Deep Knee Bends: Measuring Knee Stress with a Mechanical Model
AbstractAre you interested in things like prosthetic limbs and artificial joints that can help people with disease or injury to lead a normal life? Or maybe you're interested in sports medicine or physical therapy? Either way, this project could be a good match for you. Find out how the tension on the knee joint changes as a function of angle by building a simple mechanical model.
The goal of this project is to estimate the relative amount of strain put on the patellar tendon by various bending angles of the knee joint.
Andrew Olson, Ph.D., Science Buddies
This project is based on the following 2007 California State Science fair project, a winner of the Science Buddies Clever Scientist Award:
- Lefley, A.G., 2007. "Pliés to Perfection: The Force on Your Patellar Tendon," [accessed June 12, 2007] http://www.usc.edu/CSSF/History/2007/Projects/J0217.pdf.
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This project is based on a 2007 California State Science Fair project by Annie Lefley (Lefley, 2007). Annie is a ballet dancer, and she noticed that grand pliés (in which the knee is almost fully bent) put more strain on her knees than demi pliés (where the knee is only partially bent). Although ballet is much more graceful, you can think of it as the difference between a deep-knee bend and a half-knee bend.
Annie wanted to find out how much the strain on the knee changed as the angle of the knee became more acute. To find out, she built a cool mechanical model of the knee joint (see Figure 2, below). In order to understand the model, you should study up on the anatomy of the human leg and knee joint.
Figure 1 shows two views of the knee joint from the 1918 edition of Gray's Anatomy of the Human Body. On the top is the view of the joint from the front, and on the bottom is a sagittal section (cut-away view, showing the knee at mid-section). The knee is the joint where the femur (thigh bone) and tibia (shin bone) come together. The patella (knee cap) "floats" in front of the joint. It is held in place by tendons (which connect the ends of muscles to bones) and ligaments (which connect bones to one another). For the purposes of this experiment, the two main attachments for the patella are:
- The quadriceps tendon (on the top), which connects to the main thigh muscle
- The patellar ligament (on the bottom), which connects to the tibia
Figure 1. Two illustrations of the human knee joint. On the top is an anterior (front) view. On the bottom is a sagittal (vertical) section through the middle of the knee
When you bend your knee to squat down (the ballet dancers out there among you can perform a plié, the rest of us can stick to what we know!), you can feel the tension increase in your quadriceps muscle (on the front of your thigh). The quadriceps tendon attaches to the patella, which is in turn anchored to the tibia by the patellar ligament. The tension of the quadriceps muscle holds the femur in position as the knee is bent.
In the mechanical model, a spring is used to represent the quadriceps muscle. A spring has an interesting property discovered by the physicist Robert Hooke, and now named for him (Hooke's Law). Here is the property: the restoring force (F) produced by the spring is proportional to the distance by which the spring has been lengthened (x). In equation form, this looks like:
F = −kx.
The equation says that the force (F) of the spring is equal to the spring constant (k, a measure of the stiffness of the spring) times the distance (x) that the spring has been stretched. The minus sign says that the force is exerted in the opposite direction of the stretching. In other words, if you stretch the spring out, the spring force is pulling back in the other direction. You'll see how you will be using Hooke's Law in just a minute, after we've described how the model of the knee works.
In Annie's model (see Figure 1) the femur, tibia and foot are represented by pieces of wood. The joints between the bones are hinges. The quadriceps muscle is represented by a spring. The patella is represented by a strip of metal, and the quadriceps tendon is represented by nylon guitar string. As the angle of the knee joint is changed, the guitar tuning mechanism is used to adjust the tension on the "quadriceps" spring. When the tension is at equilibrium, the joint will maintain its position. If the tension is too low, the joint angle will increase (knee unbends). If the tension is too high, the joint angle will decrease (knee bends more).
Note: you may have noticed that the "patella" in Annie's model is not "floating" like the real one, and the real leg has more muscles than just the quadriceps. However, the point of a model is to simplify the problem and to capture its essentials.
Figure 2. Mechanical model of the knee joint constructed for this project. The model has two main parts. The first part is the supporting frame, which consists of a horizontal base and vertical supports. The second part is the model knee, which consists of three separate pieces of wood (representing the femur, tibia/fibula, and foot). The pieces are hinged together and attached to the frame via another block of wood (representing the hip) which can travel up and down along the vertical support. A single spring represents the quadriceps muscle, and the quadriceps tendon is represented by a nylon guitar string. The guitar string is attached to the spring at one end and to a guitar tuning mechanism at the other. The tuning mechanism is used to adjust the tension on the spring. (Lefley, 2007)
So how does Hooke's Law comes in to this? You can use your knowledge Hooke's Law to measure the relative force in the "quadriceps" spring as the angle of the knee is changed. First you measure the length of the spring at rest. Then, you set the angle of the knee, and use the guitar tuning mechanism to adjust the tension on the spring until you reach equilibrium (the joint is not moving one way or the other). Measure the new length of the spring. The difference between the spring's length under tension and its length at rest is the x from Hooke's Law. It is important to note from Hooke's law that force is directly proportional to displacement. So, you can still see the relative changes in force (i.e. which knee angles cause higher or lower forces) without even knowing the spring's k value. If you would actually like to calculate force instead of just measuring displacement, you will need to look up how to measure your spring's k value. Refer to the Science Buddies project Applying Hooke's Law: Make Your Own Spring Scale to learn more about that process.
Terms and Concepts
To do this project, you should do research that enables you to understand the following terms and concepts:
Anatomy of the knee and leg:
Physics of springs:
- Hooke's law,
- Spring constant
- Elastic limit
- How does the force produced by a spring change as the spring is lengthened?
- What happens if you exceed the elastic limit of the spring?
- Here is a brief introduction to Hooke's Law:
Krowne, A., 2005. "Hooke's Law," PlanetPhysics.org [accessed June 12, 2007] http://planetphysics.org/encyclopedia/HookesLaw.html.
- You might also be interested in the Science Buddies resource, Stress, Strain and Strength.
- This webpage is the source of the diagrams of the anatomy of the human knee used in the Introduction:
Wikipedia contributors, 2007. "Patellar ligament," Wikipedia, The Free Encyclopedia [accessed June 12, 2007] http://en.wikipedia.org/w/index.php?title=Patellar_ligament&oldid=109343191.
- For more advanced students, this high school physics tutorial on vectors and forces can help you to analyze the forces on the knee joint in the model:
Henderson, T., 2004. "Vectors and Motion in Two Dimensions," The Physics Classroom, Glenbrook South High School, Glenview, Illinois [accessed June 12, 2007] http://www.physicsclassroom.com/class/vectors.
- This project is based on the following 2007 California State Science fair project, a winner of the Science Buddies Clever Scientist Award:
Lefley, A.G., 2007. "Pliés to Perfection: The Force on Your Patellar Tendon," [accessed June 12, 2007] http://www.usc.edu/CSSF/History/2007/Projects/J0217.pdf.
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Materials and Equipment
To do this experiment you will need the following materials and equipment:
- Power drill and bits
- Saw (hand saw or circular saw, for cutting wood to size)
Wood for supporting frame:
- Base: 11 x 11 x 3/4 in
- 1 20 x 6-1/4 x 3/4 in (back)
- 2 20 x 2-1/2 x 3/4 in (sides)
- 2 sections of metal U-channel, 3/4 in wide, 20 in long (forms tracks for casters)
Wood for knee model:
- "Femur:" 6-1/2 x 1-1/2 x 3/4 in
- "Tibia/fibula:" 6-1/2 x 1-1/2 x 3/4 in
- "Foot:" 2 x 1-1/2 x 3/4 in
- "Hip:" 5 x 1-1/2 x 1-1/2 in (casters are attached to this piece so that it can slide up and down in U-channels)
- 4 hinges, 1-1/2 x 1/2 in
- 1 guitar string tuning mechanism, (e.g., part number GTM21 from Elderly Instruments)
- 1 nylon guitar string
- 4 nylon furniture casters, 1-1/4 in diameter
- 1 metal strip 2-1/2 x 1 x 1/8 in
- 1 sturdy cup hook or hook eye (for attaching top end of "quadriceps" spring)
- 4 #8 wood screws, 1-1/4 in long (for mounting uprights to base)
- 16 #8 wood screws, 1/4 in long (for mounting hinges)
- 10 #6 wood screws, 1-1/2 in long (for putting together the wooden uprights, attaching the metal U-channels, and attaching the guitar tuning mechanism)
- 1 spring 2-1/2 in long, 5/8 in diameter, 0.072 in gauge (or similar)
- 1 spring 2-1/2 in long, 1/2 in diameter, 0.047 in gauge (or similar)
- 2 machine screws, 6-32 x 1-1/4 in (for mounting "patella")
- 4 washers for 6-32 screws
- 2 nuts, 6-32
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Deep Knee Bends: Measuring Knee Stress with a Mechanical Model
Building the Model Knee
- Refer to Figure 2 in the Introduction to see how the model goes together.
- Mark and drill pilot holes for the four sets of casters on the "hip" piece. Then mount the casters on the "hip" piece. You will need this piece to determine the correct spacing for the vertical side supports.
The vertical support is made of three wooden pieces:
- The back of the vertical support has dimensions 20 x 6-1/4 x 3/4 in.
- The two side pieces of the vertical support have dimensions 20 x 2-1/2 x 3/4 in.
- Use wood screws to attach the U-channel pieces to the side pieces. We recommend using flat-head screws and countersinking them into the channels so that the screwheads are not in the way of the casters.
- Use wood screws to attach the side pieces to the back (drive screws through the back and into the side pieces). Use your "hip" piece with the casters mounted on it to make sure that the side supports are set at the correct distance. The casters should roll easily in the tracks made by the U-channels (see Figure 2).
- Mark and drill pilot holes for the four sets of hinges in the "hip," "femur," "tibia," "foot," and base pieces. See Figure 2 to see how each of the hinges is mounted.
- Mark and drill pilot holes for the guitar tuning mechanism (mounted on the back side of the "tibia"). You also need to drill a clearance hole for the tuning peg. It sticks through to the front side of the "tibia."
- Attach a sturdy cup hook to the top end of the "femur" piece. This will be used for attaching the top end of the spring ("quadriceps").
- Don't mount the "patella" (metal strip) until you have assembled attached the hinge that holds the "femur" and "tibia" together. You'll want the "patella" to extend just a shade beyond the joint so that the guitar string will bend over the edge of the "patella."
- Attach the four sets of hinges to assemble the leg. Before you screw the hinge down to the base, make sure to slide the "hip" piece with its four casters into the tracks of the U-channels.
- Attach one end of the "quadriceps" spring to the cup hook.
- Tie one end of the guitar string to the other end of the "quadriceps" spring.
- Attach the other end of the guitar string through the tuning peg and secure it in place.
Measuring Tension Using the Model Knee
- Check to make sure that the pieces of your model can move fairly freely. Each joint should bend fairly easily, and the "hip" piece should move easily up and down in the U-channel tracks. Make any adjustments needed to assure free motion.
- Measure the length of the spring at rest. You'll need to subtract this value from your measurements under tension to get the change in the length of the spring.
Measure the stretched length of the spring needed to hold the knee joint in place at a series of different angles (minimum of three).
- Set the desired angle by sliding the "hip" piece up or down in the track. You will need to hold it in place until the spring tension is adjusted.
- Carefully adjust the tension on the spring by turning the guitar tuning peg. You want the "quadriceps" spring to just hold the mechanical knee in place.
- Tip: to see if you are at equilibrium, you can tentatively release the "hip" piece and see if it moves. If it starts to move, quickly stop it and adjust the spring further. If the "hip" piece moves up, you need to increase the tension on the spring. If the "hip" piece moves down, you need to decrease the tension on the spring.
- When you reach equilibrium, use the protractor to measure the angle of the knee joint and use the metric ruler to measure the length of the spring. Tip: Be careful not to knock the model when making your measurements. The equilibrium is a bit unstable!
- Repeat steps 3 and 4 at least three times for each angle.
- Replace the "quadriceps" spring, and repeat the experiment.
- For each spring, make a graph to show the length of the spring (y-axis) as a function of angle of the knee joint (x-axis). Remember that according to Hooke's Law, the force in a spring is directly proportional to its displacement from equilibrium (or its length at rest). So, even though you are not calculating force directly, you can still see the relative levels of force at different knee angles by creating a plot of spring displacement vs. knee angle. If you would like to calculate force, first you will need to measure the k value for your spring (see the Background tab).
- Are the results for the two springs similar in shape?
- How does spring tension change with angle? By what factor does tension increase as the angle of the knee changes from 90° to 45°? To 30°?
- What is the angle of your knee joint when you are squatting down completely? What is the relative tension in your mechanical model at this same angle (relative to a 90° bend)?
If you like this project, you might enjoy exploring these related careers:
- For a more basic experiment on the physics of springs, see the Science Buddies project Applying Hooke's Law: Make Your Own Spring Scale.
- Advanced. Can you use vectors to analyze the forces applied to the guitar string as the "knee" angle is varied? See if you can come up with an equation to predict the force exerted on the guitar string at different angles. Solve the equation for each of the angles used in your experiment, and graph the results with your data (use a different symbol and color). How well does your equation agree with your experimental results?
- Super advanced. Maybe you're interested in a sport that involves the arms, like swinging a tennis racquet or pitching a baseball. Do you think that you could come up with a model that would help you to estimate the forces exerted on the elbow in these sports?
- Super advanced. We said in the Introduction that the point of a model is to simplify a problem and to capture its essentials. It is also a good idea to try to circle back and do a "reality check" on your results when you are done. See if you can find a mentor for this project (a physician, physical therapist, or someone with a background in bioengineering) to help you with your analysis, and maybe even help to improve your model. For example, would it make any difference to include a spring to model the lower leg muscles?
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