1. Introduction
We can observe many systems around us which are composed of subsystems on a supporting structure. Those systems can be modeled as discrete systems shown in Fig. 1. One example of such a system is a cockpit seat of a helicopter. A seat is a subsystem and a fuselage of a helicopter is a supporting structure. In many cases the mass of a subsystem is much smaller than that of a supporting structure. For dynamic analysis of such a system, the natural frequencies and the mode shapes of the system can be calculated by solving a corresponding eigenvalue problem. Although this is an exact method, the expansion of the characteristic determinant and the solution of the resulting polynomial equation can become quite tedious for large values of degree-of-freedom. Several analytical and numerical methods have been developed to compute approximately the natural frequencies and mode shapes of multi degree-of-freedom systems. They include Dunkerley’s formula(1), Rayleigh’s method(2), Holzer’s method(3), and the matrix iteration method(4). Approximate expressions for the fundamental frequency of several dynamic systems have been derived using these methods and are used frequently(5).
Fig. 1A system composed of a supporting structure and subsystems
In this research approximate expressions for the natural frequencies and the mode shapes of the above mentioned systems are derived. If the supporting structure of the system is excited by a harmonic force, both the supporting structure and subsystems vibrate harmonically. A relation between the vibration amplitudes is also derived.
2. Characteristics of the Composed Systems
2.1 Natural Frequencies
Consider a system composed of a supporting structure and a subsystem shown in Fig. 2. The mass and stiffness matrices of the system are given by
Fig. 2A system with a supporting structure and a subsystem
and
The characteristic equation is given by
Letting λ = ω2, the solution of the above quadratic equation is obtained as
Letting μ = m2/m1 and v = k2/k1, the above equation can be expressed as
If we let x = μ2(1+v)2 + 2μv(v − 1), the numerator of the third term in the bracket of Eq. (5) is of the form, . As μ approaches zero, x approaches zero and can be expressed around x = 0 as follows using the Taylor series.
Under this condition Eq. (5) can be expressed as
The above solutions become
If the term μ(1+v)2/4v is very small compared to the other terms, the above solutions approach
The above mentioned condition is satisfied when μ is very small while v is neither very small nor very large, in other words, when m2 is very small compared to m1, while k1 and k2 have magnitudes of similar orders. Finally the two natural frequencies can be expressed under this condition as
Examining the above equation, we can find that the second natural frequency is expressed in terms of the mass and stiffness of the subsystem and the mass of the supporting structure.
To obtain a more accurate fundamental frequency, Dunkerley's formula is applied. The formula gives an approximate fundamental frequency of a n-dof system with n masses m1, m2, ⋯, mn as follows.
where aij represents the flexibility influence coefficient which is defined as the deflection at point i (point where mass mi is located) due to a unit load at point j. For the system shown in Fig. 2, a11 and a22 become(6)
Inserting these values into Eq. (11), we obtain an approximate fundamental frequency
Next, consider a system composed of a supporting structure and two subsystems as shown in Fig. 3. On the analogy of the results in Eq. (10), we can expect the second and third natural frequencies as follows.
Fig. 3A system with a supporting structure and two subsystems
The fundamental frequency is obtained using Dunkerley's formula. For the system in Fig. 3, a11 and a22 are the same as those in Eq. (12), and a33 is obtained as
Inserting the aii values into Eq. (11), we obtain an approximate fundamental frequency for the system with two subsystems.
From the above results we can generalize that the natural frequencies of a system with a supporting structure and n−1 subsystems are expressed as
and
The above equations imply that the fundamental frequency is expressed in terms of the stiffness of the supporting structure and the total mass of the system, and the higher natural frequency corresponds to each subsystem and is expressed in terms of the parameters (mass and stiffness) of the subsystem and the mass of the supporting structure.
2.2 Mode Shapes
The mode shape vector of a discrete system can be obtained by solving the following equation with the previously obtained natural frequencies.
For the system in Fig. 3 the above equation becomes
where Xi represents the amplitude of mass mi. From the second and third equations of Eq. (20) we obtain
and
respectively. Then the mode shape vector becomes
If we insert the previously obtained natural frequencies , , and into Eq. (23), we obtain the corresponding mode shapes.
The above result can be generalized for n-dof systems with n−1 subsystems as follows.
Inserting the natural frequencies for ω2 in the above equation, we obtain the ith mode shape vector.
To evaluate the accuracy of the above approximate expressions, a system with three subsystems has been considered. The given parameters of the system are: m1 = 1 kg, m2 = 0.01 kg, m3 = 0.03 kg, m4 = 0.05 kg; k1 = 10 000 N/m, k2 = 6000 N/m, k3 = 1000 N/m, k4 = 10 000 N/m. These parameters satisfy the condition that the masses of subsystems are very small compared to the mass of supporting structure, and the stiffness of springs are of similar orders. The approximate natural frequencies and mode shapes are compared with the exact ones obtained by solving the corresponding eigenvalue problem in Tables 1 and 2, respectively. The approximate mode shape vectors are multiplied by some constants so that the largest components become the same to those of the exact vectors. The tables show that the errors between the exact and approximate frequencies are less than 1 %, and the approximate mode shapes are very close to the exact ones. Examining the obtained mode shapes, we can find that all masses of the system vibrate with similar amplitudes for the fundamental mode, while the mass corresponding to a higher mode vibrates with a much larger amplitude than the other masses for higher modes.
Table 1Comparison of the exact and approximate natural frequencies for a 4-dof system
Table 2Comparison of the exact and approximate mode shapes for a 4-dof system
2.3 Vibration Amplitudes in Case of Harmonic Excitation
Consider the case when the supporting structure in Fig. 3 is excited by a harmonic force F0 cosωt. The equation of motion of the system is similar to Eq. (20) and is expressed as follows.
where Xi represents the amplitude of the mass mi. Since the second and third equations of Eq. (25) are identical to those of Eq. (20), the ratios of amplitudes are identical to Eq. (21) and (22). Consequently the amplitude ratio between masses can be obtained from Eq. (23) by inserting the excitation frequency ω into the equation. Similarly the same relation can be obtained from Eq. (24) for a general n-dof system. The result implies that the amplitude ratio of a subsystem to a supporting structure is not affected by addition of another subsystem.
When the supporting structure in Fig. 2 is excited by a harmonic force, the vibration amplitude of the subsystem may be excessive. In that case it can be reduced by attaching to the subsystem a vibration absorber which is composed of a mass, ma and a spring with stiffness, ka. It can be shown that the vibration amplitude of the subsystem can be reduced to zero if the natural frequency of the absorber is equal to the excitation frequency. That is,
Next, consider a system with two subsystems on a supporting structure which is harmonically excited. Following a similar procedure as above it can be shown that the vibration amplitude of each subsystem can be reduced to zero by attaching an absorber which meets the above requirement to each subsystem. The result implies that the vibration amplitude of each subsystem on a supporting structure can be reduced by using an absorber, and the absorber can be designed independently without considering the other subsystems.
3. Conclusions
A n-dof system composed of n−1 subsystems on a supporting structure has been considered. The system is modeled as a discrete system, and it is assumed that the masses of subsystems are much smaller than the mass of the supporting structure. The natural frequencies and mode shapes of the system have been derived approximately. The fundamental frequency is expressed in terms of the stiffness of the supporting structure and the total mass of the system. The higher natural frequency corresponds to each subsystem and is expressed in terms of the parameters (mass and stiffness) of the subsystem and the mass of the supporting structure. The mode shapes are obtained from the same expression by inserting the approximate natural frequencies. Numerical simulations have proved the accuracy of the approximate natural frequencies and mode shapes.
When the supporting structure is excited by a harmonic force, the amplitude ratio between masses can be obtained from the above mode shape vector. The amplitude ratio of a subsystem to the supporting structure does not change by addition of another subsystem. A vibration absorber for each subsystem can be designed independently without considering other subsystems.
참고문헌
- Atzori, B., 1974, Dunkerley’s Formula for Finding the Lowest Frequency of Vibration of Elastic Systems, Journal of Sound and Vibration, Vol. 36, No. 4, pp. 563~564. https://doi.org/10.1016/S0022-460X(74)80122-7
- Temple, G. and Bickley, W. G., 1956, Rayleigh's Principle and Its Applications to Engineering, Dover, New York.
- Fettis, H. E., 1949, A Modification of the Holzer Method for Computing Uncoupled Torsion and Bending Modes, Journal of the Aeronautical Sciences, pp. 625~634.
- Mahalingam, S., 1980, Iterative Procedures for Torsional Vibration Analysis and Their Relationships, Journal of Sound and Vibration, Vol. 68, No. 5, pp. 465~467. https://doi.org/10.1016/0022-460X(80)90400-9
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- Rao, S. S., 2004, Mechanical Vibrations, Pearson Education, Inc., New Jersey.