Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
bartleby

Videos

Question
Book Icon
Chapter 4.3, Problem 2E
Interpretation Introduction

Interpretation:

To solve the given equation for θ, where the given integral is T = πω-asinθ. Show that sinθ = 2u1+u2 by using right angle triangle. Express T as an integral with respect to u.

Concept Introduction:

Non-uniform oscillator is moving with time period T and angular frequency ω.

The period of oscillation can be found analytically, which is shown below.

Expert Solution & Answer
Check Mark

Answer to Problem 2E

Solution:

The solution for equation T = πω-asinθ is dθ = 2du1+u2.

sinθ = 2u1+u2 is shown below.

It is proved that u± when as  θ± π.

The expression for T in the form of u is shown below.

The period of oscillation for non-uniform oscillator is T = ω2-a2.

Explanation of Solution

The period of oscillation for the non-uniform oscillator is

T = πω-asinθ

Here, ω is the angular frequency, and a is the amplitude (ω > a > 0).

The differentiation of tan-1x is

ddx(tan-x) = 11+x2

(a)

Solve the equation for θ, and express in terms of u and du.

Consider the equation.

Rearrange the equation as

θ2 = tan-1 u

θ = 2tan-1u

Differentiate the above equation.

T = πω-asinθ, sinθ = 2u1+u2 and dθ = 2du1+u2

Hence, the expression for in terms of u and du is dθ = 2du1+u2.

(b)

Show that sinθ = 2u1+u2

The figure below shows the right-angled triangle with base 1 and height or perpendicular u:

Nonlinear Dynamics and Chaos, Chapter 4.3, Problem 2E

From the above triangle,

cosθ2 = basehypotenuse         = 11+u2

From the half-angle formula,

sinθ = 2sinθ2cosθ2       = 2(u1+u2)(11+u2)       = 2u1+u2

Hence, it is proved that sinθ = 2u1+u2

(c)

Show that u± as  θ± π.

Substitute π for θ in the equation u = tanθ2.

u = tanπ2   = +

Substitute for θ in the equation u = tanθ2.

u = tan2

 = -tanπ2

= -

Hence, it is proved that u± as  θ± π.

(d)

Express T as an integral with respect to u.

Consider T = πω-asinθ, sinθ = 2u1+u2 and dθ = 2du1+u2

θ u = tanθ2
tan2=
π tanπ2=

Substitute 2u1+u2 for sinθ and 2du1+u2 for dθ in T = πω-asinθ.

T = 2πduωu2 - 2au + ω

T = π(2du1+u2)ω - a(2u1+u2)

  = 2-duω(1+u2) - (a)(2u)

  = 2-duωu2 - 2au + ω

T = 2πduωu2 - 2au + ω

Hence, the expression for T as an integral with respect to u is

T = 2πduωu2 - 2au + ω

(e)

Complete the square in the denominator of the integrand of part (d).

From part (d),

T = 2-duωu2 - 2au + ω

Simplify as,

T = 2-duω(u2 - 2aω + 1)

2ω-duu2 - 2uaω+a2ω2 + (1-a2ω2)

2ω-du(u - aω) + (1-a2ω2)

Let x = u-aω, r = 1-a2ω2

Then,

u = x + aωdu = dx+0du = dx

So, replace du = dx, x = u - aω, r = 1-a2ω2.

T = 2ω-dxx2 + r   = 2ω-dxx2 + (r)2

Put the limits.

T = 2ω1r{[tan-1()]-[tan-1(-)]}   = 2ωr[(π2)-(π2)]   = ωr

Replace r = 1-a2ω2.

T = ω1-a2ω2

  = ωω2-a2ω2.

ω2-a2

Hence, the period of oscillation for the non-uniform oscillator is T = ω2-a2.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
५ (x² + 2x-y³) (16 x + 15) dy (x+2+y2) (x+2)3 =
Q5. Manager of car dealership is trying to see how number of sale associates can affect number of final sales in his dealership. He collects the following information: Number of cars Number of working sale associates per day sold in one day 2 2 3 5 3 3 4 1 3 1 2 4 5 Calculate the correlation coefficient for this data set using the equation given on slide#77? Comment on the association of the two variables. ΣΥ) - Σ) × Σ(Υ) (X) E(Y) N 2 (Σ(x²) - 2x²) × (Σ(12) - ²) N N
Q3. The distribution for the working lifetime of light bulbs, manufactured in a company, is found to be normally distributed with a mean of 1450 hours and a standard deviation of 60 hours. a) In this distribution, find the life time of a lightbulb whose z-score is -1.8? b) Which percentage of lightbulbs have life time less than 1400 hours? c) Which percentage of lightbulbs have life time greater than 1500 hours? d) Which percentage of lightbulbs have life time between 1420 to 1500 hours?
Knowledge Booster
Background pattern image
Advanced Math
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Text book image
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning
Text book image
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Text book image
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Cengage Learning
Text book image
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Text book image
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Intro to the Laplace Transform & Three Examples; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=KqokoYr_h1A;License: Standard YouTube License, CC-BY