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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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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
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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