Two masses and 3 springs. нотот Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass m con- nected to each other and to walls on either side by springs of constant k and rest length a, as seen in the figure. The distance between the two walls is 3a. I (a) Find the lagrangian and Lagrange's equations. (b) What are the normal-mode frequencies and eigenvectors? Describe the motions. (c) Construct explicitly the modal matrix A¡¡, the normal coordinates Çk, and the diagonal form L = Σk 1 (S² – w²5²) of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values). (d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
Two masses and 3 springs. нотот Consider the longitudinal oscillations, i.e., along the axis, of a mechanical system composed of two particles of mass m con- nected to each other and to walls on either side by springs of constant k and rest length a, as seen in the figure. The distance between the two walls is 3a. I (a) Find the lagrangian and Lagrange's equations. (b) What are the normal-mode frequencies and eigenvectors? Describe the motions. (c) Construct explicitly the modal matrix A¡¡, the normal coordinates Çk, and the diagonal form L = Σk 1 (S² – w²5²) of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values). (d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
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Question 1.
![plus time spent in office hours).
1. Two masses and 3 springs.
нотот
7°F
Clear
m
(a) Find the lagrangian and Lagrange's equations.
(b) What are the normal-mode frequencies and eigenvectors? Describe the motions.
(c) Construct explicitly the modal matrix Aij, the normal coordinates Çk, and the diagonal form L = Σk ½ (5² - w²²)
of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values).
(d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the
right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
▬
Consider the longitudinal oscillations, i.e., along the axis, of a
mechanical system composed of two particles of mass m con-
nected to each other and to walls on either side by springs of
constant k and rest length a, as seen in the figure. The distance
between the two walls is 3a.
I
Q Search
D
i
I](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2F0804f28e-97ed-431f-8044-a14978c8ecfb%2Fqxx1klg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:plus time spent in office hours).
1. Two masses and 3 springs.
нотот
7°F
Clear
m
(a) Find the lagrangian and Lagrange's equations.
(b) What are the normal-mode frequencies and eigenvectors? Describe the motions.
(c) Construct explicitly the modal matrix Aij, the normal coordinates Çk, and the diagonal form L = Σk ½ (5² - w²²)
of the lagrangian (i.e. write L as an explicit sum with two terms and replace the frequencies by their values).
(d) Suppose the mass on the left is initially displaced from equilibrium a distance a to the right, the mass on the
right starts at its equilibrium position, and both masses start at rest. Compute the subsequent motion.
▬
Consider the longitudinal oscillations, i.e., along the axis, of a
mechanical system composed of two particles of mass m con-
nected to each other and to walls on either side by springs of
constant k and rest length a, as seen in the figure. The distance
between the two walls is 3a.
I
Q Search
D
i
I
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