Slide 1Dr. S. P. Harsha
Dr. S. P. Harsha
So far we dealt with discrete systems where mass, damping, and elasticity were assumed to be present only at certain discrete points in the system. There are many cases, known as distributed or continuous systems, in which it is not possible to identify discrete masses, dampers, or springs.
We must then consider the continuous distribution of the mass, damping, and elasticity and assume that each of the infinite number of points of the system can vibrate. That's is why a continuous system is also called a system of infinite degrees of freedom.
Introduction
Dr. S. P. Harsha
If a system is modeled as a discrete one. the governing equations are ordinary differential equations. Which are relatively easy to solve.
On the other hand, if the system is modeled as a continuous one, the governing equations are partial differential equations. Which are more difficult.
we can consider the vibration of simple continuous systems strings, bars, shafts, beams, and membranes.
Dr. S. P. Harsha
Dr. S. P. Harsha
Consider an elastic bar of length l with varying cross-sectional area A(x),
The axial forces acting on the cross sections of a small element of the bar of length dx are given by P and P + dP with
where σ is the axial stress, E is Young’s modulus, u is the axial displacement, and ∂u/∂x is the axial strain. If f (x, t) denotes the external force per unit length, the resulting force acting on the bar element in the x direction is
EQUATION OF MOTION OF A BAR IN AXIAL VIBRATION
Dr. S. P. Harsha
Dr. S. P. Harsha
mass × acceleration = resultant force
Dr. S. P. Harsha
Dr. S. P. Harsha
Longitudinal vibration of a bar
where ρ is the mass density of the bar. Now as
dP = (∂P/∂x) dx
Dr. S. P. Harsha
Dr. S. P. Harsha
A thin beam subjected to a transverse force is shown in Fig. Consider the free-body diagram of an element of a beam of length dx shown in Fig, where M(x, t) is the bending moment, V (x, t) is the shear force, and f (x, t) is the external transverse force per unit length of the beam. Since the inertia force (mass of the element times the acceleration) acting on the element of the beam is
EQUATION OF MOTION OF A BEAM IN TRANSVERSE VIBRATION
Dr. S. P. Harsha
MECHANICAL & INDUSTRIAL ENGINEERING DEPARTMENT INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
Since, the inertia force (mass of the element times the acceleration) acting on the element of the beam is
The force balance equation is
The moment balance equation is
EQUATION OF MOTION OF A BEAM IN TRANSVERSE VIBRATION
MECHANICAL & INDUSTRIAL ENGINEERING DEPARTMENT INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
Dr. S. P. Harsha
Use the relation , the equation becomes
As per the moment deflection relation
Equation of motion for forced lateral vibrations of a non-uniform beam as
For uniform beam
Dr. S. P. Harsha
Dr. S. P. Harsha
if the beam is given an initial displacement and an initial velocity , the initial conditions can be expressed as
If the beam is fixed at x = 0 and pinned at x = l, the deflection and slope will be zero at x = 0 and the deflection and the bending moment will be zero at x = l. Hence, the boundary conditions are given by
EQUATION OF MOTION OF A BEAM IN TRANSVERSE VIBRATION
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Consider a tightly stretched elastic string or cable of length l subjected to a equation of transverse force f(x, t ) per unit length.
The transverse displacement of the string w(x, t) is assumed to be small. Equilibrium of the force in the z direction give
net force acting on an element = inertia force acting on the element
TRANSVERSE VIBRATION OF A STRING OR CABLE
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
If P is the tension, ρ is the mass per unit length, and θ is the angle made by the deflected string with the x axis, for an element of length dx, as
Dr. S. P. Harsha
Dr. S. P. Harsha
If the string is uniform and the tension is constant.
Dr. S. P. Harsha
Dr. S. P. Harsha
If f(x, t) = 0, we obtain the free vibration equation
Hence the forced vibration equation of the non-uniform string,
wave equation
Dr. S. P. Harsha
Hence, the forced vibration equation of the non-uniform string,
If the string is uniform and the tension is constant, and for free vibration condition:
If the string is fixed at an end, say x = 0; the displacement w must always be zero and so the boundary condition is
Dr. S. P. Harsha
Dr. S. P. Harsha
If the string or cable is connected to a pin that can move in a perpendicular direction the end cannot support a transverse force.
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha
It can be solved by the method of
Separation of variables as
In this method, the solution is written as the product of a function W (x) (which depends only on x) and a function T(t) (which depends only on t) as which leads to
Dr. S. P. Harsha
Dr. S. P. Harsha
Solution of the Eq. is
Dr. S. P. Harsha
Dr. S. P. Harsha
Dr. S. P. Harsha