Two degree-of-freedom systems

4.3.2 Two degree-of-freedom systems

The simple single degree-of-freedom system can be coupled to another of its kind, producing a mechanical system described by two coupled differential equations; to each mass, there is a corresponding equation of motion (see Figure 4-4). To specify the state of the system at any instant, we need to know time t dependence of both coordinates, x1 and x2, from which follows the designation two degree-of-freedom system.

m1 m₂

Newton’s second law for each mass gives

⎟⎠

Matrix and vector notation can be incorporated into (4-25) and (4-26), which is useful for generalizing to an arbitrary number of degrees-of-freedom. The matrix formulation even makes it possible to solve the system of differential equations using software that performs matrix computations. (4-25) and (4-26) are therefore expressed as

[ ] [ ] [ ]

x F

Figure 4-4 Two degree-of-freedom system.

[ ]

_⎥ Once again, let the excitation forces and the particular solutions be expressed by rotating

vectors Putting (4-33,34,35,36) into (4-27) gives

[ ]

^M

{ }

^xˆ

[ ]

^D

{ }

^xˆ

[ ]

^K

{ }

^xˆ

{ }

^Fˆ

2 ⋅ + ⋅ + ⋅ =

−ω _p iω _{p p} . (4-37)

Solving to the homogeneous equations with the force vector FG

set equal to zero leads to the system’s eigenfrequencies. Setting, moreover, the damping matrix equal to zero, in order to obtain the undamped eigenfrequencies, the latter are found to be real. Damping, on the other hand, brings about complex-valued eigenfrequencies; the complex values contain information on both the undamped eigenfrequencies and the system damping. The eigenfrequencies ω1 and ω2 are given by the homogeneous equation

[ ] { }

^ˆ

[ ] { } { }

^{ˆ 0}

2 ⋅ + ⋅ =

−ω M x K x . (4-38)

The condition for the existence of solutions to (4-38) is that the system determinant is identically zero, i.e.,

[ ] [ ]

) 0 det(−ω²M + K =

. (4-39) For a two degree-of-freedom system, (4-39) has two solutions corresponding to two

eigenfrequencies. A system with n degrees-of-freedom has n eigenfrequencies. The eigenfrequencies of the two degree-of-freedom system are

( ) ( )

From linear algebra, it is known that there is an eigenvector corresponding to each eigenvalue (eigenfrequency). These eigenvectors are mutually independent (orthogonal), and contain information on how the system oscillates in the vicinity of their respective

eigenfrequencies. Specifically, they describe the system’s mode shapes, as noted in section 1.7. The mode shapes, x1 and x2, are obtained by substituting the eigenfrequencies, i.e., the solutions of (4-39), into (4-38), yielding

[ ] { }

^ˆ1

[ ] { } { }

^ˆ1 ⁰ 2

1 ⋅ + ⋅ =

−ω M x K x , (4-41)

[ ] { } [ ] { } { }

ˆ₂ ˆ₂ 0

2

2 ⋅ + ⋅ =

−ω M x K x . (4-42)

Example 4-7

A method that provides a respectable amount of isolation of a vibrating machine (see chapter 9) is to use a so-called double-elastic mounting; see Figure 4-5.

m1

m2

x2

x1

κ2

κ_{1 dv1} dv2

Fstör

Mass

Base

Figure 4-5 A double-layered elastic mounting can provide considerable isolation of a vibrating machine.

Suppose that the vibrating machine is represented by a point mass m2, and that its vibrations are generated by a harmonic excitation force Fexc with a circular frequency ω . To reduce the resulting vibrations in the foundation, the machine is to be isolated by incorporating a spring – mass – spring system between the machine and the foundation, as illustrated in Figure 4-5. The parameters of the model are as follows:

m₁ = 100 kg, m₂ = 500 kg, κ₁ = 5⋅10⁶ N/m, κ₂ = 1⋅10⁶ N/m, dv1 = 100 kg/s and dv2 = 200 kg/s.

a) Determine, for the isolated system, the

(i) undamped eigenfrequencies. Which frequencies, i.e., vibration frequencies generated by the machine, is the mounted machine sensitive to?

(ii) mode shapes.

b) In order to quantify the effect of vibration isolation on the foundation, it is common to compare the force on the foundation with and without the isolation system present.

Determine

(i) the force Fwithout on the foundation without the isolation system in place;

(ii) the force Fwith on the foundation with the isolation system in place;

(iii) the ratio γ = Fwithout / Fwith. Sketch, in a frequency diagram, that ratio calculated using three different choices of the spring constant κ2: 1⋅10⁶ N/m; 5⋅10⁶ N/m; and, 1⋅10⁷ N/m. In which frequency band is the vibration isolation effective?

Solution

a) In the first task, the undamped ([D] = 0) vibration isolation system’s eigenfrequencies, and corresponding mode shapes, are determined.

(i) The undamped system’s eigenfrequencies can be calculated with the aid of formula (4-40), with the third spring constant κ3 set equal to 0. Thus,

⎩⎨

(ii) The system’s undamped mode shapes are the solutions to the homogeneous system, with the circular frequency ω set to each of the eigenfrequencies in turn. The undamped ([D] = 0) homogeneous system of equations of motion is, with the assumed harmonic solution forms, exactly as given in (4-38). That, with values entered, becomes

⎪⎩

in which x1 and x2 (see Figure 4-5) indicate the coordinates of both masses. By multiplying the second of these equations by the factor

6

and substituting in one of the two numerical values for the eigenfrequency, it becomes identical to the first equation. That implies that both equations are linearly dependent, in complete agreement with the theory. A linearly dependent system, with two unknowns, has an infinite number of solutions along a straight line in the x1- x2- plane. In order to solve the system, the equation of that line must be determined. Set the amplitude of x1 to α in the second equation of the set, and solve for the amplitude of x2; that is found to be

ω ^⋅α formula above. Then, the eigenvector corresponding to eigenfrequency ωn has the form

⎪⎭

where α is an arbitrary constant. Thus, the eigenvector is a vector with a specified direction, but arbitrary length. The physical interpretation of the eigenvector’s direction is the ratio between the amplitudes of motion of the two masses in a resonant oscillation.

Putting in the eigenfrequencies as described, with α set to 1 for the sake of convenience, provides an eigenvector or mode form for each,

ωn = ω1 :

The interpretation of the first eigenvector, for example, is that if the system is excited by an excitation frequency near the first eigenfrequency, the system vibrates resonantly with the amplitude of the second mass 0.034 times that of the first. The minus sign, moreover, indicates that the masses move in opposite phase, i.e., in mutually opposing directions.

b) In the second part of the problem, the influence of the vibration isolation system on the foundation is studied; in other words, the force on the foundation, with and without isolation in place, is to be compared.

(i) Without isolators, the entire excitation force is transmitted to the foundation, i.e.,

exc without F Fˆ = ˆ ^.

(ii) With the vibration isolation system, the force on the foundation can be determined from the spring-damper relation for the spring-damper element between mass 1 and the foundation, i.e.,

{ }

^{i t}

where the displacement amplitude of mass 1, in the particular excitation case in which the machine excitation is Fexc, should be used. That displacement amplitude (of mass 1) can, in turn, be calculated by solving the non-homogeneous system of equations (4-37) with the relevant values substituted in, and with the common factor e^{iω t} cancelled out; thus,

⎪⎩

From the second of these two equations, the displacement amplitude of mass 1 is solved in terms of that for mass 2. Putting that into the first equation, the amplitude of mass 2 can then be expressed in terms of the excitation force amplitude, as

In document KTH Sound&Vibration BOOK (Page 123-128)