# CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF FREEDOM

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CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF FREEDOM
Mechanical Vibration by Palm CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF FREEDOM Instructor: Dr Simin Nasseri, SPSU © Copyright, 2010 Based on a lecture from Brown university (Division of engineering)

Types of Forcing: External Forcing Base Excitation Rotor Excitation
All of these situations are of practical interest. Some subtle but important distinctions to consider, so we will look at each. But the strategy is simple: derive Equation of Motion and put into the “Standard Form” 2

04_03_01 Base Excitation (Seismic motion) 04_03_01.jpg

04_02_08 Base Excitation (Seismic motion) 04_02_08.jpg

Base Excitation – the Earthquake Problem
Here, base supporting object is subjected to motion. How does the object respond? Draw F.B.D. and get equation of motion…. Forces in the spring, dashpot are proportional to the motion RELATIVE to the base 5

Now in the “standard form” but with a new “driving force”
(Displacement Amplitude of body)/(Displacement Amplitude of Base)

Harmonic Base Excitation
Displacement transmission ratio: 7

Base Isolation Concept:
Given an expected frequency of a driving force, Design spring/dashpot coupling to minimize response Clearly want to get in the regime “soft” springs (small k) Non-isolated Isolated “soft” spring 8

04_03_02 04_03_02.jpg 04_03_02

Design of Car Suspension for Wavy Roads:
k c m s Select springs (k) and shocks (c) to satisfy requirements of maximum car vibration amplitude when driving on a wavy road x(t) 16” 33 ft Car weighing 3000 lbs drives over a road with sinusoidal profile shown Design the suspension so that: 1. The vibration amplitude of the car is < 14” at all speeds and 2. The vibration amplitude of the car is < 4” at 55 mph 10

What is the “base excitation” here?
As car drives along at constant speed, it is as if the road is vibrating up and down underneath the car OK, but how do we represent that? 16” 33 ft Equation of road profile? Driving frequency of base is Design requirements are now: (Vibration amplitude < 14” at all speeds) at all frequencies at frequency 11

Let’s do this graphically, using our magnification plot:
1.75 0.5 will satisfy criterion #1 will satisfy criterion #2 for =0.35 12

Spring k required: Damping c required: 13

04_03_01tbl Summary: 04_03_01tbl.jpg

Harmonic Base Excitation – Motion Relative to Base
Sometimes the motion relative to the base is of interest Introducing the relative displacement z = x – y, the equation of motion becomes: Or: 15

High frequencies: base is moving but body is not, so relative motion = 1
Low frequencies: body moves with base – no relative motion

04_02_09 04_02_09.jpg

Note: device measures motions relative to its base
Design of a Seismograph: FYI Goal: Detect “Low Frequency Earthquake” tremors in the 1-5 Hz frequency range along the San Andreas fault. Constraints: Background vibrations over a wide range of higher frequencies occur with typical amplitudes of 0.1mm, so tremor amplitudes comparable to or smaller than this cannot be detected. Design a mass/spring/dashpot system (choose m, k, c) to: reliably detect tremors at a frequency of 3 Hz and having earth motion amplitudes of 0.01mm or larger, ensure that the maximum amplitude will not exceed 30 mm for earth motion amplitudes of 1.0 mm. Note: device measures motions relative to its base

Example:  when no damping Two solutions: And:
The motion of the outer cart is varying sinusoidally as shown. For what range of w is the amplitude of the motion of the mass m, relative to the cart less than 2b? when no damping Two solutions: (When ) And: (When )

We want for (0.817) for (1.414)

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