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

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External ForcingBase Excitation Types of Forcing: 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

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3 Base Excitation (Seismic motion)

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

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

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Harmonic Base Excitation Displacement transmission ratio:

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Base Isolation Concept: soft spring 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

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Design of Car Suspension for Wavy Roads: Car weighing 3000 lbs drives over a road with sinusoidal profile shown k c m s ft 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 Select springs (k) and shocks (c) to satisfy requirements of maximum car vibration amplitude when driving on a wavy road x(t)

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What is the base excitation here? Equation of road profile? ft 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? Driving frequency of base is Design requirements are now: (Vibration amplitude < 14 at all speeds)1. at all frequencies 2. at frequency

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Lets do this graphically, using our magnification plot: will satisfy criterion #2 for =0.35 will satisfy criterion #1

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Spring k required: Damping c required:

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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:

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

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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: (i)reliably detect tremors at a frequency of 3 Hz and having earth motion amplitudes of 0.01mm or larger, (ii) ensure that the maximum amplitude will not exceed 30 mm for earth motion amplitudes of 1.0 mm. Design of a Seismograph: Note: device measures motions relative to its base FYI

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Example : The motion of the outer cart is varying sinusoidally as shown. For what range of 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 )

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We want for (0.817) (1.414)

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