7 & 8 Multi Bodie Systems

Rigid bodies and joints

• Vehicle dynamics

• robots

• simple occupant analysis

• rigid bodies; parameters: centger of gravity, inertia properties, location, orientation

• joints; parameters: degrees & axis of freedom

• boundary conditions

• force-moment elements; e.g.: springs, dampers, engines

Only dynamic

Forces, moments, movements, oscillations

velocity vector, torque-time-function

a point and tripod (𝑃,𝑒⃗)

(𝑥˙,𝑦˙,𝑧˙)𝑒⃗_+(𝑟,𝑠,𝑡)𝑒⃗˙_𝑏

• X-Y-Z is an Kardan rotation sequence

• X-Y-X is an Euler rotation sequence

Natural tensorial representation velocity due to rotation

—> Transforms stresses and strains into a rotating frame

• Ω is a tensor of second order

• Ω^𝑇 = −Ω

For tensor of 2. order, is the same as shown with Ω

The inertia tensor generalizes moment of inertia to 3D rotation

characteristics

• symmetric: I = I^T

• positive-definite: ensures positive kinetic energy

• second order tensor

• the main diagonal show the moments of inertias (I_xx, I_yy ,…)

• The symmetric values are the products of inertia (I_xy = I_yx, I_yz = I_zy, …)

• Ordinary differential equations ODE for MBS with tree topology (straight forward establishing is possible)

• Differential Algebraic Equations DAE for closed kinematic chain (more difficult)

tree topology: a multibody structure in which the bodies are connected without any kinematic loops

—> Exactly one unique path between any two bodies each body has exactly one parent (e.g. robot manipulators)

closed kinematic chain: More than one path connects some bodies and Motion of one joint constrains others (e.g. Tank track wrapped around wheels)

1. defines solver settings to use for simulation

2. compute numerical partial derivatives for linearization (also gravity)

3. provides access to the world or ground frame (can have multiple world frame blocks)

• Elimination Method (less, but more complex equations)

• Augmentation Method (more, but simpler equations)

Also augmentation methods:

• Lagrange formalism (L = E_kin - E_pot)

  
  

• Hamilton formalism (H = E_kin + E_pot)

Solves an ODE by approximating the solution curve with a sequence of straight-line segments.

Starting from an initial value, the solution is advanced in small steps (fixed or variable) by moving forward in the independent variable and updating the dependent variable using the local slope given by the differential equation.

Repeating this step constructs a polygonal approximation to the true solution, with accuracy improving as the step size decreases.

small dfixed steos —> slow/ cotsly to find solution

large fixed steps —> faster but less precise and unstable

Adaptive Runge-Kutta method

advances the solution in discrete time steps using multiple slope evaluations per step

—> much higher accuracy than Euler’s method for the same step size

Stiff ODEs are not solvable with explicite methods

—> implicit methods have to be used