The equation of motion of a forced vibration with damping is given by mu"(t) + yu' (t) + ku(t)= Fo cos(wt), (1) where m, y, k are the mass, damping constant and the spring constant, respectively. Fo and w are positive constants denoting the amplitude and frequency of the external force. We know the general form of the solution for the above differential equation is U = Uc+ Up, where ue is the complementary solution and up is a particular solution for (1). To solve the differential equation (1), we can assume that up A cos(wt) + B sin(wt) and use the method of undetermined coefficients to find A and B. Then = section 3.7 of calculations up(t)= R cos(wt - 8). Prove that m(w² - w²) A where A = √m²(w² - w²)² + y²w² and w = k/m. R = Fo cos d = sin d = γω A
The equation of motion of a forced vibration with damping is given by mu"(t) + yu' (t) + ku(t)= Fo cos(wt), (1) where m, y, k are the mass, damping constant and the spring constant, respectively. Fo and w are positive constants denoting the amplitude and frequency of the external force. We know the general form of the solution for the above differential equation is U = Uc+ Up, where ue is the complementary solution and up is a particular solution for (1). To solve the differential equation (1), we can assume that up A cos(wt) + B sin(wt) and use the method of undetermined coefficients to find A and B. Then = section 3.7 of calculations up(t)= R cos(wt - 8). Prove that m(w² - w²) A where A = √m²(w² - w²)² + y²w² and w = k/m. R = Fo cos d = sin d = γω A
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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Transcribed Image Text:The equation of motion of a forced vibration with damping is given by
mu"(t) + yu' (t) + ku(t)= Fo cos(wt),
(1)
where m, y, k are the mass, damping constant and the spring constant, respectively. Fo and w
are positive constants denoting the amplitude and frequency of the external force. We know
the general form of the solution for the above differential equation is
U = Uc+ Up,
where ue is the complementary solution and up is a particular solution for (1). To solve the
differential equation (1), we can assume that up A cos(wt) + B sin(wt) and use the method of
undetermined coefficients to find A and B. Then
=
section 3.7
of
calculations
up(t)= R cos(wt - 8).
Prove that
m(w² - w²)
A
where A = √m² (w² - w²)² + y²w² and we = k/m.
R =
Fo
cos d =
sin d =
γω
A
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