The virtual work of forces acting at various points on a single rigid body can be calculated using the velocities of their point of application and the resultant force and torque.
This yields the resultant force on the system.
D'Alembert's form of the principle of virtual work edit The equations of motion for a mechanical system of rigid bodies can be determined using D'Alembert's form of the principle of virtual work.The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is Q j 0, j 1,.The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point.Displaystyle mathbf A _ialpha times (mathbf r _i-mathbf R )omega times (omega times (mathbf r _i-mathbf R )mathbf.They are equivalent to rotation matrices and rotation vectors.Displaystyle mathbf p Mmathbf V,quad mathbf L sum _i1nm_i(mathbf r _i-mathbf R )times frac ddt(mathbf r _i-mathbf R ).These m equations define the static equilibrium of the system of rigid bodies.
Finally, assume that the velocity chaos james gleick pdf V i and angular velocities i, i,1.,n, for each rigid body, are defined by a single generalized coordinate.
Dynamics of rigid rotating bodies Part 1: Centre of Gravity, Moment of Inertia, Angular Momentum and Torque Part 2: Parallel Axis Theorem and consequences).Where is the angle between the vectors P and.Chapter 12 - Rotation of a Rigid Body.If the trajectory of a rigid body is defined by a set of generalized coordinates q j, j 1,., m, then the virtual displacements r i are given by r i j 1 m r i q j q j j 1 m.Newton's formulation yields 6 M equations that define the dynamics of a system of M rigid bodies.S i T S i displaystyle mathbf S _iTmathbf S _i is the scalar product of S i displaystyle mathbf S _i with itself, while the heart of understanding pdf S i S i T displaystyle mathbf S _imathbf S _iT is the tensor product of S i displaystyle mathbf.Under a constant torque of magnitude, the speed of precession P is inversely proportional to L, the magnitude of its angular momentum: P L sin, displaystyle tau mathit Omega _mathrm P Lsin theta!The principle of virtual work is used to study the static equilibrium of a system of rigid bodies, however by introducing acceleration terms in Newton's laws this approach is generalized to define dynamic equilibrium.Therefore, as before, the orientation can be given as the rotation from the initial frame to achieve the frame that we want to describe.Kinzel, Kinematics and Dynamics, and Design of Machinery, 2nd., John Wiley and Sons, 2004.It's just play on words.Mass properties edit The mass properties of the rigid body are represented by its center of mass and inertia matrix.Displaystyle Q_jmathbf F cdot frac partial mathbf V partial dot q_jmathbf T cdot frac partial vec omega partial dot q_j,quad j1,ldots,m.