In this post, I will discuss the ninth example in our structural dynamics course covering the response of systems to harmonic excitation. The goal of this structural dynamics example is to calculate the required spring constant to achieve a specified level of isolation.
The problem statement states,
An instrument package weighing 300 lbs is mounted on a train console. When the train is running and in operation, the console experience a vibration amplitude of 0.5 in at a frequency of 40 Hz. Damping effects are negligible
A) Required spring constant of the mount to achieve 80% isolation
B) The vibration amplitude of the instrument package.
The first step is to define the variables. “W” is the weight of the instrument package, k is the structural stiffness of the flexible mount, “Y” is the vibration amplitude of the supporting base, omega is the base excitation frequency, and “t” is time.
The second step is to convert the excitation frequency from Hertz to radians per second by multiplying the frequency by two time pi.
The third step is to calculate the transmissibility ratio which is equal to dynamic magnification factor times the square root of one plus four times the damping factor squared times the frequency ratio squared. The transmissibility ratio is set equal to one minus the required percentage of vibration isolation, and we solve for the frequency ratio using this relationship.
The fourth step is to solve for the required structural stiffness. The stiffness can be expressed as the weight of instrument package times the excitation frequency squared divided by the gravitational acceleration and the frequency ratio squared.
The fifth step is to calculate the steady-state vibration amplitude due to base excitation. In a base excitation scenario, the displacement amplitude is equal to the vibration amplitude of the base times the dynamic magnification factor times the square root of one plus four times the damping factor squared times the frequency ratio squared.