(A mechanical system with two springs and two masses) m Jiga J )=(...
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(A mechanical system with two springs and two masses) Кл мееее m₁ Jiga лееееее J₁ )=( with dimensionless constants We consider an undamped system of two (linear) springs with spring con- stants k1,2> 0 and masses m1,2 > 0 as in the picture above. The time- dependent variables y₁,2 are the elongations of the corresponding masses from their equilibrium position. a) ÿ₁ Y2 Verify that in appropriately chosen units of time and length, the elongations satisfy the second-order ODE system a := kı + k2 k2 -α В y₁ y2 ż₁ Ż2 k₂ B := " ( 1 -ß Y2 B:= Y1 Y2 21 22 b) Introducing the velocities 21,2 = 1,2, rewrite the system as a first- order system of the form m. Y1 m1 M2 -α 1 B -B 3) Iz ). (Give the matrix A explicitly.) Show that if X is an eigenvalue of A then X² is an eigenvalue of the the matrix (1) c) Show that all eigenvalues of A are purely imaginary. What proper- ties of the general solution to (1) are following from this? (Are all solutions / some solutions bounded /unbounded? Do there exist non- zero constant solutions? Do there exist nonzero periodic solutions? Are there solutions having limits as t→∞o?) What does the term "eigenfrequencies" mean for our system, and what are its eigenfrequencies? (A mechanical system with two springs and two masses) Кл мееее m₁ Jiga лееееее J₁ )=( with dimensionless constants We consider an undamped system of two (linear) springs with spring con- stants k1,2> 0 and masses m1,2 > 0 as in the picture above. The time- dependent variables y₁,2 are the elongations of the corresponding masses from their equilibrium position. a) ÿ₁ Y2 Verify that in appropriately chosen units of time and length, the elongations satisfy the second-order ODE system a := kı + k2 k2 -α В y₁ y2 ż₁ Ż2 k₂ B := " ( 1 -ß Y2 B:= Y1 Y2 21 22 b) Introducing the velocities 21,2 = 1,2, rewrite the system as a first- order system of the form m. Y1 m1 M2 -α 1 B -B 3) Iz ). (Give the matrix A explicitly.) Show that if X is an eigenvalue of A then X² is an eigenvalue of the the matrix (1) c) Show that all eigenvalues of A are purely imaginary. What proper- ties of the general solution to (1) are following from this? (Are all solutions / some solutions bounded /unbounded? Do there exist non- zero constant solutions? Do there exist nonzero periodic solutions? Are there solutions having limits as t→∞o?) What does the term "eigenfrequencies" mean for our system, and what are its eigenfrequencies? (A mechanical system with two springs and two masses) Кл мееее m₁ Jiga лееееее J₁ )=( with dimensionless constants We consider an undamped system of two (linear) springs with spring con- stants k1,2> 0 and masses m1,2 > 0 as in the picture above. The time- dependent variables y₁,2 are the elongations of the corresponding masses from their equilibrium position. a) ÿ₁ Y2 Verify that in appropriately chosen units of time and length, the elongations satisfy the second-order ODE system a := kı + k2 k2 -α В y₁ y2 ż₁ Ż2 k₂ B := " ( 1 -ß Y2 B:= Y1 Y2 21 22 b) Introducing the velocities 21,2 = 1,2, rewrite the system as a first- order system of the form m. Y1 m1 M2 -α 1 B -B 3) Iz ). (Give the matrix A explicitly.) Show that if X is an eigenvalue of A then X² is an eigenvalue of the the matrix (1) c) Show that all eigenvalues of A are purely imaginary. What proper- ties of the general solution to (1) are following from this? (Are all solutions / some solutions bounded /unbounded? Do there exist non- zero constant solutions? Do there exist nonzero periodic solutions? Are there solutions having limits as t→∞o?) What does the term "eigenfrequencies" mean for our system, and what are its eigenfrequencies? (A mechanical system with two springs and two masses) Кл мееее m₁ Jiga лееееее J₁ )=( with dimensionless constants We consider an undamped system of two (linear) springs with spring con- stants k1,2> 0 and masses m1,2 > 0 as in the picture above. The time- dependent variables y₁,2 are the elongations of the corresponding masses from their equilibrium position. a) ÿ₁ Y2 Verify that in appropriately chosen units of time and length, the elongations satisfy the second-order ODE system a := kı + k2 k2 -α В y₁ y2 ż₁ Ż2 k₂ B := " ( 1 -ß Y2 B:= Y1 Y2 21 22 b) Introducing the velocities 21,2 = 1,2, rewrite the system as a first- order system of the form m. Y1 m1 M2 -α 1 B -B 3) Iz ). (Give the matrix A explicitly.) Show that if X is an eigenvalue of A then X² is an eigenvalue of the the matrix (1) c) Show that all eigenvalues of A are purely imaginary. What proper- ties of the general solution to (1) are following from this? (Are all solutions / some solutions bounded /unbounded? Do there exist non- zero constant solutions? Do there exist nonzero periodic solutions? Are there solutions having limits as t→∞o?) What does the term "eigenfrequencies" mean for our system, and what are its eigenfrequencies?
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Next we analyze the twodegreesof freedom 2DOF undamped massspring system of Figure 1221 Dynamic translations y t and y2 t shown are relative to the static equilibrium positions As usual for the purpos... View the full answer
Related Book For
Statistical Techniques in Business and Economics
ISBN: 978-1259666360
17th edition
Authors: Douglas A. Lind, William G Marchal
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