1 Overview
Generally, the ability of a structure or material to resist deformation is called stiffness.
When a structure is subjected to a static load, the ratio of the static load to the displacement is called static stiffness.
When subjected to a dynamic load, the ratio of the dynamic load to the amplitude is called dynamic stiffness.
The basic principles and applications are briefly introduced below.
2 Static Stiffness
Stiffness refers to the ability of a component to resist deformation, and it represents the relationship between the external force and the deformation generated by the object.
The greater the stiffness, the smaller the deformation of the object under the same external force.
Formula: For a spring, stiffness can be expressed as the spring constant k , i.e.: F = k \cdot \Delta x
Where F is the external force, \Delta x is the deformation, and k is the stiffness of the spring.
In mechanics of materials, static stiffness can be further subdivided into axial stiffness, shear stiffness, bending stiffness, torsional stiffness, etc., according to different structural force forms.
At the same time, we should also pay attention to distinguishing it from strength.
Strength refers to the ability of a material to resist damage or fracture.
Strength describes the maximum stress value that a material can reach when subjected to external forces. The strength of a material is usually related to the material’s own properties (such as yield strength, tensile strength, etc.) and external environmental factors (such as temperature, stress concentration, etc.).
In simple terms, stiffness is concerned with deformation, while strength is concerned with damage.
3 Positive and Negative Stiffness
When a structure displaces under the action of a load, usually, a larger displacement requires a larger applied force, so the slope of the curve is positive, which is manifested as positive stiffness.
However, in some specific cases, as the structural displacement increases, the required applied force decreases instead. This phenomenon indicates that the structure exhibits negative stiffness.
4 Dynamic Stiffness
4.1 Concept
When a structure is subjected to a static load, the ratio of the static load to the displacement is called static stiffness; static load refers to the load that does not change with time.
When subjected to a dynamic load, the ratio of the dynamic load to the amplitude is called dynamic stiffness. Correspondingly, if the load changes rapidly and obvious acceleration occurs (such as vibration), it is a dynamic load.
For ease of understanding, taking a single-degree-of-freedom elastic damping system as an example, based on the vibration system theory, its motion equation is:
m\ddot{x} + c\dot{x} + kx = f(t)
In the formula, m represents the system mass, c represents the system damping, k represents the system stiffness, f(t) represents the system excitation force, and x(t) represents the system displacement.
Assuming the system excitation force is f(t) = F_0 e^{i\omega t} , the system displacement response is x(t) = X_0 e^{i(\omega t – \phi)} , the system velocity response is \dot{x}(t) = i\omega X_0 e^{i(\omega t – \phi)} , and the system acceleration response is \ddot{x}(t) = -\omega^2 X_0 e^{i(\omega t – \phi)} .
Substituting the above results into the motion equation, the dynamic stiffness can be obtained as: K_d = k – m\omega^2 + i\omega c , where \omega is the frequency of the excitation.
It can be seen from the above formula that dynamic stiffness has the following characteristics:
1. Dynamic stiffness is a complex-valued function.
2. Dynamic stiffness changes with frequency.
3. Dynamic stiffness is related to the mass, damping, and static stiffness of the system, but has nothing to do with the magnitude of the excitation.
4. When the frequency is 0, the dynamic stiffness is equal to the static stiffness.
Dynamic stiffness is a frequency-dependent complex number, which can be described by amplitude and phase or real part and imaginary part. Its amplitude is |K_d| = \sqrt{(k – m\omega^2)^2 + (\omega c)^2} .
Plotting the curve of dynamic stiffness changing with frequency, we can see:
– At low frequencies (i.e., small \omega ), the dynamic stiffness is close to the static stiffness, and the amplitude is k . It can be considered that the dynamic stiffness is basically the same as the static stiffness.
– At the resonance frequency, the amplitude of the dynamic stiffness decreases significantly, indicating that the resonance frequency is mainly controlled by damping. At the resonance frequency, the structure is easily excited by the external world, the deformation of the structure is the largest, and the dynamic stiffness is the smallest.
– In the high-frequency range, when the frequency of the external force is much higher than the natural frequency of the structure, the structure is not easy to deform (i.e., the deformation is small). At this time, the dynamic stiffness of the structure is relatively large, that is, the ability to resist deformation is strong.
4.2 Application Example
Since the measurement of acceleration signals is more convenient than that of displacement and velocity signals, acceleration signals are usually used when collecting vibration signals.
When evaluating the dynamic stiffness characteristics of the structural attachment point, the Inertance at Point of Application (IPI) is usually used.
IPI analysis involves applying a unit force to the loading point within a certain frequency range, and at the same time taking this point as the response point to measure the curve of the acceleration admittance at this point changing with frequency (IPI curve).
If the curve has a peak at a certain frequency, it indicates that the dynamic stiffness characteristics of the measuring point are poor at this frequency.
The IPI calculation formula is IPI = \frac{\ddot{X}(i\omega)}{F(i\omega)}
Since IPI analysis only focuses on its amplitude and not on the phase, IPI can be expressed as |IPI| = \frac{|\ddot{X}(i\omega)|}{|F(i\omega)|}
It can be seen from the fluctuation of the IPI curve that it almost all exceeds the reference value curve in the frequency range of 0~90Hz, and peaks appear at frequencies of 27, 35, 54, 62, 78, and 83 Hz.
The reason for the peak in the acceleration admittance response may be resonance at this frequency or insufficient local dynamic stiffness at the attachment point.