An Engineer's Guide to Mathematica

8 Vibrations of Spring–Mass Systems and Thin Beams

8.1 Introduction

The time-varying response and the frequency response of single and two degrees-of-freedom systems for various loadings, initial conditions, and nonlinearities are examined by creating suitable interactive environments to explore their characteristics as a function of the parameters that govern the respective systems. Then, the determination of the natural frequencies and mode shapes for thin beams subject to various boundary conditions and loadings are examined. The following applications will be considered:

Single degree-of-freedom systems –

• effects of spectral content of periodic forces on the time-varying response;
• effects of squeeze film damping and viscous fluid damping on the amplitude response and phase response;
• stability in the presence of an electrostatic force;
• parameters that maximize the average power of a piezoelectric energy harvester.

Two degrees-of-freedom systems –

• effects of systems parameters on the amplitude response functions;
• system parameters that produce an enhanced piezoelectric energy harvester.

Thin beams –

• effects of in-span attachments on the natural frequencies and modes shapes of thin cantilever beams;
• effects of an electrostatic force on the lowest natural frequency and the stability of beams;
• effects of in-span attachments on the response of a cantilever beam to an impulse force.

For each of these topics, we shall use Manipulate to create an interactive graphic to explore ...