Advanced Engineering Mathematics LAPLACE TRANSFORM

Laplace Transform

Laplace Transform Problem 1:

Linear Transform

Laplace Transform Problem 2: Evaluate L{t}

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

Transformation Laplace Problem 4: Evaluate L{sin2 t}

Transformation Laplace Problem 2:

Inverse Transform

Linear Transform

Inverse Transform Problem 1:

Inverse Transform Problem 2:

Inverse Transform Problem 3:

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.

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

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

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

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

Applications Determining deflection of a Beam using Laplace Transform

Applications Determining deflection of a Beam using Laplace Transform