The bass drum is an essential part of the modern drumkit and the bass drum pedal, the mechanism used to strike the drum, is more than a hundred years old .
The bass drum pedal works by pushing a footboard downward to pull the drive (a chain or a metal part), which results in the attached beater being driven forward into the drumhead. After the footboard is released the spring attached to the cam brings the beater head back to the original position, where the tension of the spring affects the amount of force needed to strike and the amount of recoil after release. Hypothetically, the difference between the drives is that in case of the direct drive there is no flex or other unwanted movement, so control and response are tighter, whereas the chain drive makes the mechanism feel a bit smoother with less response transferred through the footboard .
In this project, a direct and a chain driven bass drum pedal was analysed based on their multibody models in MSC Adams View.
The project has two different objectives, an intra-device objective considering the spring and an inter-device objective considering the reaction forces in the two different devices:
- Finding the amount of spring pretension that is required to pull back the beater head fast enough to be able to operate the pedal at three different beat per minute (bpm) ratio, which are given as the tempo of different songs.
- Calculating and comparing the reaction forces in the footboard in case of the different drives to understand if there is a difference in response as it is hypothesised.
The modelling problem
Original and simplified models of the pedals
The analysed pedals are Pearl Demon Drive models with the same dimensions, the only difference is the type of the drive: one is a direct drive, while the other is a chain drive pedal.
The geometry of the direct drive pedal was downloaded from grabcad.com  and some simplifications were conducted in Autodesk Inventor to match the body limit of Adams Student version.
The chain driven model was designed by the modification of the downloaded geometry. In order to transfer them to Adams, the models were exported to Parasolid format.
The bodies and connectors in the direct drive model are the following:
As it can be seen, the model has 11 bodies and 12 joints. The body named Sphere is only needed for the contact between the beater head and the drumhead and will be mentioned later. The model has 2 degree of freedom, one rotational of the pedal and one translational of the drumhead which agrees with the expectations.
The modelling of the flexible connection in the chain drive was carried out by the cable module. The bodies and connectors in the chain drive model are the following:
This model has 12 bodies and 13 joints, but the bodies and connectors highlighted with grey are parts of the cable system, so they are not listed among the other bodies and connectors.
The modelling of the flexible connection in the chain drive was carried out by the cable module, with one pulley, and two anchors for the cable. The pulley is fixed to the axis, the diameter is 68,5 mm and its other dimensions were set to match the diameter of the original chain guiding part:
The mass and inertial properties of the pulley were set to a very small value (0,01), so the mass and inertial values of the chain holder part is dominant, since the pulley would have different properties from the chain holder which is an asymmetric body.
The cable connects the footboard tip to the chain holder at a specific point. These points were used as the anchors:
A simplified cable was used with the following parameters:
The density and the Young’s modulus were defined as in the other parts as stainless-steel to be closer the properties of a chain and to prevent the cable from stretching during the analysis. The diameter was an approximation of the chain, while it still fitted in the space that is bounded by the diameter of the chain holder and the point of the chain holder anchor. The other factors were left unchanged, because there is no initial velocity or preload in the cable and no need to change the ratio of the longitudinal stiffness (which is defined by the cross-sectional dimensions and Young’s modulus) or the damping.
Apart from the drumhead and the beater head all the parts were modelled as stainless-steel parts, with the material properties given in Adams. The beater head was defined as hard felt according to catalogue  and literature . The material of the drumhead is polyester, but based on the data from catalogues [7,8], it has the same density as the built-in properties of glass fiber plastic in Adams, hence it was used instead of defining a new material.
The drumhead was modelled as a horizontal cylinder 22 inches (55,88 cm) in diameter and 1 mm in thickness, a standard size version of bass drums . The position of the drumhead is not exactly specified so it was defined based on the standard rim size , so the surface of the drumhead is 23 mm from the end of the base (the approximately 40 mm thick rim is positioned 17 mm over the base, as it is indicated by the fixational system of the original model). The drumhead is also positioned in plane in a way so the beater head will strike it in the center more or less, but those positions don’t affect the interaction between the drumhead and the beater head, since the drumhead has only a translational degree of freedom.
Springs and Forces
The drumhead is connected to the ground through a spring-damper, where the spring provides the vibration, the “flexible” movement, while the damper represents the air damping of the drum. The stiffness and the dumping coefficient were calculated based on the lowest natural frequency of the drumhead. The lowest natural frequency of the drumhead is given as 40 Hz for musical mixing purposes  and in the literature . An estimation can be calculated based on the vibrational modes of a circular membrane as follows [12, 13], where T is the membrane tension, σ is the area density and D is the diameter:
Based on this result, the f=40 Hz was chosen as the frequency of the drumhead. The stiffness coefficient was calculated in the following way:
After the stiffness, the critical damping was computed, and a smaller value were considered:
The stiffness of the spring in the pedal that connects the cam and the base was calculated based on the wire diameter d, the coil diameter D, the number of coils N and the shear modulus G :
The effect of the damping coefficient was investigated, and it showed that if the values is smaller than 0,01-0,001 Ns/mm, then it is not changing the behaviour of the system, so the value was chosen as:
The behaviour of the pedal can be adjusted through the pretension of the spring. In real life, it can be done by a nut, which stretch the spring, hence increasing the tension. The force of the spring is calculated by the following equation according to Adams help:
- r is the distance between the two locations that define the spring damper measured along the line-of-sight between them.
- dr/dt is the relative velocity of the locations along the line-of-sight between them.
- c is the viscous damping coefficient.
- k is the spring stiffness coefficient.
- Fpre defines the reference force of the spring.
- l defines the reference length, so that when r = l, then force = Fpre.
Based on this to increase the force of the spring – that affects the amount of force needed to strike and the amount of recoil after release as it was mentioned in the introduction – a negative pretension should be defined.
The input of the simulation is a force applied by the foot on the upper part of the footboard, at the center of the grooved circle. The amount of the force applied was measured by me on a scale, based on experience recreating the same forces that were used on the pedal in real life as much as possible. Based on the measurements the force was given a size of 100N.
The force was applied on the footboard as a square wave, and the period of the wave was implemented based on songs, assuming a kick at every metronome click: a slow tempo at 80 bpm, a medium tempo at 120 bpm and a fast tempo at 170 bpm:
First, the contact between the beater head and the drumhead was a solid to solid contact, but in this way the type of the solver (RAPID/Parasolids) affected the results, even after increasing the faceting tolerance to reduce maximum error. To avoid this problem and to speed up solution time and realize smoother contact forces a different approach was followed, given the beater head is almost a perfect sphere, while the drumhead’s surface is plane: a sphere to plane contact was used. To do this a plane was defined at the surface of the drumhead, while a massless sphere was defined at the center of mass of the beater head with the same diameter (d= 50,75 mm) and it was fixed to it.
For normal forces, a continuous impact model was used for continuous positions as it was recommended during the lectures. The stiffness was defined to avoid too large penetration, the force exponent is low as it is recommended for softer materials, the maximum damping is less than 1% of the stiffness and the penetration depth for the maximum damping is 1/10 of the thickness of the drumhead . The effect of the damping coefficient was investigated, and it was concluded that 0,1 Ns/mm is the optimal value.
Friction Force was also implemented with the given values, keeping the static and dynamic parameters on the same values:
The effect of the change of these parameters was also investigated and it was concluded that they have small or no effect on the system.
Measurements and the simulation script
The following variables were measured during the simulations:
- The force magnitude of the contact, which is the strike, the drum hit
- The angle of the beater head
- The reaction force at the tip of the footboard
- The force on site of force input, this gives the time-profile of the kick
A dynamic simulation was used with two parts and the input force is deactivated between them. The length of the simulation is based on the period calculated by the tempo of the songs given previously. One part of the simulation has the time of a half period.
The effect of the time step size was investigated as well, and it turned out in case of the direct drive, after 5000, in case of the chain drive after 1000 steps for one part of the simulation (10000, and 2000 steps in total) there are no relevant changes in the results.
Simulations and analysis of results
Effect of the spring pretension
The plots below show the effect of spring pretension on the angle of the beater head com-pared to the original orientation in case of the direct and the chain drive considering different tempos.
The graphs show similar behaviour with different timing: the maximum angle is around 45 degrees at the first hit, and 40 degrees at second “ghost” hit, which lasts until half period, where the force becomes zero, as the foot is taken off the footboard.
Beside the time profiles of the beater head angle above, the minimum pretension needed to pull back the beater head to the original position before the next kick could be summarized in the following table:
The table shows that as the tempo increases the increase of the pretension is very similar in both drives.
Comparison of reaction forces
The reaction force in the tip of the footboard was compared in all three tempos. The pretension of the spring was set to the minimum pretension of the direct drive at the given tempo, which was always higher than the one of the chain drive, to ensure the return of the beater head to the original position with both drives.
Based on the graphs, it can be stated that there are no fundamental differences in the reaction force in case of the different drives as it was hypothesised, although, there are small differences. The plots show that the drum hits – which cause sudden increases of the reaction force – occurs later in case of the chain drive than it does in case of the direct drive at every tempo. The reason behind this is the difference of the mass and inertial proportions of the two drives and the size of the lever arm of the force on the rotational axis, since the lever arm is smaller in case of the chain drive than in case of the rigid drive (33,5 mm instead of 48 mm), but the lever arms are changing during the movement. To better quantify this difference, the input force, the pedal spring and the gravity was deactivated and a constant M=10000 [Nmm] was applied on the footboard at the heel joint. The angular acceleration of the beater head was measured, and it showed that the cable system has a larger moment of inertia at the starting, at the middle and at angle before the drum hit as well, which answers the delay between the drum hits.
It also shows that while in case of the direct drive, the moment of inertia is increaisng a lot as the beater moves from 0 to 40° (27 %), it only increases around 5 % in case of the chain drive. This aligns with the fact that the lever arm of the chain drive changes much less during the rotation than the lever arm of the direct drive.
In the time profiles of the reaction forces it can be seen too, that in case of the chain drive, mainly at the change of the input force, some oscillations occur, whose amplitudes are changing with the tempo, which can be caused by the flexible behaviour of the cable. The graphs also clearly capture the first, intentional hit and the second, “ghost” hit, which occurs when the player does not rise his foot right after the first hit, which can be seen in the video below .
And also in the videos of the models at the middle tempo:
The first objective of the study was to find the amount of spring pretension that is required to pull back the beater head fast enough to be able to operate the pedal at three different tempos. Through the analysis, the amount of pretension was determined for both direct and chain drive pedal at all three tempos.
The second objective was to compare the two pedals based on the reaction forces in the footboard to understand if there is a difference in response as it is hypothesised. The results showed that, although there are some small differences in occurrence of the drum hit and oscillation the profile of the reaction force is similar with both drives at all three tempos.
Subsequent studies should address the limitations of the project, which are mainly the behaviour of the drumhead without membrane movements, and the shortage of adequate measures that can be used in the modelling, for example for the force on the pedal and the contact characteristics.
 Dmitri Kartofelev, Anatoli Stulov, “Propagation of deformation waves in wool felt”, Acta Mech 225, 3103–3113, 2014
 Harvey Fletcher, Irvin G. Bassett, “Some experiments with the bass drum”,The Journal of the Acoustical Society of America 64, 1570 (1978)