Concept of computerized accelerometers

Accelerometers have been widely used in almost every area of science and engineering. They are supposed to be physical instruments for measuring the acceleration of a moving object. Although huge technological advance is made in hardware of accelerometers over more than one century, accelerometers have been persistently designed and fabricated mechanically under the framework of forward problems of damped mass–spring systems. We show that accelerometers are essentially inverse ill-posed source problems from the mathematical point of view, implying that small measurement errors of equivalent displacements are inherently amplified significantly such that accelerations output from accelerometers can become extremely noisy, numerically incorrect and physically meaningless. The ill-posedness of accelerometers has been always implicitly circumvented approximately for more than one century. As a result, accelerometers theoretically can only produce approximate outputs of acceleration. Here we present the concept of computerized accelerometers, because inverse ill-posed problems, as in the case of accelerometers, cannot be rigorously solved mechanically. The acceleration can only rigorously be reconstructed computationally as a regularized solution to the inverse ill-posed source problem of acceleration from equivalent displacements. The concept of computerized accelerometers theoretically warrants precise measurement of acceleration without approximation, is valid for nonlinear time-dependent damping as well and provides a turning point for accelerometers to become a fully rigorous computerized physical instrument. Simulated examples have confirmed that least squares reconstruction of acceleration can be too extremely noisy to be physically meaningful and shown that elastic acceleration is incorrect by more than 100%.