Taiji and Mechanics

When Master Chen demonstrates push hands and he “pushes me out”, I sometimes feel subjected to an overwhelming push or pull. More often I feel subjected to a combination of a force and a strong twist. This combination seems to be even more difficult to counter act.

The branch of mechanics that seems suitable to describe Taiji push hands is statics. Statics deals with the requirement for stability of structures like buildings and bridges. (What happens after we have lost our balance is not the topic of this discussion.)

Statics proclaims that for a (human) body to be at rest, all forces acting on it have to be cancelled.  This is simple when we are dealing with a sufficiently small object (a particle): all forces act on a single point. Remember, forces have a vector nature: they have a magnitude as well as a direction (push or pull) on an action line.  So for a particle to be at rest, the (vector) sum of all forces has to be zero.

On a real body however forces are applied on different locations or more precisely the action lines of these forces do not all go through the center of mass. When a force does not go through our center of mass, it will not only try to move us along the action line, it will also try to rotate us. This rotational effect associated with a force is called a torque (or more accurately the moment of a force). For a real body to stay at rest, again all forces have to cancel each other (or the vector sum of all forces has to be zero), but all torque has to cancel as well. An example of a pure torque is a simple couple.  A couple is formed when two opposing but equal in magnitude forces have parallel but different action lines.  A pure torque will rotate us around our center of mass. The center itself will not move.

Torque also has a vector nature: the magnitude is the product of force size and the moment arm (distant from the center of mass) and the action line is the axis of rotation.

Let’s review the forces that act on us during push hands. We can distinguish:

•    The forces in each contact point with the opponent. Remember in Taiji we are encouraged to establish as many points of contact as practical.
•    Our weight. ( We are told to be “stingy” when we apply this force on the opponent)
•    The reaction forces from the floor.  These are the forces we manipulate to counter act the former ones.
The reaction forces of the floor can be further subdivided in vertical and horizontal components. The vertical components counteract our weight and whatever vertical force (lift or downward push) the opponent imparts on us. How these vertical reaction forces are distributed between both feet depend on the position of our center of mass and whatever torque that has to be countered.

To counteract the horizontal forces imparted on us by the opponent we rely on the friction of our feet with the floor. These are the horizontal reaction forces. Note the vertical reaction forces with the floor cannot cancel a horizontal push.  For example if we stand on a slippery surface, an opponent can always move us with easy.
About forces.
•    A force can act anywhere on its action line without changing effect.
•    If we move a force laterally and apply it on a different location it will have a different effect unless we add a torque to compensate for the move.
•    Every force between two bodies creates an opposite but otherwise equal reaction force.  If we apply a force on an opponent we receive an equal but opposite force in return. The same happens between us and the floor.

How do we create a strong force (push)?
We are told that in Taiji a strong push is generated through structure and not by means of momentum or muscle strength. We can describe this in simple mechanics terms:
•    A connection is established between the hand   (in firm contact with the opponent) and the rear foot.
•    This connection consists of a number of joints connected by solid bone segments.
•    We can visualize these solid segments zigzagging along an imaginary straight line.
•    The more the “zigzag” approaches the straight line, the more the connection becomes like a single rod capable of withstanding a large force in the longitude direction. This happens when the angles between the bone segments approach 180 degrees.
•    A strong push is created by the final straightening of all joints in a “synchronized” and “proportional” manner.
•    Through simple vector analysis we understand that the force in the longitudinal direction, generated by two bone segments straightening becomes exceedingly large as the angle approaches 180 degrees. At the same time the lengthening becomes exceedingly small. Therefore all joints have to straighten at the same time (synchronized) to achieve the desired cumulative effect.  The small and weak joints (the arm and shoulder) are subjected to the same force as the strong joints (hip and waist) and will collapse if the timing is off (the effect is lost).

About torque.

•    The effect of a torque on our opponent is independent of where we apply it. (I am not sure if this property is relevant to Taiji.)
•    To impart a torque on our opponent we require a minimum of two contact points. The larger the distance between the contact points the easier it is to apply a large torque on our opponent. After all the magnitude of a torque is determined by the size of the forces times the distance between them.
•     If we have multiple contact points we have the potential of imparting torques in different directions. For example, depending on the nature and position of the contacts, four contacts allow us to apply up to 6 different instances of torque (six different combinations of two contact points) on the opponent. Does this explain why we are encouraged to establish multiple contacts with the opponent?

•    Even if both a force and a torque can be represented as vectors, they are different entities and have different effects on an object. If we apply a single force through the center of mass of an object (say we find ourselves in space), the object will start moving in the direction of the force without rotation. If we apply a pure torque on the same object, it will start spinning around an axis parallel to the torque, but the center of mass will stay in place.

•    Every torque between two bodies creates an opposite but otherwise equal torque as reaction. If we attempt to twist our opponent then we will be subjected to the same torque in the opposite direction. We will have to cancel this reactive torque with our feet against the floor, lest we lose our balance.
How do we create a torque?

In practical life an electrical motor provides a (pure) torque to a load via a drive shaft. To make the analogy more relevant to push hands, imagine that the load is larger than the torque delivered by the motor and the rotor does not rotate (stalled rotor). The motor is at rest because the motor feet are bolted down to the floor and the reaction forces provide a torque equal to the torque delivered to the load. (If the bolts were to snap the motor stator would break free and start spinning. Torque to the load would be lost).

What is the human equivalent of this electrical motor? There are the two structures that are mentioned often in Taiji: the hip and the waist.
The hip joints are ball joints spaced by the pelvis. The largest muscles in the body act upon them. Ball joints allow forces to act in virtually every direction. Therefore the hip seems the ideal structure to create powerful torque in different directions. (It has been compared by Master Chen to the yoke of a universal joint connected to a drive shaft)

The waist as well has a large number of “freedoms” and is powered by the strong core muscles. It can twist and turn in every direction.
The torque created by the hip or the waist (or combination) is transmitted by the upper body (the drive shaft) via the contact points (minimum two as mentioned before) to the opponent.

It seems to me that for a torque to be effective the opponent has to be “trapped” somehow. For example he is prevented to reposition his foot disallowing him to counter the torque he is subjected to.  Another example is when the torque overwhelms a smaller and vulnerable joint. As a result the opponent will let his structure collapse to save the joint.