### Credits:

3L-0T-0L-0D (9 Credits)

### Objective:

This course aims to setting-up initial-boundary value problems for some important and fundamental structural members viz. bars, strings, membrane and plates. Analytical and approximate solutions to these problems for various loading and boundary conditions are discussed and analyzed.

### Course Content:

Vibrations in bars, strings, thin and thick beams, membranes and thin plates. Derivation of equations of motion for these structures. Computation of frequencies and mode shapes. Modal analysis. Analysis of damped and forced vibrations. Computational and experimental demonstrations.

### Lecturewise Breakup

I. Introduction : (1 Lectures)

• Linear versus nonlinear vibrations; linear vibrations: principle of superposition in modal and frequency domain.

II. Introduction to mode shapes and frequency for undamped systems.(2 Lectures)

III. Hamilton's principle and Euler-Lagrange equations.(3 Lectures)

IV. Axial and torsional vibrations in bars, transverse vibrations in strings (5 Lectures)

• Derivation of equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality, RayleighRitz method.

V. Incorporation of forcing and damping terms in above systems (bars and strings).(3 Lectures)

VI. Approximate methods : (Lecture 22–24) (4 Lectures)

• Galerkin and finite difference.

VII. Travelling wave solution, transient problem, reflections from the boundaries and simple impact problems. (3 Lectures)

VIII. Euler-Bernoulli and Timoshenko beams (5 Lectures)

• Derivation of equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality

IX. Membranes (circular disc, rectangular and spherical) :(3 Lectures)

• Equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality, Helmholtz equation.

X. Thin plates :(4 Lectures)

• Equation of motion, boundary and initial conditions, mode shapes, frequencies, orthogonality.

• Select any one from (i) Elementary theory of modal analysis/testing, (ii) Shell theory , and (iii) Non-proportional damping.

XII. Computational (using FEM) or Experimental demonstrations. (5 Lectures)

• Select any one from (i) Elementary theory of modal analysis/testing, (ii) Shell theory , and (iii) Non-proportional damping.

### References:

1. Elements of Vibration Analysis, L. Meirovitch, 2nd edition, McGraw Hill Education (India), 1986

2. Methods of Analytical Dynamics, L. Meirovitch,, Dover publications, 2010.

3. Vibration of Continuous Systems, S. S. Rao, John Wiley & Sons, 2007.

4. Vibration and Waves in Continuous Mechanical Systems, P. Hegedorn and A. DasGupta, Wiley, 2007.

5. Wave Motion in Elastic Solids, K. F. Graff, Dover Publications, 1991.

6. Mechanics of Continua and Wave Dynamics, L. Brekhovskikh and V. Goncharov, SpringerVerlag, 1985.