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### MODELING A HORSE’S GAIT

Dr. Eng. Alejandro Gutiérrez S.
Dept. of Mechanical Engineering
University of Santiago of Chile

Introduction:

There are three different states or forms of support currently recognized for when a horse is in motion; these are the step, trot and canter. Each is distinguished by the number of limbs that are in contact with the ground at a given time and the duration of suspension in the air. In a walk for example, at least one limb is raised with the other three supporting and therefore no time is spent suspended. The trot on the other hand is composed two legs supporting the horse in diagonal pairs, with a brief drop and rise when alternating pairs. Finally, a gallop is distinguished by containing a suspension phase during which all four limbs are off of the ground. This basic observation with regards to a limbs contact with the ground has given rise to a mathematical model that seeks to quantify the effects caused by a horseshoe within the complex system of a horse’s gait. The model presented simply involved mechanical factors such as weight, impact strength, position, speed, and acceleration; this means other factors were not considered such as race, specialty, and age. The whole problem has been condensed into the kinetic interaction of related bodies somehow representing the horse as a whole.

Objective:

This study aims to quantify in comparative form the structural effect produced by the use of a shoe composed of an iron alloy and another composed of a copper alloy, the structural effect is based on knowing the forces that are generated in a horseshoe in the same state of motion, in this way it is possible to determine the comparative behavior, this method has been developed because the behavior of an iron horseshoe is widely known and therefore can be directly compared. To achieve this goal the following specific objectives have been developed.

• Physical Model
• Mathematical Model
• Simulation

The results of this model will define which of the two shoes has a better behavior from a purely structural point of view. The term better behavior is based on the following conditions:

• The forces are transmitted should be as small as possible
• The power transmitted should be as small as possible

If these criteria are met, it will be possible to determine whether one shoe is better than the other.

Methodology:

The developed model is presented in this section; the following considerations have been made in its development:
• In any of the three stages of motion for a horse, at least one extremity is used for support in some phase of the cycle. For the time being we will look at one support.
• In any of the three stages of motion for a horse, a limb makes contact with the ground at a given velocity increasing in value from a walk to a trot to canter.
• It is considered that the largest components involved are the horseshoes, the hoof up to the first joint, and the overall mass of the horse.

As a result of these hypotheses a model has been developed that can be seen in Figure 1, note that there exists strong compression forces in the study with respect to the area around the horseshoe, transforming the entire model in a system with only three degrees of freedom. Figure 1 – Modeling a horse’s gait

Key:
– Mc Horse’s mass
– Mp Mass of the hoof up to the first joint
– Mh Mass of the horseshoe
– kc Rigidity associated with the horse
– Cc Dampening associated with the horse
– kh Rigidity associated with the horseshoe
– Ch Dampening associated with the horseshoe

The values of each of the constants used have been derived from experimental measurements of elastic modulus and coefficients of restitution, which have been modified according to their geometry.

From this model, two versions arise, each one distinguished solely by the materials used in the horseshoe. Keeping all other variable constant, in this form we are able to draft comparative values in an objective form.

Given the characteristics of the model and the comparison between them, it is seen that the best test would come in the form of a fall of mass on a surface that is considered completely rigid; bodies to be joined by springs and dampers transmit forces from one element to another, they will then be quantified allowing comparative behavior to be determined, the results are made dimensionless in order to generalize them.

The resulting formulas correspond to a system of second order differential equations whose forcing function is generated upon impact with a platform that is considered to be completely rigid. Final Results:

The results obtained refer the forces generated between each of the bodies; the first being those between the ground forces and horseshoe, the second between the horseshoe and the hoof, and the third between the hoof and the rest of the horse. It again shows that the models are applicable to both types of horseshoes, considering all boundary and initial conditions similar; the remaining variable is the mechanical composition of each horseshoe.   Figure2
– Percentage change in the interaction forces of the bodies

Table 1 shows the maximum values in each of the interactions of bodies (Tables remain in their original Spanish forms, y axis being percentage forces, x axis showing time in seconds) Conclusions:

Given the results it is possible to obtain the following conclusions:

1. The copper horseshoe produces a force transmission of a lesser magnitude than that of the iron horseshoe according to the model.
2. There is a slight attenuation of the magnitude of the force as you progress through the union of elements.
3. The value of 8.75% attenuation at the peak level of force significant because the process is designed to model a cyclical action that may last a considerable time.
4. The interpretation of this result would mean that through the use of a copper based alloy in a horseshoe that reduces the initial and reactionary forces that one is preventing these forces from being felt by the horse.
5. Viewed from an energy standpoint, holding all other values constant, we find that a horse using copper based horseshoes uses less energy that one using an iron horseshoe.
6. Based on the previous conclusion, from a purely biomechanical point of view, reduced energy usage can lead to natural reserves being used to increase performance.
7. It should be noted that this is a simplified model based on a rigid surface cycle.

Comparative analysis of copper and iron alloys through Charpy impact and wear of the alloy of copper and the iron.