A mass spring system is a physical system consisting of a mass attached to a spring. The mass is free to move back and forth along a fixed axis, and the spring provides a restoring force that causes the mass to oscillate around its equilibrium position.
Natural frequencies are the frequencies at which a mass spring system will oscillate when disturbed from its equilibrium position. These frequencies are determined by the properties of the mass (such as its mass and stiffness) and the spring (such as its spring constant).
To find the natural frequencies in a mass spring system, you can use the equation:
f = (1/2π) * √(k/m)
where f is the frequency, k is the spring constant, and m is the mass. You can also use numerical methods or simulation software to calculate the natural frequencies.
Mode shapes are the shapes taken by a mass spring system when it oscillates at a particular natural frequency. They describe the displacement of the mass from its equilibrium position at different points in time.
To find the mode shapes in a mass spring system, you can use the equation:
x(t) = A * sin(2πft + φ)
where x(t) is the displacement of the mass, A is the amplitude, f is the natural frequency, and φ is the phase angle. You can also use numerical methods or simulation software to calculate the mode shapes.