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Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface, governed by the ODE Ay + By = f(t) for some a > 2. Let f(t) be (i) Suppose A=B= 1, and the force f(t) represents a push to the left of the mass over a time period from t=2π to t=2π + a eller y(0) = 1 f(t)=-[u(t-2π)-u (t-(2π +a))] such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function ult) is the unit step function. Use Laplace Transform to solve the initial value problem } - () g(0) = 0 16 (ii) Suppose A = 152,B= 2 and the force f(t) represents an impulse force. Write down the governing equations for the 32 displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at t = 27 time units later. [Solve the governing equation!]

Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface, governed by the ODE Ay + By = f(t) for some a > 2. Let f(t) be (i) Suppose A=B= 1, and the force f(t) represents a push to the left of the mass over a time period from t=2π to t=2π + a eller y(0) = 1 f(t)=-[u(t-2π)-u (t-(2π +a))] such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function ult) is the unit step function. Use Laplace Transform to solve the initial value problem } - () g(0) = 0 16 (ii) Suppose A = 152,B= 2 and the force f(t) represents an impulse force. Write down the governing equations for the 32 displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at t = 27 time units later. [Solve the governing equation!]

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface, governed by the ODE Aÿj + By = f(t) (i) Suppose A=B= 1, and the force f(t) represents a push to the left of the mass over a time period from t = 2π to t = 2π + a for some a > 2π. Let f(t) be f(t)= [u(t-2π)-u(t-(2π +a))] such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function ult) is the unit step function. Use Laplace Transform to solve the initial value problem 16 (ii) Suppose A = ellee y(0) = 1 J (¹) gro) = 0 32 B = 2 and the force f(t) represents an impulse force. Write down the governing equations for the displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at t = 27 time units later. [Solve the governing equation!]
Transcribed Image Text:Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface, governed by the ODE Aÿj + By = f(t) (i) Suppose A=B= 1, and the force f(t) represents a push to the left of the mass over a time period from t = 2π to t = 2π + a for some a > 2π. Let f(t) be f(t)= [u(t-2π)-u(t-(2π +a))] such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function ult) is the unit step function. Use Laplace Transform to solve the initial value problem 16 (ii) Suppose A = ellee y(0) = 1 J (¹) gro) = 0 32 B = 2 and the force f(t) represents an impulse force. Write down the governing equations for the displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at t = 27 time units later. [Solve the governing equation!]
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