Bushing measurements are typically provided in the frequency domain, and not in the time domain. Thus, for a given preload and frequency, the bushing is typically subjected to a sinusoidal input at a given frequency. The bushing response at steady state, for the given preload and frequency, is measured in terms of a dynamic stiffness (Kd ) and a loss angle .

Dynamic stiffness is the frequency dependent ratio between a dynamic force and the resulting dynamic displacement. The force can have two components: a displacement-dependent or spring-force component and a velocity-dependent or damping-force component.

Loss angle is defined as the phase angle between the displacement and the force. This phase difference is caused by the presence of the damping forces. The ratio of the damping force to the spring force is often called the loss tangent. It is the tangent of the loss angle.

 

These quantities are described in the following figure:

 

Example of a Steady State Response of a Nonlinear Dynamic System

Assume that a nonlinear system is provided with a sinusoidal input at a constant frequency ω0. The system responds with a variety of harmonics as you see summarized in the following figure:

The dynamic stiffness and phase angle are the response values of the system at the first harmonic ω0.

 

Calculating the Dynamic Stiffness and Loss Angle of a Bushing

This section discusses how to analytically calculate dynamic stiffness and loss angle of a bushing from the bushing's equations. Examples in this section are based on a rubber bushing though you can apply the same calculation methods to hydromount bushings.

 

Rubber Bushing Equations

In the above equations:

 

The States

For a sinusoidal input, at steady state and considering only the lowest harmonic, the inputs and states for the bushing assume the form:

X0 is the deformation due to an initial preload is the magnitude of the sinusoidal input and ω is its angular frequency. For each test, X0 and ω are known. The quantities Rx, Ix, Ry, Iy, Rw, Iw are not known and can be determined by the governing equations.

 

The State Derivatives

At steady state, the time derivatives of the states assume the form:

 

Steady State Equations for a Rubber Bushing: Equation 1

At steady state, equation (1) can be expressed:

 

Steady State Equations for a Rubber Bushing: Equation 2

At steady state, equation (2) can be expressed:

 

Steady State Equations of a Rubber Bushing: Equation 3

At steady state, equation (3) can be expressed:

 

Steady State Equations of a Rubber Bushing: Equations 4 and 5

At steady state, equations 4 and 5 can be expressed:

 

Steady State Equations of Rubber Bushing: Equation 6

In the physical test of the bushing, the bushing is provided an initial preload deformation, X0, and a dynamic oscillation, u, is superposed on top of it.  Therefore at steady state, the average slope for an oscillation, u, about a preload, X0, is:

The static response of the bushing for a dynamic oscillation, u, is therefore:

 

Dynamic Stiffness is the ratio of the force change per oscillation cycle divided by the displacement change. The static force F0 at the operating point cancels out during these calculations. We can equivalently achieve this by just subtracting F0 from the force, therefore:

 

Substituting the expressions for y, ẇ, X and x, one gets:

 

Dynamic Stiffness and Loss Angle of a Bushing

You can compute the dynamic stiffness and loss angle as: