The principle of superposition, a powerful mathematical technique for analyzing certain types of complex problems in many areas of science and technology, has important applications in creating centripetal force, club head speed and golf swing modeling of SGM. The principle of superposition states that problem solutions can be added together to obtain composite solutions.
The trebuchet is a compound machine that makes use of the mechanical advantage of a lever to throw a projectile far away. They are typically large constructions, and use a long arm to throw a projectile. It was a common powerful siege engine until the advent of gunpowder. A trebuchet consists primarily of a long beam attached by an axle suspended high above the ground by a stout frame and base, such that the beam can rotate vertically through a wide arc. A sling is attached to one end of the beam to hold the projectile.
Engineering of Trebuchet
The trebuchet, although designed by engineering hundreds of years ago, uses complex physics to ensure a high-velocity, long-range launch of a projectile. It is a catapult that makes use of a falling counterweight, a swinging arm and a sling to throw a projectile.
The device consists of a framework that supports a freely revolving launch arm. On one side of the arm, a heavy counterweight is attached; on the other, a long sling containing the projectile. When the trigger is released, the counterweight falls, the weighted short arm is driven by gravity into an accelerating pendulum motion, causing the lighter, long arm of the lever to revolve around the fulcrum at the opposite arc, which in turn, pulls the sling and its contents into a whipping motion at the end of the long arm, forcefully propelling the side of the arm with the sling into the air. The sling then spins in a circular motion around the end of the arm(as the center of a circle and the sling is radius) , further increasing the speed of the projectile before it is released.
A trebuchet uses a lever for its throwing arm, which is attached to a sturdy base by an axle. Trebuchets throw projectiles (these usually sit inside a sling at one end of the throwing arm) by quickly rotating the arm along its axle. Since the sling is attached to the longer end of the arm and opposite to where the force is applied at the fulcrum of the lever, the trebuchet takes advantage of leverage—also called mechanical advantage—to throw a projectile.
One key to the trebuchet’s success is the positioning of the projectile, or payload, sling, and counterweight on the throwing arm. The throwing arm is essentially a basic lever – one can understand how it works through the principle of mechanical advantage. As seen in the diagram below, the counterweight rests much closer to the fulcrum than the sling. During a launch, the counterweight drops, exerting torque on the lever. Through mechanical advantage, a slower counterweight-end velocity creates a much faster payload-end velocity. As a result, a counterweight with a large mass, which can exert greater force on the lever, creates incredibly fast motion on the payload end of the lever.
Superposition in Trebuchet: The peak of two circles
Along with using simple machines, trebuchets rely on gravity, potential energy, and rotational acceleration to loosen projectiles from a sling. For other words, there are two circles form a bigger wave: constructive interference. The first circle: the center is fulcrum and throwing arm ( from fulcrum to the point of connecting with the sling) is radius; the second circle: the center is the meeting point of throwing arm and sling, and radius is the sling. As the counterweight falls down, the throwing arm and the sling both then spin in a circular motion; the throwing arm around the fulcrum and the sling around the end of the arm.
The sequence is: the counterweights’ falling down triggers the turning of the throwing arm and then the throwing arm leads the sling to move up and rotate at same time — actions of both circular motions accumulate a huge wave to throw the projectile far away.
The theory of superposition, which states that the solutions to individual parts of a problem can be added to solve composite problems, is explained in most books on advanced calculus or differential equations.
Projectile speed is multiplied even more by the addition of a sling on the end of the throwing arm. When launched, the force of the launcharm’s motion pulls the sling taut, and the circular direction of the throwing arm’s motion results in circular motion of the taut sling.
To understand the principles of circular motion: for an equal force applied, a longer length of string produces a greater velocity of the weight at its end. F= m*V*V/r
The end of the sling containing the projectile moves much faster than the end of the throwing arm due to the principles of circular motion. The projectile is thus launched incredibly quickly due to the principles of mechanical advantage and circular motion.
As the trigger is released, the balance structure is destroyed, and the counterweight force accelerates and creates the angular velocity; then the long arm and the sling create the tangential velocity. As the falling force becomes vertical with the ground, both the long arm and the sling gain the maximum scale of pulling force or say the max resultant force by which the max tangential velocity is created . It is the same for the impact zone of golf swing and many sports like baseball and tennis have this maximum resultant force at some point.
Superposition principle applied on golf swing
If you turn the photo of turning the trebuchet upside down, you will get the golf swing drawing of arms and club.
The only difference of trebuchet and golf swing is the power source: for trebuchet is from counterweight which is set up as potential energy (static) and for golf swing the power is from backswing during which the elastic force is restored (dynamic).
Both trebuchet and golf swing comply with superposition of two circles:
- For trebuchet: the both centers of the circles are the fulcrum of the lever and the end of the lever’s long arm; the radius are the long arm and the sling.
- For golf swing: the both centers of the circles are the spine of the golfer and the wrists of the golfer; the both radius are the left arm and the golf club.
- For more detail we can say there are three circles of superposition for golf swing, the third circle center is the humeral head of shoulder.
Update of Trebuchet’s engine power for golf swing
For golf swing, we don’t have the trebuchet’s counterweight power and the lever system, so the power source of this machine is different from the golf swing even though the system of long arm and sling is adapted by golf swing. But there is a machine named Onager is more like a mechanical structure of golf swing: instead of the counterweight, the power source of Onager is powered torsion or elastic force, just like golf swing applies the torsion force restored in the torso during the backswing.
Considering both trebuchet and Onager, there are many ways to go for increasing the clubhead speed during swing:
- Restore more torsion force during backswing. Sure it’s easier to say than do. There will be details on other places later, but here I just can say we should focus on the torso action during backswing instead of the arm’s position.
- The throwing arm can be compared to our shoulders and arms: it should be strong and as long as possible;
- The sling is compared to the club in hands: it is with the action of arms, never over the arms before superposition; the wrist is the point of the throwing arm meeting the sling, so there is no action of wrists during downswing and impact zone;
- The constructive interference should happen at impact zone: the pulling forces on one line and the two peaks of circle speeds superposition. At this point we should focus on the power source: the centripetal force of the bigger circle ( neither counterweight nor powered torsion);
- Feeling carefully of the forces in torso and following the rhythm are crucial for letting superposition happen;
- The concept of three circles is even better: the torso turning is first circle, then arms circle from left shoulder and third one is the club from wrists; this way means the superposition of three tangential speeds of circles.
The power source of generating golf clubhead speed is the elastic force restored in the twisted torso.