Dmitry,
Force appropriation aims to excite a mode at it's undamped natural frequency and in its undamped normal mode shape.
The main thing is that there must be AT LEAST TWO exciters. For 'soft tuning' (explained below) you could use one exciter (hammer, shaker etc) and measure 2 or more sets of FRFs with the exciter in different positions. For 'hard tuning' you need two or more exciters on the structure at the same time and these exciters must be capable of producing a sine wave at a specified amplitude and phase (ie a hammer won't do it).
We are trying to find the vector of forces (one force for each exciter) which will a) excite the mode in question, and b) suppress the response of the other modes close to it. The vector of forces must achieve the 'phase resonance condition'. That is, the forces must be monophase (either in phase or 180 degrees out of phase). Also the responses must be monophase and in quadrature with the applied forces.
There are different ways of finding this 'appropriated force vector' but in my opinion the best method is the Modified Mode Indicator Function (MMIF). Briefly the MMIF is minimisation problem (The maths is actually pretty straightforward). It minimises the real part of the response with respect to the total response. The eigenvalues and eigenvectors of this cost function tell you a) where the modes are (from the eigenvalues) and b) what the appropriated force vectors are for those modes (from the eigenvectors). Just consider it as a black box. You put FRF data in one end and you get eigenvalues and eigenvectors out the other end. A little bit of interpretation of this output data by the user is required.
So now we have our appropriated force vector for the mode we are interested in. We have 2 options:
'soft tuning'. We simply multiply the matrix of FRF values at the undamped natural frequency by the appropriated force vector. This gives a response vector which, with ideal data, will be the undamped normal mode shape.
'hard tuning'. We apply the appropriated force vector to the structure in the form of a sine wave at the undamped natural frequency. We then measure the response which, with ideal data , will be the undamped normal mode shape.
The practicalities of hard tuning are not as simple as they may seem. You know the appropriated force vector. But what you need know is "What voltages do I apply to my shakers to produce that force vector?". This can be done, but it is a lot less straigtforward than soft tuning.
So why bother with hard tuning at all? Well, the estimate of the undamped natural frequency obtained from the MMIF eigenvalues is only as accurate as the frequency resolution of the original FRF. Small errors in the natural frequency can lead to large errors in the quality of the tuned mode. With hard tuning we can nudge the frequency up and down by tiny amounts until we get the best result.
A crucial element in the force appropriation analysis is the exciter positions. The exciters must be able to excite and isolate the mode in question. In theory, if you have 2 close modes then you can perfectly tune them with 2 exciters if they are in the right place, for 3 close modes you need 3 exciters. In practice, careful choice of exciter positions combined with good quality FRF data is enough to get pretty decent results.
This method is now well established particularly in the aerospace industy. Other sectors are now starting to use it as well. Many people only test with single exciters and so force appropriation is not an option. Some high end modal analysis software includes the ability to perform hard tuning. Soft tuning can be done easily with just a few lines of MATLAB code.
Michael