RELATION BETWEEN STATIC STIFFNESS AND FREQUENCY RESPONSE FUNCTION : APPLICATION TO THE HIGH SPEED MACHINE-TOOL L. Heuzé*, P. Ray*, G. Gogu*, C. Barra**, O. Vidal** *
Laboratoire de Recherches et Applications en Mécanique Avancée (LaRAMA) Institut Français de Mécanique Avancée (IFMA) Université Blaise Pascal, Clermont II (UBP) Campus de CLERMONT-FERRAND / Les Cézeaux BP 265 – 63175 Aubière, France. ** PCI – SCEMM, Recherche et Développement Rue Copernic 42030 Saint Etienne, France.
ABSTRACT. The study of behaviour of a machine-tool needs to know the values of dynamic and static stiffnesses. The purpose of this article is to be able to estimate these values with the same device, handy enough. Different methods of determination of static stiffness are presented and results are compared between them. Measure results of the last method are processed by a software developed with Matlab. KEYWORDS : dynamics, machine-tool, stiffness.
1. Introduction Requirements relative to the increase of productivity lead to the development of a new concept of machine-tool. By nature, those machines need an increased stiffness in order to guarantee the best machining quality, however the machine is tested. Indeed, seeing the constraints imposed by high speed machining in term of time, work speed and acceleration need update design knowledge on the dynamic behaviour of a machine-tool testing device (Fig. 1).
Figure 1 : Machine tool testing device and its modeling
Machine-tool modeling needs to know the values of static and dynamic stiffnesses. The aim is to estimate these values with a handing device. A direct measurement of the static stiffness to the nose of the machine broach implies to carry out a rigid structure around the broach allowing the force application in various positions of the broach in the machine tool working space. Such a structure is very costly and some difficulties exist to orient the force in all directions. That’s why we propose to determine the static stiffness by using the transfer function of the frequency response (FRF). The FRF based methods has the advantage of providing quickly the rigidity of the studied structure if the frequencies are well known. We present the advantages and the limits of each method applied on a simple structure and we also present an application on a high-speed machine tool. These methods permit us to establish the maps of rigidity in the three directions of the working space.
2. Problem During the analysis of transfer functions [LAL 86][MCO 95], we study particularly the compliance H=X/F (X : displacement ó structural response, F : excitation load). To get the displacement, signal from the accelerometers needs a double integration in the time domain. The FFT analyzers which we use can proceed to this transformation of the signal. The structural response is supposed with the same frequency as the excitation load. Also, the double integration is done by a double division by the pulsation ω =2π.f in the frequency domain. This approximation is correct for high enough frequencies (f > 20 Hz). It’s not usable for very low frequencies, where the double integration leads to infinite
compliance. Also, we have to use analytic methods using the valid part of the compliance curve, in order to calculate its value at ω = 0 for the static stiffness. 3. Analysis of the transfer function From the dynamic compliance curve, the static stiffness can be calculated since resonance modes are not coupled. The method supposes that the structural behaviour for low frequencies is ruled by the first mode [LAU 98][MUS 98][LAU 99]. Also, the modal parameters of the mode are dominating for the static characteristics of the structure, in particular the modal stiffness close to the static stiffness. This method was tested on a simple structure whose properties are known : the beam with a fixed end. It was also applied on a high-speed machine tool.
4. Modal sommation method Using the same location for the accelerometer and the vibrator (point named "Driving Point"), we can calculate the static rigidity without having to move the transducer. The method assumes that any FRF made up of µ modes for µ degrees of freedom can be broken down into µ FRFs each of one degree of freedom [COX 99]. For each frequency peak of the response graph X/F modal parameters can be taken. For d points of the structure used to apply forces (mark j) and to measure the response (mark i), we suppose the excitation as periodic and with the following form : d
F (t ) = ∑ F j ⋅e jω t j =1
From spatial coordinates xi, we consider extended coordinates qi defined by : µ
xi (t ) = ∑ uik qk (t ) k =1
Thus, using the Caughey condition [PED 97][PIR]
[] c[ M ]− 1 [] k = [] k[ M ]− 1 [] c , the equations of movement can be uncoupled and become : ••
•
[M ]x + [] c x + [] k x=F
i ∈ {1;...; d }
t
••
•
t t t [] u[ M ][] u q + [] u [] c [] u q+ [] u [] k [] u q= [] u F (t ) ••
•
⇒ m l ql + c l q l + k l q l = Q l =
d
∑u
jl
F j e jω t
j =1
l ∈ {1;...; µ}
So, we obtain :
ql =
Ql Ql ⋅e jω t = ⋅e jω t 2 Dl (ω ) k l + jω ⋅c l − ml ⋅ω
then
xi =
uil
F j ⋅e jω t Dl
∑ ∑ µ
l =1
d
u jl
j =1
We know that
xi =
d
∑H
ij
F j e jω t .
j =1
Also, for a couple of points i (response) and j (excitation), we can calculate Hij = Xi / Fj by identification :
∑ µ
H ij (ω ) =
uil ⋅
l =1
u jl Dijl (ω )
5. Non-linear fitting From the Partial Fractional Form used and presented by Piranda [PIR][PIR 94][DEL 99], the FRF can be written as : µ
H ( jω ) = u + jω ⋅v +
∑ l =1
tl jω − s l
Given the experimental FRF curve yi for the accelerometer on the position i, we have to minimize the difference :
f ( q j , ω k ) = y i (ω k ) − H ( jω k )
with
{q} = t (u, v, t1 ,..., t µ , s1 ,..., s µ )
Even if yi is a discrete function (due to the resolution of the FFT analyzer), we consider f as continuous and differentiable. In the neighbourhood of the point qj0 , we use a Taylor development of first order : 2µ + 2
f (q j , ω ) = f (q j 0 ,ω ) +
∑ j =1
∂f ( q j 0 , ω ) ∂q j
dq j + ε (ω )
We try to minimize f by varying q. Hence, we have to resolve a matricial problem by an iterative method. After convergence, we obtain q which defines entirely the fitting function.
6. Applications
6.1 Application on the fixed-free beam The fixed-free beam studied is made of aluminium (Fig.2).
Figure 2 : Fixed-free beam The theory permit us to calculate its static rigidity as 1335 N.m-1. By a random excitation, we get the following transfert function curve, representing the dynamic compliance, obtained thanks to the software Pulse® (Fig.3):
[dB/1,00 ((m/s²)/N)s²]
Frequency Response H1(Signal 1,Signal 5) - Input (Magnitude) Working : Input : Input : FFT Analyzer
-20
-40
-60
-80
-100
-120
-140 0
100
200
300
400 [Hz]
500
600
700
800
Figure 3 : Transfert function of a fixed-free beam
Each method determines the value of static rigidity of the beam with a precision of around 10%.
6.2 Application on the machine tool 6.2.1 Presentation The same experimental approach is applied to the structure of a machine testing device. The device was provided by the company PCI (Fig. 4). The broach carrier of the device is moved by two linear motors put on the opposite sides of the block. The movement is controlled by two sliding guidances in the Y direction.
command board
linear motor broach carrier
sliding guidance
Figure 4 : Testing device Meteor
Loads are applied on the nose of the broach for different positions of the block Measures of displacement are done in the same location and for different directions.
6.2.2
Applications
The measures are done in the three spatial directions, with intervals of 50 mm, for positions of the broach varying between extremal positions. The figure 5 illustrates the experimental method to determine the FRF in the direction X.
y response
x excitation z
Figure 5 : Measure in the direction X
The results indicate that stiffness on the broach is nearly constant for X and Z directions, what is logical in term of device design. Due to the absence of guidance in the Y direction to maintain strains, there is an evolution in this direction.
Static rigidity
Direction Z
Direction Y Direction X
75
175
275
375
475
Position (mm)
Figure 6 : Static rigidity curves depending on position of the broach
7. Conclusions Measures of the static rigidity of a machine tool is a difficult operation, but analytic methods like the modal sommation method or the non-linear fitting are suitable for our application. Indeed, their construction is strictly releated to the transfert functions theory. They are effective on simple problems. However, it’s necessary to note that the determination of the modal parameters on a resonance mode at low frequencies is very sensitive. In so critical situations, informations from acceleromters are not sure. Also, the study has to be more precise for low frequencies to avoid problems.
References [COX 99] COX, A., Mise en ouvre de la mesure des rigidités statiques et dynamiques UGV, Rapport de Master of Engineering – IFMA – Université de Bath, Mai 1999. [DEL 99] DELUCIS, N., Conception d’une structure de tests dynamiques sur une machine-outil UGV. Techniques d’extraction de rigidités à partir de fonctions de transfert, Rapport de Projet de Fin d’Etudes – IFMA, Juin 1999. [LAU 98] LAUROZ, R., Compte-rendu des mesures effectuées sur Meteor 5, Rapport interne IFMA, Juillet 1998. [LAU 99] LAUROZ, R., RAY, P., GOGU, G., "Relation entre rigidité statique et rigidité dynamique : applications aux machines UTGV", 14e Congrès Français de Mécanique, Toulouse 1999. [LAL 86] LALANNE, M., BERTHIER, P., DER HAGOPIAN, J., Mécanique de vibrations linéaires. Editions Masson, 2e édition, 1986. [MCO 95] McCONNELL, K.G., Vibration testing : Theory and Practice, Wiley Interscience, 1ere édition, 1995. [MUS 98] MUSSET, H., Etude et modélisation dynamique d’un banc machineoutil équipé de moteurs linéaires, Rapport de Fin d’Etudes – IFMA, Juin 1998. [PED 97] DEL PEDRO, M., PAHUD, P., Mécanique vibratoire – Systèmes discrets linéaires, Presses Polytechniques et Universitaires Romandes, 3e édition, 1997 [PIR] PIRANDA, J., Vibrations des structures (3e partie) : Analyse Modale Expérimentale, Cours de DEA Acousto-Opto-Electronique et Mécanique des Structures, Option Mécanique ENSMM, Laboratoire de Mécanique Appliquée R. CHALEAT [PIR 94] PIRANDA, J., CHATELET, E., "Lissage de fonctions de transfert avec procédé de condensation", Revue Française de Mécanique n°1994-1, p. 19-27, 1994.
