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Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions. To investigate this, complete parts (a) through (e) below. (a) Find the synchronous solution Acos S2t + Bsin 2: to the generic forced oscillator equation my" + by' + ky = cos 1t. y(l) = cos C1 + V sin S2t. (b) Sketch graphs of the coefficients A and B, as functions of 2. for m = 1, b= 0.1, and k = 25. Choose the correct graph below. O A. OB. Oc. OD. 1.5 1.5 A 2.5 2.5 2.5 2.5 IB 1.5 1.5 B. (c) Now set b=0 in your formulas for A and Band resketch the graphs in part (b), with m= 1. and k= 25. What happens at 2= 5? Notice that the armplitudes of the syrchronous solutions grow without bound as 2 approaches 5. Choose the correct graph below. OA. OB. Oc. OD. 4- 4- 4- A B A A

Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions. To investigate this, complete parts (a) through (e) below. (a) Find the synchronous solution Acos S2t + Bsin 2: to the generic forced oscillator equation my" + by' + ky = cos 1t. y(l) = cos C1 + V sin S2t. (b) Sketch graphs of the coefficients A and B, as functions of 2. for m = 1, b= 0.1, and k = 25. Choose the correct graph below. O A. OB. Oc. OD. 1.5 1.5 A 2.5 2.5 2.5 2.5 IB 1.5 1.5 B. (c) Now set b=0 in your formulas for A and Band resketch the graphs in part (b), with m= 1. and k= 25. What happens at 2= 5? Notice that the armplitudes of the syrchronous solutions grow without bound as 2 approaches 5. Choose the correct graph below. OA. OB. Oc. OD. 4- 4- 4- A B A A

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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What happens at 2=5? O A. The uridarriped oscillator ostillates at a constant frequericy indelinitely. O B. The trequency of the oscillations becomes unbounded. OC. The undamped oscillator stretches indefinitely. OD. The amplitude of the oscillations becomes unbounded. (d) Show directly, by substituting the form Acos 2: - Bsin 12t Into the equation my" + by' + ky = cos (2t, that when b=0 there are no aynchronous solutions if 12 = Select the correct cholce below and, if neceasary, fill in the anawer box within your chcice. OA. Substituting b= 0 and 2= m Into the coefficients A and Band simplifylng results in the equation which has no solution. OB. Substituting b=0 and 2= into the cocfficients A and B and simplifying results in civision by 0. k tsin r solves the equation my' + by' + ky = cos t when b= 0 and 2= . Notice that this nonsynchronous solution grows in time, without bound. (e) Verlty that (2m2) - [2mA :sin . Fnd y't After substituting for 2, the given differential equation will be of the form my" + ky = cos =tand the solution will be of the form y(t) = What is observed in the simplification process after insertion of y"(t) and y(t) into the differential equation my" + ky = cos m The term in y"(t) cancels out, resulting in the equation cos t= cos t, thus completing the verification.
Transcribed Image Text:What happens at 2=5? O A. The uridarriped oscillator ostillates at a constant frequericy indelinitely. O B. The trequency of the oscillations becomes unbounded. OC. The undamped oscillator stretches indefinitely. OD. The amplitude of the oscillations becomes unbounded. (d) Show directly, by substituting the form Acos 2: - Bsin 12t Into the equation my" + by' + ky = cos (2t, that when b=0 there are no aynchronous solutions if 12 = Select the correct cholce below and, if neceasary, fill in the anawer box within your chcice. OA. Substituting b= 0 and 2= m Into the coefficients A and Band simplifylng results in the equation which has no solution. OB. Substituting b=0 and 2= into the cocfficients A and B and simplifying results in civision by 0. k tsin r solves the equation my' + by' + ky = cos t when b= 0 and 2= . Notice that this nonsynchronous solution grows in time, without bound. (e) Verlty that (2m2) - [2mA :sin . Fnd y't After substituting for 2, the given differential equation will be of the form my" + ky = cos =tand the solution will be of the form y(t) = What is observed in the simplification process after insertion of y"(t) and y(t) into the differential equation my" + ky = cos m The term in y"(t) cancels out, resulting in the equation cos t= cos t, thus completing the verification.
Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions. To investigate this, complete parts (a) through (e) below. (a) Find the synchronous solution Acos S2t + Bsin 2: to the generic forced oscillator equation my" + by' + ky = cos 1t. y(l) = cos C1 + V sin S2t. (b) Sketch graphs of the coefficients A and B, as functions of 2. for m = 1, b= 0.1, and k = 25. Choose the correct graph below. O A. OB. Oc. OD. 1.5 1.5 A 2.5 2.5 2.5 2.5 IB 1.5 1.5 B. (c) Now set b=0 in your formulas for A and Band resketch the graphs in part (b), with m= 1. and k= 25. What happens at 2= 5? Notice that the armplitudes of the syrchronous solutions grow without bound as 2 approaches 5. Choose the correct graph below. OA. OB. Oc. OD. 4- 4- 4- A B A A
Transcribed Image Text:Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions. To investigate this, complete parts (a) through (e) below. (a) Find the synchronous solution Acos S2t + Bsin 2: to the generic forced oscillator equation my" + by' + ky = cos 1t. y(l) = cos C1 + V sin S2t. (b) Sketch graphs of the coefficients A and B, as functions of 2. for m = 1, b= 0.1, and k = 25. Choose the correct graph below. O A. OB. Oc. OD. 1.5 1.5 A 2.5 2.5 2.5 2.5 IB 1.5 1.5 B. (c) Now set b=0 in your formulas for A and Band resketch the graphs in part (b), with m= 1. and k= 25. What happens at 2= 5? Notice that the armplitudes of the syrchronous solutions grow without bound as 2 approaches 5. Choose the correct graph below. OA. OB. Oc. OD. 4- 4- 4- A B A A
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