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

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

17. Applications y(0) = 0 at embedded end.
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.
y(x) L y x

18. Applications Determining deflection of a Beam using Laplace Transformw0 L
Wall y 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