TLDR;
This module provides an introduction to angular kinematics, explaining its relevance to analysing rotational movements in human joints and sports equipment. It covers the measurement of angles using degrees and radians, and discusses the concepts of relative and absolute angles. The module also details how to calculate these angles using trigonometric identities and the law of cosines. Furthermore, it explains angular velocity and acceleration, their units of measurement, and the relationship between linear and angular kinematic parameters. Finally, it demonstrates the application of these concepts with an example calculating knee angles during a drop vertical jump.
- Angular kinematics involves the study of motion in terms of angles, angular displacement, velocity and acceleration.
- Angles can be measured in degrees or radians, with radians being the international standard unit.
- Relative angles (joint angles) and absolute angles (segment angles) are two ways to measure angles in biomechanics.
- Angular velocity is the rate of change of angular position, and angular acceleration is the rate of change of angular velocity.
- Linear and angular kinematic parameters are related by the radius of rotation.
Introduction to Angular Kinematics [0:00]
The module introduces angular kinematics, which involves studying motion in terms of angles, angular displacement, angular velocity and angular acceleration. It highlights the relevance of angular kinematics in analysing movements involving rotation, such as joint movements in the human body and rotational motion in sports equipment. Understanding angular kinematics is crucial for analysing and optimising movements involving rotation, providing insights into joint mechanics, sports techniques and equipment design.
Angular Position, Distance and Displacement [2:07]
To understand angular kinematics, consider an object suspended by a rope. If the object moves from its mean position (0) to position 1 at an angle θ1 in an anticlockwise direction, and then to position 2 at an angle θ2 in a clockwise direction, the angular positions are θ1 and θ2 from the mean position. The angular distance from position 0 to position 2 is the sum of the two angular positions (θ1 + θ2), while the angular displacement from position 0 to position 2 is θ2. The international standard unit for measuring angles is radians.
Units of Angular Motion Measurement [4:42]
Angular motion can be measured in degrees or radians. A degree is a 360th part of a full circle. A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.
Measuring Angles: Relative and Absolute Angles [6:08]
In human movement science, relevant angles for mechanical analysis include joint angles. Angles can be measured as relative angles (the angle between the longitudinal axes of two adjacent segments) or absolute angles (the angle between a segment and the horizontal or vertical axis). Relative angles are also known as joint angles, while absolute angles are known as segment angles. Relative angles should be measured consistently on the same side of the joint, and an angle of 0 degrees between two segments indicates a fully extended position. Absolute angles should be measured consistently in the same direction from a single reference (horizontal or vertical).
Calculating Absolute Angles [8:44]
Absolute angles can be calculated from endpoint coordinates using the arc tangent or inverse tangent function. By drawing horizontal and vertical lines to form right-angle triangles, trigonometric identities can be used to find the angle between segments. If the coordinates are in meters, the resulting angle is in radians. To convert radians to degrees, use the conversion factor 1 radian = 180/π degrees.
Calculating Relative Angles [11:56]
Relative angles can be calculated using the law of cosines, which requires knowing the segment lengths. Alternatively, if absolute angles are known, the relative angle can be found by calculating the angles between each segment and the horizontal or vertical axis. For example, the knee angle can be calculated by adding 180 degrees to the angle between the leg segment and the horizontal, and then subtracting the thigh angle from it.
Angular Velocity and Acceleration [16:36]
Angular velocity is the change in angular position divided by the change in time, with the direction determined by the right-hand thumb rule. The SI unit for angular velocity is radians per second. Angular acceleration occurs when an object in motion about an axis changes its direction, and is calculated as the change in angular velocity divided by the change in time, measured in radians per second squared.
Relationship Between Linear and Angular Kinematics [19:33]
General motion is a combination of linear and angular motion. Linear displacement (s) is equal to the radius (r) times angular displacement (θ). Linear velocity (v) is equal to the radius times angular velocity (ω), and linear acceleration (a) is equal to the radius times angular acceleration (α). While these relationships are presented as scalar components, it's important to note that as vector quantities, they are governed by the principles of vector algebra.
Example: Calculating Knee Angle During a Drop Vertical Jump [21:11]
The module uses the example of a drop vertical jump to demonstrate the calculation of knee angles at three specific positions: touchdown, maximum knee flexion and toe-off. By using the coordinates of the ankle, knee and hip joints, absolute angles are calculated first, followed by the relative knee angle. The calculations involve using trigonometric relations and the formulas discussed earlier in the module.
Touchdown Phase [22:00]
During the touchdown phase of a drop vertical jump, the knee angle is calculated using the coordinates of the ankle, knee and hip joints. Since the segment lengths of the leg and thigh are unknown, absolute angles are calculated first. Horizontal lines are drawn at the ankle and knee, and the angles of the leg (θleg) and thigh (θthigh) are calculated using trigonometric relations. The knee angle is then found using the formula: θknee = θleg + 180 degrees - θthigh, resulting in a value of 138.25 degrees.
Maximum Knee Flexion Phase [26:50]
During the maximum knee flexion phase, similar calculations are performed using the given coordinates. The thigh angle is found to be approximately 163.54 degrees, the leg angle is approximately 57.17 degrees, and the knee angle is 73.63 degrees. The process involves dividing the thigh and leg segments into right-angle triangles to calculate the respective angles.
Takeoff Instant [28:37]
At the takeoff instant, the thigh and leg segments are again analysed by drawing vertical lines to create right-angle triangles. The thigh angle (θthigh) is calculated to be 92.29 degrees, and the leg angle (θleg) is 85.71 degrees. The resulting knee angle (θknee) is approximately 173.41 degrees.
Calculating Angular Velocity and Acceleration in the Jump [30:17]
If the time difference between the touchdown and maximum knee flexion phases is known (t1), the average angular velocity can be calculated as (θknee at max flexion - θknee at touchdown) / (time at max flexion - time at touchdown). Similarly, if the angular velocities at touchdown and maximum knee flexion are known, the angular acceleration can be calculated using the time interval.
Summary [32:04]
The module covers linear and angular kinematics, methods for finding them, and biomechanical considerations for kinematic analysis. It discusses how to analyse different time points during a movement or activity and calculate linear and angular parameters, using the example of a drop vertical jump.
