Quick Look
Dynamic Signal Measurement Basics
Signal Processing
Modal Analysis
Modal Analysis is the study of the natural deformation a structure takes when excited by
some form of input.
The cantilever beam, mounted on one of its ends, is an interesting flexible structure to
study using modal analysis techniques.
We show the first 4 mode shapes for the beam. In our experiment, we will demonstrate
the fundamental measurements that we must make for modal analysis. Force stimulus is
applied through hammer blows while accelerometers mounted to the beam give the
response signal by measuring acceleration.
Modal analysis is the study of the natural deformation a structure takes when excited by
some form of input. The inherent mass, stiffness, and damping of a structure govern the
overall deformation response. The mass, or inertia, of a structure is a measure of the
structure’s resistance to a change in momentum; stiffness is a structure’s natural
resistance to deformation; damping is a structures natural tendency to return to an
equilibrium state after being excited by an input.
When an input excites a structure at a resonant frequency, the stiffness force and the
inertial force cancel and cause the structure to have a low effective mass. The resulting
structural response at resonance requires very little driving force, and structural dynamics
theory shows that each resonant frequency has an associated characteristic shape.
When a structure is excited at multiple resonant frequencies, the mode shapes associated
with each resonant frequency sum together to form the characteristic shape of the
structure.
If a system is linear, the response of the structure to any combination of input forces is
equal to the sum of the responses from each individual input force. In order for a system
to be time invariant, the modal parameters (natural frequency, damping, and mode shape)
must be independent of time. If a system is observable, the input and output
measurements contain enough information to accurately characterize the behavior of the
system. Structures with loose components are not completely observable due to
nonlinear behavior. If these assumptions are valid for a structure, the structural responses
will produce results that are predicted by linear structural dynamics theory, and analysis
will produce the modal parameters and mode shapes of the structure.
These are the five fundamental steps necessary for modal analysis.
Too few measuring points can cause spatial aliasing. Separation between points is so
large that you cannot measure high frequency modes accurately, aliasing to the lower
modes of vibration.
Too many points may lead to an unnecessary computational burden.
As a general rule-of-thumb, spacing should be at least as small as /4, where  is the
wavelength of the highest significant resonant frequency. You can measure by setting
the stimulus to the highest resonant frequency, and changing the distance between the
stimulus and the accelerometer until the phase shift goes through 360 degrees.
The force signal represents the input to the beam structure. The accelerometer output
represents the output or response of the structure. The dynamic signal analyzer measures
the input and response signals at different locations along the beam, and computes the
frequency response of the structure at these locations.
You can use each position along the beam to measure a unique frequency response.
When plotted in 3D space, response magnitude verses frequency verses position, you can
see the first three modal shapes along resonant frequencies.
We used an algorithm to search for peaks to locate resonant frequencies. Here we show
the frequency response magnitude at one position along the beam. You can easily identify
three resonant frequencies, but you must use accurate peak-searching techniques to
reduce measurement and resolution problems.
Here we demonstrate how a smooth cut made at the resonant frequencies, followed by
curve fitting, produces the first two smooth mode shapes of the vibrating cantilever beam.
Now we will take a few actual dynamic measurements from our experimental setup. We
made dual-channel analysis is made using the hammer force as the system stimulus and
the accelerometer signal as the system response.