Consider the spring-mass-damper system (SMD) mounted on a massless cart as shown in Figure 1. The mathematical model of the system is given by: dx m+b -+ky=b+kx dt dt d'y dy dt Where m is the mass of the cart, x = u (input of the system), b is damping coefficient, and k is spring constant. If m=1 kg, b=2 N-s/m, and -10 N/m. Answer the following questions: Macart 111 Figure 1: SMD on massless cart (2-b) If the input x(t) = 2t+40 sin(t), solve the differential equation (i.e, find the general solution of the resulted non-homogeneous differential equation). What will be the solution assuming zero initial conditions?

Consider the spring-mass-damper system (SMD) mounted on a massless cart as shown in Figure 1. The mathematical model of the system is given by: dx m+b -+ky=b+kx dt dt d'y dy dt Where m is the mass of the cart, x = u (input of the system), b is damping coefficient, and k is spring constant. If m=1 kg, b=2 N-s/m, and -10 N/m. Answer the following questions: Macart 111 Figure 1: SMD on massless cart (2-b) If the input x(t) = 2t+40 sin(t), solve the differential equation (i.e, find the general solution of the resulted non-homogeneous differential equation). What will be the solution assuming zero initial conditions?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the spring-mass-damper system (SMD) mounted on a massless cart as shown in Figure 1. The
mathematical model of the system is given by:
d'y dy
m +b+ky=b+kxx
dt²
dt
Mascles cart
D
Where m is the mass of the cart, x = u (input of the system), b is damping coefficient, and k is spring
constant. If m=1 kg, b=2 N-s/m, and k-10 N/m. Answer the following questions:
dx
m
dt
Figure 1: SMD on massless cart
(2-b) If the input x(t) = 2t + 40 sin(t), solve the differential equation (i.e, find the general solution of
the resulted non-homogeneous differential equation). What will be the solution assuming zero
initial conditions?
(2-c) Derive the transfer function of the system
(Y(s))
X(s).
assuming zero initial condition.
(2-d) If the input x(t) = 2t + 40 sin(t), what is X(s)? what is Y(s)? and what is y(t)? compare with the
result from 2-b.

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Step 1: Analysis and Introduction

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Step 2: Find the value of X(s) and x'(t)

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Step 3: Part b) - Laplace Transformation

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Step 4: Part b) - Inverse Laplace Transformation

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Step 5: Part c) and part d)

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(2-c) Derive the transfer function of the system
(Y(3)
X(s)
assuming zero initial condition.
(2-d) If the input x(t) = 2t + 40 sin(t), what is X(s)? what is Y(s)? and what is y(t)? compare with the
result from 2-b.

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