SDOF Dynamics & Vibration Isolation

2.1 Introduction to Single-Degree-of-Freedom Systems

In structural dynamics and Noise, Vibration, and Harshness (NVH) engineering, the Single-Degree-of-Freedom (SDOF) system is the most fundamental analytical model. Any structural system, no matter how complex, can be decomposed into a set of independent SDOF systems using modal analysis. By understanding the behavior of a simple spring-mass-damper system, engineers can gain deep insights into more complex phenomena such as vehicle suspension design, engine mount isolation, aerospace structural response, and acoustic-structure interactions.

An SDOF system is characterized by a single coordinate that completely describes the motion of the system at any instant. The fundamental physical elements of an SDOF system are:

Mass ($m$): The inertia element that stores kinetic energy.

Spring ($k$): The stiffness element that stores potential energy.

Damper ($c$): The dissipative element that converts mechanical energy into thermal energy.

Below is a premium schematic representation of a standard SDOF spring-mass-damper system under external harmonic excitation, showing the free-body coordinate $x(t)$ and the components in a high-fidelity, responsive vector format:

2.2 Free Vibrations & Damping Classifications

Free vibration occurs when a system is disturbed from its static equilibrium position and allowed to oscillate under the influence of its own internal forces, without any external dynamic excitation. The nature of the free vibration is heavily governed by the damping mechanism present in the system.

2.2.1 Viscous Damping

In viscous damping, the damping force is linearly proportional to the velocity of the mass but acts in the opposite direction. This is mathematically expressed as:

$$F_d = -c \dot{x}(t)$$

where $c$ is the viscous damping coefficient (expressed in $\text{N}\cdot\text{s/m}$ or $\text{kg/s}$). Viscous damping is the most common model used in engineering analysis due to its mathematical convenience. The behavior of a viscously damped system is categorized into three regimes based on the damping ratio $\zeta$ (zeta):

Underdamped ($\zeta < 1$):

The system oscillates with an exponentially decaying amplitude. The frequency of oscillation is the damped natural frequency $\omega_d$, which is slightly lower than the undamped natural frequency $\omega_n$. This is the most common case for structural systems, as they typically possess very low inherent damping (typically $\zeta < 0.05$ for steel and concrete structures).

Critically Damped ($\zeta = 1$):

Critical damping is the threshold at which the system returns to its equilibrium position in the shortest possible time without any oscillation. The critical damping coefficient is given by:

$$c_c = 2 \sqrt{k m} = 2 m \omega_n$$

Engineers design suspension systems (like vehicle shock absorbers) and analog measuring instruments (like galvanometers) to be close to critical damping to avoid overshoot and ensure fast settling times.

Overdamped ($\zeta > 1$):

The system does not oscillate. Due to the high resistance of the damper, the system returns slowly to equilibrium. The response decays exponentially, but slower than the critically damped case because the large damping force acts as a drag on the motion.

2.2.2 Coulomb (Dry Friction) Damping

Coulomb damping occurs due to dry friction between sliding surfaces. The damping force is constant in magnitude but changes direction to always oppose the motion:

$$F_d = -\mu N \operatorname{sgn}(\dot{x})$$

where:

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$\mu$ is the coefficient of dynamic friction.

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$N$ is the normal force.

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Cast as the signum function $\operatorname{sgn}(\dot{x})$, which is $+1$ for positive

velocity and $-1$ for negative velocity.

Key characteristics of Coulomb damping include:

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Linear Decay: Unlike viscous damping where the amplitude decays exponentially,

the amplitude of a Coulomb-damped system decays linearly at a rate of $$\Delta x = \frac{4 F_f}{k}$$ per full cycle of motion, where $F_f = \mu N$.

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Sticking Zone: The oscillation stops when the mass reaches a position where the

spring force is insufficient to overcome the static friction force ($|kx| \leq \mu_s N$). This leads to a permanent offset from the absolute equilibrium position.

2.2.3 Hysteretic (Structural) Damping

Hysteretic damping, also known as structural or material damping, represents the energy dissipation within the material itself due to internal friction, molecular slip, and micro-plastic deformations. Experiments show that for most structural materials (e.g., metals under cyclic stress), the energy dissipated per cycle is:

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Independent of the frequency of excitation.

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Proportional to the square of the amplitude of vibration.

To model this, we define a complex stiffness parameter:

$$k^* = k(1 + i\eta)$$

where $\eta$ is the structural loss factor. Under harmonic excitation, this represents a phase lag between the force and the resulting displacement, representing energy loss. We can define an equivalent viscous damping coefficient $c_{eq}$ that dissipates the same amount of energy per cycle at a specific operating frequency $\omega$:

$$c_{eq} = \frac{k \eta}{\omega}$$

This formulation highlights a fundamental challenge: the equivalent viscous damping is inversely proportional to frequency, which means a viscous model calibrated at resonance will over-predict damping at higher frequencies and under-predict damping at lower frequencies.

2.3 Forced Vibration & Dynamic Amplification

When an SDOF system is subjected to a continuous harmonic force $F(t) = F_0 \sin(\omega t)$, the response consists of two parts:

Transient Response: This is the free vibration component which decays exponentially

due to damping. Its frequency is the damped natural frequency $\omega_d$.

Steady-State Response: This is the persistent motion driven by the excitation force.

It occurs at the excitation frequency $\omega$ and lags behind the force by a phase angle $\phi$.

The steady-state displacement can be expressed as:

$$x_{ss}(t) = X \sin(\omega t - \phi)$$

The ratio of the steady-state amplitude $X$ to the static deflection $X_{static} = F_0/k$ is called the Dynamic Amplification Factor (DAF), denoted by $M$:

$$M = \frac{X}{X_{static}} = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$$

where $r = \omega / \omega_n$ is the frequency ratio.

2.3.1 Behavior as a Function of Frequency Ratio $r$

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Quasi-Static Region ($r \ll 1$):

At low frequencies, the response is controlled by the stiffness of the spring. The amplitude $X \approx F_0/k$, the DAF $M \approx 1$, and the phase angle $\phi \approx 0$. The mass moves in phase with the force.

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Resonant Region ($r \approx 1$):

Near the natural frequency, the spring and inertia forces tend to cancel each other out. The amplitude is limited solely by damping. At $r = 1$, the DAF becomes:

$$M_{res} = \frac{1}{2\zeta}$$

The phase angle is exactly $90^\circ$ ($\pi/2\text{ rad}$), meaning the excitation force is perfectly aligned with the velocity, feeding maximum energy into the system. The peak of the DAF curve occurs at a slightly lower frequency ratio:

$$r_{peak} = \sqrt{1 - 2\zeta^2} \quad (\text{for } \zeta < 1/\sqrt{2})$$

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Inertia-Controlled Region ($r \gg 1$):

At high frequencies, the mass cannot keep up with the rapid changes in force direction. The response is dominated by the mass inertia. The amplitude drops rapidly:

$$X \approx \frac{F_0}{m \omega^2}$$

The DAF $M \to 0$ asymptotically, and the phase angle $\phi \to 180^\circ$ ($\pi\text{ rad}$). The mass moves in opposition to the applied force.

2.4 Support Excitation & Vibration Isolation

Vibration isolation is the practice of inserting a resilient element (isolator) between a vibrating body and its support structure to reduce the transmission of dynamic forces or displacement. There are two primary types of isolation scenarios:

Source Isolation (Active Isolation):

Reducing the dynamic force transmitted from a vibrating machine (e.g., engine, pump) to its supporting foundation.

Receiver Isolation (Passive Isolation):

Protecting a sensitive instrument (e.g., optical microscope, lithography tool) from the vibration of its supporting foundation (e.g., floor vibrations).

Both scenarios are governed by the same mathematical parameter: Transmissibility ($T_d$ or $T_f$). Consider a system excited by a base displacement $y(t) = Y \sin(\omega t)$. The force transmitted to the foundation is due to the tension/compression of the spring and the viscous drag of the damper:

$$F_T(t) = k(x-y) + c(\dot{x}-\dot{y})$$

The transmissibility ratio (the ratio of transmitted force amplitude to excitation force, or absolute displacement amplitude $X$ to base displacement amplitude $Y$) is given by:

$$T = \sqrt{\frac{1 + (2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}}$$

2.4.2 The Critical Crossover Frequency ($r = \sqrt{2}$)

An analysis of the transmissibility equation reveals a fundamental rule of vibration isolation:

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If $r < \sqrt{2}$, then $T > 1$. The vibration is amplified by the support.

Higher damping reduces the peak amplification.

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If $r = \sqrt{2}$, then $T = 1$ for all values of the damping ratio $\zeta$. All curves intersect.

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If $r > \sqrt{2}$, then $T < 1$. The vibration is isolated. The isolation efficiency

is defined as:

$$\text{Isolation Efficiency } I_E = 1 - T$$

Below is a premium vector plot illustrating the transmissibility curves for various damping ratios, highlighting the crossover point and the regions of amplification and isolation:

2.4.3 Damping Trade-Off in Isolation Design

Vibration isolation design involves a critical trade-off:

At Resonance ($r \approx 1$): Damping is essential. Without damping,

the transmissibility reaches extreme levels ($T \to \infty$ for $\text{undamped}$), leading to mechanical failure, excessive rattle, and high stresses.

In the Isolation Range ($r > \sqrt{2}$): Damping degrades isolation performance.

At high frequencies ($r \gg 1$), the transmissibility behaves as:

$$T \approx \frac{2 \zeta r}{r^2} = \frac{2\zeta}{r}$$

This shows that for a given high frequency ratio $r$, the transmissibility is directly proportional to the damping ratio $\zeta$. Thus, a heavily damped system will transmit *more* force than a lightly damped system.

This trade-off is summarized in the table below:

Frequency Range	Damping Effect	Ideal System Attribute	Practical Solution
:---	:---	:---	:---
Resonance ($r \approx 1$)	Limits peak displacement & force	High damping (High $\zeta$)	Elastomeric mounts, liquid-filled hydro-mounts, or auxiliary dampers.
High Frequency ($r > \sqrt{2}$)	Increases force transmission	Low damping (Low $\zeta$)	Steel coil springs with minimal damping, or active damping.

2.5 Rotating Unbalance

One of the most frequent sources of forced harmonic excitation in rotating machinery (turbines, pumps, combustion engines, fans, electric motors) is rotating unbalance. This unbalance occurs when the mass center of the rotating part (rotor) does not coincide with the geometric axis of rotation.

Let the total mass of the machine casing and rotor be $M$, and let a small unbalanced mass $m$ be located at an eccentricity (radial distance) $e$ from the axis of rotation. When the rotor turns at an angular velocity $\omega$, the centrifugal force generated is:

$$F_c = m e \omega^2$$

This rotating vector force can be resolved into vertical and horizontal components. If the machine is constrained to move only in the vertical direction, the exciting force is:

$$F(t) = m e \omega^2 \sin(\omega t)$$

Note a crucial difference from direct force excitation:

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In direct excitation $F(t) = F_0 \sin(\omega t)$, the force amplitude $F_0$ is constant.

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In rotating unbalance, the force amplitude $F_0(\omega) = m e \omega^2$ is proportional to

the square of the operating speed.

2.5.1 Casing Response Characteristics

The steady-state vibration amplitude $X$ of the machine casing is:

$$\frac{M X}{m e} = \frac{r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$$

This dimensionless ratio $\frac{M X}{m e}$ behaves as follows:

Low Speed ($r \ll 1$):

The response is near zero. The unbalance force is too small to overcome the spring stiffness.

Resonant Speed ($r \approx 1$):

The response is amplified, limited only by damping. At $r=1$:

$$\frac{M X}{m e} = \frac{1}{2\zeta}$$

High Speed ($r \gg 1$):

The ratio $\frac{M X}{m e} \to 1$. This means the casing amplitude stabilizes at:

$$X \approx \frac{m e}{M}$$

At this point, the casing and unbalanced mass pivot around the center of mass of the combined assembly. The center of gravity remains stationary, representing a self-centering effect.

2.6 Electrical-Mechanical Analogies

In NVH engineering and acoustic-structure interaction, drawing analogies between mechanical systems and electrical circuits is a powerful technique. It allows mechanical engineers to use established circuit analysis techniques (e.g., loop analysis, nodal analysis, Laplace transforms) to solve complex vibration problems.

There are two primary analogies used:

2.6.1 Force-Voltage (Impedance) Analogy

In the impedance analogy, force is analogous to voltage (effort variable) and velocity is analogous to current (flow variable). This analogy maintains the mathematical structure of the governing differential equations.

Mechanical Quantity	Electrical Analogy	Governing Relationship
:---	:---	:---
Force ($F$)	Voltage ($V$)	effort variable
Velocity ($\dot{x}$)	Current ($I$)	flow variable
Mass ($m$)	Inductance ($L$)	$F = m \frac{d\dot{x}}{dt} \leftrightarrow V = L \frac{dI}{dt}$
Damper ($c$)	Resistance ($R$)	$F = c \dot{x} \leftrightarrow V = R I$
Spring Stiffness ($k$)	Reciprocal Capacitance ($1/C$)	$F = k \int \dot{x} dt \leftrightarrow V = \frac{1}{C} \int I dt$

Using this analogy, a parallel mass-spring-damper mechanical system translates to a series LRC circuit. The mechanical impedance is defined as the ratio of complex force to complex velocity:

$$Z_m(\omega) = \frac{F(\omega)}{\dot{x}(\omega)} = c + i \left( m\omega - \frac{k}{\omega} \right)$$

2.6.2 Force-Current (Mobility) Analogy

In the mobility analogy, force is analogous to current (flow variable) and velocity is analogous to voltage (effort variable). This analogy is topologically direct: mechanical elements connected in parallel correspond to electrical elements connected in parallel, which is highly intuitive.

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Mass ($m$) maps to Capacitance ($C$).

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Damper ($c$) maps to Reciprocal Resistance ($1/R$).

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Spring Stiffness ($k$) maps to Reciprocal Inductance ($1/L$).

Mechanical mobility is the reciprocal of mechanical impedance, defined as the ratio of velocity to force:

$$Y_m(\omega) = \frac{\dot{x}(\omega)}{F(\omega)}$$

2.7 State-Space Formulation of SDOF Systems

For modern multi-physics simulations, control design (e.g., active suspension), and numerical integration, the second-order differential equation of motion is rewritten as a system of first-order differential equations in state-space form.

We define the state vector $\mathbf{z}(t)$ as:

$$\mathbf{z}(t) = \begin{Bmatrix} x(t) \\ \dot{x}(t) \end{Bmatrix}$$

Differentiating the state vector with respect to time:

$$\dot{\mathbf{z}}(t) = \begin{Bmatrix} \dot{x}(t) \\ \ddot{x}(t) \end{Bmatrix}$$

Using the equation of motion $m \ddot{x} + c \dot{x} + k x = F(t)$, we express the acceleration as:

$$\ddot{x}(t) = -\frac{k}{m} x(t) - \frac{c}{m} \dot{x}(t) + \frac{1}{m} F(t)$$

We can now write the state equation in matrix form:

$$\dot{\mathbf{z}}(t) = \mathbf{A} \mathbf{z}(t) + \mathbf{B} u(t)$$

where:

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System Matrix ($\mathbf{A}$):

$$\mathbf{A} = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{c}{m} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\omega_n^2 & -2\zeta\omega_n \end{bmatrix}$$

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Input Vector ($\mathbf{B}$):

$$\mathbf{B} = \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}$$

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Input Control Signal ($u(t)$): $u(t) = F(t)$

To complete the state-space description, we define the output equation:

$$\mathbf{y}(t) = \mathbf{C} \mathbf{z}(t) + \mathbf{D} u(t)$$

If we want to measure the displacement $x(t)$ and velocity $\dot{x}(t)$, then:

$$\mathbf{C} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \mathbf{D} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$$

If our output of interest is the acceleration $\ddot{x}(t)$ (which is standard in NVH since accelerometers are the primary transducers), we substitute the expression for acceleration:

$$\mathbf{C} = \begin{bmatrix} -\omega_n^2 & -2\zeta\omega_n \end{bmatrix}, \quad \mathbf{D} = \begin{bmatrix} \frac{1}{m} \end{bmatrix}$$

2.8 Shock Response Spectra (SRS)

A shock is a transient excitation characterized by high acceleration, short duration, and sudden energy transfer. Typical examples include pyrotechnic shocks in aerospace staging, automotive crashes, and dropping electronics.

Because shock events are transient and non-harmonic, the frequency-domain transmissibility curve is insufficient. Instead, we use the Shock Response Spectrum (SRS).

2.8.1 Definition of SRS

An SRS is a graphical representation of the maximum peak response of a set of SDOF systems of varying natural frequencies to a given shock input.

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The x-axis represents the natural frequency ($f_n$) of the SDOF systems.

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The y-axis represents the maximum absolute value of response (displacement, velocity, or acceleration)

experienced by each SDOF system.

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A damping ratio of $\zeta = 0.05$ (equivalent to a quality factor $Q=10$) is standard for SRS calculations.

2.8.2 Types of SRS

Primary Shock Spectrum:

Plots the maximum response that occurs *during* the shock pulse event.

Residual Shock Spectrum:

Plots the maximum response that occurs *after* the shock pulse has terminated (free vibration phase).

Maximax Shock Spectrum:

Plots the absolute maximum response across both the primary and residual phases. This is the primary spectrum used for design envelope definition.

2.9 Random Vibrations of SDOF Systems

Many real-world excitations are not deterministic or harmonic. Road surface roughness, turbulent air flow over a fuselage, and wave action on offshore structures are random. These processes are described statistically using probability density functions and spectral density functions.

2.9.1 Power Spectral Density (PSD)

A random force $F(t)$ is characterized in the frequency domain by its Power Spectral Density (PSD), denoted as $S_{FF}(\omega)$ (expressed in $\text{N}^2/\text{Hz}$). The PSD describes how the mean-square value of the signal is distributed across frequency.

The response of an SDOF system to a random input is given by:

$$S_{XX}(\omega) = |H(\omega)|^2 S_{FF}(\omega)$$

where $H(\omega)$ is the complex frequency response function.

2.9.2 Miles' Equation

If the input force PSD is relatively flat (white noise) with a constant value $S_{FF}(\omega_n) = S_0$ near the system's natural frequency, the root-mean-square (RMS) displacement response $\sigma_x$ can be estimated using Miles' Equation:

$$\sigma_x \approx \sqrt{\frac{\pi f_n Q S_0}{2 k^2}}$$

where $Q = \frac{1}{2\zeta}$ is the quality factor.

Miles' Equation is widely used in NVH engineering for preliminary structural design and stress analysis of components mounted on vibrating structures (e.g., bracket designs in engine bays).

2.10 Tuned Mass Dampers (TMD)

A Tuned Mass Damper (TMD), also known as a vibration absorber, is a passive device attached to a primary structure to suppress vibration at a specific frequency. It consists of a secondary mass ($m_d$), a spring ($k_d$), and a damper ($c_d$).

2.10.1 Operating Principle

When the primary structure is excited at its natural frequency, the TMD is tuned to match that frequency. The TMD oscillates out of phase with the primary structure, generating a force that counters the excitation. This splits the single resonant peak of the primary SDOF system into two smaller peaks separated by a trough at the tuned frequency.

TMDs are widely used in skyscrapers (e.g., Taipei 101 uses a 660-tonne pendulum TMD to mitigate wind-induced vibrations), bridges, and automotive crankshafts (torsional vibration dampers).