Explanation Given: The given damped forced motion is, d 2 y d t 2 + c m d y d t + k m y = F 0 m cos ω t The steady state solution of above equation can be written in form, y p ( t ) = F 0 H cos ( ω t − υ ) Here, cos υ = m ( ω 0 2 − ω 2 ) H , sin υ = c ω H , ω 0 = k m and H = m 2 ( ω 0 2 − ω 2 ) + c 2 ω 2 Approach: For a damped forced motion given by, d 2 y d t 2 + c m d y d t + k m y = F 0 m cos ω t The frequency for which amplitude is maximized is same as minimizing the frequency dependence in H . Calculation: Consider the given value of H . H = m 2 ( ω 0 2 − ω 2 ) + c 2 ω 2 Partially differentiate above equation with respect to ω . ∂ H ∂ ω = 1 2 2 m 2 ( ω 0 2 − ω 2 ) ( − 2 ω ) + 2 c 2 ω m 2 ( ω 0 2 − ω 2 ) + c 2 ω 2 Substitute 0 for ∂ H ∂ ω in above equation

Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)

4th Edition

ISBN: 9780321985811

Author: Stephen W. Goode, Scott A. Annin

Publisher: Pearson (edition 4)

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 8.5, Problem 31P

To determine

To prove: The amplitude of the steady state solution is a maximum when ω=ω02−c22m2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 30P

Chapter 8.5, Problem 32P

Students have asked these similar questions

Let 2 A = 4 3 -4 0 1 (a) Show that v = eigenvalue. () is an eigenvector of A and find the corresponding (b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a) may be useful. (c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.

(c) Let 6 0 0 A = -10 4 8 5 1 2 (i) Find the characteristic polynomial of A and factorise it. (ii) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (iii) Is A diagonalisable? Give reasons for your answer.

most 2, and let Let P2 denote the vector space of polynomials of degree at D: P2➡ P2 be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative p'(t) 2at+b, that is, D(p) = p'. (a) Prove that D is a linear transformation. (b) Find a basis for the kernel ker(D) of the linear transformation D and compute its nullity. (c) Find a basis for the image im(D) of the linear transformation D and compute its rank. (d) Verify that the Rank-Nullity Theorem holds for the linear transformation D. (e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.

Chapter 8 Solutions

Differential Equations And Linear Algebra, Books A La Carte Edition (4th Edition)

Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - Prob. 5TFR Ch. 8.1 - Prob. 6TFR Ch. 8.1 - True-False Review For Questions a-j, decide if the...Ch. 8.1 - Prob. 8TFR Ch. 8.1 - Prob. 9TFR Ch. 8.1 - True-False Review For Questions a-j, decide if the...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To prove: The amplitude of the steady state solution is a maximum when ω = ω 0 2 − c 2 2 m 2 .

Solution Summary: The author explains that the amplitude of the steady state solution is a maximum when omega =sqrt

Concept explainers

Videos

To prove: The amplitude of the steady state solution is a maximum when ω=ω02−c22m2.

Want to see the full answer?

Chapter 8 Solutions