## 1 Introduction  
When two solid surfaces come into contact, it is the asperity peaks that touch, and the real area of contact is substantially less than the nominal contact area [1]. Applying a pressure to the surfaces causes a small approach of the mean lines of their roughness. The interfacial stiffness (per unit area) is then defined as the rate of change of contact pressure with approach of the mean lines of the roughness of the contacting surfaces [2]. If the real contact area is low, then a low nominal contact pressure is all that is required to deflect the asperities. Increasing the nominal contact pressure brings more asperities into contact and the interface becomes stiffer; hence interface stiffness is a nonlinear variable. Interfacial stiffness is important in tribology because it affects machine element deflection, wear, and friction, since stiffer interfaces tend to less deflection. Machining accuracy, for example, depends on the stiffness of the joints in the machine tool assembly [2]. Several studies have focused on the interfacial stiffness of rough contact from both theoretical and experimental perspectives. An early example from [PERSON] [3] provided a theoretical model for elastic contact of a smooth curved surface. Later, [PERSON] and [PERSON] (GW model) [4] presented a statistical model for the relation between the contact area of the rough elastic mating surfaces and applied pressure. The model is limited by two assumptions: firstly asperities are treated as a distribution of hemispherical capped peaks, and secondly those peaks are treated as independent such that the deformation of one peak does not affect its neighbors. The relationship between the contact area and applied pressure also presented by models such as the Whitehouse-Archard-Onion model (WAO model) [5; 6] and the Buh-Gibson-Thomas model (BGT model) [7]. These models have been widely used to predict contact stiffness. [PERSON] et al. [8] used [PERSON]'s theory to measure the stiffness ratio of stationary contact. Digital image correlation (DIC) [9; 10; 11] has been used to measure stiffness in joints. In this method, a series of images prior to applying load and during loading are taken from the interface and surrounding area. The comparison of the images shows the relative displacement. A plot of the applied load measured with a load cell against the relative surface displacement is created; the gradient gives the contact interface. Although it has advantages such as a full field capability and high resolution, the main limitation is that a surface near the contacting surfaces must be accessible for the camera to capture images. In many items of engineering machinery optical access is limited and this limits application of DIC outside a laboratory setting. Another method is to make direct measurements of the displacement of the interface under applied load. [PERSON] et al. [12] used relative displacement transducers between the fixed frame and the stack of rings to measure direct displacement of the interface. [PERSON] et al. [13] also directly measured displacement of a smooth surface using strain gauges and spring-loaded pins. For example, [PERSON] et al. [14] used a tenso-metric bridge to measure the applied load and an indicator sensor to measure the surface approach. They compared their results to theoretical predictions from the Greenwood-Williamson model in the range of elastic deformation for quasi-isotropic surfaces. However, this technique is restricted to low contact stiffness (\(1\) GPa/\(\upmu\)m) [15]. It is also possible to measure contact stiffness from a structure's resonance frequency. This contact frequency method uses a relationship between the contact stiffness and the natural frequency of the mating surfaces [16]. One of the disadvantages of this method is the need for vibration response equations of the experimental tools [17]. The approach used here, also a vibrational method, is through the reflection or transmission of ultrasound [18; 19; 20; 21; 22; 23].