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Torque Required for Cat Righting Reflex

A Strength of Materials Problem Written and Solved for MET406

By

Kristina Lawyer

On

1 December 2014

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Problem Statement Cats have an innate ability to orient themselves when falling, known as the Cat Righting Reflex. How much torque is required to execute this reflex for the average cat falling from a height of 3ft?

Given ℎℎ,ℎ = 3

Find ,

Assumptions 1. The cat’s mass is 12lbs,  = 12. 2. The cat will be modeled as 6 rectangles and 1 circle as shown in

Figure 2. The cat is 11” tall (ground to top of head) and 15.675” long.

3. The head and body of the cat is 3” wide (into the page) while the four legs and tail are 1.5” wide.

4. Angular acceleration is uniform and entirely about the x-axis. 5. Density is uniform.

Method 1. Determine the amount of time it takes the cat to fall the given

distance. 2. Find the angular acceleration required for the cat to turn 180° in

that amount of time. 3. Calculate the Centroid and Moment of Inertia of the cat. 4. Find the torque needed for the cat righting reflex.

Figure 1: Sequential Photos of the Cat Righting Reflex

MET406 Topic(s) This problem utilizes the concepts of centroids and moments of inertia from chapter 6 of the MET406 textbook.

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Solution Step 1:

 =  +  + 1 2



3 = 0 + 0    +

1 232.2

 

 →  = 0.432

Step 2:

 = ∆ ∆ =

  0.432 = 7.278

 

 = ∆ ∆ =

7.278 0.432 2

= 33.719  

Step 3:

Figure 2: Geometric Model of Cat for Calculating Moment of Inertia

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4 5 6 7

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The location of the centroid of each of the seven parts is estimated from the model. Xi is the distance from the reference y-axis (dotted line) to the centroid and yi is the distance from the reference x-axis (dotted line) to the centroid. Ai is the area of each part.

Part Ai (in 2) xi (in) yi (in) Aixi (in

3) Aiyi (in 3)

1 8.553 14.025 9.350 119.956 79.970 2 68.063 6.188 6.600 421.137 449.213 3 4.991 0.413 12.375 2.059 61.767 4 4.235 11.825 1.925 50.079 8.152 5 4.235 8.360 1.925 35.405 8.152 6 4.235 2.420 1.925 10.249 8.152 7 4.235 0.550 1.925 2.329 8.152 Total 98.514 641.213 623.559

The centroid of the composite shape (whole cat) is found by

 = ∑ 

= 641.213 98.514 = 6.509

 = ∑ 

= 623.559 98.514 = 6.330

where  is the distance to the composite centroid from the reference y-axis and  is the distance to the composite centroid from the reference x-axis.

Part Ixi (in 4) fi (in) Aifi

2 (in4) Ixi + Aifi 2 (in4)

1 5.821 7.518 483.458 489.279 2 171.574 0.319 6.934 178.508 3 15.224 6.094 185.371 200.595 4 5.231 5.318 119.785 125.016 5 5.231 1.853 14.546 19.777 6 5.231 4.087 70.729 75.960 7 5.231 5.957 150.267 155.498 Total 1244.633

Part Iyi (in 4) di (in) Aidi

2 (in4) Iyi + Aidi 2 (in4)

1 5.821 3.022 78.133 83.955 2 868.594 0.272 5.052 873.647 3 0.283 6.047 182.538 182.822 4 0.427 4.403 82.085 82.512 5 0.427 4.403 82.085 82.512 6 0.427 4.403 82.085 82.512 7 0.427 4.403 82.085 82.512 Total 1470.469

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The distance from the composite centroid to each part’s centroid along the x-axis is fi ( =  − ). The distance from the composite centroid to each part’s centroid along the y-axis is di.

The total moment of inertia is found by

 =  +  +  +  + ⋯ = 1244.633

 =  +  +  +  + ⋯ = 1470.469

Step 4:

 = 

Torque is the product of the mass moment of inertia and the angular acceleration. The mass moment of inertia can be estimated with the product of the area moment of inertia and the density. The total volume of the cat is estimated by

 = 38.553 + 68.063 + 1.54.991 + 4.235 + 4.235 + 4.235 + 4.235 = 262.745

 =  = 1244.633 12

262.74533.719 = 1916.7 −   159.7 − 