The outcome of your golf stroke is basically determined by the movement of the club head just prior to impact with the ball. Considering this, we should be able to see how your golf swing could be improved based on a multibody analysis.

Here, I will show you how I went about modeling various body parts, a golf club, and the connections among them using the Multibody Dynamics Module.

A simple way to simulate a golf swing is by using a *two-link model*, where the arm and club are the two links connected together by a hinge joint. In this model, the arm rotates about a fixed point, located at the base of the neck, and the club rotates about the wrist joint relative to the arm. The two-link model does not allow a sufficiently long backswing and is not actually a true representation of a real-life golf swing.

A better representation is the *three-link model*, which also includes the shoulder as a separate link. Adding one more link eliminates the problem related to the backswing. Hence, we will use this three-link model in our analysis.

*Diagram of the two-link and three-link swing models.*

This analysis focuses on maximizing the club head speed just prior to impact with the ball, by understanding the mechanics of a golf swing. The torque profile, applied by different body parts (shoulder, arms, and wrist) is assumed. It is limited by the maximum torque capacity of the respective parts. Among all applied torques, the wrist torque has quite an important role to play in getting the strike right.

*Modeled geometry of the three-link swing model.*

While simulating the downswing of the club, the entire swing can be divided into two phases. In the first phase, arm and club rotate about the fixed point as a rigid assembly. In this phase, the arm and club are folded to minimize the inertia about the center of rotation, which allows the development of maximum angular velocity for the given arm-torque capacity. Here, the wrist is cocked to the maximum possible angle (the amount it can be cocked before you become uncomfortable or the angle is detrimental to your swing) and the applied wrist torque tries to hold back the club in this position against the other two torques.

In the second phase, the wrist torque starts helping the shoulder and the arm torque by pushing the club forward to increase the club head speed to its maximum. The instance when the wrist torque changes its role is a crucial parameter in determining the stroke quality. To see its effect on the club head speed, we vary the wrist torque parametrically.

*Time history of torque applied by the shoulder, arm, and wrist for (t_w = 0.19 s).*

The driving torque, applied by the shoulder, arm, and wrist, has a maximum capacity and can vary within the defined range. The applied shoulder torque is assumed to start at its maximum positive value, after a short build-up time. The applied arm torque, which acts on the arm and reacts on the shoulder, builds linearly with time to its maximum positive value with the specified rate. The applied wrist torque, which acts on the club and reacts on the arm, is fully negative to start and switches to its maximum positive value at the specified time (t_w).

On the arm and wrist joint, the rotation is not fully free. It is limited in the forward and backward directions by the ligaments, muscles, joint shape, or a combination of all these. In our golf-swing analysis, rotation limit in the backward direction is more important and this limiting value may vary from person to person.

In the beginning of the downswing, due to inertial forces on the body parts, these rotations try to go below the limiting value. Hence, additional torque is applied by the equivalent stiffness and damping of the stop. This makes the *effective* torque applied by the arm and wrist more than what is actually *applied*.

*Golf club head speed during the downswing for different wrist torque switch times (t_w).*

Above, I have plotted the club head speed for various wrist torque switch times (t_w) for the entire duration of approximately 0.25 seconds. It can be observed that for t_w = 0.15 s, we reach the maximum speed before impact — this leads to early hitting. On the other hand, for t_w = 0.23 s, the club head speed couldn’t even reach its maximum value.

For t_w = 0.19 s, the club head speed is higher than the other two cases and close to the optimum value for the given geometrical parameters and muscle strength.

*Comparison of the golf club trajectory for different values of t_w (results are displayed in the increasing values of t_w).*

*Motion of links and the trajectory of arm joint, wrist joint, and the golf club head.*

Maximum arm torque throughout the swing and very high arm speed in the beginning of the downswing can cause an early release, with the club head reaching its maximum speed before actually hitting the ball.

We can also deduce that for the given torque capacity, it’s potentially advantageous to have a long arm swing as well as a large wrist-cock limit angle. Furthermore, the extent to which the wrist can *hold back* the release is limited by its torque capacity. Therefore, your golfing skills are also strongly associated with the delayed release and the late hit.

In the downloadable model, we also consider the shaft flexibility by dividing the club into two parts: the grip and the shaft. These are connected through a hinge joint with finite stiffness and damping. You can see that the effect of the shaft flexibility to the swing is negligible compared to other parameters.

- Check out our Golf Swing model in the Model Gallery for more information. If you log into your COMSOL Access account, you can download the MPH-file and documentation for this model.
- Check out our archived webinar on Multibody Dynamics Simulation
- Explore the Multibody Dynamics Module and Structural Mechanics Module

A swashplate is a device that is used to transmit the pilot’s commands from the *non-rotating* fuselage to the *rotating* rotor hub and blades. The fact that the rotor blades are rotating at a very high speed makes the swashplate mechanism’s task more challenging. The mechanism consists of two main parts: a stationary and a rotating swashplate. The stationary swashplate is able to tilt in all directions and move vertically. The rotating swashplate is mounted on the stationary swashplate by means of a bearing, and is allowed to rotate with the main rotor mast.

A swashplate mechanism controls the *cyclic* and *collective* pitch of the rotor blades. The cyclic pitch of the rotor blades is used to change a helicopter’s roll and pitch. To tilt the helicopter forward, the difference of lift around the blades should be at a maximum along the left-right plane, creating a torque that, due to the gyroscopic effect, will tilt the helicopter forward instead of sideways. This is accomplished by tilting the swashplate assembly through pushrods. Collective pitch of the rotor blades, responsible for the average lift force, can be changed by moving the swashplate assembly vertically without tilting it.

The Multibody Dynamics Module provides pre-defined features to establish various types of joints between components. In this model, different components of a swashplate mechanism are connected together using *Ball*, *Hinge*, *Prismatic*, *Cylindrical*, *Planar*, and *Reduced Slot* joints. The connections in a swashplate mechanism are positioned in such a way that components form a closed loop — the swashplate mechanism is thereby a good example of a closed loop multibody system. If you were to model all the components as rigid, the mechanism would become overly constrained and difficult to simulate. This is not the case if you model it in COMSOL Multiphysics; the constraint elimination process removes the redundant constraints from the system.

*The animation above shows the workings of a helicopter swashplate mechanism.
Here, von Mises stresses are plotted on the surfaces of flexible rotor blades (the deformation is
magnified for better visualization).*

A transient analysis is performed to analyze the workings of the swashplate mechanism. Our specific interest in this case is finding out the effect of cyclic and collective pitch, given through the swashplate mechanism to the rotor blades, on the angle of attacks of rotor blades that in turn affects the generation of lift force. In terms of flexible rotor blades, *lead-lag* and *flapping* phenomena are also demonstrated. An eigenfrequency analysis is performed to find the rigid body modes and eigenmodes of a swashplate mechanism with flexible rotor blades.

*The torsional mode of flexible rotor blades of a helicopter swashplate assembly. The eigenfrequency
corresponding to this mode is close to 185 Hz.*

- Upcoming webinar:
*Multibody Dynamics Simulation with COMSOL Multiphysics*, July 2, 2013 - Check out our Helicopter Swashplate Mechanism model for more information. If you log into your COMSOL Access account you can download the documentation for this model as well.
- Explore the Multibody Dynamics Module

A double pendulum is a simple physical system comprised of two arms that are connected to each other through a hinge joint. In spite of being a simple system, a double pendulum also exhibits rich dynamic behavior with a strong sensitivity to initial conditions, making it an example of a chaotic system. A double pendulum is also a highly nonlinear, under-actuated mechanical system.

A double pendulum model is used in control systems to measure the effectiveness of stabilizing algorithms. Many real-life physical structures can be approximated with a double pendulum to gain more insight about the system behavior. Some typical applications include analyzing the sway motion of a payload in an overhead crane, or optimizing a tennis stroke, where the two arms of the double pendulum are the racquet and forearm.

A hinge joint, also known as a pin joint, is a type of joint that connects two components in such a way that they are only allowed to rotate about each other. This allowed rotation is in a specified direction, known as the *axis of rotation*, and at a specified location, known as the *center of rotation*. The axis of rotation and the center of rotation are, in general, attached to one of the components, and move in space with that component in transient simulations. All other types of possible motion, namely translation, or rotation between the two components, are constrained by this type of joint. As a consequence of constraining these other possible motions, forces, and moments corresponding to the constrained degrees of freedom are transferred from one component to another component though this joint.

The Multibody Dynamics Module provides a pre-defined feature to establish a hinge joint between two components. This *Hinge Joint* feature is used in the double pendulum model to connect both the pendulum arms in our example model.

The aim of this video is to demonstrate how to use a *Hinge Joint* feature in COMSOL Multiphyics to model a double pendulum and perform a transient analysis. Here, pendulum arms are subjected to a gravity load and their resultant oscillatory motion is studied. Our specific interest is in computing the forces experienced by the hinge joint, and tracking the locus of the bottom tip of the pendulum. Here both the pendulum arms are assumed to be flexible components that give us the freedom to evaluate the stresses generated in the arms.

*This video will demonstrate the modeling of a double pendulum using the Multibody Dynamics Module available for use with the Structural Mechanics Module.*

The primary aim of this model is to perform a flexible multibody analysis. Here, we demonstrate features available in the Multibody Dynamics Module to model interactions between mechanical components, such as joint forces and motion tracking. We will perform a transient analysis of the double pendulum motion, and track the locus of the double pendulum’s bottom tip, as it oscillates under gravity load.

Choose 3D as the spatial dimension, and select Multibody Dynamics from the Structural Mechanics branch. Choose the ‘Time Dependent’ study type, and click the Finish button.

Right-click the geometry node, and select the “Insert sequence from file” option. Browse to the model’s Model Library folder and Import the geometry sequence from the double pendulum file.

For the Finalization method, make sure to form an “Assembly” and right-click the Geometry node to build the sequence. To help with the joint definition later on, create explicit selections on the hinge pin and the barrel hinge. For better visualization of the pin boundaries, we can hide the barrel hinge. Choose boundary as the geometric entity level and select a boundary on the pin surface. Then check the “group by continuous tangent” option to add the pin’s other boundaries. Un-hide the barrel hinge and create a second explicit to repeat the selection and creation process for the inner boundaries of the barrel hinge. Hide the hinge pin this time, select boundary and click on one interior boundary of the hinge barrel. Check the “group by continuous tangent” option to select all the interior boundaries.

A third explicit is created, and the same boundary selection process is used to create a group of boundaries around which the pendulum will rotate. Under the Materials node, open the Material Browser and from the Built-In library, add Aluminum to the model for both components of the pendulum.

We now begin the physics setup for the Flexible Multibody problem. From the Multibody Dynamics node, create an attachment using the group of boundaries from the first explicit. Create a second attachment with the second explicit’s boundaries. Now these two attachments can be used to create and define a hinge joint found under the Multibody Dynamics node in the joint menu. In the joint setting window, you can view the joint sketch by expanding the Sketch tab. In the Attachment Selection window, choose Attachment 1 as the source and Attachment 2 as the destination attachments respectively. We keep the centroid of source as the center of the joint and specify the joint axis to be along the y axis.

Create a rigid connector boundary condition using the explicit three boundaries for the pendulum to rotate around. Prescribe displacements in all x, y, and z directions, then constrain the rotation around the x and z axes, limiting rotation to the y direction. From the Multibody dynamics node, add a body load and add both domains to the selection. The force of the body load is equal to the weight of both components. Keep the default mesh settings for this model, and build the mesh.

Go to the Step 1 time dependent node and in the “times” edit field, define a range to solve for 20 seconds, with a step size of 0.025 seconds. Right click Study1 and click compute.

*Required analysis can now be performed on the solved model. The default plots are created to view the displacement and the component velocity. You can view the results at any of the 800 results solved in this model. Right click ‘Results’ and add a ’1D plot group’, then right click 1D plot group and add a ‘global’ plot. Click replace expression and go to Multibody Dynamics, hinge joints, hinge joint 1, joint force, and choose the x component. You can now copy this expression and paste it in the fields below, replacing the suffix x with y and z respectively. Click plot to view the joint forces at the different times. Create another ’1D plot group’, with a ‘point’ plot this time and add any point on the bottom face to the selection. In the expression field type ‘z’. In the expression field for the x-axis of this plot, type ‘x’. Click plot to view the bottom tip displacement.Here is an animation showing the bottom tip displacement over the twenty second interval.*

- Check out our Dynamics of Double Pendulum model for more information. If you log into your COMSOL Access account you can download the documentation for this model as well. In this tutorial model, the additional features available on
*Hinge Joint*, e.g., constraints, locking, spring and damper, and prescribed motion are also demonstrated. - Explore the Multibody Dynamics Module

The objective of multibody analysis is to find critical areas of a system to perform more detailed component-level structural analysis. Multibody analysis also gives insight into the system dynamics, forces experienced by segments of the structure, and stresses generated in flexible components leading to failure due to large deformation or fatigue.

In mechanical systems, components are interconnected in such a way that only a certain type of motion is allowed between them. One way to model such types of connections is to establish physical contact, or unilateral contact, between the two components. The benefit of using the unilateral contact approach lies in the fact that it is very generic and can model any type of connection. However, this approach only works if you use actual physical geometry of the components and fails to work with the abstract form of the system. Moreover, this approach, in spite of being generic, is computationally difficult to solve in general and the difficulty level increases significantly in cases where friction needs to be accounted for on the contact surfaces.

When it comes to complex mechanical systems, a computationally-efficient technique is needed to model the connections. Here comes the mathematical representation of different types of connections known as *Joints*. An approximation is taken in cases of flexible components where the connecting surfaces are assumed rigid.

These *Joints* provide an efficient way to connect two or more components by constraining their motion without solving for the physical contact between them. A *Joint* feature of the Multibody Dynamics Module requires the center of the joint and the axis of the joint as an input. These inputs can be either extracted from the physical geometry or can be given directly.

A multibody analysis, using rigid body dynamics approach, is used to find motion and stability of mechanical systems. The stresses and deformation in a component can be estimated later using the forces experienced by the component. This approach works for small elastic deformations. However, in the presence of large deformations and material nonlinearities, a *Flexible Multibody Analysis* is the only way to get accurate results, and hence optimize the design. In a *Flexible Multibody Analysis*, components of a system that are likely to have large deformations are modeled as flexible and other components are assumed rigid.

*In the righter-most section of the above picture you can see the deformation and von-Mises stress distribution
in the rotor blades of a helicopter swashplate mechanism. In the bottom-left you can see the time variation of the
lift force applied on each rotor blade. (Note: In this model, the rotor blades are the only flexible parts in the
entire mechanism.)*

Establishing connections using *Joints* and performing a *Flexible Multibody Analysis* is easy through the new Multibody Dynamics Module that was released on May 3^{rd} with COMSOL Version 4.3b.

A library with eight different types of *Joints*, namely, Prismatic, Hinge, Cylindrical, Screw, Planar, Ball, Slot, and Reduced Slot are included in this module. Transient, frequency-domain, eigenfrequency, and stationary multibody dynamics analyses can be performed.

Sometimes the allowed motion between two components is not free due to the restrictions imposed by other physical objects. Here, the ability to limit and conditionally lock the motion is something that helps in modeling complex systems. In the field of robotics, the motion between two components is in general a known function of time. Many times, *Joints* are spring-loaded instead of being free to improve the system stability. All this is possible to model with this new module.

The Multibody Dynamic Module expands on previous capabilities of COMSOL Multiphysics and the Structural Mechanics Module, and it is the first dedicated product we have released that deals with the modeling of a mechanical assembly. Tune into our free Multibody Dynamics Simulations webinar on June 13^{th} to learn more.