Advanced Engineering Mathematics

1. Advanced Engineering Mathematics LAPLACE TRANSFORM

2. Laplace Transform

3. Laplace Transform Problem 1:

4. Linear Transform

5. Laplace Transform Problem 2: Evaluate L{t}

6. Transformation Laplace Problem 3: Evaluate L{e-3t}

7. Transformation Laplace Problem 4: Evaluate L{sin2 t}

8. Transformation Laplace Problem 2:

9. Inverse Transform

10. Linear Transform

11. Inverse Transform Problem 1:

12. Inverse Transform Problem 2:

13. Inverse Transform Problem 3:

14. Axis of symmetry Deflection of curve Applications Deflection of Beams Beam is assumed as a homogeneous, and has uniform cross sections along its length Deflection curve can be derived from differential equation based on elasticity concept.

15. L 0 x y(x) y Applications Deflection of Beams Elasticity theory: bending moment M(x) at a point x along the beam is related to the load per unit length w(x)

16. L 0 y(x) x y(x) y Applications Deflection of Beams

17. L 0 x y(x) y Applications Deflection of Beams • y(0) = 0 at embedded end. • y’(0) = 0 (deflection curve is tangent to the x-axis at embedded end) • y”(L) = 0, bending moment at free end is zer0. • y”’(L) = 0, shear force is zero at a free end. EIy’’’ = dM/dx is the shear force.

18. w0 Wall L y Applications Determining deflection of a Beam using Laplace Transform x A beam of length L is embedded at both ends. In this case the deflection y(x) must satisfy:

19. Applications Determining deflection of a Beam using Laplace Transform

20. Applications Determining deflection of a Beam using Laplace Transform