Abstract
A system with one degree of freedom is far from reality, because we do not take into account all the degrees of freedom. In order to be close to the reality, it is necessary to use a system with several degrees of freedom. Efforts in this chapter are concentrated to the study of multi-degrees of freedom system, whether for a free undamped and forced damped system, by detailing the modal superposition method as well as a coupled coordinates. We finish the chapter with hydrodynamic study using Hozner method as well as some applications.
TopIntroduction
A system with one degree of freedom is far from reality, because we do not take into account all the degree of freedom. In more complicated structures, it is necessary to use a system with several degrees of freedom, to obtain a satisfactory dynamic model. A real system, generally comprises several masses, connected together by elements of the spring and damper type, in general, there is six degree of freedom for each node three displacements and three rotations. This chapter is devoted solely to the development of all equations concerning construction of an analytical as well as mathematical model of free, forced damped and undamped multi-degrees of freedom vibration linear system linear at the end the hydrodynamic study is detailed using Hozner method .
Figure 1. Six space degree of freedom
In the case of an undamped free vibration the equation of motion of a single degree of freedom system (SDOF) is: 
is it supposed.
U(
t) =
Gest = A cos 𝜔
t + 𝛽 sin 𝜔
tWe found
With
On the other hand, for a system with several degrees of freedom we have to manipulate matrices and vectors because the equation of motion becomes.
TopUndamped Free Vibration
For each degree of freedom, the conditions of equilibrium, including the effects due to the acceleration and those due to the other external actions of the system (Newton), will be established at considered point. Figure 2a and 2b show an example of a system with several degrees of freedom, in which the beam is infinitely rigid.
Figure 2. (a) Two-degree free system; (b) Model of an infinitely rigid beam
To write the motion equations, we use the principle of DALAMBERT (Newton's second law) in other words, for each degree of freedom, Newton's law is applied.
The equation (1) is coupled equation system, which means that each equation contains several unknowns in this case there are two unknowns U1 and U2.
Equation (1) can be written in matrix form as follows:
(2)(3)TopEigenvalues And Eigenvectors
The principle of calculating eigenvalues and eigenvectors will be detailed in the next chapters.
To solve equation (1), assume the following solution (the method of separating variables):{U(t)} = {𝜑} sin(𝜔t + 𝜃) (4) With {U(t)} Displacement vector ;
Key Terms in this Chapter
Impulse Force: It is a part of the fluid mass, which reacts by inertia to the translation of solid.
Coupled Coordinates: Translation and rotation coordinates.
Multi-Degrees of Freedom: Is a system with several degrees of freedom.
Orthogonality: A relation, which allows decoupling equations.
Eigenvectors: Mathematical vector which when operated on by a given operator gives a scalar multiple of itself.
Oscillation Force: It a part of the fluid mass, which starts to oscillate under the action of the earthquake.
Hydrodynamic: Is a branch of physics that deals with the motion of fluids and the forces acting on solid bodies containing or immersed in fluids and in motion relative to them.
Modal Superposition Method: Deformed structure can be expressed as a linear combination of all modes.
Normalization: Usually means to scale a variable.
Eigenvalues: Mathematically each of a set of values of a parameter for which a differential equation has a non-zero solution (an eigenfunction) under given conditions.