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Single Degree of Freedom Systems: Equivalent Mass and Equivalent Stiffness

We have seen that the form of the equation of motion for a simple spring{mass system is

    \[m\ddot{x} + kx = 0\]

Similarly, for a simple pendulum (undergoing small motions), the equation of motion can be written as

    \[m l^2 \ddot{\theta} + mgl \theta = 0\]

which has a very similar form. In general, for any linear undamped single degree of freedom system the equation of motion can be expressed in the form

    \[m_{EFF} \,\, \ddot{q} + k_{EFF} \,\,q = 0\]

where q is the chosen coordinate in the problem. m_{EFF} and k_{EFF} are known as the effective mass and effective stiffness respectively. Note that once m_{EFF} k_{EFF} are known, the natural frequency of the system can be found easily as

    \[p = \sqrt{\frac{k_{EFF}}{m_{EFF}}}.\]

Effective Mass

So far we have found m_{EFF} and k_{EFF} using Newton’s Laws or energy methods. There are other ways to determine these quantities which may be easier in some situations. One method used to determine m_{EFF} for a system is to note that the kinetic energy of the SDOF system can be written as

    \[T = \frac{1}{2} \bigl[m_{EFF}\bigr] \dot{q}^2.\]

where again q is the chosen coordinate. Once the expression for the kinetic energy of the system is found, m_{EFF} can be found by inspection.

Example

While in many situations it is appropriate to ignore the mass of a spring in a vibration analysis, any real spring will have some mass which may, in certain situations, need to be taken into account. Estimate the effect that the mass of the spring M_S has on the natural frequency of the simple spring–mass system shown below.

Begin by estimating the kinetic energy of the spring making some simplifying assumptions:

  1. The mass of the spring is uniformly distributed along the length of the spring.
  2. The velocity of any point on the spring is proportional to its distance from the fixed end.

Equivalent Stiffness – Stiffness and Flexibility Approaches

To determine the effective stiffness k_{EFF}, it can similarly be shown that the potential energy of the system is

    \[U = \frac{1}{2} \bigl[k_{EFF} \bigr] q^2\]

so that k_{EFF} can also be found by inspection once the potential energy expression has been determined. However, there are also other approaches, stiffness and flexibility methods, that can be used to determine k_{EFF} based on statics. These approaches are similar and are based on the idea that the force F applied to a system (in the positive coordinate q direction) and the resulting deflection \Delta (of the coordinate) are related by

    \[F = k_{EFF} \, \Delta.\]

In the stiffness approach, the system is given a unit displacement and the force F required to maintain this deformed configuration is determined through statics. The effective stiffness is then given by

    \[k_{EFF} = \frac{F}{1} = F.\]

In the flexibility approach, a unit load is applied to the system and the resulting deflection \Delta is determined using statics. The effective stiffness is then given by

    \[k_{EFF} = \frac{1}{\Delta}.\]

In both of these approaches it is important to realize that the force F and the displacement \Delta are always measured in the positive coordinate direction. Also note that if the coordinate is a rotation(measured in radians), the same approach can be used replacing the force F with a torque T. The details are summarized on the following page.

For single degree of freedom systems, the process of finding the effective stiffness can be generalized as either a stiffness or flexibility approach, both of which are static in nature and are based on the definition

    \[\boxed{k_{EFF} = \frac{\text{Force applied in the direction of motion}}{\text{Displacement in the direction of motion}}}\]

Stiffness Approach

Linear Displacements: Displace the system such that it has a unit displacement in the positive coordinate direction. Then, using statics, determine the force F required in the positive coordinate direction to maintain this deflected state. Then, since k_{EFF}=\frac{F}{\Delta} and \Delta = 1,

    \[k_{EFF} = F\]

Angular Displacements: Displace the system such that it has a unit angular displacement (\Phi = 1) in the positive coordinate direction. Then, using statics, determine the torque T required in the positive coordinate direction to maintain this deflected state. Then, since \hat{k}_{EFF}=\frac{T}{\Phi} and \Phi = 1,

    \[\hat{k}_{EFF} = T\]

Flexibility Approach

Linear Displacements: Apply a unit load in the positive coordinate direction and determine, through statics, the resulting deflection \Delta in the positive coordinate direction. Then, since k_{EFF} = \frac{F}{\Delta} and F = 1,

    \[k_{EFF} = \frac{1}{\Delta}\]

Angular Displacements: Apply a unit torque in the positive coordinate direction and determine, through statics, the resulting angular deflection \Phi in the positive coordinate direction. Then, since \hat{k}_{EFF}=\frac{T}{\Phi} and T = 1,

    \[\hat{k}_{EFF} = \frac{1}{\Phi}\]

Examples
  1. Springs in Parallel

2. Springs in Series

3. Mass at End of Cantilever (Lateral Motions)

4. Mass at End of Cantilever (Axial Motions)

5. Mass at End of Cantilever (Torsional Motions)

Occasionally we need to modify these approaches slightly when a “unit” force or displacement would significantly change the geometry of the system resulting in a nonlinearity. In such cases we apply “small” forces or displacements instead of unit ones. Another way to think of this is to still apply a unit load, but the “unit” of the load is smaller (i.e. apply 1μN instead of 1 N). A similar approach could be taken for displacements.

Example 1

In the diagram below a spring supports a mass and makes an angleαwith the direction of motion. Determine the effective stiffness of the spring in this situation. Assume that only small motions occur.

Example 2

Determine the natural frequency for the system shown below. The double pulley set has a mass moment of inertia J_O about its center O. A mass M is supported by a light inextensible cable.

Example 3

Determine the effective stiffness k_{EFF} of the system shown below. Neglect the mass of the bar AB.

Example 4

Determine the effective stiffness k_{EFF} of the system shown below. Neglect the mass of the bar AB.

Example 5

A block of mass m is supported by a cable and pulley system as shown below. Pulleys A and B are pinned to the block while pulley C is attached to a rigid support by a spring of stiffness k. Neglecting the mass of the pulleys and treating the cable as inextensible, determine

  1. the effective stiffness of the cable and pulley system,
  2. the natural frequency of free vibrations.

Example 6

In the system shown below, a rotor with polar moment of inertia J_P is supported by a combination of shafts and gears as shown. Shafts 1 and 2 have torsional stiffnesses \hat{k}_1 and \hat{k}_2 respectively while gears 1 and 2 have N_1 and N_2 teeth respectively.

Determine the effective stiffness \hat{k}_{EFF} for this system and the natural frequency of free vibrations. Neglect the masses of the gears.

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