System Response
The response of an analytical model to an imposed disturbance can be expressed in either the time-domain (response value versus time) or in the frequency domain (amplitude and phase versus frequency).
The time domainresponse generally involves the solution of a set of differential equations. The frequency domainanalysis is done with a set of transfer functions that is the ratio output/input in the Laplace transform (“s”) form.
We shall see that mobility, admittance, impedance, and transmissibility are convenient transfer-functionrepresentations. For example, transmissibility is important in vibration isolation, and mechanical impedance isuseful in tasks such as cutting, joining, and assembly that employ robots.
Experimental determination of transfer function (i.e., frequency-domain experimental modeling) is often used in modal testing—i.e., testing for natural “modes” of response—of a mechanical system. This requires imposing forces on the system and measuring its response (motion).
The system must be designed to limit both the forces transmitted from the system to the foundation, and the motions transmitted from the support structure to the main system. In these cases, the vibration isolation characteristics of the system can be expressed as transfer functions for force transmissibility and motion transmissibility.
We will see that these two transmissibility functions are identical for a given mechanical system and suspension.
Following modeling techniques for response analysis and design of a mechatronic system:
1. State models, using state variables representing the state of the system in terms of system variables, such as position and velocity of lumped masses, force and displacement in springs, current through an inductor, and voltage across a capacitor. These are time-domain models, with the independent variable t (time).
2. Linear graphs—a model using a graphic representation. This is particularly useful as a tool in developing a state model. The linear graph uses through variables (e.g., forces or currents) and across variables (e.g., velocities or voltages) for each branch (path of energy flow) in the model.
3. Bond graphs—another graphical model (like the linear graph), but using branches called bonds to represent power flow. The bond graph uses flow variables (e.g., velocities or currents) and effort variables (e.g., forces or voltages). A state model can be developed from the bond-graph representation as well.
4. Transfer-function models—a very common model type. Uses output/input ratio in the Laplace transform form (i.e., in the “s-domain”). Here the Laplace variable s is the independent variable.
5. Frequency-domain models—a special case of (4) above. Here we use the Fourier transform instead of the Laplace transform. Simply stated, s=jw in the frequency domain. Here, frequency w is the independent variable.
Model Development
Development of a suitable analytical model for a large and complex system requires a systematic approach.
Tools are available to aid this process. The process of modeling can be made simple by following a systematic sequence of steps.
The main steps are summarized below:
1. Identify the system of interest by defining its purpose and the system boundary.
2. Identify or specify the variables of interest. These include inputs (forcing functions or excitations) and outputs (response).
3. Approximate (or model) various segments (components or processes or phenomena) in the system by ideal elements, which are suitably interconnected.
4. Draw a free-body diagram for the system with isolated/separated elements, as appropriate.
5. Write constitutive equations (physical laws) for the elements.
6. Write continuity (or conservation) equations for through variables (equilibrium of forces at joints; current balance at nodes, fluid flow balance, etc.)
7. Write compatibility equations for across (potential or path) variables. These are loop equations for velocities (geometric connectivity), voltage (potential balance), pressure drop, etc.
8. Eliminate auxiliary variables that are redundant and not needed to define the model.
9. Express system boundary conditions and response initial conditions using system variables. These steps should be self-explanatory, and should be integral with the particular modeling technique that is used.
