Mechanical Dynamics :: Analyzing Motion (SimMechanics™)

As explained in the Representing Motion chapter, kinematics describes the motion of bodies, while dynamics explains the motion in terms of forces and torques. By Newton's laws of motion, the accelerations of the bodies' positions are directly related to the forces and torques applied to the bodies.

You can predict accelerations if you are given the applied forces/torques, or relate known accelerations to the forces/torques that cause them, as this section explains. The section concludes by presenting Newton's laws of dynamics for translational and rotational motion.

The books of Goldstein [1] and José and Saletan [5] present rigid body mechanics in great detail.

About the SimMechanics Machine State To perform inverse dynamics, trimming, and linearization tasks, you might need to look at and manipulate the mechanical state of your SimMechanics machine or model.

The machine state components arise from individual joint primitives in your machine's Joint blocks.
These components represent relative degrees of freedom between one body and another or between a body and ground.

See the mech_stateVectorMgr command reference for more information about constructing and interpreting the machine state.

Forward and Inverse Dynamics

Dynamical equations such as Newton's laws of motion relate cause and effect. In mechanics, the cause is a set of forces and torques applied to the bodies of a mechanical system; the effect is the set of resulting motions. Dynamical equations allow you to analyze motion in either direction:

In forward dynamics, you apply a given set of forces/torques to the bodies to produce accelerations. SimMechanics simulation integrates the accelerations twice to yield the velocities and positions as functions of time.
A set of initial conditions is needed to specify the initial positions and velocities and produce a complete solution for the motion. Initial conditions must be checked for consistency with constraints.
Inverse dynamics starts with given motions as functions of time and differentiates them twice to yield the forces and torques needed to produce the given motions. The given motion functions of time must be checked for consistency with constraints.

You can use SimMechanics analysis modes to analyze mechanical motion in both cases. The mode you choose can depend on the topology of your system.

Analysis Mode	Type of Analysis
Forward Dynamics	Forward dynamics (any topology)
Trimming	Forward dynamics (steady-state motion)
Inverse Dynamics	Inverse dynamics (open topology)
Kinematics	Inverse dynamics (closed topology)

Applying the Motion Modes

For more about motion modes, see these other sections.

Simulating and Analyzing Mechanical Motion in the Introducing SimMechanics Software chapter is an overview of the SimMechanics analysis modes.
Choosing an Analysis Mode in the Running Mechanical Models chapter contains detailed steps to implement these modes in your model.
The case study Finding Forces from Motions in this chapter applies inverse dynamics to SimMechanics models.

Forces, Torques, and Accelerations

Newton's second law of motion relates the force on a body, its mass, and the acceleration it experiences as a result of that force. The equivalents for rotational motion are the Euler equations.

Newton's Equations for Translational Dynamics

Let F_A be the net force acting on a body A that has a constant mass m_A and a center of gravity (CG) position x_A. Newton's second law, valid for an inertial observer, relates the force on A to the translational acceleration of its CG.

Equivalently, the linear momentum p_A = m_Av_A relates to force as F_A = dp_A/dt.

In forward dynamics, the force F_A is given and the motion x_A(t) is found by integration, supplemented by initial position and velocity. In inverse dynamics, the motion x_A(t) is given and the force on the body is found. In both cases, the mass must be known.

Euler's Equations for Rotational Dynamics

Rotational motion requires a pivot, the fixed center of rotation, and the angular velocity vector ω with respect to that pivot. If r is the position, with respect to the pivot, of any point in a body, the velocity v of that point is v = ω X r.

The equivalent of the mass of a body in rotational dynamics is the inertia tensor I, a 3-by-3 matrix.

The body's mass density ρ(r) is a function of r within the body's volume V. The indices i, j range over 1, 2, 3, or x, y, z. Thus

The angular momentum of a body is L = I·ω. The equivalent of the force on a body in rotational dynamics is the torque τ, which is produced by a force F acting on the body at a point r as τ = r X F.

The analog to Newton's second law for rotational motion, as measured by an inertial observer, just equates the torque τ_A applied to a body A, defined with respect to a given pivot, to the time rate of change of L_A. That is, τ_A = dL_A/dt. It is easiest to take the pivot as the origin of an inertial coordinate system such as World. Unlike the case of translational motion, however, where the mass m_A remains constant as the body moves, the inertia tensor I_A changes as the body rotates, if it is measured in an inertial frame. There is no simple way to relate dL_A/dt to the angular acceleration dω/dt.

The common solution to this difficulty is to work in the body's own rotating frame, where the inertia tensor is constant, and take the body's CG as the pivot. Diagonalize the inertia tensor. Since I is real and symmetric, its eigenvalues (I₁, I₂, I₃) (the principal moments of inertia) are real. Its eigenvectors form a new orthogonal triad, the principal axes of the body. But this frame fixed in the body is not inertial, and the torque-angular acceleration relationship is modified from its inertial form into the Euler equations:

The components of the rotational vectors here are projected along the principal axes that move with the body's rotation.

Linearizing the Dynamical Equations

To study a system's response to and stability against external changes, you can apply small perturbations in the motion or the forces/torques to a known trajectory and force/torque set. SimMechanics software and Simulink provide analysis modes and functions for analyzing the results of perturbing mechanical motion. The later sections of this chapter, demonstrate their use:

You can perturb Newton's and Euler's laws with a small additional force ΔF and torque Δτ and determine the associated perturbations in motion, Δx and Δω. You can also perturb the system inversely, making small changes to the motion and determining the forces and torques necessary to create those changes.

The perturbed Newton's and Euler's equations are