9.5 Multiple degree of freedom systems

Degrees of Freedom Concept

The number of degrees of freedom (DOF) in a system determines how many independent coordinates you need to fully describe its configuration. This concept sits at the heart of vibration analysis because it dictates the complexity of your mathematical model and the number of equations you'll need to solve.

Definition of DOF

A degree of freedom is one independent coordinate required to describe a system's position. Each DOF represents a possible motion the system can undergo, and it can be either translational (linear motion) or rotational (angular motion). The total number of DOFs directly determines how many equations of motion you need to write.

Examples in Mechanical Systems

• A simple pendulum has 1 DOF: the angle of swing.
• A double pendulum has 2 DOF: the angle of each arm.
• A robotic arm typically has 6 DOF: 3 translational and 3 rotational, allowing it to reach any position and orientation in space.
• An automobile suspension is often modeled with 7 DOF:
  • 4 for the vertical motion of each wheel
  • 3 for the vehicle body (heave, pitch, and roll)

As DOF count increases, the model captures more realistic behavior, but the math gets significantly more involved.

Constraints and DOF Reduction

Constraints are physical limitations that restrict a system's motion, reducing the number of independent coordinates you need. There are two main types:

• Geometric constraints arise from fixed connections, sliding joints, or rigid links that physically prevent certain motions.
• Kinematic constraints impose relationships between velocities or accelerations, such as a no-slip rolling condition.

Each constraint removes one DOF from the system. Identifying constraints correctly is critical because it simplifies your analysis and reduces computational effort without sacrificing accuracy.

Multiple DOF Systems

Single DOF models are useful for building intuition, but real structures almost always require multiple degrees of freedom to capture their actual behavior. MDOF systems extend those foundational concepts using matrix formulations and, in most practical cases, numerical solution methods.

Characteristics of MDOF Systems

• They possess multiple natural frequencies, each associated with a distinct mode shape.
• Motion in one coordinate is often coupled to motion in others through off-diagonal terms in the system matrices.
• The equations of motion take matrix form: $[M]\{\ddot{q}\} + [C]\{\dot{q}\} + [K]\{q\} = \{F(t)\}$
• Analytical closed-form solutions are rarely possible beyond 2 or 3 DOF, so numerical methods become essential.

Advantages over Single DOF Models

MDOF models capture things that single DOF models simply cannot:

• Interactions between components (e.g., how a floor's vibration affects adjacent floors in a building)
• Localized effects and stress concentrations
• Higher-order vibration modes that may be excited during operation
• Enough detail to optimize complex designs for weight, stiffness, and damping simultaneously

Real-World MDOF Applications

• Tall buildings subject to wind or seismic loads, where each floor can sway independently
• Aircraft wings undergoing aeroelastic flutter, coupling bending and torsion
• Automotive chassis designed for ride comfort and handling
• Spacecraft attitude control systems with multiple rotating and translating components
• Turbines and engines with many interconnected moving parts

Equations of Motion

The equations of motion mathematically describe how an MDOF system evolves over time. Deriving them correctly is the foundation for every analysis that follows.

Derivation Methods

Several approaches exist, and the best choice depends on the system's complexity:

• Newtonian mechanics: Apply $F = ma$ directly to each mass using free body diagrams.
• Lagrangian mechanics: Use energy expressions and generalized coordinates.
• Virtual work principle: Equate virtual work done by all forces to zero.
• Hamilton's principle: A variational approach for conservative systems.
• D'Alembert's principle: Converts a dynamic problem into a static equilibrium problem by introducing inertial forces.

Lagrange's Equations

Lagrange's method is especially powerful for systems with constraints or complex geometry because it avoids the need for free body diagrams. The general form is:

$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}_i}\right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} = Q_i$

where $T$ is kinetic energy, $V$ is potential energy, $q_i$ are generalized coordinates, and $Q_i$ are generalized non-conservative forces.

The procedure:

1. Choose a set of independent generalized coordinates $q_i$.
2. Write expressions for the system's total kinetic energy $T$ and potential energy $V$ in terms of those coordinates.
3. Compute each partial derivative in the Lagrange equation.
4. Substitute and simplify to obtain one equation per DOF.

Newton's Second Law Approach

For systems with simple geometry and few DOFs, applying $F = ma$ directly to each mass can be the most straightforward method.

1. Draw a free body diagram for each mass, showing all spring, damper, and external forces.
2. Write $F = ma$ (or its rotational equivalent $\tau = I\alpha$) for each DOF.
3. Express spring and damper forces in terms of the generalized coordinates and their derivatives.

For a single DOF element, the equation takes the familiar form:

$m\ddot{x} + c\dot{x} + kx = F(t)$

This approach becomes cumbersome as the number of DOFs grows, which is why Lagrange's method is generally preferred for larger systems.

Mass Matrix

The mass matrix $[M]$ encodes how mass and inertia are distributed across the system's degrees of freedom. It directly influences the natural frequencies and mode shapes.

Formation of the Mass Matrix

The mass matrix is constructed using energy methods or the principle of virtual work. For an $n$-DOF system:

$[M] = \begin{bmatrix} m_{11} & m_{12} & \cdots & m_{1n} \\ m_{21} & m_{22} & \cdots & m_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ m_{n1} & m_{n2} & \cdots & m_{nn} \end{bmatrix}$

• Diagonal elements ($m_{ii}$) represent the mass or moment of inertia associated with each DOF.
• Off-diagonal elements ($m_{ij}$, where $i \neq j$) represent inertial coupling between DOFs. These appear when acceleration of one coordinate produces an inertial force along another.

Consistent vs. Lumped Mass

There are two common strategies for building mass matrices:

• Lumped mass: Concentrate the total mass at discrete node points. This produces a diagonal mass matrix, which is computationally cheap and easy to invert.
• Consistent mass: Distribute mass throughout each element using the same shape functions as the stiffness formulation. This yields a full (non-diagonal) matrix that's more accurate, especially for higher-frequency modes.

The trade-off is accuracy versus computational cost. For lower modes of vibration, lumped mass is often sufficient. For higher modes or when precision matters, consistent mass is the better choice.

Properties of Mass Matrices

• Symmetric: $m_{ij} = m_{ji}$, a consequence of the reciprocity principle.
• Positive definite: All eigenvalues are strictly positive, which physically means the system always has positive kinetic energy for any nonzero velocity.
• Invertible: This is required for solving the eigenvalue problem in modal analysis.

Stiffness Matrix

The stiffness matrix $[K]$ describes the system's resistance to deformation. Along with the mass matrix, it determines the natural frequencies and mode shapes.

Concept of System Stiffness

Stiffness measures the force required to produce a unit displacement. The relationship between forces and displacements across the entire system is:

$\{F\} = [K]\{x\}$

where $\{F\}$ is the force vector and $\{x\}$ is the displacement vector. Stiffness depends on both material properties (like Young's modulus) and geometric configuration (like cross-sectional area and length).

Assembly of the Stiffness Matrix

For systems with multiple elements, you build the global stiffness matrix by combining individual element stiffness matrices:

1. Derive the local stiffness matrix for each element in its own coordinate system.
2. Transform each local matrix to the global coordinate system using transformation matrices.
3. Assemble by adding each element's contribution to the appropriate locations in the global matrix, based on how the elements are connected (their node numbering).
4. Apply boundary conditions by modifying rows and columns corresponding to constrained DOFs.

The result is a square matrix of size $n \times n$, where $n$ is the total number of system DOFs.

Global vs. Local Coordinates

• Local coordinates describe an element's behavior in its own reference frame, aligned with the element's geometry.
• Global coordinates place all elements in a single, system-wide reference frame.
• Transformation matrices convert between the two. If $[T]$ is the transformation matrix, then the element stiffness in global coordinates is $[K_{global}] = [T]^T [K_{local}] [T]$.

This transformation ensures compatibility (displacements match where elements connect) and equilibrium (forces balance at shared nodes) across the entire structure.

Damping Matrix

The damping matrix $[C]$ represents energy dissipation in the system. Damping is often the hardest property to model accurately because real dissipation mechanisms are complex and frequency-dependent.

Types of Damping

• Viscous damping: Force proportional to velocity. The most commonly used model because it leads to linear equations.
• Coulomb damping: Constant-magnitude friction force that opposes motion. Common in joints and sliding contacts.
• Hysteretic (material) damping: Energy lost due to internal friction within the material during cyclic deformation.
• Structural damping: Dissipation at joints, connections, and interfaces.
• Aerodynamic damping: Energy transfer between a vibrating structure and the surrounding fluid.

Rayleigh Damping

Rayleigh damping is a widely used approximation that expresses the damping matrix as a linear combination of the mass and stiffness matrices:

$[C] = \alpha [M] + \beta [K]$

The coefficients $\alpha$ and $\beta$ are chosen to match known (or assumed) damping ratios at two specific frequencies. This is convenient because it preserves the ability to decouple the equations using modal analysis. The trade-off is that damping ratios at other frequencies are only approximately correct.

Proportional vs. Non-Proportional Damping

• Proportional damping (including Rayleigh damping) allows the equations of motion to be fully decoupled into independent modal equations. This makes the math much simpler.
• Non-proportional damping means the damping matrix cannot be diagonalized by the undamped mode shapes. The modal equations remain coupled, requiring complex eigenvalue analysis.

Many real systems have non-proportional damping (e.g., a structure with localized dampers at specific joints), but proportional damping is often assumed as a reasonable approximation to keep the analysis tractable.

Free Vibration Analysis

Free vibration analysis studies how a system oscillates after an initial disturbance, with no external forcing applied. It reveals the system's fundamental dynamic properties: natural frequencies and mode shapes.

Natural Frequencies

Natural frequencies are the frequencies at which the system naturally tends to oscillate. You find them by solving the eigenvalue problem:

$(K - \omega^2 M)\Phi = 0$

This equation has nontrivial solutions only when:

$\det(K - \omega^2 M) = 0$

Solving this determinant (the characteristic equation) yields $n$ values of $\omega^2$ for an $n$-DOF system. Each $\omega_i$ is a natural frequency. The lowest natural frequency ($\omega_1$) is called the fundamental frequency and typically dominates the system's response.

Mode Shapes

Each natural frequency $\omega_i$ has a corresponding mode shape $\Phi_i$, which describes the relative displacement pattern of the system at that frequency. Mode shapes are the eigenvectors of the eigenvalue problem above.

Key properties of mode shapes:

• They are orthogonal with respect to both the mass and stiffness matrices: $\Phi_i^T [M] \Phi_j = 0$ and $\Phi_i^T [K] \Phi_j = 0$ for $i \neq j$.
• They describe relative motion, not absolute amplitudes. You can scale them by any constant.
• Visualizing mode shapes helps you understand which parts of the structure move the most at each frequency.

Modal Analysis Techniques

• Analytical methods: Solve the characteristic equation directly. Feasible for small systems (2-3 DOF).
• Numerical methods: For larger systems, algorithms like QR iteration, subspace iteration, or Lanczos methods extract eigenvalues and eigenvectors efficiently.
• Finite element analysis (FEA): Discretizes continuous structures into many elements, then solves the resulting large eigenvalue problem computationally.
• Experimental modal analysis: Measures actual frequency response data from a physical structure and extracts modal parameters through curve fitting.

Forced Vibration Response

Forced vibration analysis predicts how the system responds to external excitation during operation. The total response combines the system's inherent dynamic properties with the characteristics of the applied load.

Harmonic Excitation

When the forcing function is sinusoidal, the response has two parts:

• A transient component that decays over time due to damping.
• A steady-state component that oscillates at the forcing frequency.

Resonance occurs when the forcing frequency matches one of the system's natural frequencies. At resonance, response amplitudes grow dramatically, limited only by damping. The amplitude magnification factor quantifies how much the steady-state response exceeds the static deflection, and it depends on both the damping ratio and the frequency ratio (forcing frequency divided by natural frequency).

Impulse Response

An impulse is an idealized instantaneous force. The system's reaction to it, called the impulse response function (IRF), contains information about all of the system's modes. The IRF is a time-domain signature of the system's dynamics.

A useful relationship: the Fourier transform of the impulse response function gives the frequency response function (FRF). This connects time-domain and frequency-domain descriptions of the same system.

Frequency Response Functions

A frequency response function (FRF) describes the ratio of the system's output (displacement, velocity, or acceleration) to its input (force) as a function of frequency. FRFs are complex-valued, containing both magnitude and phase information.

• Peaks in the FRF magnitude correspond to the system's natural frequencies.
• The width of each peak relates to the damping at that mode.
• FRFs are the primary tool used in vibration testing to identify modal parameters experimentally.

Numerical Solution Methods

For most practical MDOF systems, closed-form analytical solutions don't exist. Numerical methods provide the computational tools to solve these problems, balancing accuracy, stability, and efficiency.

Time Domain Analysis

Time domain methods directly integrate the equations of motion step by step through time. Common algorithms include:

• Newmark-$\beta$ method: An implicit method with adjustable parameters that control accuracy and stability. Widely used in structural dynamics.
• Runge-Kutta algorithms: General-purpose ODE solvers, well-suited for nonlinear problems.
• Central difference method: An explicit method that's simple to implement but requires small time steps for stability.

Time domain analysis handles nonlinear systems and transient loads naturally, but it can be computationally expensive for long-duration simulations.

Frequency Domain Analysis

Frequency domain methods transform the equations of motion using Fourier or Laplace transforms, converting differential equations into algebraic ones. This approach is very efficient for:

• Steady-state response to harmonic or periodic excitation
• Systems where the frequency content of the input is well-defined

The limitation is that frequency domain methods assume linearity (superposition must hold), so they're not suitable for strongly nonlinear systems.

Modal Superposition Technique

Modal superposition is one of the most powerful tools for solving MDOF problems efficiently. The procedure:

1. Solve the free vibration eigenvalue problem to find natural frequencies and mode shapes.
2. Form the modal matrix $[\Phi]$ from the mode shape vectors.
3. Transform the physical coordinates to modal coordinates: $\{q\} = [\Phi]\{\eta\}$.
4. Substitute into the equations of motion. If damping is proportional, the equations decouple into $n$ independent single-DOF equations.
5. Solve each modal equation independently.
6. Transform back to physical coordinates by summing the modal contributions.

A major advantage is modal truncation: you can often ignore higher-frequency modes that contribute little to the response, dramatically reducing computation time. This technique works best for lightly damped, linear systems.

Vibration Control Strategies

Once you understand a system's vibration behavior, the next step is often to control or reduce unwanted vibrations. Control strategies fall into three categories based on how they interact with the system.

Passive Control Methods

Passive devices dissipate or redirect vibrational energy without requiring external power. Examples include:

• Tuned mass dampers (TMDs): A secondary mass-spring-damper system tuned to a structure's problematic natural frequency. The Taipei 101 skyscraper uses a 730-ton TMD to reduce wind-induced sway.
• Vibration isolators: Rubber mounts or spring systems that decouple a vibrating source (like an engine) from its surroundings.
• Constrained layer damping: A viscoelastic layer sandwiched between structural layers, converting vibrational energy to heat through shear deformation.

Passive methods are simple, reliable, and require no maintenance. Their main limitation is that they're tuned to specific conditions and can't adapt if the excitation changes.

Active Control Systems

Active systems use sensors to measure vibrations, a controller to compute the required response, and actuators to apply counteracting forces. They can adapt in real time to changing conditions.

• Active suspension systems in vehicles adjust damping and stiffness continuously for optimal ride quality.
• Active vibration cancellation in precision manufacturing keeps tool vibrations below tolerance thresholds.

The downsides are complexity, power requirements, and the risk of instability if the control algorithm is poorly designed.

Semi-Active Control Approaches

Semi-active systems split the difference between passive and active. They adjust system properties (like damping or stiffness) in real time but don't inject energy into the system. This makes them inherently stable.

• Magnetorheological (MR) dampers change their damping coefficient when a magnetic field is applied to the MR fluid. They're used in seismic protection of buildings and in some vehicle suspensions.
• Adaptive engine mounts in aircraft adjust stiffness to avoid resonance across different operating speeds.

Semi-active systems offer better performance than passive devices while requiring far less power than fully active systems.

Experimental Modal Analysis

Theoretical models always involve assumptions and simplifications. Experimental modal analysis (EMA) measures the actual dynamic behavior of a physical structure, providing data to validate and refine those models.

Impact Testing

Impact testing is the simplest form of EMA. You strike the structure with an instrumented hammer and measure the response with accelerometers.

• The hammer's force transducer records the input force.
• Accelerometers at various locations record the response.
• Dividing the response by the input (in the frequency domain) gives the FRF.

Impact testing is portable and requires minimal equipment, making it ideal for smaller structures and field testing. The challenge is ensuring that the impact excites all modes of interest, which can be difficult for large or heavily damped structures.

Shaker Testing

For more controlled excitation, electrodynamic or hydraulic shakers are attached to the structure. Shakers can deliver various input signals:

• Sine sweep: Slowly varies frequency to map out the FRF with high resolution.
• Random excitation: Broadband input that excites all frequencies simultaneously.
• Burst random: Short bursts of random signal, useful for reducing leakage in FFT processing.

Shaker testing provides better control over input force and frequency content than impact testing, but requires more setup time and equipment. It's preferred for larger structures and when high-frequency modes need to be captured.

Data Acquisition and Processing

Collecting clean, reliable data requires attention to several details:

• Sampling rate must satisfy the Nyquist criterion: at least twice the highest frequency of interest.
• Measurement duration must be long enough to capture the system's response, especially for lightly damped modes that ring for a long time.

Common signal processing steps include:

• Windowing (e.g., Hanning, exponential) to reduce spectral leakage in the FFT
• Averaging multiple measurements to improve the signal-to-noise ratio
• Filtering to remove noise outside the frequency range of interest

Once clean FRFs are obtained, modal parameter estimation algorithms (such as polyreference least-squares complex frequency or rational fraction polynomial methods) extract the natural frequencies, damping ratios, and mode shapes from the data.