Mechanics of Spring-Powered Trebuchet

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1 Physics 382, Student (2016) Advanced Lab v3.3, p.1 Mechanics of Spring-Powered Trebuchet C. Lewis, K. Prasher, T. Yang, M. Roark, and M. J. Madsen Wabash College, 301 W. Wabash Ave, Crawfordsville, IN (Dated: March 15, 2016) During the Middle Ages, the trebuchet became the dominant siege weapon. The designers that were the architects of destruction manipulated gravity in order to launch massive projectiles at their foe with devastating effects. We wished to study the mechanics of a counter-weight trebuchet at a much smaller scale and modify the system to construct a spring-powered trebuchet. We modified the flexible fulcrum used for the siege machines and ran analyses to better understand the motion of a trebuchet and understand the interaction of the forces throughout the entire system.

2 Physics 382, Student (2016) Advanced Lab v3.3, p.2 I. INTRODUCTION We have observed the impressive and destructive power of a trebuchet. One trade off of these magnificent siege machines is their massive size needed to launch the large projectiles and the great counter-weights that are needed with the assistance of gravity to launch the projectiles. Our purpose was to modify the design of the trebuchet to reduce the size requirements, but maintain it s impressive strength. In reducing the size of the machine, we are greatly reduced the weight and forces that could be applied to the system. The large counter-weight that most trebuchets used was a potential problem with this new design, which resulted in the replacement of the counter-weight with a spring to generate the necessary force to launch the projectile [1]. This greatly reduced weight for our system while maintaining high durability. The arm of the trebuchet needed to be redesigned after the reduction of weight. The arm needed to maintain a 5:1 length ratio between the throwing end and the counter-weight end, or the end that will be connected to the spring. The purpose of this was to produce an efficient fulcrum while also increasing the torque applied to the projectile, without putting equal amounts of force on the arm. While maintaining a focus on high strength and low weight, the optimal design was produced through a triangular truss design. The truss design has been used consistently through construction for high strength and low weight proportions for objects such as bridges radio towers. This hollow design allowed air to flow through it, due to a lower surface area than a standard solid arm. It reduces the drag coefficient which allowed for a more efficient transfer of energy into the projectile [3]. The triangular truss shape also provided a 30% reduction in air drag over the rectangular shape [2]. During the construction of the throwing arm, simple trials, and strain analyses using Inventor provided a deeper understanding of the interacting forces. Carbon fibre was originally the optimal material to provide the light weight to high strength relationship but too expensive [6].It, also, proved to be too flexible for optimal transfer of energy to the projectile not allowing for maximum distance. The optimal design was calculated and construction began. Materials continued to be a cause for a redesign of the throwing arm. In the end, a light-weight, spring-powered trebuchet was constructed and tested for it s range ability.

3 Physics 382, Student (2016) Advanced Lab v3.3, p.3 II. SETUP A. Carbon Fibre Design Fig. 1 shows our initial truss design consisting of thin, carbon fibre rods that were connected via rubber plugs along the 3 main beams. The initial setup of this truss revealed that cross braces would not be possible. Following initial tests, the flex of the carbon fibre truss was too large to transfer energy efficiently into the projectile. FIG. 1: The above design is an inventor generated model of the original truss design, constructed in carbon fiber. After running stress-strain analysis using Inventor, we found the relationship between truss side height and displacement of the truss when a Newton force was applied to it about a 10 inch pivot point which shown in Fig. 2. This strain analysis also showed that the addition of a X pattern of cross beams would provide minimum deflection. Unfortunately neither the X or V pattern was possible for construction with the material available for the carbon fiber. To justify the expenditure of higher quality materials, thorough analysis of the design was needed to provide good basis that it would function as our model suggested. B. Hollow Tapered Design From Fig. 3, the main part of the trebuchet was the launch arm with a truss system. There were 11 triangles with equal spaces, and the ratio of the longer arm to shorter arm

4 Physics 382, Student (2016) Advanced Lab v3.3, p Displacement (cm) Truss Spacing (cm) FIG. 2: The above plot shows the relationship between the side truss height and the truss displacement when a constant force was applied about a constant pivot point. was 5:1, The total length of the launch arm was 60 inches. Several triangles within the arm such as the pointed attached to the springs were reinforced with thicker wood. The length of side of triangles was decreasing from the shorter arm to the longer arm. The largest was 8.5 inches and the interval of each space was 7.5 inches. There were three diagonal struts across 4 spaces, which made the throwing arm more stable with higher strain constant. The pivot of the trebuchet was located at 1 foot from the end of the shorter arm. The base of the trebuchet was made of a square of wood and there were two pyramid structures on each left and right side to support the trebuchet. The length of the sling was 5 feet; it would release the load when the other side of the sling takes off from the hook. At the end of the longer arm, there was hook to control the releasing angle. The power of the trebuchet consisted of two springs with length of 0.35 meters, and each of them had spring constant with 280 ± 25N/m(95%CI, t dist.). The final design for the arm made in Inventor is shown in Fig. 3. With the pivot point at 12 inches in from the large end, more braces had to be input to support this. This design was modeled with a 3-dimensional frame in Inventor to better understand the forces. The frame constructed for this can be seen in Fig. 4. The corner of the arm was modeled with ANSI AISC pipes made of aluminum 6061-AHC with a dimension of 3/4 x Inventor did not have angle brackets that would orient properly for this purpose, so pipes were the best alternative. The triangular struts throughout the arm were constructed of wood, however, due to the an-isotropic nature of wood due to the grains, Inventor has no accurate strain

5 Physics 382, Student (2016) Advanced Lab v3.3, p.5 FIG. 3: This is a representation of the throwing arm generated in Inventor. This shows the dimensional construction design for the throwing arm. The variables listed above are a simplification of the dimensions. X1 though X10 are 7.2 inches long. X10 is hidden due to over-constraining and follows on the end of X9. X11 and X12 are products of the pivot point for the arm and are equal to 3.5 and 1 inch respectively. analysis data. This was therefore modeled as ABS plastic in ANSI AISC HSS square tubes with a dimension of 1 1/4 x 1 1/4 x 3/16. ABS was the best substitute for wood. The crossbeams on each side are modeled and constructed with ANSI AISC rectangular flat bars made of Aluminum 6061-AHC with a dimension of 3/4 x 3/8. Fig. 5 shows Inventor s strain analysis program on the hollow tapered throwing arm was run through. This model was run through a static analysis with 6 fixed points and an applied force of Newtons of applied force. The 6 pivot points are a result of the extra triangular struts used to support the pivot point of the arm. The applied force was an arbitrary number used constantly throughout all trials to obtain relative results for comparison. In this analysis, gravity was set in the z-direction. This was due to the fact that when the arm is transferring the force to the projectile where the arm is vertical. The alignment of gravity has dramatic effects on the strain analysis, and the design needs to be rigid enough to not fail upon release of the projectile. This truss design produced a maximum strain result of a inch deflection due to the applied force. The data provides sufficient evidence that this design will be usable for the throwing arm, however, it needs to

6 Physics 382, Student (2016) Advanced Lab v3.3, p.6 FIG. 4: This is a representation of the throwing arm generated in Inventor. This shows the frame construction generated for strain analysis to understand the forces that act upon the arm and obtain an optimal design for construction. The three long beams are modeled in Aluminum 6061-AHC pipes. The triangular sections throughout the arm are modeled as ABS square tubes, and the three crossbeams on each side are modeled as aluminum 6061-AHC flat bars. be noted the difference in materials between the frame design and actual construction. III. MODEL Fig. 6 shows the system set up that will be used for the Lagrangian model as well as construction. This diagram shows the functionality of our trebuchet, with the tension of the spring providing the force to launch the projectile. The model of the trebuchet is based on a Lagrangian Equation of motion with two generalized coordinates in the system. The position of the of load is in both x and y components is x(t) = ( L 2 sin θ(t) L 3 cos (ψ(t) ( π 2 θ(t)))), (1) y(t) = ( L 2 cos θ(t) + L 3 sin (ψ(t) ( π 2 θ(t)))), (2) where θ is the angle between the shorter arm and the vertical line, and ψ is the angle between the longer arm and sling from Fig. 6A. In order to solve for the Lagrangian equation of the projectile, the kinetic energy and the potential energy needed to be determined. The total potential energy of our concerned system consisted of the potential energy of the spring and the gravitational energy to get U = 1 2 k s2 + mgy(t), (3)

7 Physics 382, Student (2016) Advanced Lab v3.3, p.7 FIG. 5: The figure shows the Inventor Strain Analysis representation produced, showing the deflection of the arm under a controlled and constantly applied force to one end. where k is the spring constant, s is the length change of the spring, m is the mass of load and g is gravitational constant of 9.8m/s. The kinetic energy was calculated to be T = 1 2 m(x (t) 2 + y (t) 2 ). (4) Now, the Lagrangian equation can be created by L = T U. (5) In the Lagrangian equation, the motion equation can be obtained by L q = d L dt q (6) where q is the generalized coordinates of θ and ψ. To check the model, the domain check was used by setting the spring constant to 0. When there is no spring in the system, and the load was put at the vertical, the load fell on it s own. As the Fig. 8 shown, we can see the angle change during the falling of the trebuchet. As the Fig. 7 shown, it indicates the angular velocity of these tow angles. As the launch

8 Physics 382, Student (2016) Advanced Lab v3.3, p.8 FIG. 6: This figure contains all three perspectives of the trebuchet system with the appropriate parameterized measurements. Fig. A contains a side view of the trebuchet. θ is the angle of the short end with respect an imaginary y-axis along the post. ψ is the angle of the long arm below another imaginary x-axis, from the arm. Fig. B shows the top view of our trebuchet system with appropriate parameterized measurements. Fig. C shows the front view of the trebuchet system with appropriate parameterized measurements. arm rotating, θ was increasing with an increasing angular velocity; and ψ was decreasing with a slightly decreasing angular velocity. When the spring reached the equilibrium point, θ would decrease, along the angular velocity of θ decreasing to zero. As the angle ψ, also, reached the equilibrium point, it opened to release the ball. FIG. 7: The plot above shows the angular velocity of θ and ψ.

9 Physics 382, Student (2016) Advanced Lab v3.3, p.9 FIG. 8: The plot above shows domain check of the model. The angle ψ is increasing and θ is decreasing without spring. IV. DATA This siege machine was modeled with a Lagrangian equation of motion to predict the energy of the system. Preliminary results from the first 2 designs were produced via swinging the arm free of the base to track the movement of the projectile, as well as run durability tests of the throwing arms. The first aluminum arm design proved to be stronger and more effective than the carbon fiber design. However, it still failed and needed to be reinforced. The final arm design can be seen in Fig: 4. Once the siege machine was fully constructed we began full scale testing. The results of this are shown in Fig:9. The lack of a visible relationship between the data points originates from testing performed on multiple dates. Although this was a relatively small trebuchet, it had to be dismantled for storage. This led to minor changes in construction and alignment, as well as inconsistencies in video recording for data analysis. The trials run on February 26th were run with the least amount of changes. Here the only changes made were release angle and the implementation of a safe release mechanism. While this was a minor change that would offer more consistency, however it did affect the system. The siege machine did achieve an average release velocity of ± 0.25 m/s 95% C.I. T Dist. There is a list of concerns that we believed to be the source of inconsistencies for data collection method. The inconsistent construction of the base, while being minor, potential caused slight changes in the axle alignment, which affected the launching ability. Also, due to the construction of the throwing arm, it had a unnatural curve. This may have increased

10 Physics 382, Student (2016) Advanced Lab v3.3, p Distance m Launch Angle Degrees FIG. 9: The plot above shows all distances achieved in correlation to their release angle. the drag of the arm compared to an ideal, straight design. We also designed a safe, consistent release mechanism that would provide consistent method of launching. While it worked well, past an angle of 25 degrees from the horizontal, the sling for the projectile slid behind the chain that locked the arm in place while the pouch was loaded. As the arm was released, there was a minor interaction between the chain and the pouch which potentially altered the release of the ball altering the flight path of the projectile. These concerns require further investigation to know their true effect on the system and potentially remove or minimize their effects. V. ANALYSIS Using the above ideal model, we can accurately determine the motion of the trebuchet and the forces need to launch projectiles of different masses at different distances. This model allowed us to start initial tests of the truss design to determine stronger designs that will only allow minimal axial displacement of the arm about a pivot point. The first truss had high flex and had a high relative displacement value. The second design with aluminum struts, wooden inserts, and aluminum cross braces was much more rigid and seemed more promising in terms of throwing distance. This arm still failed under stress. The third throwing arm design included several reinforced wooden inserts, to withstand greater torque. We determined the flexible design did not efficiently transfer the energy into the

11 Physics 382, Student (2016) Advanced Lab v3.3, p.11 projectile and needed the more rigid arm. The optimal triangular truss design included an X pattern for the cross members. However, due to the restrictions of our materials, we were only able to produce and elongated, alternating V pattern. This is not the optimal design, however, it does offer higher strength properties as compared to without the cross members. This coupled with our reinforced members provided a strong arm that did not fail under strain. This can be seen in the deflection of the lofted arm design in Fig:5. Also, using the plot of the motion of the projectile during the launching process, we were able to determine the angle of release from the slope of the linear portion of tracked motion. This angle was adjustable to get closer to the ideal 45 angle of release, however, our system produced a maximum angle of release around 27. This data is shown in Fig: 9. This data is figured with an angle uncertainty of ±0.16 (95% C.I. Rectangular P DF ) along with a distance uncertainty of ±0.14m 95% (C.I. Rectangular P DF ). Using Fig: 6, we are able to use the relationship between the two angles to ensure that our model agrees with the motion of the trebuchet throwing arm and sling. VI. CONCLUSION During this experiment, we were able measure and analyze truss designs for trebuchets, as well as produce a functional siege machine. Using different designs, we were able to determine a design that had the smallest deflection distance from strain analysis. Also using strain analysis, we were able to find the relationship of the side truss height to reduce the deflection distance. This provided a guided plan for construction. From the Lagrangian, the theoretical motion and the relationship plots of the angles agree with the motion of the initial tests which provides ample evidence that our equation of motion is a good model and allows for the calculation of the forces acting upon or within the trebuchet. For future research, more research could be done on the optimal design and the effects of materials, shape, and spacing. These findings should also be held in precision during construction. Given an optimally constructed arm, the base can be constructed around the arm to produce balance and control to handle the strength of the launch method. More research and testing should be done to find the optimal angle for the release pin for the top side of the sling with relationship to angle of the truss arm to the horizontal. Our design had dramatic effects on the range and efficiency of the trebuchet, however, there are always ways for improvement.

12 Physics 382, Student (2016) Advanced Lab v3.3, p.12 ACKNOWLEDGMENTS We acknowledge the help and support of the Wabash College Physics Department. [1] Denny, M. (2005). Siege engine dynamics. European Journal of Physics Eur. J. Phys., 26(4), [2] Durfee, R. H. (1986). Review ofhttps://preview.overleaf.com/public/nrpzwdrdkybb/images/3cbd7c66cef7ee9bc Triangular Cross Section Truss Systems. Journal of Structural Engineering J. Struct. Eng., 112(5), [3] Functional Materials for Smart Gossamer Spacecraft. (2006). Recent Advances in Gossamer Spacecraft, [4] Gyula Greschik, Truss Beam with Tendon Diagonals: Mechanics and Designs, AIAA Journal, Vol. 46, No.3,pp (2008). [5] Krenk, S., & Hgsberg, J. (n.d.). Statics and mechanics of structures. [6] Schutze, R.. Lightweight Carbon Fibre Rods and Truss Structures, Materials & Design, Vol 18, pp (1997). [7] Zhang, D., Huang, Y., Zhao, Q., Li, F., Li, F., & Gao, Y. (2014). Structural Performance of a Hybrid FRP-Aluminum Modular Triangular Truss System Subjected to Various Loading Conditions. The Scientific World Journal, 2014, Retrieved January 25, 2016.