Continuous Transverse Vibration

In this module the equation of motion of continuous systems or distributed mass systems will be derived using both d'Alembert principle and extended Hamilton's principles. Different one-dimensional systems such as longitudinal vibration of rod, transverse vibration of string, torsional vibration of rod and transverse vibration of Euler-Bernoulli beams will be considered in this module.

Introduction to Continuous systems

In the previous modules we have studied about discrete mass system, which are modelled as single, two or multi-degrees of freedom systems. In these cases the system has a definite number of lumped masses, stiffness elements and damping elements. For example the cantilever beam with a tip mass as shown in Figure 10.1 is modelled as a single degree of freedom system with a spring and a mass. The stiffness k of the system was calculated using the following equation.
 

Here W is the weight of the attached mass, δ is the deflection of the beam with length L, Young's modulus E and moment of inertia I. The natural frequency can be calculated using the formula

 
Where m is the attached mass. In this calculation we have neglected the mass of the beam. Hence it may be observed that by considering a point mass at the tip we obtained one natural frequency of the system. Instead of modelling this system as a single-spring mass if one consider the beam to be consist of several masses, then the system can be modelled as a multi-degree of freedom system as shown in figure 2(a). But as the dimension of each elemental mass considered in the above case is arbitrary, one may consider the beam as a continuous system with infinite number of distributed mass and stiffness and hence has infinite number of natural frequencies as shown in fig 2 (b).
 


• So in contrast to the discrete mass system, in distributed mass or continuous system the system has infinite number of natural frequencies and corresponding to each natural frequency, the system will have a distinct mode shape.

• It may be observed that the response of the continuous system depends time and space coordinate (location). But in case of discrete system the response is only a function of time. Hence while the equation of motion of discrete systems are written in terms of ordinary differential equations, in case of continuous system they are written in terms of partial differential equations.

• It may be noted that all the real systems are continuous system.

• A continuous system for analysis purpose can be reduced to a finite number of discrete models.
             
Each discrete model can be reduced to an eigen value problem.

• In case of continuous system the solution yields infinite number of Eigen values and Eigen functions where as in discrete system the Eigen values and Eigen vectors are finite.

• The concept of orthogonality is applicable to both discrete and continuous systems.

• The eigen value problem in case of discrete system takes the form of algebraic equations while in continuous systems differential equations and sometimes integral equations are obtained. Eigenvectors of the discrete system becomes eigenfunction of the continuous system.

• The response of the system will depend on the boundary conditions. There are two different types of boundary conditions viz., geometric boundary conditions and natural boundary conditions.

• Geometry boundary conditions also known as essential or imposed boundary conditions result from conditions of purely geometric compatibility. For example in case of a clamped-clamped beam in both the ends deflections and slopes are zero (Fig. 3).
 

Strings are mostly used in musical instruments and many other applications of domestic and industrial in nature. A string of length L is shown in Figure 4(a), which is subjected to tension T. Let at time t = 0, the string is pulled in the lateral direction (y direction) as shown in figure 4 (b) and left. Hence the lateral deflection u along the string is a function of the space variable x and time t i.e.,

 
If the lateral deflection is small, the change in tension T due to the deflection is negligible. Figure 4(c) shows the free body diagram of an elemental length dx of the string. When the string is vibrating, in the y direction inertia force is acting. Considering the forces in the vertical direction, applying Newton 's second law one may have
 


Here m is the mass per unit length of the string. Now assuring small deflections u and slope q , the equation 1 reduces to following equation.



Now substituting the slope   in equation 2 one may write
 
         

The above equation is known as Wave equation. One may use Hamilton 's principle to derive the equation of motion of this system also.

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