Table of Contents
- Author: Johnny Liu, CEO at Dowway Vehicle
- Published: July 6, 2026
- Category: Rotordynamics, Mechanical Engineering, NVH & Suspension Design
About the Author
Johnny Liu leads Dowway Vehicle as CEO. With over twenty years in car structural dynamics, chassis setup, and noise, vibration, and harshness (NVH) testing, Johnny focuses on putting fresh rotordynamics and vibration control ideas into practical use for electric drivetrains and vehicle suspension systems.
In basic mechanics, stiffness is simple: it is just the force you need to make a unit of displacement. Under a static load, this is a fixed number. But when we look at rotating parts and moving structures, we face a different beast: dynamic stiffness. This concept belongs to the world of forced vibration.
If a spring-like part meets a harmonic force $P$ with an amplitude $A$ and a frequency $\omega$, we write it as:$$P = A \sin(\omega t) \quad \text{— (1)}$$
This force causes a vibration at that same frequency. The movement $y$ (with an amplitude $Y$) in the direction of the force is:$$y = Y \sin(\omega t) \quad \text{— (2)}$$
We define the dynamic stiffness ($k_d$) of this part as:$$k_d = \frac{A}{Y} \quad \text{— (3)}$$
Simply put, dynamic stiffness is the ratio of the peak harmonic force at one point to the peak displacement caused by that force in the same direction.
Because forced vibration frequency matches the force frequency, for single-frequency cases, dynamic stiffness is just force amplitude divided by displacement amplitude. Both numbers change with frequency. This means dynamic stiffness is not constant. It changes based on where you apply the force, how the structure is built, and the frequency of the force itself.
The Math: Dynamic Stiffness on a Mass Support
Let us look at a simple setup to see how mass and stiffness work together. Picture a one-degree-of-freedom system where a dynamic force acts on a support made of a spring and a mass.
From basic force balance, the equation of motion for this mass-spring support under force $P$ is:$$P – m\ddot{y} = k y \quad \text{— (4)}$$
Here, $k$ is the static stiffness of the spring, and $m$ is the support mass.
Now we plug in our force equation (1) and displacement equation (2). Since the second derivative of displacement is $\ddot{y} = -\omega^2 Y \sin(\omega t)$, we get:$$(k – m\omega^2) Y \sin(\omega t) = A \sin(\omega t) \quad \text{— (5)}$$
Solve this for the displacement amplitude $Y$:$$Y = \frac{A}{k – m\omega^2} \quad \text{— (6)}$$
Put equation (5) into (6) to find the final dynamic stiffness ($k_d$) of this mass support:$$k_d = \frac{A}{Y} = k – m\omega^2 \quad \text{— (7)}$$
This math shows a clear fact: the dynamic stiffness of a mass-supported system is the static spring stiffness ($k$) plus the dynamic inertia stiffness of the mass ($-m\omega^2$).
Unlike ideal springs where stiffness never changes, a real support changes with frequency $\omega$.
- At low frequencies: The mass term $m\omega^2$ is tiny. Dynamic stiffness stays positive and close to static stiffness.
- At high frequencies: The mass term $-m\omega^2$ grows fast. When $\omega > \sqrt{k/m}$, the dynamic stiffness turns negative. Here, inertia forces run the show.
Frequency Curves: Stiffness vs. Flexibility
For a simple system under harmonic force, we track both the dynamic stiffness curve ($k_{D0}$) and its opposite, the dynamic flexibility curve ($W_0$).
We write these as:$$k_{D0} = k \sqrt{(1-\lambda^2)^2 + (2\xi\lambda)^2} \quad \text{— (8)}$$$$W_0 = \frac{1}{k \sqrt{(1-\lambda^2)^2 + (2\xi\lambda)^2}} \quad \text{— (9)}$$
Here:
- $k$ is the static stiffness.
- $\lambda = \frac{\omega}{\omega_n}$ is the frequency ratio (force frequency divided by natural frequency).
- $\xi$ is the damping ratio.
By plotting equations (8) and (9) against the frequency ratio $\lambda$, we get two clear curves.
Figure 1: Forced Vibration Dynamic Flexibility ($W_0$) vs. Frequency Ratio ($\lambda$)
Figure 2: Forced Vibration Dynamic Stiffness ($k_{D0}$) vs. Frequency Ratio ($\lambda$)
The Three Frequency Zones
The frequency ratio $\lambda$ changes dynamic stiffness in different ways depending on your frequency range. We divide this behavior into three zones: the quasi-static zone, the resonance zone, and the inertial zone.
We can estimate the dynamic stiffness in these zones using equations (10), (11), and (12):
1. The Quasi-Static Zone (Static Stiffness Dominant)
When the force frequency is low and stays far below the natural frequency ($\lambda \ll 1$), we can ignore damping ($\xi \approx 0$). We estimate the stiffness as:$$k_{D0} \approx k (1 – \lambda^2) \quad \text{— (10)}$$
In this zone, static stiffness $k$ is the main factor. The dynamic stiffness drops slowly as a simple parabola as frequency rises.
2. The Resonance Zone (Damping Dominant)
When the force frequency gets close to the natural frequency ($\lambda \approx 1$), spring forces and mass forces cancel out. Damping becomes the main controller. We estimate the stiffness as:$$k_{D0} \approx 2\xi k \quad \text{— (11)}$$
Here, the dynamic stiffness of the structure drops to its lowest point. Without enough damping ($\xi$), the movement $Y$ will spike.
3. The Inertial Zone (Mass Dominant)
When the frequency goes far past the natural frequency ($\lambda \gg 1$), mass forces take over. We estimate the stiffness as:$$k_{D0} \approx k\lambda^2 \approx m\omega^2 \quad \text{— (12)}$$
In this zone, the dynamic stiffness depends almost entirely on the physical mass $m$. The spring stiffness $k$ does almost nothing to stop high-frequency forces.
Summary of Frequency Zones
| Frequency Zone | Frequency Ratio ($\lambda$) | Approximation Formula | Dominant Parameter | Stiffness Behavior |
|---|---|---|---|---|
| Quasi-Static Zone | $\lambda \ll 1$ | $k_{D0} \approx k(1 – \lambda^2)$ | Static Stiffness ($k$) | Stays close to static stiffness; drops slowly. |
| Resonance Zone | $\lambda \approx 1$ | $k_{D0} \approx 2\xi k$ | Damping Ratio ($\xi$) | Drops to its lowest point; controlled by damping. |
| Inertial Zone | $\lambda \gg 1$ | $k_{D0} \approx m\omega^2$ | Mass ($m$) | Controlled by mass; drops below zero as frequency rises. |
Frequently Asked Questions (FAQ)
Q1: What is the main difference between static and dynamic stiffness?
Short Answer: Static stiffness resists steady forces; dynamic stiffness resists vibrating forces and changes with frequency.
Detailed Explanation: Static stiffness is a fixed value. It shows how a part resists a steady load. Dynamic stiffness changes with frequency because mass and damping forces join the fight when the load vibrates.
Q2: Why does dynamic stiffness turn negative at high frequencies?
Short Answer: Because high-frequency inertia forces outgrow the spring forces and push the mass in the opposite direction.
Detailed Explanation: Once the frequency goes past the natural frequency ($\omega > \sqrt{k/m}$), the mass’s inertia becomes stronger than the spring’s pull. This phase shift makes the mass move opposite to the spring force, which mathematically shows up as negative stiffness.
Q3: How does this guide vehicle suspension and motor mount design at Dowway Vehicle?
Short Answer: It helps us keep motor mounts soft for high-frequency hums but stiff for low-frequency driving forces.
Detailed Explanation: Electric car motors make high-frequency hums. We design mounts to work in the inertial zone for these hums, making the dynamic stiffness low so vibration does not reach the cabin. But for low-frequency torque when you hit the gas, the mount works in the quasi-static zone, keeping the engine stable with high static stiffness.