University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

146

Pages:

5
Author:

Jabin F.

Modes of Oscillations How particles oscillate/move in normal modes? To know relation b/w displacement of particles, solve equations: V = ω²T x₁ = A₁sin(ωt + φ₁) x₂ = A₂sin(ωt + φ₂) ... and find relation b/w x₁, x₂, ... Number of frequencies of normal modes is equal to DOF of system. If system is free to translate or rotate, then corresponding to each translation & rotation, one frequency will be zero. Example 1: Diagram of a system with multiple masses connected by springs. DOF = 8 Translation of the system is possible along the line of motion. So, one frequency will be zero. Example 2: Diagram of a system with two masses connected by a spring, attached to a rigid support. DOF = 2 System is attached to rigid support. So it can't be translated along the line of motion. So, two non-zero frequencies. Example 1: Diagram of a system with multiple masses connected by springs. DOF = 3 Translation along the line of oscillation is possible. So, one frequency is zero and two are non-zero. Example 2: Diagram of a system with multiple masses connected by springs. Oscillation in plane: all particles remain in the xy-plane. DOF = 6 For this system, translation in the xy plane is possible, and rotation in the xy plane is also possible. So, three frequencies are zero, and three frequencies are non-zero. Note: If a system has N particles and is free to move in 3D space, the system will have at least (3N-6) non-zero frequencies. P-27 and Q-19: Problem statement: Two masses 3m and m are suspended vertically by identical massless springs, each of stiffness constant k. The mass m is suspended from the mass 3m, and the mass 3m is suspended from a rigid support. Only vertical motion is permitted. Find the frequencies of small oscillations of this system. Solution: Note: When the spring-mass system oscillates exactly in the vertical direction, gravity can be ignored. Equations: T = (1/2) * 3m * x₁² + (1/2) * m * x₂² V = (1/2) * k * (x₁ - x₂)² Solving for the frequencies: ω₁ = √(k/3m) ω₂ = √(4k/3m)