Notes on Knee Biomechanics: Vector Addition, Cross Product, and Torque

Knee biomechanics: pulley mechanism and net muscle forces

• The patella acts like a pulley, redirecting the quadriceps force around the front of the knee and changing its angle of attack on the tibia.
  • In full knee extension, the resultant force vector from the quadriceps tendon and the patellar ligament tends to point up and a bit anteriorly, aiding knee extension by increasing the extensor moment arm.
  • As the knee flexes, the two tensile forces (quadriceps tendon and patellar ligament) become less aligned, altering the net force on the patellofemoral joint.
  • Patellofemoral pain can arise for multiple reasons; one contributor may be increased compressive force behind the patella against the femur in certain ranges of knee flexion (e.g., osteoarthritis can contribute).
• Problem setup (example): constant magnitudes Fquadriceps = Fpatellar = 300 N, but with changing directions as the knee flexes. Goal: determine how the net force changes with knee flexion.
  • Intuition: the net force is the vector sum of the two forces; the angle between them changes with knee flexion, altering the resultant magnitude and direction (compression behind the patella).
  • Conceptual takeaway: using vector composition helps understand how training loads might be achieved with less pain by keeping the same muscle stimulus while reducing compressive loading.
• Vector composition: two common ways to visualize/compute the resultant
  • Parallelogram method (tail-to-tail): as the knee flexes and the lines of action change, the resultant (purple) vector changes in magnitude/direction while the magnitudes of the two red vectors stay fixed.
  • The length of the resultant vector can increase in compression direction as flexion increases, illustrating how the joint experiences varying contact forces even with the same muscle force magnitudes.
• Coordinate system setup for 2D force composition
  • Important to choose a convenient coordinate system to minimize trig calculations.
  • Strategy suggested: align one basis vector with one of the force vectors (either align x with the quadriceps tendon or align y with the patellar ligament).
  • Example setups:
    1) Set x along the quadriceps tendon vector; then the quadriceps force has components (Fq,x, Fq,y) = (300, 0) in that rotated frame, and the patellar ligament requires decomposing into x and y components relative to that axis.
    2) Set y along the patellar ligament vector; then the patellar force has components (Fpx, Fpy) = (0, -300) in that rotated frame, and the quadriceps vector must be decomposed into x and y relative to that axis.
  • Rationale: choosing a coordinate system that aligns with one force reduces the amount of trig you have to do for that force and simplifies the other force’s decomposition.
• A practical example in the lecture (non-unique coordinate choices): use standard x to the right and y up for the page, with the patellar ligament aligned with the y-axis in one setup.
  • If the quadriceps tendon makes a conventional angle of 30° with the chosen x-axis, then its components are:
  • F{ ext{quad},x} = Fq \, ext{cos}(30^ ext{o}) = 300 \, ext{cos}(30^ ext{o}) = 150\, ext{\sqrt{3}} \, ext{N} \\rightarrow F_{ ext{quad},x} = 150\sqrt{3} \approx 259.81\ ext{N}
  • F{ ext{quad},y} = Fq \, ext{sin}(30^ ext{o}) = 300 \ times \sin(30^ ext{o}) = 150\ ext{N}
  • The patellar ligament is taken at angle -90° relative to the same x-axis (i.e., vertical downward):
  • F{ ext{patella},x} = Fp \, \cos(-90^ ext{o}) = 300 \times 0 = 0
  • F{ ext{patella},y} = Fp \, \sin(-90^ ext{o}) = 300 \times (-1) = -300\ ext{N}
  • Net force components (sum of both forces):
  • Fx = F{ ext{quad},x} + F_{ ext{patella},x} = 150\sqrt{3} \ ext{N} \approx 259.81\ ext{N}
  • Fy = F{ ext{quad},y} + F_{ ext{patella},y} = 150 + (-300) = -150\ ext{N}
  • Net force magnitude and direction:
  • Magnitude: |\mathbf{F}| = \sqrt{Fx^2 + Fy^2} = \sqrt{(150\sqrt{3})^2 + (-150)^2} = \sqrt{22500(3+1)} = \sqrt{90000} = 300\ ext{N}
  • Conventional angle (relative to +x): \theta = \arctan\left( \frac{Fy}{Fx} \right) = \arctan\left( \frac{-150}{150\sqrt{3}} \right) = \arctan\left(-\frac{1}{\sqrt{3}}\right) = -30^ ext{o}
  • Therefore, the net force points in quadrant IV (positive x, negative y) at an angle of -30° from the +x axis.
  • Sign convention for angles and torque direction:
  • Clockwise (CW) rotation is considered negative, counterclockwise (CCW) positive, when using the standard right-handed coordinate system with +z out of the page.
  • This aligns with the idea that a conventional angle rotation clockwise yields a negative torque about the +z axis.
• Transition to cross product (vector product) and torque
  • Motivation: Cross product directly yields torque as a measure of rotational effect from a force about a point (e.g., joint center).
  • 2D cross product intuition: In 3D terms, torque is the z-component of r × F when r = (Rx, Ry, 0) and F = (Fx, Fy, 0).
  • In 2D form, the scalar torque about the z-axis is:
    • \tauz = Rx Fy - Ry F_x
  • Vector form: \mathbf{\tau} = \mathbf{r} \times \mathbf{F} = (Rx Fy - Ry Fx) \hat{\mathbf{z}}
  • The cross product is sensitive to the order: r × F is not the same as F × r. For torque calculations, use r × F to get the correct sign/direction.
  • Radius of rotation (lever arm) concept: The vector from the joint center to the muscle insertion on the bone; this is the r in the cross product. The force vector F is the line of pull of the muscle.
  • Why use cross product here: It naturally encodes the perpendicularity between lever arm and force, and the sign/direction corresponds to rotation sense about the axis perpendicular to the plane of motion.
  • Units: Torque has units of Newton-meters (N·m). It results from multiplying a length (m) by a force (N).
• Coordinate system nuances for torque direction
  • With standard orientation (x right, y up, z out of page), the z-axis direction is determined by the right-hand rule: x × y = z (positive z out of the page).
  • If x and y are oriented differently, z may point into the page (negative z) depending on the order of basis vectors; this affects the sign of the computed torque.
  • Example orientation checks:
  • If x is to the right and y is up, positive z is out of the page.
  • If x is off the page (toward you) and y is to the right, positive z points into the page or out of the page depending on the exact orientation; use the right-hand rule to verify.
• Application: torque about a joint in a practical problem (soccer kick example)
  • Setup: radius vector r from the joint center to the tibial tuberosity (attachment point); force vector F from the quadriceps tendon acting on the tibia (leg being kicked).
  • The cross-product form to compute knee-extension torque about the axis perpendicular to the sagittal plane is:
  • \tauz = rx Fy - ry F_x
  • In the discussed scenario, plugging in the given components yielded a torque about the z-axis of approximately \tau_z = -700\ ext{N·m} (negative z, i.e., into the page), indicating a clockwise rotation when viewed with the standard orientation.
• Conceptual takeaway: The cross product provides a consistent, general framework to assess rotational effects (torques) in biomechanics, applicable to open-chain and closed-chain analyses, and helps connect anatomical lever arms (bone geometry) with the forces generated by muscles.
• Planes of motion and axes (quick recap linked to the cross product discussion)
  • Frontal/contal plane movements (e.g., shoulder abduction) are driven around an anterior-posterior axis (perpendicular to the plane).
  • In the shoulder abduction example, the movement looked at is in the frontal plane; the axis of rotation is anterior-posterior, and the torque direction about this axis can be analyzed via a cross-product approach for the distal segment.
  • Skeltonized note: The same torque calculation approach (r × F) applies across joints and planes, with the axis of rotation chosen perpendicular to the plane of motion.
• Practical tips for solving vector/torsion problems in biomechanics
  • Decide on a coordinate system that minimizes trig work, often by aligning one force with an axis (x or y).
  • Express both r (radius of rotation) and F (force) in that coordinate system.
  • Use the 2D cross-product formula for torque about the z-axis: \tauz = Rx Fy - Ry F_x, or the general 3D form if needed.
  • Check signs with the right-hand rule and the observed direction of rotation (CW vs CCW) to ensure consistency between the calculated torque and the physical intuition.
• Quick worked summary of key equations used in these notes
  • Force components of quadriceps when the conventional angle is 30°:
  • F{ ext{quad},x} = Fq \cos(30^ ext{o}) = 300 \cos(30^ ext{o}) = 150\sqrt{3} \approx 259.81 \text{ N}
  • F{ ext{quad},y} = Fq \sin(30^ ext{o}) = 300 \sin(30^ ext{o}) = 150 \text{ N}
  • Patellar ligament components at -90°:
  • F{ ext{patella},x} = Fp \cos(-90^ ext{o}) = 0
  • F{ ext{patella},y} = Fp \sin(-90^ ext{o}) = -300 \text{ N}
  • Net force components:
  • Fx = 150\sqrt{3} \text{ N}, \quad Fy = -150 \text{ N}
  • Net force magnitude and angle:
  • |\mathbf{F}| = \sqrt{(150\sqrt{3})^2 + (-150)^2} = 300 \text{ N}
  • \theta = \arctan\left( \frac{Fy}{Fx} \right) = -30^ ext{o}
  • Torque (2D cross-product form):
  • \tauz = Rx Fy - Ry F_x
  • Sign convention: positive z is out of the page; negative z is into the page.
• Summary of educational aims illustrated in these notes
  • Understand how the patella functions as a pulley to modify quadriceps torque and knee joint contact forces.
  • Learn how to set up coordinate systems to simplify vector addition of muscle forces.
  • Practice decomposing forces into components and computing resultant forces both graphically (parallelogram) and analytically (component addition).
  • Introduce the cross product as a systematic method to compute torque in 2D biomechanics problems, with emphasis on right-hand rule conventions and axis orientation.
  • Apply the concepts to practical biomechanics problems (e.g., knee loading, shoulder abduction, soccer kicking) and interpret the sign/direction of torque in terms of rotation direction and axis.
• Final note on applications beyond exercise science
  • The same vector composition and cross-product framework underpin many biomechanical analyses, including gait analysis, joint load estimation, prosthetics design, and rehabilitation planning, where you need to balance training load with joint health and pain management.