Introduction to dynamic analysis#
So far we have focused on kinematics: the study of motion (position, velocity and acceleration) regardless of its causes. In this lesson we move on to dynamics, which relates that motion to the forces and torques acting on the mechanism.
Dynamic analysis is essential in machine design: it lets us size links and joints so that they do not fail, and choose actuators (motors, cylinders) able to produce the desired motion.
Types of problems in mechanism analysis#
Kinematic problem: some positions, velocities and accelerations are known and the rest are sought. This is the kinematic analysis of lesson 2.
Direct (forward) dynamics problem: forces and masses are known, and the resulting motion (\(\theta(t)\), \(\dot{\theta}(t)\), \(\ddot{\theta}(t)\)) is sought. It requires integrating the differential equations of motion: this is dynamic simulation.
Inverse dynamics problem: the motion (positions, velocities and accelerations of all links) and the masses are known, and the forces producing it are sought: the input torque, actuator forces and joint reactions. This is the usual problem in machine design and in robot trajectory planning: what motor do I need for the machine to move the way I want?
In this lesson we mostly solve the inverse dynamics problem, also called kinetostatic analysis.
Classification by constraints#
The difficulty of writing the equations of motion of a mechanical system grows with the number and complexity of the constraints between its bodies. In increasing order of difficulty:
Free rigid bodies:
Links in an open kinematic chain:
Links in a closed kinematic chain:
We first set up the equations of a free rigid body (the Newton-Euler equations). Then we apply them to each link of a mechanism, replacing the constraints with the reaction forces exerted by the joints.
Lesson contents#
Particle dynamics: review of linear momentum, angular momentum and equations of motion of a particle.
Rigid body dynamics: Newton’s and Euler’s equations of a rigid body, in 3D and in the plane.
Computing inertias in 2D: computing moments of inertia of planar links.
D’Alembert’s principle: inertia forces and torques; dynamic equilibrium; quasi-static analysis; superposition principle.
Analytical force method: the analytical (vector) force method: free-body diagrams and the system of equations of the whole mechanism.
Graphical methods: graphical methods: two-force and three-force members, force polygons and equivalent inertia force.
Friction forces: friction forces: sliding, rolling, pivoting, friction cone and self-locking.
Solved examples: step-by-step solved examples.