• The vibrating systems, which require two coordinates to describe its motion, are called two-degrees-of -freedom systems.
• These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of freedom of the system.
• Unlike single degree of freedom system, where only one co-ordinate and hence one equation of motion is required to express the vibration of the system, in two-dof systems minimum two co-ordinates and hence two equations of motion are required to represent the motion of the system.
For a conservative natural system, these equations can be written by using mass and stiffness matrices.
• One may find a number of generalized co-ordinate systems to represent the motion of the same system.
While using these co-ordinates the mass and stiffness matrices may be coupled or uncoupled.
When the mass matrix is coupled, the system is said to be dynamically coupled and when the stiffness matrix is coupled, the system is known to be statically coupled.
• The set of co-ordinates for which both the mass and stiffness matrix are uncoupled, are known as principal co-ordinates. In this case both the system equations are independent and individually they can be solved as that of a single-do system.
• A two-dof system differs from the single dof system in that it has two natural frequencies, and for each of the natural frequencies there corresponds a natural state of vibration with a displacement configuration known as the normal mode. Mathematical terms associated with these quantities are eigenvalues and eigenvectors.
• Normal mode vibrations are free vibrations that depend only on the mass and stiffness of the system and how they are distributed. A normal mode oscillation is defined as one in which each mass of the system undergoes harmonic motion of same frequency and passes the equilibrium position simultaneously.
• The study of two-dof- systems is important because one may extend the same concepts used in these cases to more than 2-dof- systems. Also in these cases one can easily obtain an analytical or closed-form solutions. But for more degrees of freedom systems numerical analysis using computer is required to find natural frequencies (eigenvalues) and mode shapes (eigenvectors).
Figure below shows two masses m1 and m2 with three springs having spring stiffness k1, k2 and k3 free to move on the horizontal surface. Let x1 and x2 be the displacement of mass respectively.
As described in the previous lectures one may easily derive the equation of motion by using d'Alembert principle or the energy principle (Lagrange principle or Hamilton 's principle)
Using d'Alembert principle for mass m1 from the free body diagram shown in figure
and similarly for mass m2
Important points to remember
• Inertia force acts opposite to the direction of acceleration, so in both the free body diagrams inertia forces are shown towards left.
• For spring m2 assuming x1> x2 , The spring will pull mass m2 towards right by k2 (x2- x1) and it is stretched by x2- x1 (towards right) it will exert a force of k2 (x2- x1) towards left on mass m2 . Similarly assuming x1> x2 the spring get compressed by an amount x2- x1 and exert tensile force of k2 (x2- x1).
One may note that in both cases, free body diagram remain unchanged.
Now if one uses Lagrange principle,
So, the Lagrangian
The equation of motion for this free vibration case can be found from the Lagrange principle
Noting that the generalized co-ordinate q1 = x1 and q2 = x2
which yields
Normal Mode Vibration
Again considering the problem of the spring-mass system in figure with m1 = m, m2 = 2m, k1 = k2 = k3 = k,the equation of motion can be written as
We define a normal mode oscillation as one in which each mass undergoes harmonic motion of the same frequency, passing simultaneously through the equilibrium position. For such motion, we let
Hence,
or, in matrix form
Hence for nonzero values of A1 and A2 (i.e., for non-trivial response)
Now substituting ω2 = λ, equation yields
Hence,
So, the natural frequencies of the system are
Now from equation 1, it may be observed that for these frequencies, as both the equations are not independent, one cannot get unique value of A1 and A2. So one should find a normalized value.
One may normalize the response by finding the ratio of A1 to A2. From the first equation, the normalized value can be given by
and from the second equation of the normalized value can be given by
the two normal modes of this problem are:
In the 1st normal mode, the two masses move in the same direction and are said to be in phase and in the 2ndmode the two masses move in the opposite direction and are said to be out of phase. Also in the first mode when the second mass moves unit distance, the first mass moves 0.731 units in the same direction and in the second mode, when the second mass moves unit distance; the first mass moves 2.73 units in opposite direction.
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