## Abstract

The work presented here shows that the natural frequencies of constrained systems may be obtained from asymptotic models of corresponding systems where the constraints are replaced by artificial mass or moment of inertia of very large positive and negative values. This offers a convenient alternative to the current practice of using artificial elastic restraints of large stiffness, a concept introduced by Courant in 1943, to remove a limitation on the choice of admissible functions. Recent publications show that in order to control the error caused by approximating constraints with restraints of large stiffness, it is necessary to use both positive and negative stiffness values. However, the negative stiffness introduces instability near the lower modes of vibration and the magnitude of negative stiffness parameter used must be greater than the highest critical stiffness to ensure bounded results are obtained. The use of positive and negative artificial inertial parameters overcomes this problem as they do not introduce instability near the lower modes, allowing the natural frequencies of constrained systems to be delimited to any desired accuracy.

## 1. Introduction

The Rayleigh–Ritz method is widely used in static and dynamic structural analysis and forms the basis of finite element programmes for structural analysis. It gives upper bound results for the natural frequencies. The proof of this bounded nature of the solution was first reported by Lord Rayleigh (1894, 1899). According to Rayleigh's principle, equating the maximum potential energy associated with vibration to the maximum vibrational kinetic energy gives an upper bound estimate of the fundamental natural frequency for any assumed displacement form that does not violate the geometric constraints and continuity conditions. Therefore, the best possible estimate of the fundamental natural frequency may be obtained from a given set of functions by taking the displacement as a series with weighting coefficients, which are found by minimizing the expression for the frequency. From the theorem of separation (Gould 1966; Rayleigh 1894) it has been shown that this minimization gives upper bounds for natural frequencies of higher modes too. Simple proofs of the theorem of separation and of the bounded nature of the Rayleigh–Ritz method are given by Gould (1966). Rayleigh introduced the notion of minimizing the expression for the frequency in the form of a ratio, now known as the Rayleigh quotient (Rayleigh 1894). Ritz (1908) presented a convenient minimization procedure in which the potential and kinetic energy terms are expressed as quadratic functions of undetermined displacement coefficients leading to a simple eigenvalue matrix equation.

An essential requirement of the Rayleigh–Ritz method is that the displacement functions used must be admissible, in that they must not violate any geometric constraint and must be continuous within the solution domain. For vibration and stability analysis, this means the geometric conditions such as zero translation and rotation at the supports and continuity of translation and rotation at connections must be satisfied by each of the functions used.

This limitation may be overcome by using the Lagrangian multiplier method but it results in two sets of equations: the constraint equations and the modified minimization equations. This makes the formulation of the stiffness matrix more complicated than in a typical Rayleigh–Ritz type formulation. In addition, continuous constraints such as line supports do not result in explicit constraint equations that can be readily used and are usually approximated by a series of point supports.

A more convenient method of handling constraints was introduced by Courant (1943) who proposed the idea of replacing rigid supports with partial elastic restraints of very high stiffness. Partial restraints do not have any admissibility requirements but they control the translation or rotation through the strain energy terms associated with their deformation. This method is frequently used in vibration analysis (Courant 1943; Gorman 1989; Cheng & Nicolas 1992; Yuan & Dickinson 1992, 1994; Lee & Ng 1994; Amabili & Garziera 1999) and has also been adopted in other mathematical procedures to solve problems in a vast range of disciplines, in what is known as the penalty functions method (Zienkiewicz 1974, 1977; Gavete *et al.* 2000; Pannachet & Askes 2000).

The drawback with this approach is that it is not possible to determine the error caused by approximating a rigid constraint with a restraint of large stiffness and, hence, a suitable stiffness value is normally found by trial and error until the solution shows numerical convergence (Zienkiewicz 1974, 1977). In addition, the upper bound nature of the Rayleigh–Ritz solution would be lost if a constraint is replaced with a restraint, resulting in an upper bound solution of a model that is more flexible than the original system.

However, recent publications show that these problems may be overcome by using positive and negative values for the stiffness of artificial restraints in asymptotic models (Ilanko & Dickinson 1999; Ilanko 2002*a*,*b*, 2003). Using positive values for the stiffness of artificial restraints underestimates the natural frequencies and critical loads and overestimates the static deflection, while large negative stiffness values have an opposite effect—provided the magnitude of negative stiffness values is greater than the highest critical stiffness value. This means the difference between the results corresponding to positive and negative stiffness values gives the maximum possible error due to violation of any geometric constraints. The magnitude of the stiffness may be increased until the difference between the results for positive and negative stiffness values becomes acceptably small.

There still remains a problem with the above approach. When using negative stiffness values some of the frequencies vanish as the negatively restrained system goes through critical states. For a system with *h* negative restraints, there are *h* such critical states. It is therefore necessary to ensure that the magnitude of any negative stiffness parameter used is higher than the highest critical stiffness. This is particularly important when calculating the lower natural frequencies. This problem does not arise when using the method presented here, in which positive and negative artificial inertial restraints (mass or moment of inertia) are used instead of stiffness restraints.

The idea that adding a large inertial term effects a constraint may be inferred from Lord Rayleigh's statement in his famous treatise, ‘The theory of sound’, where he states that constraint conditions may be practically reached by ‘supposing the kinetic energy of any motion violating a constraint to increase without limit’. Based on the theorem of separation (Rayleigh 1894; Gould 1966), several theorems have been derived to prove that the natural frequencies of a constrained system are bound by the natural frequencies of systems where the constraints are replaced by inertial restraints of positive and negative mass or moment of inertia, and to show that as the magnitude of the inertial parameter approaches infinity the natural frequencies of the constrained system are approached by the frequencies of the inertially restrained systems. Since the direction of approach depends on the sign of the inertial parameter, it is possible to delimit the natural frequencies of the constrained system using asymptotic models with positive and negative inertial restraints.

## 2. Theoretical derivations

### (a) The natural frequencies of an *n* degree of freedom system with one additional positive or negative artificial mass or inertia

Consider an *n* degree of freedom system. This will be referred to as the ‘original system’ and labelled System *A*. Figure 1*a* shows a typical spring–mass system, which may be used as an illustration. If we now add to this system, a body of mass *m*_{1} which provides inertial resistance along coordinate *q*_{i} forming a modified system *A*_{1} (see figure 1*b*), the maximum kinetic energy of the system, expressed in terms of natural frequency *ω* will change by(2.1)Here it is assumed that there is already a mass or moment of inertia of finite magnitude associated with *q*_{i}. The case where an artificial mass is attached along a previously unrestrained coordinate *q*_{i} will be discussed later. For convenience, the term ‘mass’ will be used in a generic sense and may mean either a mass or a moment of inertia. If the coordinate is a rotational displacement the ‘mass’ would actually be a moment of inertia.

Since for any physical system the energy terms may not be infinite or indefinite, from equation (2.1), as *m*_{1}→±∞,(2.1a)(2.1b)Here denotes the constrained system (see figure 1*d*). In the limiting case where the added artificial mass is infinite, whether the system has a zero natural frequency or becomes constrained depends on the sign of the added mass and the mode number as explained later.

When the author proposed the idea of using a mass to enforce constraints at the First International Symposium on Vibrations of Continuous Systems, there were reservations on its applicability to enforce continuity conditions (Ilanko 1997). Although it is not necessary for the artificial inertial parameter to have a physical meaning, a mass connecting two degrees of freedom using a pulley system as shown in figure 1*c* would serve as an illustration. In this case the mass *m*_{1} resists the relative displacement of two coordinates *q*_{i} and *q*_{j} and the additional maximum kinetic energy term will take the form(2.2)For this case, as *m*_{1}→±∞,(2.2a)(2.2b)Schematic representations of the constrained systems corresponding to figure 1*b*,*c* are shown in figure 1*d*,*e*, respectively. When the added mass is positive the modified system is labelled System *A*_{1+}. It is well known that increasing the mass or adding mass to a system cannot increase the Rayleigh quotient, and hence the natural frequencies of the modified system *A*_{1+} will not be greater than the natural frequencies of the original system *A* (Rayleigh 1894).

Denoting the *m*th natural frequencies of Systems *A* and *A*_{1+} by *ω*_{m,A} and respectively, the above statement may be expressed as(2.3a)Similarly, decreasing the mass or adding negative mass to a system cannot decrease the natural frequencies. However, one exception should be mentioned. When removing mass or adding negative mass, one degree of freedom would be lost and one frequency would become infinite and cease to exist for any further decrease in mass if the net mass associated with a degree of freedom becomes zero. Therefore, the inequality statement for *A*_{1−} needs to be conditional. That is, if exists then(2.3b)The existence of is discussed later. Denoting the *m* th natural frequency of System by , and applying Rayleigh's theorem of separation to Systems *A* and gives(2.4)Similarly, applying Rayleigh's theorem of separation to Systems *A*_{1+} and yields(2.5a)Furthermore, if and exist(2.5b)From equations (2.3*a*), (2.4) and (2.5*a*),(2.6a)Also from equations (2.3*b*), (2.4) and (2.5*b*), if and exist(2.6b)Now it is possible to determine the effect of adding a mass of infinite magnitude.

Since there are *n*−1 natural frequencies for the constrained system and *n* natural frequencies for *A*, from equation (2.6*a*), there are at least *n*−1 natural frequencies for *A*_{1+} between and *ω*_{n,A}.

Furthermore, from equations (2.1*b*) or (2.2*b*) and (2.6*a*), as(2.7a)This means that each natural frequency of the constrained system will be approached by the natural frequency of the system with an artificial mass corresponding to the next higher mode as the mass approaches infinity. The approach is, from above, giving an upper bound result. The lowest natural frequency of System *A*_{1+} will approach zero as given by equation (2.1*a*) or (2.2*a*). That is, as(2.7b)Therefore, for a system modified by the addition of a positive mass, for any finite value of *m*_{1}, there exist *n* natural frequencies.

From equation (2.6*b*), it is clear that there exist at least (*n*−1) natural frequencies for System *A*_{1−}, which are bounded by the first (*n*−1) natural frequencies of the original system *A* and the constrained system .

Also from equations (2.1*b*) or (2.2*b*) and (2.6*b*), as(2.7c)Thus, for an *n* degree of freedom system modified by adding a negative mass as the mass approaches negative infinity, each of the first (*n*−1) natural frequencies asymptotically approaches the natural frequency of the constrained system from below. It should be noted that this is true for *m*<*n*, and the highest mode will not approach a constrained system mode. There may be *n* frequencies for small magnitudes of negative mass but as soon as the net mass associated with *q*_{i} (or *q*_{i}−*q*_{j} in the case of a system similar to the one in figure 1*c*) becomes zero, one of the frequencies will cease to exist. Therefore, for the system with an added negative mass, only *n*−1 natural frequencies are certain to exist.

If it is necessary to enforce a constraint along a coordinate that has no associated mass prior to application of the artificial mass, then as soon as an artificial positive mass is added, the system becomes an *n*+1 degree of freedom system. Following the preceding arguments for this system demonstrates that for any value of added mass there will exist at least *n* natural frequencies and modes and, as the magnitude of the added mass approaches infinity, *n* natural frequencies of the asymptotic model would approach that of the constrained system. These results may be stated in the form of the following theorems.

*If a mass or a moment of inertia of positive or negative value* *(inertial parameter)* *is added to an n-degree of freedom system where n*>1, *so as to resist the motion along a coordinate which is already associated with a mass or moment of inertia*, *then for the resulting system A*_{1}, *there exist at least* (*n*−1) *natural modes and frequencies*.

*As the magnitude of the added inertial parameter approaches infinity*, (*n*−1) *natural frequencies and modes of the modified system would asymptotically approach those of a corresponding constrained system*, *from above in the case of positive inertial terms and from below otherwise*.

*If a mass or a moment of inertia of positive or negative value is added to an n-degree of freedom system*, *so as to resist the motion along a coordinate which is not already associated with a mass or moment of inertia*, *then for the resulting system A*_{(1)}, *there exist at least n natural modes and frequencies*.

*As the magnitude of the added inertial parameter (mass or moment of inertia) approaches infinity*, *n natural frequencies and modes of the modified system would asymptotically approach those of a corresponding constrained system*, *from above in the case of positive inertial terms and from below otherwise*.

### (b) The natural frequencies of an *n* degree of freedom system with *h* additional positive or negative inertial terms

We can now seek to generalize the above theorems for a system with *h* additional inertial terms. Since the proof is by induction, it is necessary to state the general theorems first.

*If h artificial inertial restraints of positive or negative mass are added to an n degree of freedom system A where h*<*n*, *so as to resist the motion along h coordinates* *(each of which is already associated with a mass or moment of inertia)*, *then for the resulting system A*_{h}, *there exist at least* (*n*−*h*) *natural frequencies and modes*.

*Furthermore*, *as the h inertial parameters approach infinity*, (*n*−*h*) *natural frequencies and modes of System A _{h} would asymptotically approach those of the corresponding constrained system* ,

*from above in the case of positive inertial terms and from below otherwise*.

*If h artificial inertial restraints of positive or negative mass are added to an n degree of freedom system A*, *so as to resist the motion along h coordinates* *(none of which are already associated with a mass or moment of inertia)*, *then for the resulting system A*_{(h)}, *there exist at least n natural frequencies and modes*.

*Furthermore*, *as the h inertial parameters approach infinity*, *n natural frequencies and modes of System A*_{(h)} *would asymptotically approach those of the corresponding constrained system* , *from above in the case of positive inertial terms and from below otherwise*.

### (c) Proof by mathematical induction

Let us denote the *n* degree of freedom system subject to *r* inertial restraints (where *n*>*r*>1) having positive or negative stiffness values by *A*_{r}.

If theorems 2.3*a* and 2.3*b* are true for *A*_{r}, then(Statement 1a)and(Statement 1b)Applying theorem 2.1*a* to system *A*_{r}, and using (Statement 1*a*), we can state that adding one more inertial restraint to *A*_{r} will result in a new system *A*_{r+1} for which there exist (*n*−*r*−1) natural frequencies and modes. That is,(Statement 2a)Applying theorem 2.1*b* to with an extra inertial restraint, we can state that as the magnitude of the mass of the (*r*+1)th inertial restraint (newly added) approaches infinity, the resulting system frequencies and modes would approach that of . From statement 2.1*b*, as the inertial parameters for the *r* restraints approach infinity, the frequencies and modes of *A*_{r} would approach those of . Therefore, if the magnitude of the inertial parameter of all *r*+1 inertial restraints were to approach infinity, the natural frequencies and modes of *A*_{r+1} would asymptotically approach that of the *n* degree of freedom system subject to *r*+1 constraints . That is,(Statement 2b)From theorems 2.1*a*, 2.1*b*, 2.3*a* and 2.3*b* are true for *h*=1. From statements 2.2*a* and 2.2*b* they are true for *h*=*r*+1 if they are true for *h*=*r*. Hence, by induction, theorems 2.3*a* and 2.3*b* are true for any *h*>0. Theorems 2.4*a* and 2.4*b* may be proven in a similar manner.

### (d) Applicability of theorems 2.1*b* and 2.3*b* for continuous systems

The above arguments hold for continuous systems, with the exception of any reference to the highest mode and the highest mode number *n*. For continuous systems, since a highest mode does not exist, the condition that *m*<*n*, does not apply and when using negative inertial values for additional inertial restraints, all natural frequencies and modes will be bound on both sides by the natural frequencies of the corresponding constrained systems.

However, when finding the natural frequencies and modes using the Rayleigh–Ritz procedure, continuous systems are effectively discretized and the highest mode number *n* corresponds to the number of terms used in the Rayleigh–Ritz formulation for the displacement.

## 3. Illustrative examples

As an illustrative example, first consider a five degrees of freedom spring–mass system subject to two additional inertial elements of mass *m*_{1} and *m*_{2}, as shown in figure 2*a*. This system will be referred to as System *B*_{2}. The magnitudes of the stiffness of the springs are denoted by , , , and . Let the displacements of the five masses , , , and be *q*_{1}, *q*_{2}, *q*_{3}, *q*_{4} and *q*_{5}, respectively. Of the additional inertial restraints, the mass *m*_{1} resists the displacement of the second mass (*q*_{2}) while *m*_{2} restrains the relative displacement between masses 3 and 5 (i.e. (*q*_{3}−*q*_{5})).

The eigenvalue equation for the system is(3.1)where the stiffness and mass matrices are(3.2a)(3.2b)These equations were solved for various combinations of stiffness and mass parameters, and in all cases the results confirmed the existence of natural frequencies and modes and their convergence towards the frequencies of the corresponding constrained system, as predicted by the theorems presented earlier. Only some sample results are presented here, for the case of and for *i*=1, 2, …, 5, and for various values for *m*_{1} and *m*_{2} in the range of −10^{4} to +10^{4} kg. The results are shown graphically in figures 3 and 4, and numerically in tables 1 and 2. Similar results were reported in an earlier work (Ilanko 2002*a*) where the constraints were effected by using additional springs of positive and negative stiffness of large magnitude. The results indicate that the use of inertial restraints eliminates potential problems with instability when finding natural frequencies of lower modes.

### (a) Results for one additional constraint

Figure 3 and table 1 show the variation of natural frequencies with *m*_{1}, for *m*_{2}=0. For *m*_{1}<−0.1 kg, there exist only four natural frequencies and modes and, as *m*_{1} takes very large negative values, all of them converge to the natural frequencies of , the five degrees of freedom system subject to the constraint *q*_{2}=0 (actually a four degree of freedom system) shown in figure 2*b*. The results for the constrained system are also given in table 1 in the row where the mass is ±∞. It is important to note that the convergence to the frequencies of the constrained system for negative *m*_{1} is from below. It may also be noted from table 1 and figure 3 that as *m*_{1} takes very large positive values the first four natural frequencies converge to the natural frequencies of the constrained system from above, while the lowest natural frequency continues to decrease and asymptotically approaches zero. By comparing the results for positive and negative values of mass, the maximum deviation of the frequencies of the inertially restrained system from those of the constrained system may be found. For example, the first natural frequency of the constrained System is 14.073 rad s^{−1} and it is bracketed by the natural frequencies of *B*_{1} (*B*_{1} is a special case of *B*_{2} where *m*_{2}=0) for *m*_{1}=−100 and 100 kg, which give the lowest natural frequencies as 14.054 and 14.093 rad s^{−1}, respectively. By increasing the magnitude of mass to 10^{4} kg the lowest frequencies of *B*_{1} are estimated as 14.072 and 14.076 rad s^{−1} for negative and positive values, respectively. These results are in agreement with the predictions from theorems 2.1*a* and 2.1*b*. These results were also obtainable using artificial restraints with positive and negative stiffness (Ilanko 2002*a*). However, the use of artificial inertial terms does not introduce instability of the system at lower modes. For a system with *h* elastic restraints of negative stiffness it is necessary to use stiffness values that are higher in magnitude than the highest critical stiffness since there are *h* modes of instability associated with the first *h* frequencies. For a system with *h* constraints modelled by artificial inertial terms only, the highest *h* frequencies vanish and the first (*n*−*h*) frequencies always exist. Therefore, in finding the natural frequencies of lower modes it is not necessary to determine the critical masses. This makes the proposed method superior to the current asymptotic modelling method using artificial stiffness.

### (b) Results for two additional constraints

Results for various combinations of *m*_{1} and *m*_{2} were obtained and in all cases when very large values were used for the magnitudes of *m*_{1} and *m*_{2}, three natural frequencies were found to converge to those of System , shown in figure 2*c*. This constrained system has three modes and frequencies, as two modes of the original system (*B*) have been suppressed by the constraints. The results of increasing both *m*_{1} and *m*_{2} at the same rate (i.e. a common mass factor *m* was increased where *m*_{1}=*m*_{2}=*m*) are shown in figure 5 and table 2. For positive values of *m*, there are five natural frequencies but two of these natural frequencies become imaginary for *m*<0.1 kg. Three frequencies still exist for all values of added mass as predicted by theorem 2.3*a*. The results clearly show the convergence of these frequencies towards the frequencies of the constrained system, thus confirming theorem 2.3*b*. Once again, only the highest two frequencies cease to exist for negative added mass of large magnitude. The lowest (*n*−2) frequencies can be traced unambiguously until they approach the natural frequencies of the constrained system.

### (c) Results for a system with constraints that do not change the number of degrees of freedom

Figure 5*a* shows a two degrees of freedom spring–mass system *C*. The addition of two constraints as shown in figure 5*b* will not change the number of degrees of freedom as these constraints are not directly associated with any existing mass. Its asymptotic model (System *C*_{(2)}) with added masses is shown in figure 5*c*. The following values were used in calculations: , *i*=1,2…5 and for *i*=1,4.

Results obtained for various values of added mass *m*=*m*_{1}=*m*_{2} are shown in figure 6. The natural frequencies of the constrained system are also shown in this figure (see also table 3). The results show that at least two natural frequencies exist for the inertially restrained system and that they asymptotically approach the frequencies of , as predicted by theorems 2.4*a* and 2.4*b*.

### (d) Results for a continuous system

The first three natural frequencies of a cantilever beam of length *L*, flexural rigidity *EI* and mass per unit length *m*, and carrying a particle of mass *m*_{0} at its tip (see figure 7*a*) were determined by solving the governing partial differential equation exactly. The results were also obtained by using the Rayleigh–Ritz method with the following eight-term series for the vibratory deflection of the beam(3.3)where(3.3a)The maximum potential energy is given by(3.4)The maximum kinetic energy function is(3.5)The Rayleigh–Ritz minimization equation yields the following eigenvalue equation(3.6)where(3.7a)and(3.7b)Results from both methods were found to agree to three decimal places. The variation of the non-dimensional natural frequency parameter given by against a dimensionless mass parameter *μ*=*m*_{0}/(*mL*) is shown in figure 8. The exact natural frequency parameter for a propped cantilever is also shown in the same figure. The numerical results are given in table 4. For convenience, the cantilever with mass and the propped cantilever are labelled *D*_{1} and , respectively (see figure 7). The results in figure 8 and table 4 confirm the following equations predicted by theorems 2.1*a* and 2.1*b*:The following convergence may also be noted:These are in agreement with the predictions of theorem 2.1*b*. Numerical tests on continuous systems with rotational constraints confirm that these too can be asymptotically modelled using positive and negative artificial moments of inertia. The results are not reported here but an interactive programme to generate these is freely available in a zipped form (asymptot.zip) from the vibration resources website maintained by the author at http://www.geocities.com/ilanko/vibration.htm (Ilanko 2000).

## 4. Conclusions

A mathematical proof has been presented to show that if *h* artificial inertial terms of positive or negative values are added to an *n* degree of freedom system (*A*) where *h*<*n* in such a way that the added inertial terms are directly associated with a degree of freedom then, for the resulting system (*A*_{h}), there exist at least (*n*−*h*) natural frequencies and modes. Furthermore, as the *h* inertial parameters approach infinity the (*n*−*h*) natural frequencies and modes of System *A*_{h} would asymptotically approach those of the *n* degree of freedom system, subject to *h* constraints . The error due to asymptotic modelling of rigid constraints and connections may be determined by using both positive and negative values for the artificial inertia terms in the asymptotic models. Alternatively, the magnitude of inertial parameters required to keep the error due to asymptotic modelling within any desired tolerance may be determined. Unlike the case of asymptotic modelling with artificial elastic restraints of positive and negative stiffness, the use of positive and negative inertial terms does not introduce instability at lower modes. Therefore, asymptotic modelling with positive and negative terms may be conveniently employed in natural frequency calculations using the Rayleigh–Ritz procedure without the need to select admissible functions which satisfy geometric constraint conditions.

## Footnotes

- Received October 8, 2004.
- Accepted February 28, 2005.

- © 2005 The Royal Society