IMPROVED ANALYSIS OF A PROPPED CANTILEVER UNDER LATERAL VIBRATION

Continuous systems can be analysed as lumped masses connected by massless elements. This reduces the structure’s degree of freedom and therefore simplifies the analysis. However, this over simplification introduces an error in the analysis and the results are therefore approximate. In this work, sections of the vibrating beam were isolated and the equations of the forces causing vibration obtained using the Hamilton’s principle. These forces were applied to the nodes of an equivalent lumped mass beam and the stiffness modification needed for it to behave as a continuous beam obtained. The beam’s stiffness was modified using a set of stiffness modification factors φ1to φ4. It was observed that by applying these factors in the dynamic analysis of the beam using the Lagrange’s equation, we obtain the exact values of the fundamental frequency irrespective of the way the mass of the beam was lumped. From this work we observed that in order to obtain an accurate dynamic response from a lumped mass beam there is need to modify the stiffness composition of the system and no linear modification of the stiffness distribution of lumped mass beams can cause them to be dynamically equivalent to the continuous beams. This is so because the values of the modification factors obtained for each beam segment were not equal. The stiffness modification factors were obtained for elements at different sections of the beam


INTRODUCTION
Most imposed loads on structures are dynamic in nature. They either vary in time or in space. The structure therefore vibrates frequently under the effect of these loads. Structural and mechanical systems in general consist of structural components which have distributed mass and elasticity [1]. All bodies possessing mass and elasticity are capable of vibration [2,3].The dynamic analysis of structures can be done using the Newton's equation of motion. This is possible for very simple structures with few degrees of freedom. But as the degrees of freedom increase the resulting equations become very cumbersome and an energy method is preferred [4].
The earliest energy method for such analysis is the Lagrange's equations. These equations were formulated for lumped masses connected by massless elements [5]. The masses are assumed to be concentrated at specified points known as nodes. When used to model a system with a continuous distribution of mass, this method will give an approximate result. The accuracy of the results will however increase with an increase in the number of lumped masses and uniformity in their spacing. An increase in the number of lumped mass increases the size of the resulting matrix and hence the size of the required computational analysis. A better energy method is the Hamilton's principle. This is an extension of the principle of minimum potential energy. This method enables us to formulate partial differential equations for the analysis of the structure as uniform systems (i.e. structures with uniformly distributed masses) [6,7]. The results are exact. Its major drawback is that it is very difficult to formulate the differential equations of complex structural systems using this method.
The prevalence of computers has increased the use of numerical methods in structural analysis [8]. Finite difference method, Ritz method, Rayleigh-ritz method and finally the finite element method is today widely used in such analysis. The finite element method is the most popular and like the Ritz approach incorporates the use of shape function. These shape functions are used to formulate inertia matrix known as the consistency matrix [9]. If the shape function is truly representative of the deformed shape of the structure, the consistency matrix should be equal to the equivalent mass matrix. The equivalent mass matrix for a segment of a continuous system is one that returns precisely the dynamic properties of the original segment in discretized form [10]. The power of the finite element method is further enhanced by its ability to subdivide the structure into finite elements, the smaller the size of the finite elements the better the results from their analysis. This has made it a widely applied tool in researches structural analysis [11,12,13]. However increasing the number of elements however increases the size of the matrices to analyze and therefore increases the computational cost.
Despite the rapid advances in these numerical approaches, the lumping of continuous masses has persisted due to its visual appeal and its simplification of the analysis. Mass lumping distorts the mass distribution and leads to a less representative inertia matrix [14]. This introduces an error in the analysis which ultimately affects the values of the computed natural frequencies of the structure. It has limited number of coordinates and may not fully account for the structural characteristics of a system accurately [15]. Despite these limitations It is still widely used in introductory topics in structural dynamics and in advanced research involving complex systems [16 -19].
Efforts have been made in time past to generate better equivalent mass matrices for analysis of continuous systems [20,21,22]. Also Ericson and Parker [23] suggested that varying the stiffness of the structural system would lead to better analysis results. This was implemented for longitudinally vibrating bars using a set of two stiffness modification factors [24]. In this work, the variation of the structure's stiffness distribution as a means of nullifying the effect of lumping of continuous mass in laterally vibrating beams was explored.

Mathematical Theory
The partial differential equation governing the free lateral vibration of a beam is given by [26] +̈= 0 Where EI is the flexural rigidity of the beam and µ is its mass per unit length. For a harmonic vibration and by applying the boundary conditions for the beam we obtain four equations that can be solved numerically to give us the roots from which the natural frequency of vibration is computed.
The mode shape is obtained as [25] ∅ ( ) = cosh + 2 sinh − cos + 4 sin Where the constants Aj and Bj can be determined from the initial conditions. .

METHODS
The vibrations of structural systems are governed by two essential components; the structure's mass distribution and the structure's stiffness [24,27]. These properties are represented by the structure's inertia matrix and stiffness matrix. If we alter the mass distribution we will expect a corresponding change in the stiffness distribution. The lumping of continuous mass at specified nodes alters the mass distribution (inertia matrix). There is need to find the corresponding modification in the stiffness distribution needed to restore the vibration characteristics of the system. This as in [24] was done by equating two equations. One is the force equilibrium equation while the other is the equation of motion of a vibrating system.
The force equilibrium equations and the equations of motion are force equations. Force equilibrium equations have been largely applied in statics [28]. Just as in [24]. It can also be applied in dynamics if the equations for the vector of fixed end moments/forces {F} are formulated. The structure with continuous mass distribution was analyzed using the Hamilton's principle and the equations for the fixed end forces {F} and nodal displacements {D} formulated for any arbitrary segment of a laterally vibrating beam at time t = 0. These were used to get the vector of nodal forces {P} causing the vibration.
The equations of motion were used to simulate the lumped mass beam. For a vibrating element of the real beam (beam with continuous mass) and that of a corresponding element of a lumped mass beam to be equivalent then their nodal deformation {D} and forces acting on their nodes {P} must be equal [24]. Figure 1 shows a propped cantilever beam under inertia forces. A segment of the beam shown is being restrained by the fixed end forces F1 to F4.
From the D'Alembert principle the forces on the vibrating beam can be calculated from its inertia force [29]. For an elementary part of the beam at a distance z from the origin this force is ̈. (See Figure 1) ` Using the principle of virtual work and the flexibility method we determine the fixed end forces F1 to F4 of the isolated element of the excited beam of Figure 1b to be Where 1 = 3 3 (

Fig. 1 -(a) A fixed-pinned beam under lateral vibration due to the inertial forces ̈ (b) A segment of the beam under longitudinal vibration due to inertial forces
For us to be able to evaluate these equations (for the fixed end forces F1, F2, F3 and F4) there is need to derive an expression for Aj for a fixed-pinned beam. Consider a uniform propped cantilever beam under the action of its self weight as shown in Figure  2. µ is the mass per unit length of the beam and g is the acceleration due to gravity.
From the equation of elastic curve and by considering the initial boundary conditions and the equation for the static deflection of the uniform beam under its self weight, let the initial deflection of the beam (at time t = 0) be where b is a dimensionless constant equal to 3 48 . Then from [19] by substituting equations (3) Equation (14) is an expression for the constant Aj for a propped cantilever beam under an initial lateral displacement caused by its self weight. It can be substituted into Equations (7) to (10) to obtain the values of the fixed end forces F1, F2, F3 and F4. With these equations the force equilibrium equations for segments of a vibrating beam can be written and the inherent nodal forces in the system that is causing motion calculated. An arbitrary segment of a vibrating element is identified by means of the normalized distances 1 and 2 of its nodes from an origin. Having obtained the fixed end forces, the vector of nodal forces {P} is obtained from the force equilibrium.
If a segment of a vibrating beam is isolated it will be in equilibrium with the application of the force vector {P} (see Figure 3). The force vector {P} represents the effect of the removed adjourning elements on the isolated segment. When the continuous bar is represented by a lumped mass bar just like the real segment the equivalent segment is supported by the same nodal forces P1, P2 P3 and P4 and has the same nodal displacements as the continuous/real bar.

The equation of motion for the lumped mass vibrating beam is taken as [24]
Where [m] is the inertial matrix, {u} is a vector of nodal displacement and kd is the stiffness of the lumped mass segment under consideration. The proposed stiffness matrix for the lumped mass segment kd is written as Where 1 , 2 , 3 and 4 are the stiffness modification factors for lateral vibration. They are to help redistribute the stiffness of the lumped mass segment in such a way as to annul the effect of the discretization of the beam mass due to the lumping of its distributed mass on selected nodes.
By rearranging (15) we obtain Equation (17) is a mathematical expression for calculating the four stiffness modification factors 1 , 2 , 3 and 4 for a segment of a beam under lateral vibration. µ is the mass per unit length of the beam. ω is the fundamental frequency of the vibrating mass, u11, u21, u31, u41 are the values of nodal displacements u1, u2, u3 and u4 for the first mode of vibration.
Equation (6) was used to evaluate the total displacements u1 to u4 at the nodal points of a segment of the vibrating beam. Though the equation represents the summation of an infinite series, an evaluation of the first few terms provides values of very good precision.
For this beam there are two possible cases, and the method of obtaining the stiffness modification factors depends on the case being considered.

a)
When ξ1 is greater or equal to zero and ξ2 is less than 1 In this case the segment of the fixed-pinned beam under consideration is not positioned to the far right of the beam (the end that is pinned). Hence the process of calculating the stiffness modification factors will be as outlined above.

b)
When ξ2 is equal to 1 In this case the segment under consideration is located at the far right of the fixed-pinned beam. This implies that the segment is fixed at the left end and pinned at the right end hence its stiffness matrix is different from that of (16). The proposed stiffness matrix for this beam segment is therefore By substituting (18) into (15) and putting P4 = 0 because of the hinged end we obtain a set of three equations with four unknowns ( 1 4 ). To solve it there is need to know the value of one of the unknowns. By assuming 4 = 1 we obtain the values of the other three as Note that 4 = 1.
Equations (17) and (19) provide a way of calculating the stiffness modification factors ϕ1 to ϕ4 for a element of a fixed-pinned beam under lateral vibration are calculated. Using these equations the values of stiffness modification factors at different values of 1 and 2 for the lateral vibration of a fixed-pinned beam can be obtained. A numerical demonstration of these steps are presented Table  1

RESULTS
For the propped cantilever beam of Figure 4a, the lumped mass is at a distance L1 from the fixed end. By solving it for different values of L1 using the steps outlined in Table 1 and comparing results with that from the finite element model we obtain the results presented in Table 2. For the finite element model the inertia matrix used was the popular consistency matrix derived from the shape functions.   From the results in Table 2 we observe that the values of the natural frequency obtained from the lumped mass beam varied with the relative values of L1 and L2. It varied more with increase in the difference between L1 and L2 and gave the best prediction of the fundamental frequency when L1 was equal to L2. The results in Table 2 also shows that using the Lagrange's equations it is possible to obtain a near accurate value of natural frequency by a careful section of the relative values of L1 and L2. For instance at L1/L= 0.7 the value of natural frequency obtained was 15.35Hz which has an error of only 0.45%. This is not so with the finite element at the best value is only available when L1 = L2. However with the application of the stiffness modification factors the obtained values of fundamental natural frequency was exact irrespective of the relative values of L1 and L2.
The natural frequency values obtained from a finite element model continued to decrease steadily with an increase in the ratio of L1 to L2. It gave the maximum value at L1/L = 1/10, L2/L = 9/10. The trend is better appreciated in the plot of natural frequency against L1/L presented in  analysis with Lagrange's equation without ϕ gives its best results when the lumps are evenly spaced. This is also true of the finite element model as its best result of 14.94Hz with an error of 3.1% was obtained when L1 was equal to L2. The error margins increased continuously as the difference between L1 and L2 increased. The plot of natural frequencies obtained with the application of the stiffness modification was a horizontal line showing that it was not affected by the relative values of L1 and L2.
To further study the effect of mass lumping and the stiffness modification factors on the accuracy of results the beam of Figure 4b was analyzed using the Lagrange's equation and the finite element model. The result is presented in Table 3. To produce the results of Table 3, the lumped mass of Figure 4b were moved at L/12 and the fundamental natural frequency calculated at all possible positions of the lumped masses. The results in the Table show that Figure 6 shows the interaction plots for the L1/L and L2/L when using Lagrange's equation on the lumped masses. In Figure 6 only the values of L1 and L2 were considered because if we know the values of L1 and L2 the values of L3 becomes defined and can be obtained from subtracting L1 and L2 from the total length of the beam. We first observe that each of the curves tend to have a U shape. This shows that for almost all values of L1/L there are for each two possible values of L2/L that will give a fundamental frequency value of the beam to a good precision. The curves are not parallel and this shows that there is some level of interaction between the values of L1 and L2 on the natural frequency values obtained. The curves all appear to converge between the values of L2/L = 1/3 and L2/L = 5/12. At this value the value of L1/L tend to have minimal effects on the calculated natural frequency.  From Figure 8 it would be observed that there seems to be not interaction between L1/L and L2/L between values of L2/L= 1/3 and ½. At these values any change in the values of L1/L does not have significant effects on the calculated values of the fundamental frequency. This is evident in the near parallel nature of the lines of points of equal L1/L in Figure 8.

Tab. 3 -Natural frequency of a lumped mass propped cantilever of
From the Tables 2 and 3 it would be observed that the natural frequencies obtained from the use of Lagrange's equations on the propped cantilever that had its mass lumped had some measure of errors. When the stiffness of the system was however modified using the stiffness modification factors, the use of Lagrange's equations was able to predict accurately the fundamental frequencies irrespective of the position of lumped mass.

CONCLUSION
This work improved the values of fundamental frequencies obtained in the analysis of continuous systems as having discrete masses connected by mass-less elements using a propped cantilever as a case study. Even though the stiffness modification factors is a product of some rigorous mathematical manipulation, its implementation is largely simplified by the use of Matlab software. Hence the calculation of the stiffness modification can be automated. From this work we can infer that 1) To obtain an accurate dynamic response from a lumped mass beam there may be need to modify the stiffness composition/distribution of the system. 2) There is no linear modification of the stiffness distribution of a lumped mass beam under lateral vibration that can cause it to be dynamically equivalent to the continuous beam. This is so because the values of 1 , 2 , 3 and 4 obtained for each segment as shown in Table  1 are not equal. 3) A careful selection of the relative positions of the lumped masses can lead to results with very good accuracy. 4) Having more lumps will lead to a better results just as breaking a structure into more elements in finite element analysis will give a better result.
This work lays the foundations for precise analysis of structures by lumping its distributed mass at select nodes. The mass lumping is an idealization that simplifies the analysis of the structure while the stiffness modification factors helps in keeping the results of the analysis accurate.
In the formulation of the equations of end forces only the deformation due to bending moments were considered. The effects of shearing and axial stresses on deformation were ignored in order to simplify the analysis. Further work including their effects is recommended.