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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)

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**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

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Rotating unbalance Rotor excitation 04_04_01 04_04_01.jpg

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**Unbalanced. More mass one side than other.**

Rotor Excitation Rotating machinery not always perfectly balanced e.g. your car’s wheels; Imperfectly machined rotating disks; Turbine engine with cracked turbine blades w Unbalanced. More mass one side than other. Leads to a sinusoidally-varying force k c q=wt m M w e x(t) Engineering Model: Effective unbalanced mass m Effective eccentricity e In a machine of total mass M 4

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**Rotor Excitation q=wt w -kx -cv -FMm FmM R1 R2**

x(t) M is total mass INCLUDING eccentric mass Non-rotating part of the machine has position x, Rotating piece has position: FBDs for the two masses: -kx -cv -FMm FmM R1 R2 Equations of motion for the two masses (vertical direction only): Standard Form but with 5

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**Steady State Solution (same as before)**

Standard Form but with Steady State Solution (same as before) Same form as relative base excitation…….

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High frequencies: imbalance force grows at same rate as magnification factor decreases, leading to amplitude ratio =1 (not usually a problem) Low frequencies: not moving fast enough to generate a force large enough to drive the mass

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**Force Transmission to Base**

k c q=wt m w e x(t) With vibrating machinery, forces exerted on the supporting structures can become large near resonance. Equipment is thus constructed on isolating mounts (springs and dashpots – dashpots suppress resonance) k c kx cv What is the force transmitted? Draw FBD for base structure: Position, velocity are always 90o out of phase, so transmitted force is Since max of over all q is we have Maximum transmitted force: 8

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And we found earlier Force required to stretch the spring by a distance me/M

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**i.e. m = 10 kg Here, “Total mass of the device” = 10 kg**

Example: “Total mass of the device” = 10 kg i.e. m = 10 kg Determine the two possible values of the equivalent spring stiffness k for the mounting to permit the amplitude of the force transmitted to the fixed mounting due to the imbalance to be 1500 N at a speed of 1800 rpm Here,

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**i.e. we can choose a stiff spring and run the machine BELOW the natural freq**

or we can choose a soft spring and run the machine ABOVE the natural freq

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**04_04_01tbl Summary: =e R=e (eccentricity)**

04_04_01tbl.jpg Force transmitted to the base:

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