Spring-coupled masses

saurabhgoyal

please explain two body oscillators with some example i am very confused in it.

edison

By two body oscillators if you mean coupled oscillations. Then examples are

1) two masses at the two ends of a spring and thus coupled by means of spring

2) In a lattice or solid the neighboring atoms are coupled to each other and thus the motion of one influences the motion of other neighboring atoms.

3) Similarly we talk about coupled harmonic oscillator.

Moreover Coupled Oscillations occur when two or more oscillating systems are connected in such a manner as to allow motion energy to be exchanged between them. Coupled oscillators occur in nature (e.g., the moon and earth orbiting each other) or can be found in man-made devices (such as with the pacemaker).

Coupled oscillations

The harmonic oscillator, and the more complicated systems for which it stands as a simple model, has a single degree of freedom. More complicated systems have more degrees of freedom, for example two masses and two springs. In such cases, energy is converted between the respective inertias of each degree of freedom and the several restoring forces in the system. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks mounted on a common wall will tend to synchronise. The apparent motions of the individual oscillations typically appears very

Spring-coupled masses

Consider the two degree of freedom dynamical system pictured in folowing Figure. In this system, two point objects of mass

are free to move in one dimension. Furthermore, the masses are connected together by a spring of spring constant

, and are also each attached to fixed supports via springs of spring constant

.

**Spring Coupled masses**

Let

and

be the displacements of the first and second masses, respectively, from the equilibrium state. If follows that the extensions of the left-hand, middle, and right-hand springs are

,

, and

, respectively. The kinetic energy of the system takes the form

$egin{displaymath} K = rac{m}{2} ,(dot{q}_1^{,2} + dot{q}_2^{,2}), end{displaymath}$

whereas the potential energy is written

$egin{displaymath} U= rac{1}{2}left[k',q_1^{,2} + k,(q_2-q_1)^2+ k',q_2^{,2} ight]. end{displaymath}$

The above expression can be rearranged to give

$egin{displaymath} U= rac{1}{2}left[(k+k'),q_1^{,2} -2,k,q_1,q_2 + (k+k'),q_2^{,2} ight]. end{displaymath}$

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Coupled oscillations

Spring-coupled masses