Introduction of D’alemcerts
History of D’Alembert
In July 1739 he made his first contribution to the field of mathematics, pointing out the errors he had detected in L'analyse démontrée (published 1708 by Charles René Reynaud) in a communication addressed to the Académie des Sciences. At the time L'analyse démontrée was a standard work, which d'Alembert himself had used to study the foundations of mathematics. D'Alembert was also a Latin scholar of some note and worked in the latter part of his life on a superb translation of Tacitus, from which he received wide praise including that of Denis Diderot.
In 1740, he submitted his second scientific work from the field of fluid mechanics Mémoire sur la réfraction des corps solides, which was recognized by Clairaut. In this work d'Alembert theoretically explained refraction.
In 1741, after several failed attempts, d'Alembert was elected into the Académie des Sciences. He was later elected to the Berlin Academy in 1746 [1]
When the Encyclopédie was organized in the late 1740s, d'Alembert was engaged as co-editor (for mathematics and science) with Diderot, and served until a series of crises temporarily interrupted the publication in 1757. He authored over a thousand articles for it, including the famous Preliminary Discourse. D'Alembert "abandoned the foundation of Materialism"[2] when he "doubted whether there exists outside us anything corresponding to what we suppose we see."[3] In this way, D'Alembert agreed with the Idealist Berkeley and anticipated the Transcendental idealism of Kant.
In 1752, he wrote about what is now called D'Alembert's paradox: that the drag on a body immersed in an inviscid, incompressible fluid is zero.
What are D’Alembert Lagrange Principal Equations?
Definition,
For a N-degree of freedom system, regardless whether it is rigid or flexible, d’Alembert-Lagrange’s principal equations are obtained by summing all the forces and all the moments(with respect to a point) in the system.
Symbolic Expressions:
Sum of forces (3 equations at most):
(fi- miai) = 0
Sum of moments (3 equations at most):
{Mi+ rix (fi- miai)} = 0
Semantic Questions relate to
The d’Alembert-Lagrange Principal Equations:
Is a floating structure in an equilibrium condition?
Or
Is a free-free substructure, partitioned from an assembled system, in its equilibrium state?
The answer is not in the blowing wind,
but in the mathematical expressions of the d’Alembert-Lagrange Principal Equations and their physical meaning.
A bottom-up approach to D’Alembert-Lagrange’s Principal Equations
Condition:
This model does not capture the boomerang motions
D'Alembert's principle of inertial forces :
D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment. The inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this "inertial force and moment" and the external forces. The advantage is that, in the equivalent static system' one can take moments about any point (not just the center of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero). Even in the course of Fundamentals of Dynamics and Kinematics of machines, this principle helps in analyzing the forces that act on a link of a mechanism when it is in motion. In textbooks of engineering dynamics this is sometimes referred to as d'Alembert's principle.
Example for plane 2D motion of a rigid body
For a planar rigid body, moving in the plane of the body (the x–y plane), and subjected to forces and torques causing rotation only in this plane, the inertial force is
F= -mr
Where r is the position vector of the centre of mass of the body, and m is the mass of the body. The inertial torque (or moment) is
T= - I( angle)
where I is the moment of inertia of the body. If, in addition to the external forces and torques acting on the body, the inertia force acting through the center of mass is added and the inertial torque is added (acting around the centre of mass is as good as anywhere) the system is equivalent to one in static equilibrium. Thus the equations of static equilibrium
Sum of All Forces In X Direction is Zero.
Sum of All Forces In Y Direction is Zero.
And Sum of Torqe,Moment Is Zero.
hold. The important thing is that is the sum of torques (or moments, including the inertial moment and the moment of the inertial force) taken about any point. The direct application of Newton's laws requires that the angular acceleration equation be applied only about the center of mass.
Application 1: mass properties of complex structures
(Example - Bar element)
It is not trivial at all for computing the rotatory inertia and its coupling matrix with the translational mass properties!
Translational mass matrix of a bar element:
Summation operator for the case when the moment is computed around node 1
The above matrix correctly captures the rotatory inertia when the rotational motion is measured with respect to node 1 and the corresponding coupling matrix with the Translational mass. Complex cases yield the correct mass
properties. Computations of the rotatory inertia for complex systems, e.g., automobile, satellites, etc., with the present procedure is straightforward and consistent. All one needs for computing the inertia properties of complex systems are the translational mass matrix and the moment summation operator!
Small application: Generation of Diagonal Mass Matrix
Application 2: Solvability for unconstrained systems under quasi-static equilibrium states
This equation is indefinite and consequently requires a delicate care for its solution!
Partitioned equations of motion for structures:
Eliminate q to obtain:
This equation is regular and consequently can be solved in a routine manner!
Application 3: Consistent starting vector for an iterative solution of partitioned system equations
Step 1: Solve for a least-square value of the interface forces:
Step 2: Project the new iterate to be orthogonal to the interface rigid-body modes: in order to minimize the residual:
Future Potential Applications
A. Least-squares nominal interface forces that may provide a preliminary design modification or control strategy for systems with constraints.
B. Augmented solution of (q, ) for dual control strategy development, i.e., for principal motions and deformational motions in tandem.
C. Filtering of mean motion signals from output signals.
D. Advanced multi-physics modeling
Discussions
1. The d’Alembert-Lagrange principal equations consists of 6 rigid-body motions regardless how large the flexible mechanical structural systems may be, and they provide the mean motions of the overall system dynamics.
2. The rotatory inertia and its coupling terms with the translational mass properties are obtained as part of the derivational process of the d’Alembert- Lagrange principal equations presented herein.
3. The d’Alembert-Lagrange principal equations constitute the key solvability condition for systems partially constrained or in completely free-free state.
4. For an iterative solution of coupled multi-physics problems, the solution of the d’Alembert-Lagrange principal equations provides a consistent starting vector, thus accelerating the iterative process.
5. There remains a challenge to expand the usage of the d’Alembert-Lagrane principal equations, some of which have been outlined herein.
Bibiliography
2)http://www.physicsforums.com/showthread.php?t=130381
3)http://en.wikipedia.org/wiki/D'Alembert's_principle
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