Authors: Florian Nießl Simon-Hauke Wichmann, Rodrigo Castedo-Hernandez
Supervisor: Dr.-Ing. Axel Greim
Introduction and Motivation
The clapper bell is a traditional and timeless instrument that has been used for centuries to announce important events and signal the passage of time. However, the striking impact between the clapper and the bell can put stress on the bell tower, potentially leading to damage over time. To ensure that these cherished structures remain standing, it is crucial to understand the dynamics of the transient impact between the clapper and the bell.
Dr.-Ing. Axel Greim of Dr. Schütz Ingenieure in Kempten has developed a software tool to model the impact of the clapper on the bell tower structure. A key component of the software's accuracy is the determination of the contact time between the clapper and the bell. To provide this crucial data, we will simulate the transient impact between the two components using ANSYS software.
We will investigate how different variables such as the size and velocity of the clapper and bell, and the material properties, can affect the contact time. By providing accurate contact time information, we can assist Dr.-Ing. Greim in perfecting his software and advising land owners in Bavaria on the best way to preserve the longevity of their structures.
Geometric Model
Initial Model
At the beginning we created an initial model characterised by a diameter of one metre at the bottom of the bell and a height of 0.7 metres. The shape of the bell was created from a revolution of a profile made by ourselves for a generic bell.
The pdf below shows the dimensions of each part of the assembly as well as the thickness of the bell at the impact point.
Once the initial model was created, it was decided to split the model (Fig.1) in order to make the meshing process easier, as will be shown later. The slice seen is 50º.
|
|---|
| Figure 1: Split geometry. |
Simulation Models
In order to evaluate how different parameters from the simulation are relevant to the contact time between bell and clapper, we decided to vary some aspects of the geometry such as the size of the assembly and the size of the clapper ball.
Three different bell sizes were used for the simulation by scaling the original model and three different ball diameters were used for each bell size. The sizes of the clapper balls were chosen by increasing and decreasing the diameter of the initial model of each bell by 25%. Figure 2 shows the different shapes for each type of clapper. They are designed to strike the bell at the same point, regardless of the size of the spherical part.
|
|---|
| Figure 2: Different shapes for the clapper. |
In the tables below (Tab. 1-3) it is shown the volume, mass and diameter of the clapper and bell for each simulation. The bell's diameter is the diameter of the bottom part of the bell (the dimension that is one meter on the initial model) and the clapper's diameter is the dimension of the clapper's ball.
Table 1: Details of the small bell (0.8m).
Bell diameter 0.8 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% | |||
| Bell | Clapper | Bell | Clapper | Bell | Clapper | |
| Volume [mm³] | 1,7056e+007 | 5,7961e+005 | 1,7056e+007 | 8,4221e+005 | 1,7056e+007 | 1,2784e+006 |
| Diameter [mm] | 800 | 72 | 800 | 96 | 800 | 120 |
| Mass [kg] | 148,39 | 4,5789 | 148,39 | 6,6534 | 148,39 | 10,1 |
Table 2: Details of the medium bell (1.2m).
| Bell diameter 1.2 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% | |||
| Bell | Clapper | Bell | Clapper | Bell | Clapper | |
| Volume [mm³] | 5,7564e+007 | 1,9562e+006 | 5,7564e+007 | 2,8425e+006 | 5,7564e+007 | 4,3147e+006 |
| Diameter [mm] | 1200 | 180 | 1200 | 144 | 1200 | 180 |
| Mass [kg] | 500,81 | 15,454 | 500,81 | 22,456 | 500,81 | 34,086 |
Table 3: Details of the large bell (1.6m).
| Bell diameter 1.6 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% | |||
| Bell | Clapper | Bell | Clapper | Bell | Clapper | |
| Volume [mm³] | 1,3645e+008 | 4,6369e+006 | 1,3645e+008 | 6,7366e+006 | 1,3645e+008 | 1,0227e+007 |
| Diameter [mm] | 1600 | 144 | 1600 | 192 | 1600 | 240 |
| Mass [kg] | 1187,1 | 36,631 | 1187,1 | 53,219 | 1187,1 | 80,797 |
The geometry for these three different bell sizes can be seen in detail in the next pdf file. The thickness of the bell at the point of impact and the diameter of the bell at the point of impact (in the middle of the thickness) are also shown.
Simulation
Method
Before we could start the simulation, we had to think about a method for a realistic simulation. Both explicit and implicit methods are used for time-dependent simulations. The choice of method depends on the required time span of the simulation.
|
|---|
| Figure 3: Guide for implicit and explicit time integration methods. [5] |
Implicit integration methods determine the equilibrium at the unknown time step t_i_+_1. The advantage of these methods is, that their computations are highly stable and effective. Because of this unrestricted stability, the time step size can be large. Implicit methods are best suited to static equilibria, for large time steps and slow linear processes. [1]
|
|---|
| Figure 4: Illustration of implicit time step methods. [1] |
The basic equation of motion solved by an implicit transient dynamic analysis, is as follows
| (1) | m\ddot{x}+c\dot{x}+kx=F(t) |
These equations can be viewed as a set of 'static' equilibrium equations, which also take into account inertial and damping forces at all times. [2]
Explicit integration methods consider the equilibrium at the current time step t_i. The variables of the next time step t_i_+_1 can be calculated by solving the system of equations. The stability of explicit methods depends strongly on the size of the time step. Because of it's conditional stability, this method is only used for very small time steps. Typical applications for explicit methods are drop tests, impact simulations and problems with large deformations. [1]
|
|---|
| Figure 5: Illustration of explicit time step emethods. [1] |
The impact of the clapper is a short-duration time event in the range of milliseconds. For this reason, we used Ansys Explicit Dynamics for our simulation.
|
|---|
| Figure 6: The Ansys Explicit Dynamics module. |
The partial differential equations solved in an Explicit Dynamics analysis express the conservation of mass, momentum, and energy in Lagrangian coordinates.
The partial differential equations expressing the conservation of momentum are as follows. These relate the acceleration to the stress tensor \sigma_i_j.
| (2) | \rho\ddot{x}=b_x+\frac{\partial\sigma_x_x}{\partial x}+\frac{\partial\sigma_x_y}{\partial y}+\frac{\partial\sigma_x_z}{\partial z}\\ \rho\ddot{y}=b_y+\frac{\partial\sigma_y_x}{\partial x}+\frac{\partial\sigma_y_y}{\partial y}+\frac{\partial\sigma_y_z}{\partial z}\\ \rho\ddot{z}=b_z+\frac{\partial\sigma_z_x}{\partial x}+\frac{\partial\sigma_z_y}{\partial y}+\frac{\partial\sigma_z_z}{\partial z} |
Energy conservation is represented as follows
| (3) | \dot{e}=\frac{1}{\rho}(\sigma_x_x\dot{\varepsilon}_x_x+\sigma_y_y\dot{\varepsilon}_y_y+\sigma_z_z\dot{\varepsilon}_z_z+2\sigma_x_y\dot{\varepsilon}_x_y+2\sigma_y_z\dot{\varepsilon}_y_z+2\sigma_z_x\dot{\varepsilon}_z_x) |
The solutions are based on the input values at the end of the previous time step and are solved for each element of the model. [2]
Material
We used stainless steel for the clapper and typical bronze for the bell. It should be noted, that in reality each bell is unique and the material parameters are slightly different for each bell due to the casting process. The following table shows the parameters we used for the simulation.
Table 4: Material parameters for the simulation.
| Stainless Steel | Bell Bronce | |
|---|---|---|
| Density [kg/dm³] | 7.9 | 8.7 |
| Young's Modulus [MPa] | 190000 | 92000 |
Poisson's Ratio | 0.30 | 0.30 |
| Bulk Modulus [MPa] | 158330 | 76667 |
| Shear Modulus [MPa] | 73077 | 35385 |
Our input parameters were the Young's modulus and the Poisson's ratio. The bulk modulus and the shear modulus are derived from these. In reality there are no plastic deformations in the system, so we performed an isotropic linear elastic simulation. However, wear occurs during the contact between the clapper and the bell.
Meshing
We used an element size of 50mm and three levels of refinement at the impact regions and at the clapper ball. Mesh refinement is a post-processing step in the mesh generation that subdivides the elements of the selected topology.
When the entire model has such a fine mesh, the simulation time increases exponentially, so we only refined in the critical areas to achieve sufficient accuracy in our results.
For an accurate meshing and to avoid the problem of hanging nodes in the mesh, parts that should share a mesh, must be grouped into one part in the "Ansys Design Modeler".
In addition, the quality of the elements plays an important role in the simulation. The aspect ratio evaluates the quality of the trihedral elements. Elements with an aspect ratio close to one have a high element quality, while tetrahedrons with a larger aspect ratio have a lower element quality.
|
|---|
| Figure 7: Visualization of the tetrahedral element quality. [4] |
In the figure below you can see the quality of the mesh. On the left side of the x-axis you can see the elements with a large aspect ratio, these have a low element quality. As you move along the right hand side, the quality of the elements improves until you reach the highest quality element with a value of one on the x-axis.
|
|---|
| Figure 8: Element quality of the generated mesh. |
 |
|
|
|---|---|---|
Figure 9: Meshing of bell geometry with contact area refinement. | ||
Boundary Conditions
We used several boundary conditions for our analysis:
- the top of our bell is a fixed support, just as in reality the crown of the bell is mounted on a yoke
- the pin is also defined as a fixed support as well for the clapper
- the clapper ball can only move vertically and towards the bell (x-z-plane)
- frictionless contact between the sides of the clapper eye and the pin mounting
- frictionless contact between the clapper and the pin
- frictionless contact between the clapper and the striking surface
|
|
|
|---|---|---|
| Figure 10: Boundary conditions for the analysis. | ||
The fixed support boundary condition prevents the scoped object from moving or deforming.
|
|
|
|---|---|---|
| Figure 11: Definition of the frictionless contact areas. | ||
The frictionless contact setting models a one-sided contact, where the normal pressure is zero when separation occurs. When this contact type is enabled, sliding between external nodes and/or faces is possible without any friction and separation between bodies can occur. [3]
Initial Conditions
To simulate the striking of the bell in our simulation, the clapper is given an initial angular velocity that rotates around the centre of the pin.
|
|---|
| Figure 12: Adjustment of the initial velocity as an angular velocity. |
Results Ansys
Contact detection is performed using the trajectory of nodes and faces involved in a frictionless or frictional contact. If the trajectory of a node or a face intersects, a contact is detected. The faces or nodes of the contact can be initially separated or coincident. Trajectory based contact detection has no constraints concerning on analysis time.[2]
|
|---|
| Figure 13: Contact detection with the trajectory of nodes and faces [2] |
In Ansys we defined a body element where the contact force is determined.
|
|---|
| Figure 14: Contact force sensing body element. |
Results and Discussion
Contact Time
To obtain the contact time between the clapper and the bell, we wrote a Python script that accurately captures the first point of impact during each simulation. The contact time was defined as the difference in time from when the contact force first becomes non-zero to when it returns to zero. The figure below shows the contact force plotted over time, from which we derived the contact time.
|
|---|
| Figure 15: The contact time derived from the contact force over time. |
The results of our simulations indicate that the minimum observed contact time between the clapper and the bell was 0.45ms, while the maximum contact time is 1.59ms. Our findings also show a correlation between the size of the bell and the contact time, with larger bells resulting in longer contact times. In terms of the influence of clapper scaling, a linear increase in contact time is observed as the scaling increases, whereas the velocity of the clapper has a relatively constant effect on the contact time.
|
|---|
| Figure 16: Results for the contact times for all three bells (increasing bell diameter from left(0.8m), middle(1.2m), right(1.6m)). |
The following three tables show the results (contact time in milliseconds) that we receive after truncating the prolonged contact time (see prolonged contact time) and disregarding the second contact:
Table 5: Velocities for the different configurations of the small bell (0.8m).
| Bell diameter 0.8 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% |
|---|---|---|---|
| Velocity 0.75 [m/s] | 0.450 | 0.577** | 0.750* |
| Velocity 1.0 [m/s] | 0.450 | 0.577** | 0.752* |
| Velocity 1.25 [m/s] | 0.451 | 0.577 | 0.753* |
Table 6: Velocities for the different configurations of the small bell (1.2m).
| Bell diameter 1.2 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% |
|---|---|---|---|
| Velocity 0.75 [m/s] | 0.632 | 0.868 | 1.212* |
| Velocity 1.0 [m/s] | 0.632 | 0.868 | 1.201* |
| Velocity 1.25 [m/s] | 0.631 | 0.868** | 1.197* |
Table 7: Velocities for the different configurations of the large bell (1.6m).
| Bell diameter 1.6 m | Clapper Scaling 75% | Clapper Scaling 100% | Clapper Scaling 125% |
|---|---|---|---|
| Velocity 0.75 [m/s] | 0.910 | 1.190 | 1.581* |
| Velocity 1.0 [m/s] | 0.921 | 1.190 | 1.576* |
| Velocity 1.25 [m/s] | 0.920 | 1.190 | 1.587* |
*trunceted contact time for comparison
**second contact occured
Discussion
The impact can be simplified as a linear single-degree-of-freedom spring-system which explains the lack of correlation between the impact time and the velocity. For the spring-system the period of the system is the inverse of the eigenfrequency which is dependent on the stiffness (here mainly due to the bell) and the mass of the system (mainly the mass of the clapper). With that one can reason that the velocity of the clapper would have very little impact on the contact time. [6]
| (4) | \omega_n = \sqrt{\frac{k}{m}} |
| (5) | T_n = \frac{2\pi}{\omega_n} |
However, one can observe a rise in peak contact force with an increase in velocity. As depicted in the following figure, a case study of a bell with a diameter of 0.8m and a clapper scaling of 100% illustrates this correlation. The peak contact force increases from approximately 10.5 kN to 17.5 kN as the velocity increases from 0.75 m/s to 1.25 m/s. The discrepancy in impact time between these simulations is a result of the change in velocity, as the initial separation between the clapper and the bell remains constant. Consequently, a rise in velocity results in a reduction of the time required for the clapper to traverse the distance between these components.
|
|---|
| Figure 17: Reduced peak contact force between the clapper and the bell with a reduction in velocity. This example is for a bell with the diameter 0.8 m and a clapper scaling of 100%. |
Prolonged Contact Time
The dynamic response of the bell also results in an extended contact between the clapper and the bell. The figure below displays the contact force and directional velocity of the clapper and bell. Upon impact, the bell experiences acceleration, after which both the clapper and bell undergo deceleration as the bell behaves as a spring. At a certain point, the bell is accelerated inward, leading to a subsequent increase in the contact force.
|
|---|
| Figure 18: Directional Velocity of the clapper and the bell at the point of impact and the contact force between these two elements. |
This type of contact is not feasible in actuality, as our observations indicate that it arises from an excessive scaling of the clapper. Thus, this highlights the limitations of the clapper's size and should be avoided in practice. In some simulations, a similar response leads to a second contact immediately following the first impact. This, too, is a result of the dynamic response and can be mitigated by reducing the size of the clapper.
Figure 19 depicts the simulations in which a second contact and prolonged contact occurred. It is emphasized that the clapper scaling is the most effective means of regulating these phenomena.
|
|---|
Figure 19: Contact time of all simulations with valid contact (green), second contact (orange) and infeasable contact (red), increasing bell diameter from left(0.8m), middle(1.2m), right(1.6m). |
To enable comparison of the contact time across various simulations, the prolonged contact time was artificially truncated at the nearest minimum to the contact time when disregarding the dynamic response of the system.
|
|---|
| Figure 20: Shortened contact time for comparison. |
Material
The material parameter that varies the most in reality is the Young's modulus. To determine its effect on the contact time, we ran simulations for different combinations of Young's moduli. We varied the Young's modulus of the clapper (stainless steel) from 180000 MPa to 200000 MPa. We did the same with the Young's modulus of the bell (bronze), varying it from 90000 MPa to 95000 MPa. We can conclude that the material has a very small effect on the contact time. For our variation of the parameters, the variation of the impact time is in the range of 2 to 3.5%.
Conclusion
From the results it can be concluded that there is a linear relationship between the clapper scaling and the contact time. Furthermore, the impact velocity does not significantly affect the contact time. However, excessive scaling of the clapper may result in prolonged contact time due to the dynamic response of the bell, highlighting the importance of considering the feasible clapper size in the design process. The material parameters, on the other hand, have a limited effect on the contact time.
Outlook
In addition to analysing the impact of the bell and clapper geometry on the contact time, it is also important to investigate the influence of the mesh on the simulation results. For example, one could examine the effect of making the mesh uniform throughout the circumference of the bell to rule out any filtering out of the dynamic waves that travel around the bell and eventually impact the dynamic response at the point of impact. The meshing strategy employed in the simulation can significantly impact the accuracy and reliability of the results, particularly for complex dynamic systems such as the bell and clapper. Therefore, it is essential to carefully consider the meshing approach and to perform sensitivity analyses to assess the robustness of the simulation results. This will help ensure that the simulation accurately captures the relevant physical phenomena and provides valuable insights into the behaviour of the bell and clapper system.
In the future this kind of work might be used to develop a model to estimate the contact time between the clapper and the bell for a variety of shapes and sizes. To achieve this objective, one could take the following steps:
- Develop a geometric model of the bell and clapper that includes relevant dimensions such as height, diameter, and wall thickness.
- Use the geometric model to derive a set of simplified geometric measurements that can be used as input to a gerneralized function. This may involve selecting key dimensions that are most relevant to the impact time.
- Define this function in terms of the simplified geometric measurements for the bell and clapper. The function should be designed to give an estimation of the contact time between the two elements.
- Validate the function by comparing its predictions to experimental data or results from simulation studies.
- Refine the function as needed to improve its accuracy and predictive power. This may involve adjusting the set of geometric measurements used as input or fine-tuning the function itself.
References
[1] G. Müller, Structural Mechanics (lecture notes).Munich, 2022
[2] ANSYS,Inc., Ansys Explicit Dynamics Analysis Guide. Canonsburg (USA), July 2022
[3] ANSYS,Inc., Mechanical User's Guide. Canonsburg (USA), July 2022
[4] Ö. Sunar, Is Your Mesh Good Enough?. [Online]. https://www.mechead.com/mesh-good-enough/ (11.02.2023)
[5] ANSYS,Inc., Discussion of Implicit and Explicit Methods and Timestep Size. [Online]. https://courses.ansys.com/wp-content/uploads/2021/02/Lesson3_DiscussionOfTimeStepSize.pdf (11.02.2023)
[6] Sisemore, C., Babuška, V. (2020). Single Degree-of-Freedom Systems. In: The Science and Engineering of Mechanical Shock. Springer, Cham. https://doi.org/10.1007/978-3-030-12103-7_3, Chapter 3.1 SDOF Governing Equations