OPERATING DEFLECTION SHAPES FROM TIME VERSUS FREQUENCY DOMAIN MEASUREMENTS

The vibration parameters of a structure are typically derived from acquired time domain signals, or from frequency domain functions that are computed from acquired time domain signals. For example, the modal parameters of a structure can be obtained by curve fitting a set of frequency response functions (FRFs), or by curve fitting a set of (time domain) impulse response functions. Similarly, the operating deflection shapes of a structure can be obtained either from a set of time domain responses, or from a set of frequency domain responses. Two of the most commonly asked questions about vibration are: 1. What is the deformation (deflection shape) of a machine or structure under a particular operating condition? 2. How much is the machine or structure actually moving at certain points? Time domain responses can be used to answer both of these questions, for linear as well as non-linear vibration. On the other hand, frequency domain responses can be used to answer these questions for specific frequencies. NOMENCLATURE t = time variable (seconds). ω = frequency variable (radians/second). n = number of measured DOFs. m = number of modes. [M] = (n by n) mass matrix (force/unit of acceleration). {x''(t)} = acceleration response n-vector. [C] = (n by n) damping matrix (force/unit of velocity). {x'(t)} = velocity response n-vector. [K] = (n by n) stiffness matrix (force/unit of displacement). {x(t)} = displacement response n-vector. {f(t)} = excitation force n-vector. {X(jω)} = discrete Fourier transform of the displacement response n-vector. {F(jω)} = discrete Fourier transform of the excitation force n-vector. [H(jω)] = (n by n) Frequency Response Function (FRF) matrix. {xf(t)} = forced response n-vector. {uk} = complex mode shape (n-vector) for the kth mode. pk = pole location for the kth mode = σk + jωk σk = damping of the kth mode (radians/second). ωk = frequency of the kth mode (radians/second). Ak = a non-zero scaling constant for the kth mode. [Rk] = the (n by n) residue matrix for the kth mode = Ak{uk}{uk} tr tr denotes the transpose. * denotes the complex conjugate. INTRODUCTION The vibration parameters of a structure are typically derived from acquired time domain signals, or from frequency domain functions that are computed from acquired time signals. Using a modern multichannel data acquisition system, the vibration response of a structure is measured for multiple points and directions (DOFs) with motion sensing transducers. Signals from the sensors are then amplified, digitized, and stored in the system's memory as blocks of data, one data block for each measured DOF of the structure. A key requirement of the multichannel system is that it be able to simultaneously sample or digitize the vibration signals as it converts them from analog signals (voltages) to digital data (numbers). If the acquisition system is also an FFT-based system (an FFT Analyzer), then one additional requirement must be met in order to compute valid frequency domain functions. To prevent aliasing (false frequencies in their frequency spectra) the frequency content of the time domain signals must be bounded to satisfy the Nyquist criterion. That is, the maximum frequency in the analog signals cannot exceed one half of the sampling frequency used to digitize them.