Lesson 11: Solving Mechanical System Dynamics Using Laplace Transforms

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Presentation transcript:

Lesson 11: Solving Mechanical System Dynamics Using Laplace Transforms ET 438a Automatic Control Systems Technology lesson11et438a.pptx

Learning Objectives After this presentation you will be able to: lesson11et438a.pptx Learning Objectives After this presentation you will be able to: Draw a free body diagram of a translational mechanical system, Write a differential equation that describes the position of a mechanical system as it varies in time. Use Laplace transforms to convert differential equations into algebraic equations. Take the Inverse Laplace transform and find the time response of a mechanical system. Examine the impact of increased and decreased damping on a mechanical system.

Using Laplace Transforms to Solve Mechanical Systems lesson11et438a.pptx Using Laplace Transforms to Solve Mechanical Systems Example 11-1: Write the differential equation for the system shown with respect to position and solve it using Laplace transform methods. Assume f(t) = 50∙us(t) N, M= 1 Kg, K=2.5 N/m and B=0.5 N-s/m. The mass slides on a frictionless surface. x(0)=0. + Draw a free body diagram and label the forces fI(t) f (t) - fI(t) - fB(t) - fK(t) = 0 f(t) M= 1 Kg fB(t) f(t)=fI(t)+fB(t)+fK(t) fK(t)

Laplace Solution to Mechanical Systems (1) lesson11et438a.pptx Laplace Solution to Mechanical Systems (1) Solve for X(s)

Laplace Solution to Mechanical Systems (2) lesson11et438a.pptx Laplace Solution to Mechanical Systems (2) Solve using inverse Laplace transform and partial fractions expansion Use Quadratic formula to find factors in denominator

Laplace Solution to Mechanical Systems (3) lesson11et438a.pptx Laplace Solution to Mechanical Systems (3) Factored form s1 s2 Partial Fractions Expansion- Find A

Laplace Solution to Mechanical Systems (4) lesson11et438a.pptx Laplace Solution to Mechanical Systems (4)

Laplace Solution to Mechanical Systems (5) lesson11et438a.pptx Laplace Solution to Mechanical Systems (5) Find B

Laplace Solution to Mechanical Systems (6) lesson11et438a.pptx Laplace Solution to Mechanical Systems (6) Find C

Laplace Solution to Mechanical Systems (7) lesson11et438a.pptx Laplace Solution to Mechanical Systems (7) Reverse angle rotation Expanded Laplace relationship Remember

Laplace Solution to Mechanical Systems (8) lesson11et438a.pptx Laplace Solution to Mechanical Systems (8) From Laplace table Where Ans

Time Plot of System Position lesson11et438a.pptx Time Plot of System Position Damping of system is B=0.5

Effects of Increased Damping lesson11et438a.pptx Effects of Increased Damping Large damping value reduces oscillations and overshoot but slows response

lesson11et438a.pptx End Lesson 11: Solving Mechanical System Dynamics Using Laplace Transforms ET 438a Automatic Control Systems Technology