysis, one mode focuses on the translation of the tub and another shows the rotational motion of the tub about the vertical axis. Each of these eigenmodes highlights the corresponding deformation of the housing, which is relatively small compared to the motion of the tub.

*On the left: The translational tub mode. On the right: The rotational tub mode.*

The next set of figures shows the displacement magnitude of the tub with the angular position of the unbalanced clothes for the full time duration. Here, the color red represents the initial time of the trajectory and the color blue illustrates the final time.

To identify the vibrations induced in the housing during the machine’s spinning cycle, we can perform a transient analysis. The graph below depicts the housing deformation in multiple directions at a point on the right side wall.

Next, we can analyze the *normal acceleration* (a measure of noise emitted by the side walls) of the housing at a point in the middle of the right side wall. These results are illustrated in the graph below on the left. Meanwhile, the graph on the right depicts the frequency spectrum of this acceleration.

We can note that the frequencies of the side wall vibration are predominantly within the range of 0-30 Hz, with a peak of around 1.67 Hz, which can be considered the *excitation frequency*. This corresponds to the frequency expected from the inertial mass ratio of the drum and its load at the set rotational velocity. Yet, it is interesting to note that the frequency of the side wall vibration also occurs at higher values than the excitation frequency, which leads to an additional emission of noise.

With this in mind, the washing machine can be designed to mainly emit noise at lower, non-audible frequencies. The choice of materials and design must also ensure that the washing machine is structurally strong enough to limit the vibration magnitude at all frequencies in order to prevent the failure of its components.

- Download the model from the Model Gallery: Vibration in a Washing Machine Assembly

Whether traveling a far distance between work and home or simply caught in a continuous wave of traffic, many of us spend a great deal of time on the road each day. On days where your evenings are free, this might be thought of as a time to relax and listen to a favorite CD or radio station. However, on days where your schedule is a bit more hectic, the lengthy commute can become a source of frustration as you race to make it on time to your destination. No matter the situation, an element that you rely on to make your car ride more enjoyable is comfort within your vehicle.

While on the road, you may have noticed the sensation of vibrations that can sometimes arise from your seat. The root of these vibrations can be traced to a number of sources, including road conditions, speed, engine vibrations, and the design of the vehicle’s seats. While not only a source of discomfort, prolonged exposure to such oscillations can be hazardous to one’s health, potentially resulting in fatigue or pain. With the growing concern behind the impact of these vibrations, some vehicles have begun to implement vibration isolators within the design of their seats in an effort to minimize this effect.

*A vehicle’s seat can be a source of vibrations. (“Sedile in pelle di un’Alfa Romeo Giulietta” by Pava — Own work. Licensed under Creative Commons Attribution Share-Alike 3.0, via Wikimedia Commons).*

With this biomechanical model built using the Multibody Dynamics Module, we can simulate the human body’s response to such vibrations, thereby helping to optimize the design of vibration isolators as well as analyze ride quality in vehicles.

The Biomechanical model of the human body in a sitting posture makes this analysis possible. An important element in the design of this model is addressing the complexity of the human body and how the different body parts are connected. In this example, we focus on the vibrational impact in six different areas of the body: the head, the torso, the viscera, the pelvis, the thighs, and the legs. Each element is treated as a lumped mass and defined as a rigid body.

To approximate the connections between varying body parts, we apply translational and rotational dampers and springs on the relative motion between the two connected body parts — a connection that is modeled with the elastic version of a fixed joint. This provides the translational and rotational stiffness and damping values between the connected body parts.

A fixed joint is used to model the connections between the body parts directly touching the seat (the legs, the thighs, and the pelvis) and the seat itself, which is the source of the vibration. To model the seat’s cushioning effect, elasticity on the joints is included when needed.

Note that rather than modeling the seat itself, a base motion node is used where the input excitation is 1 m/s^{2} in the vertical direction at three different locations.

We begin with an eigenfrequency analysis designed to determine the damped and undamped natural frequencies of vibration.

The figure below illustrates a rotational eigenmode on the undamped model. Considerable rotational movement is noted in the head and the torso. In comparison, little movement is found in other parts of the model.

We then shift our analysis to the translational eigenmode of the damped model. In the first major translational eigenmode, the results indicate a downward movement in the head, the pelvis, and the viscera, with no significant movement noted in the other body parts. This is illustrated in the figure below.

The second major translational eigenmode (shown below) notes displacement in the downward direction of the head, the torso, and the pelvis, whereas the viscera moves in the upward direction.

This example then features a frequency response analysis around the natural frequencies to analyze three different elements: vertical transmissibility, rotational transmissibility, and apparent mass.

Let’s first focus our attention on vertical transmissibility. *Vertical transmissibility* refers to the ratio between the head’s vertical acceleration and the seat’s input acceleration. When compared with the excitation frequency, the results show that the primary resonance is visible in the range of 4-6 Hz and the secondary resonance in the range of 8-10 Hz.

*Vertical transmissibility vs. excitation frequency.*

*Rotational transmissibility* is the ratio between the head’s angular acceleration and the seat’s input acceleration. With regards to this form of transmissibility, it is important to avoid high values as this can enhance discomfort as well as affect one’s vision. The plot below depicts its variation with the excitation frequency.

*Rotational transmissibility vs. excitation frequency.*

Finally, *apparent mass* refers to the ratio of the seat’s force to the seat’s input acceleration. Rather than depicting the end point characteristics of the model, this element conveys the driving point characteristics.

*Apparent mass vs. excitation frequency.*

In this blog post, we have introduced you to a biomechanical model of the human body, particularly highlighting its application within the automotive industry. We have showed you how, with the *Multibody Dynamics* interface, you can model various parts of the human body — and their connections — as well as analyze its dynamic response to whole body vibrations.