hwk10_soln

hwk10_soln - Homework 10.1 A fixed-fixed string having a...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 10.1 A fixed-fixed string having a tension of P and mass/length ρ is excited by a force/length of f ( x,t ) = f g ( t ), as shown below with f being constant in both x and t . If g ( t ) = sin Ω t , the particular solution for the EOM for this system at x = L/ 3 can be written as u ( L/ 3 ,t ) = U (Ω) sin Ω t a) Determine an expression for U (Ω) using modal uncoupling. This result will be in terms of an infinite summation. b) Identify the modes that do NOT contribute to U (Ω). c) Make a hand sketch of U (Ω) vs. Ω for Ω up through the first four resonance frequencies that appear in the response at x = L/ 3. (Note that not all modes will contribute to this response.) Clearly indicate all frequencies of resonance and anti-resonance in this frequency range. f g t ( ) u x , t ( ) SOLUTION The natural frequencies and mass-normalized modal functions for the fixed-fixed string are given by: ω j = jπ s P ρL 2 ˆ U ( j ) ( x ) = r 2 ρL sinjπ x L The following modal coordinate transformation using these modes: u ( x,t ) = ∞ X j =1 ˆ U ( j ) ( x ) p j ( t ) produces the following uncoupled modal EOMs: ¨ p j + ω 2 j p j = Z L ˆ U ( j ) ( x ) f ( x,t ) dx = ˆ f j sin Ω t where ˆ f j = f Z L L/ 2 ˆ U ( j ) ( x ) dx = f r 2 ρL Z L L/ 2 sinjπ x L dx =- f r 2 ρL L jπ cosjπ x L x = L x = L/ 2 = f r 2 ρL L jπ cosj π 2- cosjπ The particular solutions of the modal equations are: p j ( t ) = ˆ f j ω 2 j- Ω 2 sin Ω t = P j sin Ω t Therefore, u ( L 3 ,t ) = ∞ X j =1 ˆ U ( j ) ( L 3 ) P j sin Ω t = U (Ω) sin Ω t where U (Ω) = f r 2 ρL 2 L π ∞ X j =1 ( cosj π 2- cosjπ ) sinj π 3 j ( ω...
View Full Document

This document was uploaded on 12/23/2011.

Page1 / 7

hwk10_soln - Homework 10.1 A fixed-fixed string having a...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online