2. Answer the following questions. (A) [50%] By Newton's second law of motion (my" = Ftotal(t)), the motion in the mass-spring system with a damper and periodic driving (external) force is governed by the oscillator DE my" +cy' +ky = Fo cos(wt) Fdriving (t) (1) where y = y(t) represents the position of the mass at any time t, my" is the inertial force, cy' is the damping force, ky is the restoring force, and the driving force Fdriving (t) is to make the mass move. Often, DE (1) is written in the form: y" +2λy' +wy = focos(wt) (2) с k D о k m +√2 where is called the damping ratio and wo = is called the (system) natural frequency. These parameters, A and wo, characterize the system itself. On the right-hand side of the DE, fo = is called the amplitude of the driving force, and w is called the angular frequency of the driving force. The latter parameters characterize the driving force. (i) (Simple harmonic motion (SHM)). Considering the undamped motion with zero driving force in DE (1), find the mass position (or displacement), y(t), at any time t. (ii) Let the constant parameters c₁ and c₂ in the general solution (found in part (i)) be defined by c₁ = Rcos, c₂ = R sin, with R = √ √₁₁+&² = tan¹ 0≤≤2TT The constant R is called the amplitude of the SHM (maximum displacement of the mass), and is the called the phase angle. Show that the mass displacment at any time can be written as follows: ° +√2 y(t) Rcos(woto) (3) (iii) Suppose that m = ½ (kg), k = 16 (N/m), the mass is initially displaced by + m (i.e., to the right of the equilibrium) (see Figure), and is initially given an outward velocity of +√2 (m/sec) (to the right). Determine the equation describing the mass motion (as defined in (3)) along with its amplitude, period (p = 2x/wo sec) of the periodic motion, and the frequency (cycle/sec). (B) [50%] (Resonance) Consider the undamped mass-spring system that undergoes the non-zero driving force fo cos(wpt) (i.e., w = wo). Answer the following questions. (i) Write the non-homogeneous DE that describes the mass motion. (ii) Suppose that the mass is initially at its equilibrium position and starts at rest. Show that the mass-spring system (when w=wo) experiences the so-called resonance phenomenon; that is, the mass displacement at any time t is given by y(t)= fot sin(wot) 2wo (iii) Considering fo = 1 and wo = 4, sketch the curve that represents the mass displacement at any time. Remark. Note that the sine wave oscillates (or grows) between the lines ±fot. 2wo (iv) To eliminate the resonance, suggest a mathematical modification to the DE obtained in part (i). What is the name of the force or the piece of equipment that has to be implemented into the system? Remark. You need not justify how the newly added term will eliminate the resonance; however, you are encouraged to investigate it by solving the new auxiliary (characteristic) equation. Also, feel free to ask the TA in the Lab. (C) (Beat) (Optional and will not be graded) Assuming wwo, undamped motion and non-zero driving force, and again (mass is initially at its equilibrium position and starts at rest), show that undamped system experiences the so-called beat phenomenon. You may refer to the beat frequency (when wwo) in Lab #6 where y(t)= 2fo w-w² sin wo-w 2 sin wo+w 2 outer wave inner wave 1

2. Answer the following questions. (A) [50%] By Newton's second law of motion (my" = Ftotal(t)), the motion in the mass-spring system with a damper and periodic driving (external) force is governed by the oscillator DE my" +cy' +ky = Fo cos(wt) Fdriving (t) (1) where y = y(t) represents the position of the mass at any time t, my" is the inertial force, cy' is the damping force, ky is the restoring force, and the driving force Fdriving (t) is to make the mass move. Often, DE (1) is written in the form: y" +2λy' +wy = focos(wt) (2) с k D о k m +√2 where is called the damping ratio and wo = is called the (system) natural frequency. These parameters, A and wo, characterize the system itself. On the right-hand side of the DE, fo = is called the amplitude of the driving force, and w is called the angular frequency of the driving force. The latter parameters characterize the driving force. (i) (Simple harmonic motion (SHM)). Considering the undamped motion with zero driving force in DE (1), find the mass position (or displacement), y(t), at any time t. (ii) Let the constant parameters c₁ and c₂ in the general solution (found in part (i)) be defined by c₁ = Rcos, c₂ = R sin, with R = √ √₁₁+&² = tan¹ 0≤≤2TT The constant R is called the amplitude of the SHM (maximum displacement of the mass), and is the called the phase angle. Show that the mass displacment at any time can be written as follows: ° +√2 y(t) Rcos(woto) (3) (iii) Suppose that m = ½ (kg), k = 16 (N/m), the mass is initially displaced by + m (i.e., to the right of the equilibrium) (see Figure), and is initially given an outward velocity of +√2 (m/sec) (to the right). Determine the equation describing the mass motion (as defined in (3)) along with its amplitude, period (p = 2x/wo sec) of the periodic motion, and the frequency (cycle/sec). (B) [50%] (Resonance) Consider the undamped mass-spring system that undergoes the non-zero driving force fo cos(wpt) (i.e., w = wo). Answer the following questions. (i) Write the non-homogeneous DE that describes the mass motion. (ii) Suppose that the mass is initially at its equilibrium position and starts at rest. Show that the mass-spring system (when w=wo) experiences the so-called resonance phenomenon; that is, the mass displacement at any time t is given by y(t)= fot sin(wot) 2wo (iii) Considering fo = 1 and wo = 4, sketch the curve that represents the mass displacement at any time. Remark. Note that the sine wave oscillates (or grows) between the lines ±fot. 2wo (iv) To eliminate the resonance, suggest a mathematical modification to the DE obtained in part (i). What is the name of the force or the piece of equipment that has to be implemented into the system? Remark. You need not justify how the newly added term will eliminate the resonance; however, you are encouraged to investigate it by solving the new auxiliary (characteristic) equation. Also, feel free to ask the TA in the Lab. (C) (Beat) (Optional and will not be graded) Assuming wwo, undamped motion and non-zero driving force, and again (mass is initially at its equilibrium position and starts at rest), show that undamped system experiences the so-called beat phenomenon. You may refer to the beat frequency (when wwo) in Lab #6 where y(t)= 2fo w-w² sin wo-w 2 sin wo+w 2 outer wave inner wave 1

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.7: Applications

Problem 18EQ

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